Supian SUDRADJAT

editura universitaţii bucureşti

Referenţi ştiinţifici: Prof. univ. dr. Vasile PREDA

Prof. univ. dr. Ion VĂDUVA

@ editura universităţii din bucureşti Şos. Panduri, 90-92, Bucureşti-050663;Telefon/Fax: 410.23.84

E-mail:editura_unibuc@yahoo.com Internet:www.editura.unibuc.ro

Descrierea CIP a Bibliotecii Naţionae a Romaniei SUDRADJAT, SUPIAN Mathematical programming models for portfolio selection / Supian Sudradjat – Bucureşti: Editura Universittăţii din Bucureşti, 2007 ISBN 978-973-737-351-9 51-7:336.71+336.717

This work would not have been possible without the advice and help of many people. Foremost, I wish to express my deep gratitude to: - Professor Vasile PREDA, - Prof. univ. dr. Ion VĂDUVA I would also like to thank all the people who helped me during the course of my studies. Above all, - Rector of Bucharest University Romania, - Rector of Padjadjaran University Bandung Indonesia, - H.E. Nuni Turnijati Djoko,(the Ambasador of the Republic of Indonesia in Bucharest Romania), - Purno Wirawan - Islah Abdullah Dedication

To: - my dear parents,

Halimah and the late Ojon SUPIAN - my wife Deti SUDIARTI, and - my childrens

Sudradjat ISMAIL HASBULLAH, Sudradjat MUHAMMAD IKHSAN, and Sudradjat FITRIYANTI

Preface

Gratitude to the Almighty, the only God, for completeness of this book

entitled “Mathematical Programming Models For Portfolio Selection” so it can be publish

as planed. The subject of this book is in close connection to some mathematical techniques

applications in financial modeling. More specifically, multicriteria portfolio optimization

started with the Markowitz mean-variance model. Basically, Harry Markowitz introduced the

theory of modern portfolios, which originates in a quadratic programming problem applied

for evaluating a portfolio of assets. The resulting model, namely the mean-variance model, is

one of the most used quadratic programming models. Then, Markowitz’s model was

extended in various directions. Recently, some authors implemented dynamic investments

models in order to study long-term effect and improve the performance.

Constructing a dynamic financial model consists of three basic components: 1) a

stochastic differential system of equations for describing the model’s relevant random

quantities development (alternative scenarios are therefore generated); 2) a decision simulator

for finding investor position at each moment and 3) a dynamic optimization model.

In the classical approach of portfolios selection, expected utility theory is applied

based on a set of axioms related to investor’s behavior and on order relation between

deterministic and random events from the set of possible choices. The specific characteristics

of axioms characterizing the utility function take into account the assumption that a

probability measure could be defined on random results. If, in addition, one assumes that the

origins of these random results are not very well known, then the probability theory proves

itself inadequate due to the lack of experimental information. In these situations, the decision

problem could be addressed on uncertainty basis, using different mathematical instruments.

Furthermore, the preferences function describing investor’s utility could be modified with

respect to uncertainty degree.

The portfolio selection problem on uncertainty assumption could be transformed into

a decision problem in fuzzy environment. Fuzzy theory was intensively used from 1960 for

solving many problems, including financial risk management problems. The concept of fuzzy

random variable is a proper extension of classical random variable. Using fuzzy approach, the

experts’ knowledge and subjective opinions of investors could be easier fit in a portfolio

selection model.

The main goal of this book is to examine the methods for solving statistical problems

involving fuzzy element in the random experiment and it aims to be a starting point in

constructing a portfolio selection model of Markowitz type. There are presented models

which involve stochastic dominance constraints on the returns of portfolios and necessary

conditions for possible constraints programming, which are solved by transforming them into

multi-objective linear programming problems.

In the first chapter there is underlined the importance of the topic proposed in this

book, and then, some important results from the literature are presented. Also, in this chapter

are slightly detailed the other chapters of the book, and some results are highlighted.

In the second chapter, “Some classes of stochastic problems”, the relationships

between efficiency sets for some multi-objective determinist programming problems are

presented. These results will be used later in analyzing the concept of efficient solution for a

multi-objective stochastic programming problem. We have to note here the results obtained

in Sections 2.4, 2.6. and 2.7, which extend the results of Cabalero, Cerda, Munoz, Rez,

Stancu-Minasian and White.

In third chapter, “Portofolio optimization with stochastic dominance constraints, it is

considered the construction of a portfolio with finite assets whose returns are described by a

discrete distribution. A portfolio optimization model with stochastic dominance constraints

on the returns is presented. Optimality and duality of these models are studied and, also,

equivalent optimization models are constructed using utility functions.

In forth chapter, “The dominance-constrained portfolio”. We remark the results from

Sections 4.3, 4.4 and 3.6, extending the results of Dentcheva, Ruszczynski, Rothschild,

Stiglitz and Ogrzczak.

In fifth chapter, “Portfolio optimization using fuzzy decision”. In this chapter we

introduce with fuzzy linear programming models and interactive fuzzy linear programming.

Also represents a generalization of Chapter 4. Here optimization problems with stochastic

dominance constraints, using fuzzy decisions. The fuzzy linear programming problems and

fuzzy multi-objective programming problems are thoroughly treated. We remark again the

important results of Sections 5.4, 5.5, 5.6, 5.7 and 5.8. and the extensions of some results

belonging to Markowitz, Klirr, Zuan, Gasimov, Lai and Hwang, and in Section 5.9, we

studied about multiobjective fractional programming problem under fuzziness.

In sixth chapter, “A possibilistic aprroach for portfolio selection problem“ there is

considered a programming problem with possible constraints, which will be solved by

transforming it into a multi-objective programming problem. The results from Sections 6.22,

6.3.3, 6.4 and 6.5, extend some results given by Chen, Inuiguchi, Ramik, Majlender, Yhou

and Li.

In seventh chapter, “Atzbergerţ’s extension of Markowitz portfolio selection”,

represent one basic manner by which Markowitz’s theory for portfolio selection can be

extended to account for non-gaussian distributed returns. We then discuss how a model

incorporating information about the performance of the assets in different market regimes

over the holding period can be developed.

Most of the original results presented in this book were presented in very important

conferences and workshops. Also, we have to note the large list of references considered

elaborating this book.

I wish to acknowledge the teachers, colleagues, and reviewers who contributed to

earlier editions of this book and further to extend my appreciation for the guidance and

suggestions donated during its revision.

Gratitude is particularly due to Prof. DR. Vasile PREDA, Prof. DR. Ion VĂDUVA,

Acad. Marius IOSIFESCU, Dr. Ion M. STANCU-MINASIAN, DR. Miruna BELDIMAN,

DR. Roxana CIUMARA. I would also like to thank all the people who helped me. Above

all, Rector of Bucharest University Romania, Rector of Padjadjaran University Bandung

Indonesia, H.E. Nuni Turnijati Djoko (Ambasador of the Republic of Indonesia in Bucharest

Romania), Islah Abdullah, Purno Wirawan, Sam E. Marentek, Hary Irawan, Pratiwi

Amperawati, Dedin M. Nurdin.

Supian SUDRADJAT

CONTENTS Preface Chapter 1 Introduction …………..……………………….. ……….... 1 Chapter 2 Some classes of stochastic problems …………………....... 7

2.1 Introduction …………………………………………………… 7

2.2 Efficient solution concepts ………………………………… 10

2.3 Relations between the efficient sets of several deterministic

multiobjective programming problems ………………………... 13

2.4 Some relations between expected-value efficient solution,

minimum-variance efficient solutions and expected-value

standar-deviation efficient solutions ………………………... 19

2.5 Multicriteria problems …………………………………………. 20

2.6 Relations between classes of solutions for (P1), (P2) and (P3)….. 21

2.7 White’s approach multiobjective weighting factors auxiliary

optimization problem for (P1), (P2) and (P3) ……………….. 27

2.7.1 Introduction …………………………………………. 28

2.7.2 Transformations and auxiliary optimization problem

associated to (P1), (P2) and (P3) ……………….. 29

2.7.3 Non-convex auxiliary optimization problem 32

Chapter 3 Stochastic dominance …..………….…………………….. 40

3.1 Introduction …………………………………………………… 40

3.2 Stochastic dominance …………………………………………. 42

3.3 The portfolio problem ………………………………………... 44

3.4 Consistency with stochastic dominance ………………………. 45

Chapter 4 The dominance-constrained portfolio problem ………... 52

4.1 Introducere ………………………………………………….. 52

4.2 Dominance-constrained . …….………………………………… 53

4.3 Optimality and duality ………………………………………… 56

4.5 Spliting ….………………………………………………….. 60

4.6 Decomposition …….…………………………………………. 64

Chapter 5 A fuzzy approach to portfolio optimization ……………. 68

5.1 Introduction ………………………………………………….. 68

5.2 Fuzzy linear programming models ..………………………... 69

5.3 Interactive fuzzy programming ……………………………….. 76

5.3.1 Interactive fuzzy linear programming algorithm ……… 78

5.4 Portfolio problem .…………………………………………. 80

5.5 Case of fuzzy technological coefficient and fuzzy

right-hand side numbers ………………………………… 83

5.5.1 Case of fuzzy technological coefficients …………… 83

5.5.2 Portfolio problems with fuzzy technological

coefficients and fuzzy right-hand-side numbers ………. 88

5.6 The modified subgradient method …………………………….. 93

5.7 Defuzzification and solution of defuzzificated problem ………. 96

5.7.1 A modified subgradient method to fuzzy

linear programming ………………………………… 96

5.7.2 Fuzzy decisive set method ..……………………....... 98

5.8 Portfolio problem with fuzzy multiple objective …………….. 110

5.9 Multiobjective fractional programming problem under fuzziness.. 115

5.9.1 Problem formulation and the solution concept ………… 116

5.9.2 Solution algorithm ………………………………….. 122

5.9.3 Basic stability nations for problem (FMOFP) ………. 125

5.9.4 Utilization of Kuhn-Tucker conditions corresponding

to problem ……………………..……………..... 125 )( λP

Chapter 6 A possibilistic approach for a portfolio selection problems .. 128

6.1 Introduction ………………………………………………….. 128

6.2 Mean VaR portfolio selection multiobjective model

with transaction costs ..………………………………………. 129

6.2.1 Case of downside-risk ..…………………………….. 129

6.2.2 Case of proportional transaction costs model ………... 131

6.3 A possibilistic mean Var portfolio selection model …………... 131

6.3.1 Possibilistic theory. Some preliminaries ……………. 132

6.3.2 Triangular and trapezoidal fuzzy numbers …………... 133

6.3.3 Construction of efficient portfolios .……………….. 135

6.4 A weighted possibilistic mean value approach ……………….. 138

6.5 A weighted possibilistic mean variance and covariance

of fuzzy numbers …………………………………………... 142

Chapter 7 An extention of Markowitz portfolio selection ……….. 146

7.1 Introduction ………………………………………………….. 146

7.2 Gaussian mixture distribution ………………………………… 148

7.3 An extention of the Markowitz portfolio theory ……………….. 151

7.4 Portfolio selection problem (GM-PoS) ………………………... 152

Bibliography ….……………………………….……………… 154 Apendix Notations ………………………………………………………. 172

Acronyms & Abbreviations …………………………… 174 Index ………………………………………………….. 175

1

CHAPTER 1

INTRODUCTION

The problem of optimizing a portfolio of finitely many assets is a classical problem in

theoretical and computational finance. Since the seminal work of Markowitz [112] it is

generally agreed that portfolio performance should be measured in two distinct

dimensions: the mean describing the expected return, and the risk which measures the

uncertainty of the return. In the mean–risk approach, we select from the universe of all

possible portfolios those that are efficient: for a given value of the mean they minimize

the risk or, equivalently, for a given value of risk they maximize the mean. This

approach allows one to formulate the problem as a parametric optimization problem, and

it facilitates the trade-off analysis between mean and risk.

In the classical approach to portfolio selection, one often applies the theory of

expected utility that is derived from a set of axioms concerning investor behaviour as

regards the ordering relationship for deterministic and random events in the choice set.

The specific nature of the axioms that characterize the utility function is based on the

assumption that a probability measure can be defined on the random outcomes.

However, if we assume that the origins of these random events are not well known, then

the theory of probability proves inadequate because of a lack of experimental

information. In such instances, one has to approach the decision theory problem under

uncertainty using different mathematical tools. Further, the preference function that

describes the utility of the investor may itself be changing with the degree of uncertainty.

Moreover, one could postulate that the investor has multiple preference functions each of

which corresponds to a particular view on various factors that influence the future state

of the economy and the confidence with which it is held. Under these conditions, the

existing literature in the field of economic theory does not provide the investor with

sufficient tools to address the portfolio selection problem. The discussion above

highlighted potential difficulties one would encounter when addressing the portfolio

selection problem under uncertainty. It was postulated that under uncertainty the investor

2

would be confronted with multiple utility functions. Each one of these utility functions

may be attributed to a particular market view being held and can be broadly described as

capturing the investor’s level of satisfaction if it turns out to be true. For instance, a fund

manager structuring a fixed-income portfolio may have only vague views regarding

future interest rate scenarios and these can broadly be described as being “bullish”,

“bearish” or “neutral”. Such views may arise out of the subjective and/or intuitive

opinion of the decision-maker on the basis of information available at the given point in

time. Under these circumstances, one might try to characterize the range of acceptable

solutions to the portfolio selection problem as a fuzzy set (see Bellman and Zadeh [9]).

In simple terms, a fuzzy set is a class of objects in which there is no clear distinction

between those objects that belong to the class and those that do not. Further, associated

with each object is a membership function that defines the degree of membership of the

object in the set. In this respect, fuzzy set theory provides a framework to deal with

problems in which the source of imprecision is the absence of sharply defined criteria of

class membership rather than the presence of random variables. This provides the point

of departure from probability theory, where the uncertainty arises from the random

nature of the environment rather than from any vagueness of human reasoning. In the

context of choosing optimal portfolios that target returns above the risk-free rate for

certain market scenarios while at the same time guaranteeing a minimum rate of return,

fuzzy decision theory provides an excellent framework for analysis. This is because the

nature of the problem requires one to examine various market scenarios, and each such

scenario will in turn give rise to an objective function. In the face of uncertainty, one will

not be able to assign a numerical value to the probability of these scenarios occurring.

Under this constraint, it is not clear how a suitable weighting vector can be determined to

solve the multi-objective optimization problem. One way to overcome this difficulty is to

use the membership function that arises in fuzzy decision theory to serve as a suitable

preference function for finding an ordering relation for the uncertain events. In fact, one

can describe the membership function as the fuzzy utility of the investor, which

describes the behaviour of indifference, preference or aversion towards uncertainty,

Mathieu-Nicot [115]. The advantage of using the membership function is that it does not

rely necessarily on the existence of a probability measure but rather on the existence of

relative preference between the uncertain events.

3

The above arguments show how the portfolio selection problem under uncertainty can be

transformed into a problem of decision-making in a fuzzy environment Bellman and

Zadeh [9]. To do this, one has to model the aspirations of the investor on the basis of the

strength of the views held on various market scenarios through suitable membership

functions of a fuzzy set. For instance, a fund manager structuring a fixed-income

portfolio may have aspiration levels as to what the portfolio’s acceptable excess return

over the risk-free rate should be for those scenarios he/she considers more likely. The

concepts of fuzzy sets, fuzzy goals and fuzzy decision will be introduced and a fuzzy

multi-criteria optimization problem will be formulated.

As stated by Markowitz in [112,114), “The expected utility maxim appears reasonable

offhand. But so did the expected return maxim. Perhaps there is some equally strong

reason for decisively rejecting the expected utility maxim as well”.

The classical Markowitz model is

[ ])()( xRarx V=ρ ,

where )(xρ is the variance of the return, and )(xR is total return.

The mean–risk portfolio optimization problem is formulated as follows:

[ ])()(max xxx

λρμ −∈X

.

where R and X are defined in section 3.3.

Here, λ is a nonnegative parameter representing our desirable exchange rate of mean

for risk. If 0=λ , the risk has no value and the problem reduces to the problem of

maximizing the mean. If 0>λ we look for a compromise between the mean and the

risk. The general question of constructing mean–risk models which are in harmony with

the stochastic dominance relations has been the subject of the analysis of the recent

papers Dentcheva and Ruszcynski [41,42], Rothschild and Stiglitz [155], Ogryczak and

Ruszczynski [127, 128].

Portfolio selection is generally based on a trade-off between expected return and risk,

and requires a choice for the risk measure to be implemented. Usually, the risk is

evaluated by the conditional second-order moment, i.e., conditional variance or

volatility. This leads to the determination of the mean-variance efficient portfolio

introduced by Markowitz [114]. It can also be based on a safety-first criterion

4

(probability of failure), initially proposed by Roy [149] and then implemented by Levy

and Sarnat [100]. The efficient portfolio is one for which there does not exist another

portfolio that has higher mean and no higher variance, and/or has less variance and no

less mean at the terminal time T . In other words, an efficient portfolio is one that is

Pareto optimal.

Notwithstanding its popularity, mean variance approach has also been subject to a lot of

criticism. Alternative approaches attempt to conform the fundamental assumptions to

reality by dismissing the normality hypothesis in order to account for the fat-tailedness

and the asymmetry of the asset returns. Consequently, other measures of risk, such as

Value at Risk (VaR), expected shortfall, mean absolute deviation, semi-variance and so

on are used.

Another theoretical approach to the portfolio selection problem

- Stochastic dominance (Mosler and Scarsini, [121]), the concept of stochastic dominance

is related to models of risk-averse preferences Fishburn [52]. It originated from the

theory of majorization Hardly, Littlewood and Poya [70] for the discrete case, was later

extended to general distributions Quirk and Saposnik[146]; Hadar and Russel [66];

Hanoch and Levy [68]; Rothschild and Stielits [155], and is now widely used in

economics and finance (Levy [99]).

- The usual (first order) definition of stochastic dominance gives a partial order in the

space of real random variables. More important from the portfolio point of view is the

notion of second-order dominance, which is also defined as a partial order. It is

equivalent to this statement: a random variable R dominates the random variable Y if

)]([)]([ YuERuE ≥ for all non-decreasing concave functions u(·) for which these

expected values are finite. Thus, no risk-averse decision maker will prefer a portfolio

with return Y over a portfolio with return R.

- The stochastic optimization model with stochastic dominance constraints Dentcheva and

Ruszcynsk [42, 44], can be used for risk-averse portfolio optimization. We add to the

portfolio problem the condition that the portfolio return stochastically dominates a

reference return, for example, the return of an index. We identify concave non-

decreasing utility functions which correspond to dominance constraints. Maximizing the

5

expected return modified by these utility functions, guarantees that the optimal portfolio

return will dominate the given reference return.

- Fuzzy set theory, since 1960s, has been widely used to solve many problems including

financial risk management. The concept of a fuzzy random variable is a reasonable

extension of the concept of a usual random variable in the many practical applications of

random experiments, where the implicit assumption of data precision seems to be an

inappropriate simplification rather than an adequate modeling of the real physical

conditions. By using fuzzy approaches, the experts’ knowledge and the investors’

subjective opinions can be better integrated into a portfolio selection model. Bellman and

Zadeh [9] proposed the fuzzy decision theory. Ramaswamy [14] presented a bond

portfolio selection model based on the fuzzy decision theory, Sudradjat and Preda [188]

presented on portfolio optimization using fuzzy decisions. The notion of a fuzzy random

variable (see for example, Kwakernaak [91], Puri and Ralescu [145], Kruse and Meyer

[89] provides a valuable model that is manageable in a probabilistic framework. Also,

the concept of fuzzy information presented by Zadeh [216] can formalize either the

experimental data or the events involving fuzziness. The concept of a fuzzy random

variable Puri and Ralescu [145] was defined as a tool for establishing relationships

between the outcomes of a random experiment and inexact data, Ostermark [128]

proposed a dynamic portfolio management model. Watada [201] presented another type

of portfolio selection model based on the fuzzy decision principle. The model is directly

related to the mean-variance model, where the goal rate for an expected return and the

corresponding risk described by logistic membership functions.

- In standard portfolio models uncertainty is equated with randomness, which actually

combines both objectively observable and testable random events with subjective

judgments of the decision maker into probability assessments. A purist on theory

would accept the use of probability theory to deal with observable random events, but

would frown upon the transformation of subjective judgments to probabilities. Tanaka

et al [194] give a special formulation of fuzzy decision problems by the probability

events. Carlsson et al [26] studied the portfolio selection model in which the rate of

return of security follows the possibility distribution. Sudradjat, Popescu and Ghica

[187] studied on possibilistic approach a portfolio selection problem. Applying

possibilistic distribution may have two advantages: (1) the knowledge of the expert can

6

be easily introduced to the estimation of the return rates and (2) the reduced problem is

more tractable than that of the stochastic programming approach. Korner [86] pointed

out that the variability is given by two kinds of uncertainties: randomness (stochastic

variability) and imprecision (vagueness). Randomness models the stochastic variability

of all possible outcomes of an experiment. Fuzziness describes the vagueness of the

given or realized outcome. Kwakernaak [91] presented another explanation for the

difference between randomness and fuzziness. He pointed out that when we consider

an opinion poll in which a number of people are questioned, randomness occurs

because it is not known which response may be expected from any given individual.

Once the response is available, there still is uncertainty about the precise meaning of

the response.

The aim of this book is to examine methods for handling statistical problems

involving fuzziness in the elements of the random experiment, and serves as a point from

which to derive the Markowitz portfolio model in the presence of efficient solution

concepts for a stochastic multi-objective programming, develop portfolio optimization

model involving stochastic dominance constraints on the portfolio return and necessary

and sufficient conditions of optimality and duality, we develop portfolio optimization

using fuzzy decisions in concentrate on fuzzy linear programming, and finally we

consider a mathematical programming model with possibilistic constraint and we it solve

by transforming into multi-objective linear programming problem.

7

CHAPTER 2

SOME CLASES OF STOCHASTIC PROBLEMS

2.1 Introduction Stochastic programming deals with a class of optimization models and algorithms in

which some of the data may be subject to significant uncertainty. Such models are

appropriate when data evolve over time and decisions need to be made prior to observing

the entire data stream. For instance, investment decisions in portfolio planning problems

must be implemented before stock performance can be observed. Similarly, utilities must

plan power generation before the demand for electricity is realized. Such inherent

uncertainty is amplified by technological innovation and market forces. As an example,

consider the electric power industry. Deregulation of the electric power market, and the

possibility of personal electricity generators (e.g. gas turbines) are some of the causes of

uncertainty in the industry. Under these circumstances it pays to develop models in

which plans are evaluated against a variety of future scenarios that represent alternative

outcomes of data. Such models yield plans that are better able to hedge against losses

and catastrophic failures. Because of these properties, stochastic programming models

have been developed for a variety of applications, including electric power generation

(Murphy [124]), financial planning (Carino et al [23]), telecommunications network

planning (Sen et al [170]), and supply chain management (Fisher et al [51]), to mention a

few.

The widespread applicability of stochastic programming models has attracted

considerable attention from the OR/MS community, resulting in several recent books

(Kall and Wallace [77], Birge and Louveaux [16], Prekopa [138, 139]) and survey

articles (Birge [15], Sen and Higle [169]). Nevertheless, stochastic programming models

remain one of the more challenging optimization problems.

While stochastic programming grew out of the need to incorporate uncertainty in linear

and other optimization models (Dantzig [39], Beale [8], Charnes and Cooper [30]), it has

close connections with other paradigms for decision making under uncertainty. For

8

instance, decision analysis, dynamic programming and stochastic control, all address

similar problems, and each is effective in certain domains. Decision analysis is usually

restricted to problems in which discrete choices are evaluated in view of sequential

observations of discrete random variables. One of the main strengths of the decision

analytic approach is that it allows the decision maker to use very general preference

functions in comparing alternative courses of action. Thus, both single and multi-

objectives are incorporated in the decision analytic framework. Unfortunately, the need

to enumerate all choices (decisions) as well as outcomes (of random variables) limits this

approach to decision making problems in which only a few strategic alternatives are

considered.

These limitations are similar to methods based on dynamic programming, which also

require finite action (decision) and state spaces. Under Markovian assumptions the

dynamic programming approach can also be used to devise optimal (stationary) policies

for infinite horizon problems of stochastic control (see also Neuro-Dynamic

Programming by Bertsekas and Tsitsiklis [13]). However, DP-based control remains

wedded to Markovian Decision Problems, whereas path dependence is significant in a

variety of emerging applications. Stochastic programming provides a general framework

to model path dependence of the stochastic process within an optimization model.

Furthermore, it permits uncountably many states and actions, together with constraints,

time-lags etc. One of the important distinctions that should be highlighted is that unlike

dynamic programming, stochastic programming separates the model formulation activity

from the solution algorithm. One advantage of this separation is that it is not necessary

for stochastic programming models to all the same mathematical assumptions. This leads

to a rich class of models for which a variety of algorithms can be developed. On the

downside of the ledger, stochastic programming formulations can lead to very large scale

problems, and methods based on approximation and decomposition become paramount.

A whole series of production processes, economic system of different types, and

technical objective is described by mathematical models which are multi-criteria

optimization problems (Steuer [177], Chankong and Haimes [29] and Stancu-Minasian

[175]) . This situation is quite usual, because frequently it is necessary to take into

9

account simultaneously the influence of a number of contradictory external factors on

the system.

The most intensive development of the theory and the methods of detailed bibliographic

description of which is given in Zeleny [217] and Urli and Nadeau [196], are linear and

non linear multi-criteria optimization problems. Some classifications of the methods of

this type, oriented to the specific user, and multi-criteria optimization problems with

contradictory constraints were explored in are given (Salukavadze and Topchishvili

[166]). Very interesting results generalized into the general domination cone for different

classes of solutions of multi-criteria problem are given (Salukavadze and Topchishvili

[166]).

Now, one of the widely developing fields in multi-criteria optimization is its qualitative

theory; the most important results are given (Salukavadze and Topchishvili [166]). Well-

known algorithms can be modified and new theoritical results.

The objective of this chapter is to examine some properties of different classes of multi-

criteria optimization problem solutions.

Most real-life engineering optimization problems require simultaneous optimization of

more than one objective function. In these cases, it is unlikely that the same values of

design variables will results in the best optimal values for all the objectives. Hence, some

trade-off between the objectives is needed to ensure a satisfactory design.

As the system efficiency indices can be different (and mutually contradictory), it is

reasonable to use the multi-objective approach to optimize the overall efficiency. This

can be done mathematically correctly only when some optimality principle is used. We

use Pareto optimality principle, the essence of which is following. The multi-objective

optimization problem solution is considered to be Pareto-optimal if there are no other

solutions that are better in satisfying all of the objectives simultaneously. That is, there

can be other solutions that are better in satisfying one or several objectives, but they

must be worse than the Pareto-optimal solution in satisfying the remaining objectives.

10

2.2 Efficient solution concepts Consider a model in which the design/decision associated with a system is specified via

vector x. Under uncertainty, the system operates in an environment in which there are

uncontrollable parameters which are modeled using random variables. Consequently, the

performance of such a system can also be viewed as a random variable. Accordingly,

stochastic programming models provide a framework in which designs (x) can be chosen

to optimize some measure of the performance (random variable). It is therefore natural to

consider measures such as the worst case performance, expectation and other moments

of performance, or even the probability of attaining a predetermined performance goal.

Let us consider the stochastic multi-objective programming problem Caballero, et al [21]

( ))~,(),...,~,(min 1 cxzcxz qDx∈, (2.1)

where the following notations and assumptions

• there is a compact set nD R⊆ of feasible actions;

• nx R∈ is thevector of decision variables of the problem and c~ is a random

vector whose components are random continous variables, defined on the set nRE∈ . We assume given the family F of events (that is, subset of E ) and the

distribution of probability P defined on F so that, for any subset of E , E⊂A ,

F∈A , the probability P(A) is known. Also, we assume that the distribution of

probability P is independent of the decision variables nxx ,...,1 ;

• there are q objective functions )( ⋅kf with +∈R)(xfk for all Dx∈ and

c~ is a random vector whose components are random continuous variable;

• it is required to find members of the efficient (vector minimal) set E of D with

respect to the order relation ≤ on qR , where, by definition,

)()()()(,: xfyfxfyfDyDxE =→≤∈∈= (2.2)

Let )(xzk is the expected value of the kth objective function, and let )(xkσ be its

standard deviation, ,...,1 qk ∈ . Let us assume that, for every ,...,1 qk ∈ and for

every feasible vector x of the stochastic multi-objective programming problem, the

standard deviation )(xkσ is finite. In this section we will shows the definitions and

11

relations between expected value standard deviation efficient solution and efficient

solutions.

Next the following definitions by Caballero, et al [21],

Definition 2.1 [21] Expected-Value Efficient Solution. The point Dx∈ is an expected-

value efficient solution of the stochastic multi-objective problem if it is Pareto efficient to

the following problem :

( ))(),...,(min:)( 1 xzxzPE qDx∈.

Let PEE be the set of expected-value efficient solution of the stochastic multi-objective

problem.

Definition 2.2 [21] Minimum-Variance Efficient Solution. The point Dx∈ is a

minimum-variance efficient solution for the stochastic multi-objective problem if it is a

Pareto efficient solution for the problem :

( ))(),...,(min:)( 221

2 xxP qDxσσσ

∈.

Let 2σPE be the set of efficient solutions of the problem )( 2σP .

Definition 2.3 [21] Expected-Value Standard-Deviation Efficient Solution or σE

Efficient Solution. The point Dx∈ is an expected-value standard-deviation efficient

solution for the stochastic multi-objective programming problem if it is a Pareto efficient

solution to the problem

( ))(),...,(),(),...,(min:)( 11 xxxzxzPE qqDxσσσ

∈.

Let σPEE be the set of expected-value standard-deviation efficient solutions of the

stochastic multi-objective programming problem (2.1).

Now, we give the concepts of efficiency for two criteria of maximum probability. As we

will see next, in order to define these two concepts, the minimum-risk criterion (concept

of minimum-risk efficiency) and Kataoka criterion (efficiency in probability) are applied

respectively to each stochastic objective.

Definition 2.4 [21] Minimum-Risk Efficient Solution for the Levels quu ,...,1 . See

Stancu-Minasian and Tigan (180). The point Dx∈ is a minimum risk vectorial solution

for levels quu ,...,1 if it is a Pareto efficient solution to the problem:

12

( )))~,(~(,...,)~,(~(max:))(( 11 qqDxucxzPucxzPuPRM ≤≤

∈,

Let )(uPRME be the set of efficient solution for the problem (PMR(u)).

Definition 2.5 [21] Efficient Solution with Probabilities qββ ,...,1 or β -Efficient

Solution. The point Dx∈ is an efficient solution with probabilities qββ ,...,1 if there

exist qu R∈ such that ttt ux ),( is a Pareto efficient solution to problem:

))(( βPP

( )

DxqkucxzP

uu

kkk

qux

∈=≥≤ ,,1,)~,(~

,...,min 1,

β

Let )(βPPE nR⊂ be the set of efficient solutions with probabilities qββ ,...,1 for the

stochastic multi-objective programming problem (2.1).

It may be noted that these definitions of efficient solution are obtained by applying the

same transformation criterion to each one of the objectives separately (expected value,

minimum variance, etc.), and by building after word the resulting deterministic

multiobjective problem. In this sense, it is necessary to the following results.

Remark 2.1 The concepts of expected value, minimum variance, etc., weak and properly

efficient solution can be defined in a natural way.

Remark 2.2 The concepts of minimum-risk efficiency and β -efficiency require setting a

priori a vector of aspiration levels u or a probability vector β . This implies that, in both

cases, the efficient set obtained depends on the predetermined vectors in such a way that,

in general, different level and proba bility vector give rise to different efficient sets,

).()(

),()(''

''

ββββ PPPP

PRMPRM

EE

uEuEuu

≠⇒≠

≠⇒≠

Remark 2.3 The concept of expected standard-deviation efficient solution is an

extention to multiobjective case of the concept of the mean-variance efficient solution

that Markowitz [114] defines for the stochastic single objective problem of portfolio

selection. In this way, we have the two statistical moments corresponding to each

stochastic objective in the same measuring units. Since the square root function is

13

strictly increasing, the set of efficient solutions does not vary in problem if we substitute

standard deviation for variance, White [209].

Remark 2.46 The efficiency in probability criterion is a generalization of the one

presented by Goicoechea, Hansen, and Duckstein [63], who define the same concept

taking the same probability β for all the stochastic objectives and with the probabilistic

equality constraints taking the form

β=≤ )~,(~ kk ucxzP .

This notion was introduced by Stancu-Minasian [179], considering the Kataoka problem

in the case of multiple criteria.

2.3 Relations between the efficient sets of several of deterministic

multiobjective programming problems

We present some relations between the efficient sets of several problems of deterministic

multi-objective programming problems. These results will be used later for analysis of

concepts of efficient solutions for multi-objective stochastic problems.

Considered f and g be vectorial functions defined on the same set nH R⊂ with

nHf R⊂: qR→ and nHg R⊂: qR→ and let γα , be nonnull vectors with q

real components, that is, qR∈γα , and 0, ≠γα . Let us consider the following

multiobjective problems:

(PD1) ( )))(()),...,((),(),...,(min 111 xgxgxfxf qqqDxγγ

∈ (2.3)

(PD2) ( ))(),...,(min 1 xfxf qDx∈ (2.4)

(PD3) ( ))))(()),...,((min 2211 xgxg qqDx

γγ∈

(2.5)

with, qR∈γ , 0≠γ . Let 321 ,, EEE be the sets of weakly efficient, efficient, and

proper efficient points of problem )( iPD , respectively. The following theorem relates

these problems (PD1), (PD2) and (PD3) problems to each other.

14

Theorem 2.1 We assume that 0>g for every Dx∈ ,. Then:

i1 132 EEE ⊂∩

i2 wEEE 132 ⊂∪

i3 www EEE 132 ⊂∪

Proof:

( 1i ) 32 EEx ∩∈

Let us show that 1Ex∈ by reductio ad absurdum. We assume that 1Ex∉ . Then, there

exist an Dx ∈* such that )()( * xfxf kk ≤ and ))(())(( * xgxg kkkk γγ ≤ , for every

,...,1 qk ∈ , there being an ,...,1 qs∈ for which the inequality is strict,

)()( * xfxf ss < or ))(())(( * xgxg ssss γγ < .

Therefore, 2Ex∉ or 3Ex∉ , since ))(())(( * xgxg kkkk γγ ≤ , implies ≤))(( *0 xg skkγ

))(( 0 xg skkγ , contrary to 32 EEx ∩∈ .

(i2 ) wEEE 132 ⊂∪

Let 32 EEx ∪∈ . Let us see that wEx 1∈ by reductio de absurdum. We assume that

wEx 1∉ . Then, there exist a vector Dx ∈* that weakly dominates x and so verifies

)()( * xfxf kk < and ))(())(( * xgxg kkkk γγ < , for every ,...,1 qk = . Thus, 2Ex∉

and, since ))(())(( * xgxg kkkk γγ < , implies <))(( *0 xg skkγ ))(( 0 xg s

kkγ , 3Ex∉ ,

contrary to 32 EEx ∪∈ .

(i3) www EEE 132 ⊂∪

Let ww EEx 32 ∪∈ . Let us see that wEx 1∈ by reductio de absurdum. We assume that

wEx 1∉ . Then, there exist a vector Dx ∈* that weakly dominates the vector x and

therefore verifies that )()( * xfxf kk < and ))(())(( * xgxg kkkk γγ < , for every

,...,1 qk ∈ . Thus, wEx 2∉ and, since ))(())(( * xgxg kkkk γγ < , implies

<))(( *0 xg skkγ ))(( 0 xg s

kkλ , wEx 3∉ , contrary to ww EEx 32 ∪∈ .

15

Thus, (i2) can be deduced from (i3)

Now we consider the following problem

( )))(()()),...,(()(min 111 xgxfxgxf qqqDxαα ++

∈ (2.6)

where q1 :),...,( RR →= +qααα .

Let )(4 αE and )(4 αGE denote the efficient solutions set and the properly efficient

solutions set respectively for problem (2.6). We will now present some relations between

these sets and the set of efficient solutions and properly efficient solutions for problem

(PD1).

Theorem 2.2 [21] For qq RR →= +:),...,( 1 γγγ , q

1 :),...,( RR →= +qααα , with

0, ≠kk γα and qksignsign kk ,1),()( == γα , the following relation holds :

14 )( EE ⊂α .

Proof: Let )(4 αEx∈ . We assume that 1Ex∉ . In this case, there is a solution *x that

dominates the solution x, that is,

)()( * xfxf kk ≤ and ))(())(( * xgxg kkkk γγ ≤ , for every ,...,1 qk ∈ , and there

exist at least one ,...,1 qs∈ for which the inequality is strict, that is,

)()( * xfxf ss < or ))(())(( * xgxg ssss γγ <

From this point onward, since

)()( * xfxf kk ≤ , ))(())(( * xgxg kkkk γγ ≤ , implies ≤))(( *xgkkα ))(( xgkkα ,

the following inequalities are verified:

))(()())(()( *** xgxfxgxf kkkkkk λα +≤+ , for every ,...,1 qk ∈ , (2.7)

))(()())(()( * xgxfxgxf kkkkkk αα +≤+ , for every ,...,1 qk ∈ . (2.8)

From (2.7) and (2.8), we obtain

))(()())(()( ** xgxfxgxf kkkkkk αα +≤+ , for every ,...,1 qk ∈ .

In particular, for sk = , we have the results bellow:

(a) if )()( * xfxf ss < ,

))(()())(()( *** xgxfxgxf ssssss αα +<+ ,

16

and the following inequality is obtained from (2.8):

))(()())(()( ** xgxfxgxf ssssss αα +<+ ;

(b) if ))(())(( * xgxg ssss αα < ,

))(()())(()( *** xgxfxgxf ssssss αα +<+ ,

and since )()( * xfxf ss ≤ , we obtain

))(()())(()( ** xgxfxgxf ssssss αα +<+ .

Therefore, for every ,...,1 qk ∈ ,

))(()())(()( ** xgxfxgxf kkkkkk αα +≤+ ,

and there is at least a subscript ,...,1 qs∈ for which

))(()())(()( ** xgxfxgxf ssssss αα +<+ ,

which implies that the solution *x dominates the solution x; therefore, we reach a

contradiction with the hypothesis of *x being the efficient solution to problem (2.6).

Next, we prove that, in some conditions, this relationship is hold for the set of properly

efficient solution. For this purpose, we define problems ),(, μλγgfP and )(ξαP ,

obtained by applying the weighting method to problems (2.3)-(2.6) respectively as

follows:

∑=

∈+

q

kkkk

t

Dxgf xgxfP1

, )()(min:)),(( γμλμλγ ,

))()((min:))((1

xgxfP kkk

q

kkDx

αξξα +∑=

∈.

We use the results available in the literature about the relationships between the

optimal solution to the weighting problem and the efficient solutions to the multi-

objective problem. Some results, see Chankong and Haimes [29], applied to problem

(2.3) and its associated weighted problem ),(, μλγgfP , are as follow :

17

(a) If f and tqq gg ),...,( 11 γγ are convex functions, D is convex, and *x is a properly

efficient solution for the multi-objective problem (2.3), there exist some weight

vector μλ, with strictly positive components such that *x is the optimal solution

for weighted problem ),(, μλγgfP .

(b) For each vector of weights with strictly positive components, the optimal solution to

the weighted problem ),(, μλγgfP is properly efficient for the multi-objective

problem (P1).

Proposition 2.1 If f and ))(),...,(( 11 qq gg γγ are convex functions, D is a convex set

and there exist qq RR →= +:),...,( 1 ααα , )()( kk signsign γα = , for every

,...,1 qk ∈ then GG EE 14 )( ⊂α .

Proof: If f and ))(),...,(( 11 qq gg γγ are convex functions and if D is a convex set, then

the set of properly efficient solutions to problems (PD1) and (2.6) are obtained from the

associated weighted problems for strictly positive weight vectors. We will prove that any

solutions to the optimization problem )(ξαP , with 0>ξ , is a solution to problem

),(, μλγgfP for some vector 0),( >μλ .

Let )(4 αGEx∈ . Then, given the established hypotheses, there exist a vector 0>ξ for

which x is the solution for problem )(ξαP . Let us assume that, for every

0,,,...,1 ≠∈ kkqk γα . Then, we take

kk ξλ = , 0,,/ >= kkkkkk μλγαξμ ,

Since 0>ξ , we obtain that x is an optimal solution to problem ),(, μλλgfP . For some

,...,1 qi∈ if 0== ii γα , then the proof would be the same, since in problem (2.3)

the function ig is not involved and since in problem (2.6) the function ith objective

would be if .

In general, the inverse inclution does not hold, as it’s shown by the following example.

18

Example 2.1. Let us consider the following problem:

,0,

,19./

),,(max

22

,

≥≤+

yxyxts

yxyx

with 1,),(,),( === uyyxgxyxf .

The set of efficient points for this problem is 0,,14/),( 222 >=+∈ yxyxyx t R

and is represented in Fig. 2.1.

We outline the solution of the problem

,0,

,19./

,max

22

,

≥≤+

+

yxyxts

yxyx

α

with 0>α . For each fixed 0>α , the optimal solution of the resulting problem is one

of property efficient solutions to the original becriterion problem.

y

1 ε

D

3 x

Figura 2.1

Proposition 2.2 If f and ))(),...,(( 11 qq gg γγ are convex functions, then

UΩ∈

⊂α

α )(41GG EE ,

with ,1),()(:),...,( 1 qksignsignR kkq

q ==→==Ω + γαααα R .

Proof: As the previous case, the proof of the proposition is carried out by demonstrating

that any solution to the problem ),(, μλγgfP is a solution to the problem )(ξαP for

some vector nR∈α , with ,...,1),()( qksignsign kk ∈= γα , and for some 0>ξ .

19

Consider GEx 1∈ . Since f and tqq gg ),...,( 11 γγ are convex functions, there exist vector

0, >μλ such that x is a solution to problem ),( μλfugP . Because 0, >μξ we put

kk λξ = , k

kkk ξ

γμα = ,

since 0, >μξ , therefore we obtain that x is also a solution to the problem )(ξαP .

From Proposition 2.1 and Proposition 2.2, if f and tqq gg ),...,( 11 γγ are convex

functions and if )()( kk signsign γα = , 0)().( >tt kk γα , for every ,...,1 qk ∈ , the

sets of properly efficient solutions to problem (2.3) and (2.6) verify the following

properties:

a. Every properly efficient solution to problem (2.6) is properly efficient for problem

(2.3);

b. Setting qR∈γ , with nonnull components, the set of properly efficient solutions to

problem (2.3) is a subset of the union in α of the set of properly efficient solutions

for problem (2.6).

2.4 Some relation between expected-value efficient solution, minimum-variance efficient solution and expected-value standard deviation efficient solution

Consider a problem (2.1) and sets efficient solution expected value ( PEE ) minimum

variance ( 2σPEE ), and expected value standard deviation ( σPEE ) associated with the

problem. Let wPE

wPE

wPE EEE σσ

,, 2 be the sets of weakly efficient solutions associated with

the problems in Definitions 2.1-2.3, respectively.

If we consider

)()(),()( xxgxzxf kkkk σ==

And if we choose 1=γ , given that, for ,...,1 qk ∈ , its verified that +→ RR n:σ ,

then the relations between these efficient sets are deduced directly from Theorem

20

2.5 Multi-criteria problems

Consider the following model of a multi-criteria optimization problem:

( ))(),...,(min 1 xFxF q (2.8)

Dx∈ (2.9)

where D is a nonempty set of all feasible solution, mD R⊂ ; R→DFF q :,...,1 . Stated

briefly, a multi-criteria optimization problem consists in the choice of a particular

solution Dx ∈* for which all of the utility functions qkxFk ,1),( = , simultaneously

approach bigger values or at least do not decrease.

Let us recall some concepts of multi-criteria optimization problem solutions; (Zeleny

[217] and Urli and Nadeau [196], Salukavadze and Topchishvili [166]).

Definition 2.6 The solution DxP ∈ is called Pareto-optimal (or efficient) for the

problem (2.8)-(2.9) if and only if, for every Dx∈ , the system of inequalities

)()( Pkk xFxF < , qk ,1= ,where at least one inequality is strict, is inconsistent.

Definition 2.7 The solution Dxw ∈ , is called weakly efficient (or Slater-optimal) for the

problem (2.8)-(2.9) if and only if, for every Dx∈ , the system of strict inequalities

)()( wkk xFxF < , qk ,1= , is inconsistent.

Definition 2.8 The solution DxG ∈ , is called proper efficient (or Geoffrion-optimal)

for the problem (2.8)-(2.9) if and only if it is a Pareto-optimal solution for the problem

(2.8)-(2.9) and there exists a positive number 0>θ such that, for each pk ,1= , we

have

θ≤−− )]()(/[)]()([ xFxFxFxF jG

jG

kk ,

for some j such that )()( Gjj xFxF > where Dx∈ and )()( G

kk xFxF < qk ,1= , is

inconsistent.

Let ,wjE ,E G

jE denoted the sets of weakly-efficient, efficient, and proper efficient

solutions, respectively, for the problem (2.8)-(2.9). It is obvious that

GjE ⊂ E ⊂ w

jE .

21

Next we will studied some relations between the efficient sets of several problems of

deterministic multi-objective programming.

Let f and g be vectorial functions defined on the same set nH R⊆ , with

qRHf →: and qRHg +→: . Let us consider the following multi-objective problems:

(P1) ( )))(()),...,((),(),...,(min 111 xguxguxfxf qqqDx∈ (2.10)

(P2) ( ))(),...,(min 1 xfxf qDx∈ (2.11)

(P3) ( ))))(()),...,((min 0011 xguxgu s

qqs

Dx∈ (2.12)

with, HD ⊆ , qu RR →+: , ),...,( 1 quuu = and 00 >s a real number.

2.6 Relations between classes of solutions for (P1), (P2) and (P3)

We present some relations between the efficient sets of above considered deterministic

multi-objective programming problems. These results will be used later for analysis of

concepts of efficient solutions for multi-objective stochastic problems. These results

extend Section 2.4.

For 3,2,1=i , let Gii

wi EEE ,, be the sets of weakly efficient, efficient, and proper

efficient points of problem )( iP , respectively. The following theorem relates these

problems (P1), (P2) and (P3) problems to each other.

Theorem 2.3 We assume that 0>g for every Dx∈ , and for 0, 21 >tt and qk ,1=

we have )()()( 21 tutu kk <≤ implies that )()()( 0021s

ks

k tutu <≤ . Then:

(i) 132 EEE ⊂∩

(ii) wEEE 132 ⊂∪

(iii) www EEE 132 ⊂∪

Proof:

(i) 32 EEx ∩∈

22

Let us show that 1Ex∈ by reductio ad absurdum. We assume that 1Ex∉ . Then, there

exist an Dx ∈* such that

)()( * xfxf kk ≤ and ))(())(( * xguxgu kkkk ≤ , for every ,...,1 qk ∈ , there

being an ,...,1 qs∈ for which the inequality is strict,

)()( * xfxf ss < or ))(())(( * xguxgu ssss < .

Therefore, 2Ex∉ or 3Ex∉ , since ))(())(( * xguxgu kkkk ≤ , implies ≤))(( *0 xgu skk

))(( 0 xgu skk , contrary to 32 EEx ∩∈ .

(ii) wEEE 132 ⊂∪

Let 32 EEx ∪∈ . Let us see that wEx 1∈ by reductio de absurdum. We assume that

wEx 1∉ . Then, there exist a vector Dx ∈* that weakly dominates x and so verifies

)()( * xfxf kk < and ))(())(( * xguxgu kkkk < , for every ,...,1 qk = . Thus, 2Ex∉

and, since ))(())(( * xguxgu kkkk < , implies <))(( *0 xgu skk ))(( 0 xgu s

kk , 3Ex∉ ,

contrary to 32 EEx ∪∈ .

(iii) www EEE 132 ⊂∪

Let ww EEx 32 ∪∈ . Let us see that wEx 1∈ by reductio de absurdum. We assume that

wEx 1∉ . Then, there exist a vector Dx ∈* that weakly dominates the vector x and

therefore verifies that )()( * xfxf kk < and ))(())(( * xguxgu kkkk < , for every

,...,1 qk ∈ . Thus, wEx 2∉ and, since ))(())(( * xguxgu kkkk < , implies

<))(( *0 xgu skk ))(( 0 xgu s

kk , wEx 3∉ , contrary to ww EEx 32 ∪∈ .

Thus, (ii) can be deduced from (iii)

Remark 2.5 Also we see that (ii) can be deduced from (iii). It is obvious that

www EEE 132 ⊂∩

is also verified. Futhermore, as

wEE 22 ⊂ and wEE 33 ⊂

23

then

ww EEEE 3131 ∪⊂∪ .

Remark 2.6 We note that RR →+:u with ttu λ=)( or 0,)( >= pttu pλ and

0>λ satisfy condition from Theorem 2.3.

Now we consider the following problem

( )))((~)()),...,((~)(min 111 xguxfxguxf qqqDx++

∈ (2.13)

where q1 :)~,...,~(~ RR →= +quuu .

Let )~(4 uE and )~(4 uEG denote the efficient solutions set and the properly efficient

solutions set respectively for problem (2.13). We will now present some relations

between these sets and the set of efficient solutions and properly efficient solutions for

problem (P1).

Theorem 2.4 For qquuu RR →= +:),...,( 1 and q

1 :)~,...,~(~ RR →= +quuu , such that

)()()( 21 tutu kk <≤ implies )(~)()(~21 tutu kk <≤ , qk ,1= , the following relation holds :

14 )~( EuE ⊂ .

Proof: Let )~(4 uEx∈ . We assume that 1Ex∉ . In this case, there is a solution *x that

dominates the solution x, that is,

)()( * xfxf kk ≤ and ))(())(( * xguxgu kkkk ≤ , for every ,...,1 qk ∈ , and there

exist at least one ,...,1 qs∈ for which the inequality is strict, that is,

)()( * xfxf ss < or ))(())(( * xguxgu ssss <

From this point onward, since

)()( * xfxf kk ≤ , ))(())(( * xguxgu kkkk ≤ , implies ≤))((~ *xgu kk ))((~ xgu kk ,

the following inequalities are verified:

))((~)())((~)( *** xguxfxguxf kkkkkk +≤+ , for every ,...,1 qk ∈ , (2.14)

))((~)())((~)( * xguxfxguxf kkkkkk +≤+ , for every ,...,1 qk ∈ . (2.15)

From (2.14) and (2.14), we obtain

24

))((~)())((~)( ** xguxfxguxf kkkkkk +≤+ , for every ,...,1 qk ∈ .

In particular, for sk = , we have the results bellow:

(c) if )()( * xfxf ss < ,

))((~)())((~)( *** xguxfxguxf ssssss +<+ ,

and the following inequality is obtained from (2.15):

))((~)())((~)( ** xguxfxguxf ssssss +<+ ;

(d) if ))((~))((~ * xguxgu ssss < ,

))((~)())((~)( *** xguxfxguxf ssssss +<+ ,

and since )()( * xfxf ss ≤ , we obtain

))((~)())((~)( ** xguxfxguxf ssssss +<+ .

Therefore, for every ,...,1 qk ∈ ,

))((~)())((~)( ** xguxfxguxf kkkkkk +≤+ ,

and there is at least a subscript ,...,1 qs∈ for which

))((~)())((~)( ** xguxfxguxf ssssss +<+ ,

which implies that the solution *x dominates the solution x; therefore, we reach a

contradiction with the hypothesis of *x being the efficient solution to problem

(2.13).

Next, we prove that, in some conditions, this relationship is hold for the set of properly

efficient solution. For this purpose, we define problems ),( μλfugP and )(~ ξuP , obtained

by applying the weighting method to problems (2.10) and (2.13) respectively as follows:

( )∑=

∈+

q

kkkkkkDxfug xguxfP

1))(()(min:)),(( μλμλ ,

)))((~)((min:))((1

~ xguxfP kkk

q

kkDxu +∑

=∈

ξξ .

We use the results available in the literature about the relationships between the

optimal solution to the weighting problem and the efficient solutions to the multi-

25

objective problem. Some results, see Chankong and Haimes [29], applied to problem

(2.10) and its associated weighted problem ),( μλfugP , are as follow :

(c) If f and ))(),...,(( 11 qq gugu are convex functions, D is convex, and *x is a properly

efficient solution for the multi-objective problem (2.10), there exist some weight

vector μλ, with strictly positive components such that *x is the optimal solution

for weighted problem ),( μλfugP .

(d) For each vector of weights with strictly positive components, the optimal solution to

the weighted problem ),( μλfugP is properly efficient for the multi-objective

problem (P1).

Proposition 2.3 If f and ))(),...,(( 11 qq gugu are convex functions, D is a convex set

and there exist qquuu RR →= +:)~,...,~(~

1 with 0)(~).( >tutu kk , for every 0≥t and

,...,1 qk ∈ then GG EuE 14 )~( ⊂ .

Proof: If f and ))(),...,(( 11 qq gugu are convex functions and if D is a convex set, then

the set of properly efficient solutions to problems (P1) and (2.13) are obtained from the

associated weighted problems for strictly positive weight vectors. We will prove that any

solutions to the optimization problem )(~ ξuP , with 0>ξ , is a solution to problem

),( μλfugP for some vector 0),( >μλ .

Let )~(4 uEx G∈ . Then, given the established hypotheses, there exist a vector 0>ξ for

which x is the solution for problem )(~ ξuP . Let us assume that, for every

0,,...,1 ≠∈ kuqk , then we take

kk ξλ = , u~ and kμ such that )()(~ tutu kkkk μξ = for ,...,1 qk ∈ .

Since 0>ξ , we obtain that x is an optimal solution to problem ),( μλfugP . For some

,...,1 qk ∈ if 0=ku , then the proof would be the same, since in problem (2.10) the

function ig is not involved and since in problem (2.13) the kth objective would be kf .

Proposition 2.4 If f and ))(),...,(( 11 qq gugu are convex functions, then

26

UΩ∈

⊂u

GG uEE~

41 )~( ,

with ,,0)(.)(~:)~,...,~(~ 1 tktutuRuuu kkq

q ∀>→==Ω + R .

Proof: Using the Proposition 2.3, we see that the proof of this proposition is carried out

by demonstrating that any solution to the problem ),( μλfugP is a solution to the

problem )(~ ξuP with qquuu RR →= +:)~,...,~(~

1 , 0)().(~ >tutu kk , tk,∀ and for some

0>ξ .

Consider GEx 1∈ . Since f and ))(),...,(( 11 qq gugu are convex functions, there exist

vector 0, >μλ such that x is a solution to problem ),( μλfugP . Because 0, >μξ we

put

kk λξ = , k

kkk

tutuξ

μ )(.)(~ = , for +∈∈ Rtqk ,,...,1 .

Therefore we obtain that x is also a solution to the problem )(~ ξuP .

From Proposition 2.3 and Proposition 2.4, if f and ))(),...,(( 11 qq gugu are convex

functions and if tktutuuuu kkq

q ,,0)().(~,:)~,...,~(~1 ∀>→= + RR , then the sets of

properly efficient solutions to problem (2.10) and (2.13) verify the following properties:

a. Every properly efficient solution to problem (2.13) is properly efficient for

problem (2.10);

b. Setting 0\: qu RR →+ , the set of properly efficient solutions to problem (2.3)

is a subset of the union in u~ of the set of properly efficient solutions for problem

(2.6).

By combining both results, the following corollary:

Corollary 2.4 If f and ))(),...,(( 11 qq gugu are convex functions, then

UΩ∈

=u

GG uEE~

41 )~( .

Corollary 2.5 If f , and ))(),...,(( 11 qq gugu are convex functions and

0)(~).( >tutu kk , for every 0≥t and ,...,1 qk ∈ then GG EuE 14 )~( ⊂ .

27

Corollary 2.6 (Caballero et al [21]) If 0\qR∈α , ),...,( 1 qααα = , and

ttu kk .)( α= , then problem (2.13) to reduced from problem (4) in Caballero et al [21].

Corollary 2.7 If 0\qR∈γ , ),...,( 1 qγγγ = , 20 =s , ttu kk .)( γ= , qk ,1= , then

(P1)-(P3) to reduced from (1)-(3) in Caballero et al [21].

In this case we obtaint true consequence from Theorem 3.1 and Theorem 3.2 in

Caballero et al [21]. Further if we have in view Corollary 2.6, we see that we get a

consequently Proposition 3.1, Proposition 3.2 and Corollary 3.1 in Caballero et al [21].

For example, using Chankong and Haimes [29], we can applied to problem (P1) and its

associated weighted with problem ),( μλfugP . Thus we have:

Proposition 2.5 If f and ))(),...,(( 11 qq gugu are convex function, D is a convex set and

10 Ex ∈ , then there exist qq μμλλ ,...,,,..., 11 nonnegative real numbers such that 0x is

optimal solution for problem ),( μλfugP .

Proposition 2.6 Let qq μμλλ ,...,,,..., 11 nonnegative real numbers and 0x is an of

optimal solution for problem ),( μλfugP then 10 Ex ∈ .

2.7 White’s approach multiobjective weighting factor auxiliary

optimization problem for P1, P2 and P3

We consider the generation of efficient stochastic multi-objective solution using

weighting factor, q th-power approach for some non-convex auxiliary function

optimization problem. We will introduce our class of auxiliary function and give some

known standard results and we will show tree classes of non-convex auxiliary

optimization problems, giving a concavity-preserving transformation for the q th power

of concave function for range solution for auxiliary optimization problem.

2.7.1 Introduction

Consider the following class of multi-objective problems:

28

(i) there is a compact set nD R⊆ of feasible actions;

(ii) there are q objective functions )( ⋅kF with +∈R)(xFk for all Dx∈ ;

(iii) it is required to find members of the efficient (vector minimal) set E of D, where,

by definition

)()()()(,: xFyFxFyFDyDxE =→≤∈∈= , (2.16)

which ),...,( 1 qFFF = .

The most common of these auxiliary forms is the positively weighted form given by the

class of auxiliary functions

∑=

=⋅q

kkk FF

1),;( δδν , (i.e., ∑

=

=q

kkk xFFx

1)(),;( δδν ), (2.17)

with

⎭⎬⎫

⎩⎨⎧

=∈=Δ∈ ∑=

++ 1:1

q

kk

q δδδ R

or

0: >Δ∈=Δ∈ +++ δδδ .

We define ),;(minarg),( FxFMDx

δνδ∈

= .

Then using well known results in multi-objective programming (see, for example Karlin

[82]) we have the following proposition.

Proposition 2.7

a) EFM ⊆++Δ∈

),(δδU ;

b) If F is convex vector function and D is convex, then (Karlin[82])

U++Δ∈

⊆δ

δ ),( FME .

According to Karlin [82] there are two central issues arising from this class of auxiliary

functions, namely

a. this class of auxiliary functions may not be capable of generating enough points in

E,

29

b. if F is not convex vector function or D is not convex, there may be no current

method for finding ),( FM δ .

2.7.2 Transformations and auxiliary optimization problems associated to (P1), (P2),

and (P3)

We consider the transformation given by

qq ++ →= RR:),...,( 1 θθθ

Relative to (P1), (P2), and (P3), we consider the following transformated problems (2.18),

(2.19), and (2.20), respectively.

(PS1) ⎟⎟⎠

⎞⎜⎜⎝

⎛+∑∑

==∈

)))((())((min11

xguxf kkk

q

kkkk

q

kkDx

υυ θμθλ (2.18)

(PS2) ⎟⎟⎠

⎞⎜⎜⎝

⎛∑=

∈))((min

1

xf kk

q

kkDx

υθλ (2.19)

(PS3) 0,)))(((min 01

0 >⎟⎟⎠

⎞⎜⎜⎝

⎛∑=

∈sxgu s

kkk

q

kkDx

υθμ (2.20)

with 0>υ , qquuu RR →= +:),...,( 1 .

We now define for problem (2.18) an auxiliary optimization problem ),,,(1 θμλqAP as

follows for 1),(,0\ ++ Δ∈∈ μλZq ,

where ⎭⎬⎫

⎩⎨⎧

=+×∈=Δ ∑∑==

+++ 1/),(11

1q

ii

q

ii

qq μλμλ RR .

Find ),,,,(minarg),,,( 11 θμλψθμλ qxqMDx∈

= the set of optimal solutions for (2.18),

where +→ RDqx :),,,;(1 θμλψ is given by

)))((())((),,,;(11

1 xguxfqx kkqk

q

kkk

qk

q

kk θμθλθμλψ ∑∑

==

+= (2.21)

For ∞=q relation (2.21) is replaced by

⎭⎬⎫

⎩⎨⎧=∞ )))(((max)),((maxmax),,,;(1 xguxfx kkkkkkkk

kθμθλθμλψ (2.22)

30

By using the lines of White [207, 209], Bowman [19], Karlin [82], relative to the

problem (2.18) we obtain the following results:

Theorem 2.5

)1a If ∞≠q , then 11 ),,,( EqM ⊆θμλ if 1),( ++Δ∈μλ ;

)2a If a certain uniform dominance condition hold (Bowman [19]), then

11 ),,,( EM ⊆∞ θμλ for 1),( ++Δ∈μλ ;

)3a If ∞≠q , ))(( ⋅kqk fθ , qk ,1= are all convex on D and D is convex, then

U1),(

11 ),,,(++Δ∈

⊆μλ

θμλqME ;

)4a U1),(

11 ),,,(++Δ∈

∞⊆μλ

θμλME ;

)5a If D is finite, then there exists a 0\*+∈ Zq , such that

U1),(

11 ),,,(++Δ∈

=μλ

θμλqME *qq ≥∀ .

For problem (2.19) an auxiliary optimization problem ),,(2 θλqAP as follows for

2,0\ ++ Δ∈∈ λZq ,

where ⎭⎬⎫

⎩⎨⎧

=∈=Δ ∑=

++ 1/1

2q

ii

q λλ R .

Find ),,,(minarg),,( 22 θλψθλ qxqMDx∈

= , where +→ RDqx :),,;(2 θλψ is given

by

))((),,;(1

2 xfqx kqk

q

kkθλθλψ ∑

=

= . (2.23)

For ∞=q equation (2.23) is replaced by

))((max),,;(2 xfx kkkk

θλθλψ =∞ (2.25)

By using the lines of White [207, 209], Bowman [19], Karlin [82], relative to the

problem (2.19) we obtain the following results:

Theorem 2.6

)1a If ∞≠q , then 22 ),,( EqM ⊆θλ if 2++Δ∈λ ;

31

)2a If a certain uniform dominance condition hold (Bowman [19]), then

22 ),,( EM ⊆∞ θλ if 2++Δ∈λ ;

)3a If ∞≠q , ))(( ⋅kqk fθ , qk ,1= are all convex on D and D is convex, then

U2

),,(22++Δ∈

⊆λ

θλqME ;

)4a U2

),,(22++Δ∈

∞⊆λ

θλME ;

)5a If D is finite, then there exists a 0\*+∈ Zq , such that

U2

),,(22++Δ∈

=λ

θλqME *qq ≥∀ .

For problem (2.20) an auxiliary optimization problem ),,(3 θμqAP as follows for

2,0\ ++ Δ∈∈ μZq ,

Find ),,,(minarg),,( 33 θμψθμ qxqMDx∈

= , where +→ RDqx :),,;(3 θμψ is given by

∑= ))((),,;( 03

skk

qkk guqx θμθμψ (2.25)

For ∞=q equation (2.25) is replaced by

)))(((max),,,( 03 xgux s

kkkkkθμθμψ =∞ (2.19)

Finally for problem (2.20) we obtain the following results:

Theorem 2.7

)1a If ∞≠q , then 33 ),,( EqM ⊆θμ if 2++Δ∈μ

)2a If a certain uniform dominance condition hold (Bowman [19]), then

33 ),,( EM ⊆∞ θλ if 2++Δ∈μ

)3a If ∞≠q , ))(( ⋅kqk fθ , qk ,1= are all convex on D and D is convex, then

U2

),,(33++Δ∈

⊆μ

θμqME

)4a U2

),,(33++Δ∈

∞⊆μ

θμME

32

)5a If D is finite, then there exists a 0\*+∈ Zq , such that

U2

),,(33++Δ∈

=μ

θμqME *qq ≥∀ .

Remark 2.7 Some proofs of these results are given in Sudradjat and Preda [189].

2.7.3 Non-convex auxiliary optimization problem

The problem is to finding points in D which are in E or close to points in E. We also

wish to use convexity and concavity properties. For these to be meaningful, we need

appropriate convex sets within which to embed our analysis. nR is too large, because

we have stipulate merely that ∈)(xF q+R for all Dx∈ and not for all ∈x nR . When

D is convex, all that we need state is that F is defined on *D , with ∈)(xF q+R for all

*Dx∈ . In the following we use lines given by White [149].

a) Case of concave ∞≠⋅⋅ qguf kkk )),((),(

We assume that the kf are all concave on *D and look at the choice of )( ⋅kθ and

q and associated algorithms for auxiliary optimization problems ),,,(1 θμλqAP ,

),,(2 θλqAP and ),,(3 θμqAP .

Generally even if ))(( ⋅kk fθ is concave on *D it is not necessarily true that

))(( ⋅kqk fθ is concave on *D . We need to choose ))(( ⋅kk fθ so that, at least for some

q , ))(( ⋅kqk fθ are all concave on *D . The following form of )( ⋅kθ will provide an

instance which will do what is required, namely

qkatt kk ,1),)(log())(( =+= ϕϕθ

over the range

,1)( −−≥ katϕ qk ,1= , where R⊆ ka

Lemma 2.1 If )( ⋅kf are concave on *D , then ))(( ⋅kqk fθ are all concave on *D

for all 0\+∈ Zq and ),min( 21 qqq = such that

33

)]1)))(([log(minmin

)]1))([log(minmin

,12

,11

++≤

++≤

∈=

∈=

kkkDxqk

kkDxqk

axguq

axfq (2.27)

provided that

)]([min xfa kDxk ∈≥ , qk ,1= (2.28)

Proof: For 1,,1 −−≥∈≥ kazRzq ,

2

2

2

2

)()]log()1)[((log)(

k

kkq

kqk

azazqazq

dzazd

++−−+

=+ −θ

.

Thus for any given ,1−−≥ kaz

0)(2

2

≤dz

zd qkθ ,

if )log(1 kazq ++≤ .

Replacing z by )(xfk and ))(( xgu kk , we see that )(⋅qkθ is concave on *D provided

that (2.26) and (2.27) hold.

b) Case of convex )( ⋅kf , ))(( ⋅kk gu and finite D

If )( ⋅kθ are all convex on +R and )( ⋅kf and ))(( ⋅kk gu are all convex on *D ,

then ))(( ⋅kqk fθ are all convex on *D . This applies, for example, when

+∈∀= R)()())(( tttk ϕϕϕθ , qk ≤≤1 .

In this case the auxiliary optimization problem ),,,(1 θμλqAP become one of

minimizing a convex function over a finite set D, also for ),,(2 θλqAP and

),,(3 θμqAP .

This also hold in the case of ∞=q .

Let us now assume that )(xψ is ),,,;(1 θμλψ qx , ),,;(2 θλψ qx or ),,;(3 θμψ qx

respectively, and that )(xψ is convex on *D . Then consider the following algorithm,

given the qualifier ,,, θμλq , ,, θλq or ,, θμq respectively for ease of

34

exposition : )( txψ∂ is the subdifferential of ψ at txx = and φψ ≠∂ )( tx

(Rockafellar, [150]); tS is any finite non-empty subset of )( txψ∂ obtained by some

specified method; 1x is the first component of x.

Algorithm 2.1

(i) Select 0\+∈ Rε .

(ii) Set DDt = .

(iii) Set t = 1.

(iv) Assume that we have derived tD .

(v) Find tS and

)(minarg xxtDx

t ψ∈

∈

(vi) Set tttt SyxxyDxD ∈∀−≤−∈=+ ε)(:1

(vii) If φ=+1tD , set

)(minarg],...,[ 1

xxtxxx

A ψ∈

∈ ,

and stop.

(viii) If φ≠+1tD , go to step (v).

We have the following theorem.

Theorem 2.8

(i) Algorithm 2.1 terminates in a finite number of iterations.

(ii) If *ψ is the minimal value of ψ on D, then

εψψψ +≤≤ ** )( Ax , where ,, 321 ψψψψ ∈ .

Proof: (i) Let us suppose that the algorithm, is not finite.

Because D is finite, there exist a set 0\ +⊆ Zr such that

10 ≥∀= rxx rt

where 0x is some member of D and

35

11 ≥∀∈+ rDx rr tt ,

then

1),()( 01 ≥∂∈∀−≤−+ rxyxxy rr tt ψε .

Thus

1),()( 01 ≥∂∈∀−≤−+ rxyrxxy rr tt ψε .

This is possible.

(ii) Let φ=+1tD and

,\ 1+= sss DDY ts ≤≤1 ,

then

Ut

ssYD

1== .

Let sYx∈ . Then 1+∉ sDx and

ε−≥− )( sxxy for some )( sxy ψ∂∈ .

Because ,, 321 ψψψψ ∈ are convex,

εψψ −≥−≥− )()()( ss xxyxx .

Hence

εψψψ +≤≤ ** )( Ax .

Remark 2.8 Step (v) really only requires finding a feasible solution in tD . The use of

1x as an objective function is merely to facilitate this.

Remark 2.9 Step (v) is a subproblem of minimizing a linear function over those

solutions defined by a polytope, say tZ , generated by the subgradient constraints, which

are also in D. When all function are differentiable, tD is singleton gradient vector for

ψ at 1xx = . In the general case )( txψ∂ and hence tD may be found in term of the

subdifferentials of the functions ))(( ⋅kk fθ , (Rockafellar, [150]).

Remark 2.10 If D is the vertex set of a polytope *D , then any solution from tD is also

a vertex of tZD ∩* . Thus a vertex search subalgorithm such as that of Murty [124] can

36

be developed. Other procedures for enumerating the vertices of a polytope may be

adaptable for this problem (Matheiss and Rubin, [116]).

Remark 2.11 If D is integral set of a polytope, the subproblem in step (v) is an integer

linear programming problem, for which a range of algorithms exist.

Whatever method is use to solve the auxiliary problem, the convexity of ψ is helpful in

providing lower bounds, because if sx is any set of solutions generated, then

)]()([maxmin)(,

* ss

xysDxxxyx

s−+≥

∂∈∈ψψ

ψ.

Remark 2.12 If D is a subset of a polytope *D , we note that

)(min*

* xDxψψ

∈≥ . (2.22)

The determination of right-hand side of (2.22) is a convex programming problem. Lower

bounds may be useful in determining how close the best solution to date is to an optimal

solution, so that computations may be termined early if wished.

c) Case of mixture of concave and convex ∞≠⋅⋅ qguf kkk )),((),(

From a global optimization point of view this is the hardest problem of all.

Consider 1M , 2M , '1M and '

2M are non-empty subsets of ,...,1 q such that

,...,1'2

'121 qMMMM =∪=∪ , φ=∩ 21 MM , and φ=∩ '

2'1 MM ; ))(( ⋅k

qk fθ be

concave on *D for ))((;1 ⋅∈ kqk fMk θ be convex on *D for 2Mk ∈ , also that

)))((( ⋅kkqk guθ be concave on *D for '

1Mk ∈ and )))((( ⋅kkqk guθ be convex on *D

for 2Mk ∈ ;

∑∑∈∈

⋅+⋅=⋅'11

)))((())((),,,;(1Mk

kkqkkk

qk

Mkk gufq θμθλθμλψ

)))((())((),,,;('22

2 ⋅+⋅=⋅ ∑∑∈∈

kkqk

Mkkk

qk

Mkk gufq θμθλθμλψ

Then, dropping the qualifiers ,,, θμλq for ease of exposition, we have

)()()( 21 ⋅+⋅=⋅ ψψψ

where )(1 ⋅ψ and )(2 ⋅ψ are respectively concave and convex on *D .

37

The following algorithm is an extension of an algorithm of White ([206]). tS2 is any

finite subset of the sub-differential )(2txψ∂ of )(2 ⋅ψ at txx = obtained by some

specified method.

Algorithm 2.2

(i) Select 0\+∈ Rε .

(ii) Set DD =1 .

(iii) Set t = 1.

(iv) Assume that we have derived tD .

(v) Find tS and

)(minarg xxtDx

t ψ∈

∈

(vi) Set tttt SyxxyDxD ∈∀−≤−∈=+ ε)(:1

(vii) If φ=+1tD , set

)(minarg],...,[ 1

xxtxxx

A ψ∈

∈

and stop

(viii) If φ≠+1tD , go to step (v).

We have the following theorem, where )(2 xψ∂ is the subdifferential of )(2 ⋅ψ at point

x.

Theorem 2.9

(i) If U Dxx

∈∂ )(2ψ is compact, then Algorithm 2.2 terminates in a finite number of

iterations.

(ii) If *ψ is the minimal value of )(⋅ψ on D, then

εψψψ +≤≤ ** )( Bx

Proof: (i) Let us suppose that the algorithm, is not finite.

Because of the assumptions about the sub-differentials and the compactness of D, there

exist a ny R∈0 and a set 0\ +⊆ Zr such that

38

12/)( 10 ≥∀−≤−+ rxxy rr tt ε

Thus

12/)( 10 ≥∀−≤−+ rrxxy rr tt ε ,

Inequalities which is not possible.

(ii) Let φ=+1tD and ,\ 1+= sss DDY ts −=1 , then Ut

ssYD

1== .

Let sYx∈ . Then 1+∉ sDx and ε−≥− )( sxxy for some )(2txψ∂ .

Because )(2 ⋅ψ is convex,

εψψ −≥−≥− )()()( 22ss xxyxx .

Also,

)()( 11sxx ψψ ≥ .

Thus

εψψψ +≤≤ ** )( Bx .

Remark 2.13 Step (v) is a subproblem involving the minimization of a concave function

over tD . When D is a polytope, so is tD , and algorithms exist for solving such

problems (e.g. Glover and Klingman, [62]; Falk and Hoffman, [49]; Carino, [23]). We

note that in this case step (vi) adds cutting constraints which exclude the current solution

and for which the dual simplex method is useful (Hadley, [65]).

Remark 2.14 If D is the integral set of a polytope *D , then the subproblem takes an

integer programming form, for which algorithm exist.

Remark 2.15 If D is the vertices of a polytope *D , then except in special cases (e.g.

when vertices are integral as in the assignment problem) some new algorithm is

required.

39

CHAPTER 3

STOCHASTIC DOMINANCE 3.1 Introduction. The relation of stochastic dominance is a fundamental concept of decision theory and

economics , Dentcheva and Ruszczynski [41, 42], Hanoch and Levy [68], Quirk and

Saposnik [146] and Rothschild [155].

A random variable X dominates another random variable Y in the second order, which

we write as YX )2(f , if )]([)]([ YuXu EE ≥ for every concave nondecreasing function

)(⋅u , for which these expected values are finite.

A basic model of stochastic optimization can be formulated as follows:

)],([max ωϕ zZz

E∈

. (3.1)

In this formulation ω denotes an elementary event in a probability space ),,( PFΩ , z is a

decision vector in an appropriate space Z , and R→Ω×Z:ϕ . The set Z⊂Z is

defined either explicitly, or via some constraints that may involve the elementary event ω

and must hold with some prescribed probability.

The first stochastic optimization models with expected values were introduced by

Lehmann [96] and Hanoch and Levy [68]. Mathematical theory of expectation models

involving two-stage and multistage decisions has been developed by Wets [204, 205]

and Birge [16].

Models involving constraints on probability were introduced by Charnes and Cooper

[31], Prekopa [139], Dentcheva and Ruszczynski [42] discusses in detail the theory and

numerical methods for linear models with one probabilistic constraint on finitely many

inequalities.

Another way to look at problem (3.1) is to consider the set C of random variables X such

that, for some Zz∈ , one has ),()( ωϕω zX ≤ a.s. Then we can write the model as

][max XCX

E∈

.

40

In practice, however, it is almost impossible to elicit the utility function of a decision

maker explicitly. Additional difficulties arise when there is a group of decision makers

with different utility functions who have to come to a consensus.

In some applications a reference outcome Y in ),,(1 PFL Ω is available. It may have the

form Ω∈= ωωϕω ),,()( zY , for some policy z . Our intention is to have the new

outcome, X, preferable over Y. Therefore, we consider the following optimization

problem:

)(max Xf (3.2)

subject to YX )2(f , (3.3)

CX ∈ . (3.4)

Here Y is a random variable in ),,(1 PFL Ω , the set ),,(C 1 PFL Ω⊂ is convex and

closed, and R→Cf : is a concave continuous functional. Constraints (3.3) guarantees

that for any decision maker, whose utility function )(⋅u is concave and nondecreasing,

the solution X of the problem will satisfy the relation )]([)]([ YuXu EE > .

Another class of models that recently attracted much attention are mean-risk models. In

our notation they take the form

)(][max XXCX

λρ−∈

E .

In this problem λ > 0 and )(⋅ρ is a risk functional which depends on the entire

distribution of X and assigns to it a scalar measure of its variability. For example, the

expected shortfall below the mean,

[ ]+−= )][()( XXEEXρ ,

may be used as the risk functional. Here ),0max()( XX =+ . Mean-risk models are also

closely related to stochastic dominance relations. If we use an appropriate risk measure

ρ and the parameter λ is within a certain range, then the optimal outcome X is not

stochastically dominated by any other feasible outcome Dentcheva and

Ruszczynski [42], Orgryczak and Ruszczynski [127, 128, 129].

41

Other stochastic optimization models involving general risk functional were considered

by Dentcheva and Ruszczynski [42], Rockafellar and Uryasev [152]. Model (3.2)–(3.4)

correspond to a new approach in stochastic optimization problem.

3.2 Stochastic dominance

In the stochastic dominance approach, random returns are compared by a point-wise

comparison of some performance functions constructed from their distribution functions.

For a real random variable V , its first performance function is defined as the right-

continuous cumulative distribution function of V :

),( ηη ≤= VVF P for R∈η .

A random return V is said stochastically dominate another random return S to the first

order (Dentcheva and Ruszczynski [42], Lehmann [96] and Quirk and Saposnik [146]),

denoted SV FSDf , if

);();( ηη SFVF ≤ for all R∈η .

Define the function );(2 ⋅VF as

ααηη

dVFVF ∫ ∞−= );();(2 for R∈η , (3.5)

as an integral of a nondecreasing function, it is a convex function of η and defines the

weak relation of the second-order stochastic dominance (SSD). That is, the random return

V stochastically dominates S to the second order, denoted SV SSDf , if

);();( 22 ηη SFVF ≤ for all R∈η .

The corresponding strict dominance relations for FSDf and SSDf are defined in the

usual way: SV f if and only if SV f , VS f/ .

Furthermore, for ),,( PV FLm Ω∈ we can define recursively the functions

∫ ∞− −=η

ααη dVFVF kk ),();( 1 for 1,3k +=∈ mR,η . (3.6)

Furthermore, for ),,( PV FLm Ω∈ we can define recursively the functions

42

∫ ∞− −=η

ααη dVFVF kk ),();( 1 for 1,3k +=∈ mR,η . (3.6)

Figure 3.3 First order dominance

RE ∈−== +∞−∫ ηηααη

ηforXdXFXF )();();( 12

Figure 3.2 Scond-order dominnce

They are also convex and nondecreasing functions of the second argument.

Definition 3.1. A random variable ),,(1-k PX FL Ω∈ dominates in the kth order

another random variable ),,(1-k PY FL Ω∈ if

);();( ηη YFXF kk ≤ for all R∈η . (3.7)

We shall denote relation (3.7) as

YX k )(f (3.8)

and the set of X satisfying this relation as

:),,()( )(1 YXPXYA k

kk fFL Ω∈= − . (3.9)

43

By changing the order of integration we can express the function );(2 ⋅VF as the

expected shortfall (Rockafellar and Uryasev [152]): for each target value η we have

[ ]+−= )();(2 VEVF ηη , (3.10)

where )0,max()( VV −=− + ηη . The function );(2 ⋅VF is continuous, convex,

nonnegative and nondecreasing. It is well defined for all random variables V with finite

expected value.

3.3 The portfolio problem

Let nRR ,...,1 be random returns of n assets. We assume that ∞<][ jRE for all

nj ,1= . Our aim is to invest our capital in these assets in order to obtain some desirable

characteristics of the total return on the investment. Denoting by nxx ,...,1 the fractions of

the initial capital invested in assets n,...,1 respectively we can easily derive the formula

for the total return:

nn xRxRxR ++= ...)( 11 . (3.11)

Clearly, the set of possible asset allocations can be defined as follows:

X njxxxx jnn ,1,0,1...: 1 =≥=++∈= R ,

where ,..., 1 nxxx = .

In some applications one may introduce the possibility of short positions, i.e., allow some

jx ’s to become negative. Other restrictions may limit the exposure to particular assets or

their groups, by imposing upper bounds on the jx ’s or on their partial sums. One can

also limit the absolute differences between the jx ’s and some reference investments jx ,

which may represent the existing portfolio, etc. Our analysis will not depend on the

detailed way this set is defined; we shall only use the fact that it is a convex polyhedron.

All modifications discussed above define some convex polyhedral feasible sets, and are,

therefore, covered by our approach.

44

The main difficulty in formulating a meaningful portfolio optimization problem is the

definition of the preference structure among feasible portfolios. If we use only the mean

return

[ ])()( xRx E=μ ,

then the resulting optimization problem has a trivial and meaningless solution: invest

everything in assets that have the maximum expected return. For these reasons the

practice of portfolio optimization resorts usually to two approaches.

In the first approach we associate with portfolio x some risk measure )(xρ representing

the variability of the return )(xR . In the classical Markowitz model )(xρ is the variance

of the return,

[ ])()( xRarx V=ρ ,

but many other measures are possible here as well.

The mean–risk portfolio optimization problem is formulated as follows:

)]()([max xxx

λρμ −∈X

(3.12)

Here, λ is a nonnegative parameter representing our desirable exchange rate of mean for

risk. If 0=λ , the risk has no value and the problem reduces to the problem of

maximizing the mean. If 0>λ we look for a compromise between the mean and the

risk. The general question of constructing mean–risk models which are in harmony with

the stochastic dominance relations has been the subject of the analysis of the recent

papers Dentcheva and Ruszczynski [41, 42], Rothschild and Stiglitz [155], Ogryczak and

Ruszczynski [127, 128].

We have identified there several primal risk measures, most notably central semi-

deviations, and dual risk measures, based on the Lorenz curve, which are consistent with

the stochastic dominance relations.

The second approach is to select a certain utility function RR →:u and to formulate

the following optimization problem

( )[ ])(max xRuEx X∈

(3.13)

It is usually required that the function u(·) is concave and nondecreasing, thus

representing preferences of a risk-averse decision maker. The challenge here is to select

45

the appropriate utility function that represents well our preferences and whose application

leads to non-trivial and meaningful solutions of (3.13).

3.4 Consistency with stochastic dominance

The concept of stochastic dominance is related to an axiomatic model of risk-averse

preferences Fishburn [52]. It originated from the theory of majorization (Hardy,

Litltewood and Polya [70], Marshall and Olkin [109]) for the discrete case and was later

extended to general distributions (Quirk and Saposnik [146], Hadar and Russell [66],

Hanoch and Levy [68], Rothschild and Stiglitz [155]. It is nowadays widely used in

economics and finance Bawa[7], Levy [99].

In the stochastic dominance approach, random returns are compared by a point-wise

comparison of some performance functions constructed from their distribution functions.

Figure 3.3. Mean–risk analysis. Portfolio x is better than portfolio y in the mean–risk sense, but none of them is efficient.

For a real random variable V, its first performance function is defined as the

rightcontinuous cumulative distribution function of V :

RP ∈≤= ηηη forVFV ,)( .

A random return V is said (Lehmann[94], Quirk and Saposnik [144]) to stochastically

dominate another random return S to the first order, denoted SV FSDf , if

R∈= ηηη allforFF SV ),()( .

The second performance function 2F is given by areas below the distribution function F,

∫ ∞−=

ηααη dVFVF );();(2 for R∈η (3.14)

46

and defines the weak relation of the second-order stochastic dominance (SSD). That is,

random return V stochastically dominates S to the second order, denoted SV SSDf , if

);();( 22 ηη SFVF ≤ for R∈η . (3.15)

(see Hadar and Russell [66], Hanoch and Levy [99]). The corresponding strict

dominance relations FSDf and SSDp are defined in the usual way

SV FSDf ⇔ SV SSDp . and SV SSDf (3.16)

For portfolios, the random variables in question are the returns defined by (3.11). To void

placing the decision vector, x, in a subscript expression, we shall simply write It will not

lead to any confusion, we believe. Thus, we say that portfolio x dominates portfolio y

under the FSD rules, if );();( yFxF ηη = for all R∈η , where at least one strict

inequality holds. Similarly, we say that x dominates y under the SSD rules

))()(( yRxR SSDf , if );();( )2()2( yFxF ηη = for all R∈η , with at least one inequality

strict.

)();( )( ηη xFxF R= and )();( )(22 ηη xFxF R= .

Figure 3.4. The expected shortfall function.

Stochastic dominance relations are of crucial importance for decision theory. It is known

that )()( yRxR FSDf if and only if ))](([))](([ yRUxRU EE ≥ for any nondecreasing

function U(·) for which these expected values are finite. Also, ))()(( yRxR SSDf if and

only if ))](([))](([ yRUxRU EE ≥ for every nondecreasing and concave U(·) for which

these expected values are finite (see, e.g., Levy [97]).

47

For a set P of portfolios, a portfolio P∈x is called SSD-efficient (or FSD-efficient) in

P if there is no P∈y such that )()( xRyR SSDf (or )()( xRyR FSDf ).

We shall focus our attention on the SSD relation, because of its consistency with risk-

averse preferences: if ))()(( yRxR SSDf , then portfolio x is preferred to y by all risk-

averse decision makers. By changing the order of integration we can express the function

);(2 xF ⋅ as the expected shortfall (Orgyczak ang Rusczynski [129)]: for each target value

η we have

)]0),([max();(2 xRExF −= ηη . (3.17)

The function );()2( xF ⋅ ) is continuous, convex, nonnegative and nondecreasing. Its graph

is illustrated in Figure 3.4.

Following [42, 43], we introduce the following definition.

Definition 2.1 Ruszczy´nski and Vanderbei [156] The mean-risk model ),( ρμ is

consistent with SSD with coefficient 0>α , if the following relation is true

)()()()()()( yyxxyRxR SSD λρμλρμ −≥−⇒f for all αλ ≤≤0 .

for all αλ ≤≤0

In fact, as we shall see in the proof below, it is sufficient to have the above inequality

satisfied for α ; its validity for all αλ ≤≤0 follows from that.

The concept of consistency turns out to be fruitful. In [129] we have proved the following

result.

Theorem 3.1. The mean–risk model in which the risk is defined as the absolute

semideviation,

)0),()(max()( xRxx −= μδ E , (3.18)

is consistent with the second-order stochastic dominance relation with coefficient 1.

We provide an easy alternative proof here.

Proof. First, it is clear from (3.17) that the line )(xμη − is the asymptote of );()2( xF ⋅

for ∞→η . Therefore )()( yRxR SSDf implies that

)()( yx μμ ≥ . (3.19)

48

Secondly, setting )(xμη = in (4) we obtain

))()(,0max()( yRxx −≤ μδ E .

Since 0)()( ≥− yx μμ , we have

))()(,0max()()())()()()(,0max())()(,0max(

yRxyxyRyyxyRx

−+−≤−+−=−

μμμμμμμ

Taking the expected value of both sides and combining with the preceding inequality we

get

)()()()( yyxx δμμδ +−= ,

which can be rewritten as

)()()()( yyxx δμδμ +≥− . (3.20)

Combining inequalities (3.19) and (3.20) with coefficients λ−1 and λ , where

]1,0[∈λ , we obtain the required result.

An identical result (under the condition of finite second moments) has been obtained in

Ogryczak and Ruszczynski [128] for the standard semideviation, and further extended in

Ogryczak and Ruszczynski [129] to central semideviations of higher orders and

stochastic dominance relations of higher orders (see also Gotoh and Konno [64]).

Elementary calculations show that for any distribution

)(21)( xx δδ = ,

where )(xδ is the mean absolute deviation from the mean:

)()()( xxRx μδ −= E . (3.21)

Thus, )(xδ is a consistent risk measure with the coefficient 21

=α . The mean–absolute

deviation model has been introduced as a convenient linear programming mean–risk

model by Konno and Yamazaki [86].

Another useful class of risk measures can be obtained by using quantiles of the

distribution of the return R(x). Let )(xqp denote the p-th quantile (In the financial

literature, the quantity )(xqp W, where W is the initial investment, is sometimes called

the Value at Risk ) of the distribution of the return R(x), i.e.,

49

)(xδ .

We may define the risk measure

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−= zxRxRz

ppxp )()),((1max)( Eρ . (3.22)

In the special case of 21

=p the measure above represents the mean absolute deviation

from the median. For small p, deviations to the left of the p-th quantile are penalized in a

much more severe way than deviations to the right.

Although the p-th quantile )(xqp might not be uniquely defined, the risk measure

)(xpρ is a well defined quantity. Indeed, it is the optimal value of a certain optimization

problem:

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−= zxRxRz

ppxp )()),((1maxmin)( E

Xρ . (3.23)

It is well known that the optimizing z will be one of the p-th quantiles of R(x) (see, e.g.,

Bloomfield [17]). In Ogryczak and Ruszczynski [127] we have proved the following

result.

Theorem 2. The mean–risk model with the risk defined as )(xpρ is consistent with the

second-order stochastic dominance relation with coefficient 1, for all )1,0(∈p ).

Again, we provide here an alternative proof.

Proof. Let us consider the composite objective in our mean–risk model (scaled by p):

)()();( xpxpxpG pρμ −= . (3.24)

If follows from (3.23) that we can represent it as an optimal value:

( )[ ][ ]))(()),()(1(maxsup);( zxRpxRzppxpG −−−−= EμX

. (3.25)

Clearly, we have the identity

( ) ( ) ( )zxRpxRzzxRpxRzp −+−=−−− )()(,0max))(()),()(1(max .

Using this in (3.25) we obtain

[ ]),(sup[);( )2( xzFpzxpG −=X

. (3.26)

50

Figure 3.6 The absolute Lorenz curve

Therefore, the function );( xG ⋅ is the Fenchel conjugate of );(2 xF ⋅ (see Fenchel [50]

and Rockafellar[148]). Consequently, the second-order dominance )()( yRxR SSDf

implies that

);();( ypGxpG ≥

for all ]1,0[∈p . Recalling (13) we conclude that

)()()()( yyxx pp ρμρμ −≥− .

Since we also have (8), Definition 1 is satisfied with all ]1,0[∈λ .

Interestingly, the function );( xG ⋅ can also be expressed as the integral:

∫=η

α α0

)(),( dxqxpG (3.27)

(non-uniqueness of the quantile does not matter here). Indeed, it follows from (3.26) that

the quantile )(xq p , which is the maximizer in (3.26), is a subgradient of );( xG ⋅ at p (see

Fenchel [50] and Rockafellar[150]). The integral in (3.27) is called the absolute Lorenz

curve and is frequently used (for nonnegative variables and in a normalized form) in

income inequality studies (see Arnold [2], Gastwirth [59], Lorenz [106], Hardy,

Litlewood and Polya [70] and the references therein). It is illustrated in Figure 3.6.

51

CHAPTER 4

THE DOMINANCE-CONSTRAINED PORTFOLIO PROBLEM

4.1 Introduction

The problem of optimizing a portfolio of finitely many assets is a classical problem in

theoretical and computational finance. Since the seminal work of Markowitz [112, 114,

114] it is generally agreed that portfolio performance should be measured in two distinct

dimensions: the mean describing the expected return, and the risk which measures the

uncertainty of the return. In the mean–risk approach, we select from the universe of all

possible portfolios those that are efficient: for a given value of the mean they minimize

the risk or, equivalently, for a given value of risk they maximize the mean. This

approach allows one to formulate the problem as a parametric optimization problem, and

it facilitates the trade-off analysis between mean and risk.

Another theoretical approach to the portfolio selection problem is that of stochastic

dominance (see Mosler and Scarsini [121], Whitmore and Findlay [210], and Levy

[99]). The concept of stochastic dominance is related to models of risk averse

preferences, Fishburn [52]. It originated from the theory of majorization, Hardy and

Litlewood [70], and Marshall and Olkin [109] for the discrete case, was later extended to

general distributions (Quirk and Saposnik [146], Hadar and Russell [66], Hanoch and

Levy [68], Rothschild and Stiglitz [155], and is now widely used in conomics and

finance Levy [99].

The usual (first order) definition of stochastic dominance gives a partial order in the

space of real random variables. More important from the portfolio point of view is the

notion of second-order dominance, which is also defined as a partial order. It is

equivalent to this statement: a random variable R dominates the random variable Y if

)]([)]([ yuERuE ≥ for all nondecreasing concave functions u(·) for which these

52

expected values are finite. Thus, no risk-averse decision maker will prefer a portfolio

with return Y over a portfolio with return R.

Introduced a new stochastic optimization model with stochastic dominance constraints

Dencheva and Ruszczynski [41,42]. In this chapte we show how this theory can be used

for risk-averse portfolio optimization. We add to the portfolio problem the condition that

the portfolio return stochastically dominates a reference return, for example, the return of

an index. We identify concave nondecreasing utility functions which correspond to

dominance constraints. Maximizing the expected return modified by these utility

functions, guarantees that the optimal portfolio return will dominate the given reference

return.

4.2. Dominance-constrained

Consider stochastic dominance relations between random returns defined by (3.11).

Thus, we say that portfolio x dominates portfolio y under the FSD rules, if

));(());(( ηη yRFxRF ≤ for all R∈η ,

where at least one strict inequality holds. Similarly, we say that x dominates y under

the SSD rules ))()(( yRxR SSDf , if

));(());(( 22 ηη yRFxRF ≤ for all R∈η

with at least one inequality strict. Recall that the individual returns kjR have finite

expected values and thus the function ));((2 ηxRF is well defined.

Stochastic dominance relations are of crucial importance for decision theory. It is known

that )()( yRxR FSDf if and only if

))](([))](([ yRuExRuE ≥ , (4.1)

for any nondecreasing function u(·) for which these expected values are finite.

Furthermore, )()( yRxR SSDf if and only if (4.1) holds true for every nondecreasing and

concave u(·) for which these expected values are finite (see, Dentcheva and Ruszczynski

[41, 42] and Levy [99]).

A portfolio x is called SSD-efficient (or FSD-efficient) in a set of portfolios X if there

is no ∈y X such that ))()(( xRyR SSDf (or FSDyR f)( ))(xR .

53

We shall the focus to the SSD relation, because of its consistency with risk-averse

preferences: if )()( xRyR SSDf , then portfolio x is preferred to y by all risk-averse

decision makers.

The starting point consider the assumption that a reference random return Y having a

finite expected value is available. It may have the form )(zR , for some reference

portfolio z . It may be an index or our current portfolio. Our intention is to have the

return of the new portfolio, )(xR , preferable over Y . We introduce the following

optimization problem :

)(max xf (4.2)

Subject to kSSD

k YxR f)( , υ,1=k (4.3)

X∈x , (4.4)

where nkn

kk xRxRxR ++= ...)( 11 , υ,1=k .

Here R→X:f is a concave continuous functional. In particular for 0>kw , υ,1=k ,

we may use

)]([)(1

xREwxf k

kk∑

=

=υ

,

and this will still lead to nontrivial solutions, due to the presence of the dominance

constraints.

Using the line of Dentcheva and Ruszczynski [41] we obtain the following proposition.

Proposition 4.1 Assume that kY , υ,1=k has a discrete distribution with realizations

kiy , mi ,1= , υ,1=k . Then relation (3.16) is equivalent to

υ,1,,1],)[(]))([( ==∀−≤− ++ kmiYyxRy ki

ki EE . (4.5)

Proof. If relation (4.3) is true, then the equivalent representation (3.10) implies (4.5).

It is sufficient to prove that (4.5) imply that

);());(( 22 ηη kk YFxRF ≤ for all R∈η , υ,1=k .

With no loss of generality we may assume that km

k yy << ...1 , mi ,1= , υ,1=k . The

distribution function );( ⋅kYF is piecewise constant with jumps at kiy , mi ,1= ,

54

υ,1=k . Therefore, the function );( ⋅kYF is piecewise linear and has break points at

kiy , mi ,1= , υ,1=k . Let us consider three cases, depending on the value of η .

Case 4.1: If ky1≤η we have

0);());(());((0 12122 =≤≤≤ kkkkk yYFyxRFxRF η

Therefore the required relation holds as an equality.

Case 4.2: Let ],[ 1ki

ki yy +∈η for some i. Since, for any random return )(xRk , the

function ));((2 ⋅xRF k is convex, inequalities (4.5) for i and 1+i imply that for all

],[ 1ki

ki yy +∈η one has

));(()1());(());(( 1222ki

kki

kk yxRFyxRFxRF +−+≤ λλη

);();()1();( 2122 ηλλ kki

kki

k YFyYFyYF =−+≤ +

where kii

ki

yyy

−−

=+1

ηλ The last equality follows from the linearity of );(2 ⋅kYF in the

interval ],[ 1ki

ki yy + .

Case 4.3: For kmy∈η the function );(2 ηkYF is affine with slope 1, and therefore

km

km

kk yyYFYF −+= ηη );();( 22

));(());(());(( 22 ηααη

xRFdxRFyxRF k

y

kkm

kkm

=+≥ ∫ ,

as required.

Let us assume now that the returns have a discrete joint distribution with realizations

njTtr kjt ,1,,1, == and υ,1=k , attained with probabilities Ttp k

t ,1, = . The

formulation of the stochastic dominance relation (4.3) resp. (4.5) simplifies even further.

Introducing variables kits representing shortfall of )(xRk below k

iy in realization,

mit ,1, = Tt ,1= and υ,1=k we obtain the following result.

Proposition 4.2 Assume that njR kj ,1, = , υ,1=k and kY have discrete distributions.

Then problem (4.2)–(4.4) is equivalent to the problem

55

)(max xf (4.6)

Subject to ki

kit

kjt

n

jj ysrx ≥+∑

=1, mi ,1= , Tt ,1= , υ,1=k , (4.7)

),(21

ki

kkit

T

t

kt yYFsp ≤∑

=

, mi ,1= , υ,1=k , (4.8)

0≥kits mi ,1= , Tt ,1= , υ,1=k , (4.9)

X∈x . (4.10) Proof. If nx R∈ is a feasible point of (4.2)–(4.4), then we can set

+=⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∑ k

jt

n

jj

ki

kit rxys

1, mi ,1= , Tt ,1= , υ,1=k .

The pair ),( sx is feasible for (4.7)–(4.10).

On the other hand, for any pair ),( sx , which is feasible for (4.7)–(4.10), we get

⎟⎟⎠

⎞⎜⎜⎝

⎛−≥ ∑

=

kjt

n

jj

ki

kit rxys

1

, mi ,1= , Tt ,1= , υ,1=k .

Taking the expected value of both sides and using (3.21) we obtain

),,());(( 22ki

kki

k yYFyxRF ≤ mi ,1= , υ,1=k .

Proposition 4.1 implies that x is feasible for problem (4.2)–(4.4).

4.3 Optimality and duality

From now on we shall assume that the probability distributions of the returns and of the

reference outcome kY are discrete with finitely many realizations. We also assume that

the realizations of kY are ordered: km

k yy << ...1 , υ,1=k . The probabilities of the

realizations are denoted by miki ,1, =π .

We define the set U of functions RR →:u satisfying the following conditions:

)(⋅u is concave and nondecreasing;

)(⋅u is piecewise linear with break points kiy , mi ,1= , υ,1=k ;

0)( =tu for all kmyt ≥ , υ,1=k .

It is evident that U is a convex cone.

56

Let us define the Lagrangian of (4.2)–(4.4), RR →× υUnL : , as follows

( )∑=

−+=υ

υ

1

)( ))(())(()(),(k

kkkk YuExREuxfuxL . (4.11)

where ),...,( 1)( υυ uuu = .

It is well defined, because for every U∈ku and every nx R∈ the expected value

))](([ xRu kkE exists and is finite.

Theorem 4.4 If x is an optimal solution of (4.2)–(4.4) then there exists a function υυ U∈)(u , such that

)ˆ,(max)ˆ,ˆ( )()( υυ uxLuxLx X∈

= (4.12)

and

)](ˆ[))]ˆ((ˆ[ kkkk YuxRu EE = , υ,1=k , (4.13)

where )ˆ,...,ˆ(ˆ 1)( υυ uuu = .

Conversely, if for some function υυ U∈)(u an optimal solution x of (4.12) satisfies

(4.3) and (4.13), then x is an optimal solution of (4.11)–(4.13).

Proof. By Proposition 4.2 problem (4.2)–(4.4) is equivalent to problem (4.6)–(4.10). We

associate Lagrange multipliers mR∈μ with constraints (4.8) and we formulate the

Lagrangian Λ : RRRR →×× mmTn υ)( as follows:

⎟⎠

⎞⎜⎝

⎛−+=Λ ∑∑∑

== =

T

t

kit

kt

ki

k

k

m

i

ki spyYFxfsx

12

1 1

)()( ),()(),,(υ

υυ μμ

where ),...,( 1)( υυ sss = , ),...,( 1)( υυ μμμ = and ),...,( 1km

kk μμμ = .

Let us define the set

⎭⎬⎫

⎩⎨⎧

===≥+×∈= ∑=

+ υυυ ,1,,1,,1,:)(),(1

)( kTtmiysrxRsxZ ki

n

j

kit

kjtj

mTX .

Since Z is a convex closed polyhedral set, the constraints (4.8) are linear, and the

objective function is concave, if the point )ˆ,ˆ( )(υsx is an optimal solution of problem

57

(4.2)–(4.4), then the following Karush-Kuhn-Tucker optimality conditions hold true.

There exists a vector of multipliers 0ˆ ≥μ such that:

)ˆ,,(max)ˆ,ˆ,ˆ( )()(

),(

)()()(

υυυυ μμυ

sxsxZsxΛ=Λ

∈ (4.14)

and

υμ ,1,,1,0),(ˆ1

2 ===⎟⎠

⎞⎜⎝

⎛−∑

=

kmispyYFT

t

kit

kt

ki

kki . (4.15)

We can transform the Lagrangian Λ as follows:

∑∑∑∑∑= = == =

−+=Λυυ

υυ μμμ1 1 1

21 1

)()( ),()(),,(k

kit

ki

m

i

T

t

ki

ki

k

k

m

i

ki spyYFxfsx

∑∑∑∑∑== == =

−+=m

i

kit

ki

k

T

t

kt

ki

k

k

m

i

ki spyYFxf

11 12

1 1),()( μμ

υυ

.

For any fixed x the maximization with respect to )(υs such that Zsx ∈),( )(υ yields

+=⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∑

n

j

kjtj

ki

kit rxys

1

( ) ,,1,,1,,1,)]([ υ===−= + kTtmixRy tkk

i

where tk xR )]([ is the t-th realization of the portfolio return. Define the functions

RR →:kiu , mi ,1= , υ,1=k by

+−−= )()( ηη ki

ki yu ,

and let

)()(1

ημημ ∑

=

=m

i

ki

ki

k uu k , ),...,( 1km

kk μμμ = .

Let us observe that υμ

U∈kku . We can rewrite the result of maximization of the

Lagrangian Λ with respect to s as follows:

( )∑∑∑∑ ∑== === =

++=Λm

it

kki

ki

k

T

t

kt

ki

k

k

m

i

ki

sxRupyYFxfsx

11 12

11 1

)()( )]([),()(),,(max)(

μμμυυ

υυυ

( )∑∑∑∑= == =

++=υ

μ

υ

μ1 1

21 1

)]([),()(k

tkk

T

t

kt

ki

k

k

m

i

ki xRupyYFxf k .

58

(4.16)

Furthermore, we can obtain a similar expression for the sum involving kY :

+== == =∑∑∑∑∑ −= )(),(

11 12

1 1

m

l

kl

ki

kl

k

m

i

ki

ki

k

k

m

i

ki yyyYF πμμ

υυ

+== =∑∑∑ −= )(

11 1

m

i

kl

ki

ki

k

m

l

kl yyμπ

υ

.

)(1 1

kl

km

i

m

l

kl yu kμ

π∑∑= =

−=

Substituting into (4.16), we obtain

[ ]∑=

−+=Λυ

μμυυ μ

υ1

)()( )]([))](([)(),,(max)(

k

kkkk

sYuxRuxfsx kk EE ),( )(υ

μuxL= ,

(4.17)

Setting kkkuu

μ=ˆ , υ,1=k we conclude that the conditions (4.14) imply (4.12), as

required. Furthermore, adding the complementarity conditions (4.15) over mi ,1= , and

using the same transformation we get (4.13).

To prove the converse, let us observe that for every υυ U∈)(u we can define

),()ˆ()()ˆ(ˆ '' ki

kki

kki yuyu +− −=μ mi ,1= , υ,1=k .

with ')ˆ( ku− and ')ˆ( ku+ denoting the left and right derivatives of ku :

t

tuuukk

t

k

−−

=↑− η

ηηη

)(ˆ)(ˆlim)()ˆ( ' and

ttuuu

kk

t

k

−−

=↓+ η

ηηη

)(ˆ)(ˆlim)()ˆ( .

Since ku is concave, 0ˆ ≥kμ . Using the elementary functions +−−= )()( ηη ki

ki yu we

can represent ku as follows:

∑=

=m

i

ki

ki

k uu1

)(ˆ)(ˆ ημη , υ,1=k .

Consequently, correspondence (4.17) holds true for kμ , and )(ˆ υu . Therefore, if x is the

maximizer of (4.12), then the pair )ˆ,ˆ( )(υsx , with )ˆ,...,ˆ(ˆ )(1)( υυ sss = ,

+=⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∑

n

j

kjtj

kiit rxys

1

)( ˆˆ υ ,

59

is the maximizer of )ˆ,,( )()( υυ μsxΛ , over Zsx ∈),( )(υ . Our result follows then from

standard sufficient conditions for problem (4.6)–(4.10) (see,e.g.,Rockafellar [150,

Theorem. 28.1]).

We can also develop duality relations for our problem. With the Lagrangian (4.11) we

can associate the dual function

),(max)( )()( υυ uxLuDx X∈

= .

We are allowed to write the maximization operation here, because the set X is compact

and ),( )(υuxL is continuous.

The dual problem has the form

⎪⎩

⎪⎨⎧

∈∈

.

),(min)()(

)()(

υυ

υυυ

UU

u

uDu (4.18)

The set υU is a closed convex cone and )(⋅D is a convex functional, so (4.18) is a

convex optimization problem.

Theorem 4.5 Assume that (4.11)–(4.13) has an optimal solution. Then problem (4.18)

has an optimal solution and the optimal values of both problems coincide. Furthermore,

the set of optimal solutions of (3.31) is the set of functions U∈u satisfying (4.11)–

(4.13) for an optimal solution x of (4.11)–(4.13).

Proof. The theorem is consequence of Theorem 4.4 and general duality relations in

convex non-linear programming (see Beale [10, Theorem. 2.165]). Note that all

constraints of our problem are linear or convex polyhedral, and therefore we do not need

any constraints qualification conditions here.

4.4 Splitting

Let us now consider the special form of problem (4.11)–(4.13), with

∑=

=υ

1)]([)(

k

kk xREwxf , 0>kw for υ,1=k .

Recall that the random returns υ,1,,1,R == knjkj , have discrete distributions with

realizations υ,1,,1, == kTtr kjt , attained with probabilities k

tp .

60

In order to facilitate numerical solution of problem (4.11)–(4.13), it is convenient to

consider its split-variable form:

⎥⎦

⎤⎢⎣

⎡∑=

υ

1)(max

k

kk xRwE (4.19)

subject to kk VxR ≥)( , υ,1=k a.s., (4.20)

kk YV )2(f , υ,1=k (4.21)

X∈x . (4.22)

In this problem, kV is a random variable having realizations ktv attained with

probabilities ktp , υ,1,,1 == kTt , and relation (3.33) is understood almost surely. In

the case of finitely many realizations it simply means that

kt

kj

n

j

kjt vxr ≥∑∑

= =

υ

1 1 , υ,1,,1 == kTt . (4.23)

We shall consider two groups of Lagrange multipliers: a utility function υυ U∈)(u , and

vector 0, ≥∈ kTk θθ R . The utility functions )()( ⋅υu will correspond to the dominance

constraints (4.21), as in the preceding section. The multipliers kt

ktp θ , υ,1,,1 == kTt ,

will correspond to the inequalities (4.23). The Lagrangian takes on the form

+⎟⎟⎠

⎞⎜⎜⎝

⎛−+= ∑∑∑∑∑∑

== === =

kt

n

jj

kjt

k

T

t

kt

kt

n

jj

kjt

k

T

t

ktk vxrpxrpwuVx

11 111 1

)()()( ),,,(υυ

υυυ θθL

∑∑∑∑= == =

−+υυ

π1 11 1

)()(k

kl

km

l

kl

kt

k

k

T

t

kt yuvup , (4.24)

where the random variable kV is identified by its realization kT

k vv ,...,1 . (Thus we put

),...,,( 21υT

kkk vvvV = and ),...,( 1)( υυ VVV = ).

The optimality conditions can be formulated as follows.

Theorem 4.6 If )ˆ,ˆ( )(υVx is an optimal solution of (4.19)–(4.22), then there exist

U∈u and a nonnegative vector υυθ )(ˆ )( TR∈ , such that

)ˆ,ˆ,,(max)ˆ,ˆ,ˆ,ˆ( )()()(

)(),(

)()()()(

υυυυυυ θθυυ

uVxuVxTVx

LLX R×∈

= , (4.25)

61

0)(ˆ)ˆ(ˆ11

=−∑∑==

m

l

kl

kkl

kt

kT

t

kt yuvup π , υ,1=∀k (4.26)

0)ˆˆ(ˆ1

=−∑=

n

jj

kjt

kt

kt xrvθ , υ,1,,1 == kTt . (4.27)

Conversely, if for some function υυ U∈)(u and nonnegative vector υυθ )(ˆ )( TR∈ , an

optimal solution ( ))(ˆ,ˆ υVx of (4.25) satisfies (4.20)–(4.22) and (4.26)–(4.27), then

( ))(ˆ,ˆ υVx is an optimal solution of (4.19)–(4.22).

Proof. By Proposition 4.1, the dominance constraints (4.21) is equivalent to finitely

many inequalities

υ,1,,1],)[(]))([( ==−≤− ++ kmiYyxRy kki

kki EE .

Problem (4.19)–(4.22) takes on the form:

⎥⎦

⎤⎢⎣

⎡∑=

υ

1)(max

kkk xRwE

subject to ∑=

≥n

j

ktj

kjt vxr

1, υ,1,,1 == kTt

υ,1,,1],)[(]))([( ==−≤− ++ kmiYyxRy kki

kki EE

.X∈x

Let us introduce Lagrange multipliers miki ,1, =μ , υ,1=k associated with the

dominance constraints. The standard Lagrangian takes on the form:

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=Λ ∑∑∑∑∑∑

== === =

n

j

ktj

kjt

kt

k

T

t

ktj

n

j

kjt

k

T

t

ktk vxrpxrpwVx

11 111 1

)()()( ),,,( θθμυυ

υυυ

∑∑∑∑ ∑∑∑=

+= == =

+= =

−+−−m

l

kl

ki

kl

k

m

i

ki

T

t

n

jj

kjt

kit

k

m

i

ki yyxryp

11 11 11 1][][ πμμ

υυ

.

Rearranging the last two sums, exactly as in the proof of Theorem 4.4, we obtain the

following key relation. For every 0≥kμ , setting

+=

−−= ∑ )()(1

ημημ

ki

m

i

ki

k yu k

62

we have

),,,(),,,( )()()()()()( υυμ

υυυυ θθμ kuVxVx L=Λ ,

where ),...,( )(1)( υμμ

υμ kkk uuu = .

The remaining part of the proof is the same as the proof of Theorem 4.4.

The dual function associated with the split-variable problem has the form

),,,(sup),( )()()(

)(,

)()(

)(

υυυυυ θθυυ

uVxuTVx

LDX R∈∈

=

and the dual problem is, as usual,

),(min )()(

0,)(

υυ

θθ

υυu

uD

U ≥∈, (4.28)

The corresponding duality theorem is an immediate consequence of Theorem 4.5 and

standard duality relations in convex programming. Note that all constraints of our

problem (4.19)–(4.22) are linear or convex polyhedral, and therefore here we do not

need additional constraints qualification conditions.

Theorem 3.4 Assume that (4.19)–(4.22) has an optimal solution. Then the dual problem

(4.28) has an optimal solution and the optimal values of both problems coincide.

Furthermore, the set of optimal solutions of (4.28) is the set of functions υυ U∈)(u and

vectors 0)( ≥υθ satisfying (4.25)–(4.27) for an optimal solution )ˆ,ˆ( )(υVx of (4.19)–

(4.22).

We can analysis in more detail the structure of the dual function:

∑ ∑∑∑∑∑= =====∈∈ ⎭

⎬⎫

⎩⎨⎧

+−+=υ

θθ1 11111,

)()(sup),,(k

T

t

kt

kt

ktj

n

j

kjt

T

t

kt

ktj

n

j

kjt

T

t

kt

VxvupvxrpxrpvuD

TRX

)(1 1

kl

k

m

l

kl yu∑∑

= =

−υ

π

∑ ∑∑∑∑= ==∈= =

∈ ⎭⎬⎫

⎩⎨⎧

−−++=υ

πθθ1 111 1

)(])([sup)1(maxk

n

j

kl

kl

T

t

kt

kt

kt

kt

RVj

kj

m

j

kt

T

t

ktx

yuvvupxrpTX

.

63

In the last equation we have used the fact that X is a simplex and therefore

the maximum of a linear form is attained at one of its vertices. It follows that the

dual function can be expressed as the sum

∑∑=

+=

++=T

t

kT

kt

kt

kt

kuDvuDpDvuD

11

10 )(),,()(),,( θθθ

υ

(4.29)

with

∑∑= =

≤≤−=

υ

θθ1 110 )1(max)(

k

kjt

kt

k

t

ktnj

rpD , (4.30)

])([sup),( kt

kt

kt

v

kt

kt vvuuD

t

θθ −= , υ,1,,1 == kTt , (4.31)

and

∑∑= =

+ −=m

k

kl

m

l

kl

kT yuvuD

1 11 ))(),( π , υ,1=k . (4.32)

If the set X is a general convex polyhedron, the calculation of 0D involves a

linear programming problem with n variables.

To determine the domain of the dual function, observe that if kt

kyu θ<− )( 1' , then

+∞=−∞→

])([lim kt

kt

kt

vvvu

kt

θ ,

and thus the supremum in (4.22) is equal to ∞+ . On the other hand, if kt

kyu θ≥− )( 1' , then the function k

tkt

kt vvu θ−)( has a nonnegative slope for kk

t yv 1≤

and nonpositive slope ktθ− for kk

t yv 1≥ . It is piecewise linear and it achieves its

maximum at one of the break points.

Therefore

)(:),( 1' k

tkk

tkt yuRUuDdom θθ ≥×∈= −+ .

At any point of the domain,

])([max),(1

kl

kl

klmk

kt

kt yyuuD θθ −=

≤≤. (4.33)

The domain of kD0 is the entire space TR .

64

4.5 Decomposition

It follows that the dual functional can be expressed as a weighted sum of 2+T

functions (4.30)–(4.32).

In order to analyze their properties and to develop a numerical method we need to

find a proper representation of the utility function u . We represent the function

u by its slopes between break points. Let us denote the values of u at its break

points by

)( kl

kl yuu = , υ,1,,1 == kml .

We introduce the slope variables

)(' kl

kl yu−=β , υ,1,,1 == kml

The vectors ,....),...,,,...,( 221

111 mm βββββ = is nonnegative, because u is

nondecreasing. As u is concave, kl

kl 1+≥ ββ , υ,1,1,1 =−= kml . We can represent

the values of u at break points as follows

∑>

−−−=ll

kl

kl

kl

kl yyu

'

)( 1β , 1,1 −= ml .

The function (4.24) takes on the form

⎥⎦

⎤⎢⎣

⎡−−−=−= ∑

>−≤≤≤≤

ll

kl

kt

kl

kl

klml

kl

kl

klml

kt

kt yyyyuuD

'

)(max][max),( 111θβθθ .

In this way we have expressed ),( kt

kt uD θ as a functions of the slope vector

mR∈β and of +∈Rktθ . We denote

⎥⎦

⎤⎢⎣

⎡−−−= ∑

>−≤≤

ll

kl

kt

kl

kl

klml

kt

kt yyyB

'

)(max),( 11θβθβ . (4.34)

Observe that ktB is the maximum of finitely many linear functions in its domain.

The domain is a convex polyhedron defined by

kl

kt βθ ≤≤0 .

65

Consequently, ktB is a convex polyhedral function. Therefore its subgradient at a

point ),( ktθβ of the domain can be calculated as the gradient of the linear

function at which the maximum in (4.34) is attained. Let *l be the index of this

linear function. Denoting by 'lδ the 'l th unit vector in mR we obtain that:

⎟⎟⎠

⎞⎜⎜⎝

⎛−−− ∑

>−

*'

),( 1ll

kl

kl

kll yyyδ is the subgradient of ),( k

tktB θβ .

Similarly, function (3.32) can be represented as a function kTB 1+ of the slope

vectors β :

∑ ∑= >

−+ −=m

l ll

kl

kl

kl

kl

kT yyB

111

'

)()( βπβ .

It is linear in β and its gradient has the form

)( 11 ''

'kl

kl

ll

kl

n

ll yy −

<=

−∑∑ πδ .

Finally, denoting by *j the index at which the maximum in (4.30) is attained, we

see that the vector with coordinates

ktj

kt rp * , υ,1,,1 == kTt , (4.35)

is a subgradient of kD0 .

Summing up, with our representation of the utility function by its slopes, the dual

function is a sum of T + 2 convex polyhedral functions with known domains.

Moreover, their subgradients are readily available. Therefore the dual problem

can be solved by nonsmooth optimization methods (see Dentcheva and

Ruszczynski [41], Beale [8] and Bonnans and Shapiro [18]). We have developed

a specialized version of the regularized decomposition method described in

Dentcheva and Ruszczy´nski [41] and Ogryezak and Ruszczy’nski [127]. This

approach is particularly suitable, because the dual function is a sum of very many

polyhedral functions.

66

After the dual problem is solved, we obtain not only the optimal dual solution

)ˆ,ˆ( θβ , but also a collection of active cutting planes for each component of the

dual function.

Let us denote by kj0 the collection of active cuts for kD0 . Each cutting plane for

kD0 provides a subgradient (4.35) at the optimal dual solution. A convex

combination of these subgradients provides the subgradient of kD0 that enters the

optimality conditions for the dual problem. The coefficients of this convex

combination are also identified by the regularized decomposition method. Let kg0

denote this subgradient and let 0, Jjvkj ∈ the corresponding coefficients. Then

∑∑∈=

=0Jj1

0kj

kjt

kt

T

t

kt

k vrpg δ ,

where

0≥kjv , ∑

∈

=0Jj

1kjv .

For each t the subgradient of ktB with respect to k

tθ entering the optimality

conditions is

)34.4(inmaximizeris:ˆ ** lyconvv k

lkt ∈ .

Therefore

0ˆ1

0 =−∑=

Tkt

kt

k vpgl

,

Using these relations we can verify that v are the vector of optimal portfolio

returns in scenarios Tt ,1= . Thus the optimal portfolio has the weights

⎩⎨⎧

∉=∈=

.,0ˆ,,ˆ

0

0

JjxJjvx

j

jj

67

68

CHAPTER 5

A FUZZY APPROACH TO PORTFOLIO OPTIMIZATION

5.1 Introduction

In general optimization problem in which the objective function(s) and the constraints in

the space of the decision variables are linear is said to be a linear programming problem.

In addition, if there are multiple objectives, then the optimization problem will be called a

multi-objective linear programming problem.

In a recent paper Sakawa and Yana [165] very nicely demonstrate the state of the art

when we want to deal with multiple criteria problems which are - and cannot be - well-

defined. Their model is a multiple objective linear fractional programming model, with

fuzzy parameters and an uncertain goal for the objective function. The uncertainty is of

two types: (i) an uncertainty of the satisfaction with the value of an objective function;

and (ii) an uncertainty of the possibility to generate the wanted value of the objective

function. The Sakawa-Yana method handles both types of uncertainties and reduces the

problem to an ordinary multi-objective programming problem.

Another approach has been developed by Kacprzyk and Yager [74], in which they use

fuzzy logic with linguistic qualifiers to bring human consistency to multiobjective

decision making. They use rather a nonconventional solution concept, which is based on

searching for some optimal option which ”best satisfies most of 2 the important

objectives”; this differs from the traditional notion to try to find an optimal option which

best satisfies ”all the objectives”.

Both the Sakawa-Yana and the Kacprzyk-Yager papers seem to support the idea that

traditional multi-criteria decision models (MCDM), and their underlying notion of an

optimal solution, are much to o limited for actual, real-world problem-solving with

MCDM methods. The reason for this is simple: when the solution derived from a well-

formulated mathematical MCDM-model is applied to an actual problem there are some

major problems to consider (Roy [149]): (i) the set of feasible decision alternatives is

69

fuzzy, and this set changes during the problem solving process; (ii) the DM does not exist

as an active entity, and the preferences consist of badly formulated beliefs, which are

riddled with conflicts and contradictions; (iii) data on preferences are imprecise, and (iv)

a decision should be good or bad not only in relation to some model, but in relation to the

actual context. These problems have initiated active and fast-growing research on the use

of fuzzy set theory in solving multiple criteria decision problems (Carlsson [28, 29],

Takeda [191] and Zimmermann [222]).

The problem of optimizing a portfolio of finitely many assets is a classical problem in

theoretical and computational finance. Since the seminal work of Markowitz [112] it is

generally agreed that portfolio performance should be measured in two distinct

dimensions: the mean describing the expected return, and the risk which measures the

uncertainty of the return. As one of theoretical approach to the portfolio selection

problem is that of stochastic dominance (see Rockafellar and Uryasev [152].

In the context of choosing optimal portfolios that target returns above the risk-free rate

for certain market scenarios while at the same time guaranteeing a minimum rate of

return, fuzzy decisions theory provides an excellent framework for analysis. This is

because the nature of the problem requires one to examine various market scenarios, and

each such scenario will in turn give rise to an objective function. In the last section, we

will describe a multi-objective linear programming problem formulation where the

objective functions are considered to be fuzzy.

5.2 Fuzzy linear programming models

Empirical surveys reveal that linear programming (LP) is one of most frequently applied

operations research techniques is real-word problems. However, given the power of LP

one could have expected oven more applications. This might be due to the fact that LP

requiresmunch well-defined and precise data which involves high-information cost. In

real-word applications certainty, reliability and precision of data is often illusory.

Furthermore the optimal solution of LP only depends on a limited number of constraints

and, thus, much of the information collected has little inpact on the solution. Fuzz linear

programming, propused by Bellman and Zadeh, is an extention of LP with both objective

function(s) and constraints represented by fuzzy sets.

70

Now we can defined a LP problem with crisp of fuzzy resource constraints, and a crisp or

fuzzy objective as:

subject to

,0

,1,~,~max

1

≥

=≤

=

∑=

X

pibXa

XcZ

ij

m

jij

T

(5.1)

where fuzzy resources ibi ∀,~

have the same form of membership function. We may also

consider the following fuzzy inequality constraints:

subject to

,0

,1,~~

,~max

1

≥

=≤

=

∑=

X

pibXa

XcZ

ij

m

jij

T

(5.2)

even though (5.1) and (5.2) are defferent in some points of view, we can use the same

approach to handle them under the pre-assumtion of the membership functions of the

fuzzy available resources and fuzzy inequality constraints.

The difference between crisp and fuzzy constraints is that in case of crisp constraints the

decision maker can strictly differentiate between feasibility and infeasibility; in case of

fuzzy constraints he wants to consider a certain degree of feasibility in the interval

(Werners, 1987).

Now, we consider some approaches for fuzzy linear programming models.

Correspondingly we could build portfolio models as sections 5.5, 5.6 and 57.

First approach: The resources can be determined precisely, a traditional LP problem is

consider as :

subject to

,0

,~,max

1

≥

∀≤

=

∑=

X

ibXa

XcZ

ij

m

jij

T

(5.3)

where ibac iij ∀,and, are precisely given. The optimal solution of (5.3) ia a unique

optimal solution.

71

Second approach Chanas and Verdegay: A decision maker wishes to make a

postoptimization analysis. Thus, a parametric programming problem is formulated as:

subject to

,0

],1,0[,~,max

1

≥

∈+≤

=

∑=

X

pbXa

XcZ

ij

m

jij

T

θθ (5.4)

where ipbac iiij ∀,and,, are precisely given and θ is a parameter, ipi ∀, are

maximum tolerances which are always positive. The solution )(* θZ of (5.4) are function

of θ . That is, for each θ we can obtain an optimal solution.

On the other hand, the available resources may be fuzzy. Then the LP problem with fuzzy

resources becomes:

subject to

,0

,~,max

1

≥

∀≤

=

∑=

X

ibXa

XcZ

ij

m

jij

T

(5.5)

It is possible to determine the maximum tolerance ip of the fuzzy resources ibi ∀, . Then

we can construct the membership functions iμ assumed linear for each fuzzy constraints,

as follows:

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

+>

+≤≤−

−

≤

=

∑

∑∑

∑

=

=

=

=

.if0

,if1

,if1

1

1

1

1

ii

m

jjij

ii

m

jjiji

i

i

m

jjij

i

m

jjij

i

pbXa

pbXabp

bXa

bXa

μ (5.6)

Verdegay and Chanas, propuse that (5.5) and (5.6), however, are equivalent to (5.4), a

parametric LP where ipbac iiij ∀,and,, are given, by use of the λ -level cut concept.

72

For each λ -level cut of the fuzzy constraint set (5.5) becomes a traditional LP problem.

That is,

subject to

].1,0[,0,,,

,max

∈≥∀≥=

∈=

λλμλ

λ

XandiXXXX

XcZ

i

T

(5.7)

and equivalent to:

subject to

,0and]1,0[

,,)1(

,~max

1

≥∈

∀−+≤

=

∑=

X

ipbXa

XcZ

ii

m

jjij

T

λ

λ (5.8)

where ipbac iiij ∀,and,, are precisely given. Now, if we set θλ −=1 , then equation

given by (5.8) will be the same as (5.4). Then a solution table is presented to the decision

maker to determine the satisfying solution. )(* θZ , ]1,0[∈θ is the fuzzy solution

corresponding to Verdegay’s approach.

Third approach (Weners’s approach): A decision maker may want to solve a FLP

problem with a fuzzy objective and fuzzy constraints, while the goal 0b , is not given.

That is:

subject to

,0

,,~

,~max

1

≥

∀≤

=

∑=

X

ibXa

XcZ

i

m

jjij

T

(5.9)

which is equivalent to:

subject to

,0and]1,0[

,,~

,~max

1

≥∈

∀+≤

=

∑=

X

ipbXa

XcZ

ii

n

jjij

T

λ

θ (5.10)

where ipbac iiij ∀,and,, are given, but the goal of the fuzzy objective is not given.

To solve (5.10) by use of Werners’s approach, let us first define 0Z and 1Z as follows:

73

),0()maxinf( *0 ===∈

θZXcZ T

X X (5.11)

)1()maxsup(( *1 ===∈

θZXcZ T

X X (5.12)

where .0and],1,0[,,1

≥∈∀+≤= ∑=

XipbXaX ii

n

jjij θθX

Then, we can obtain Werners’s membership function 0μ of the fuzzy objective. That is:

⎪⎪⎩

⎪⎪⎨

⎧

<

≤≤−−

−

>

=

.if0

,if1

,if1

0

1001

1

1

0

ZXc

ZXcZZZ

XcZZXc

T

TT

T

μ (5.13)

The membership functions ii ∀,μ , of the fuzzy constraints are defined as (5.6). By use of

the min-operator proposed by Bellman and Zadeh, we can obtain the decision space D

which is defined by its membership function Dμ where,

),...,min( 0 pD μμμ = . (5.14)

It is reasonable to choose the decision where Dμ is maximal as the optimal solution of

(5.9). Therfore, (5.9) is equivalent to:

λmax

,0],1,0[and,

,,

0

0

≥∀∈

≥≥

Xii

i

μμλλμλμ

(5.15)

where ipbac iiij ∀,and,, are given, and ),...,min( 0 mD μμμλ == .

Let θλ −=1 . Then the problem given by (5.15) will be equivalent to:

θmax

subject to

,0and]1,0[

,,)(),( 011

≥∈

∀+≤−−≥

X

ipbXaZZZXc

iiiij

T

θ

θθ

(5.16)

74

where ipbac iiij ∀,and,, are given and θ is fraction of )( 01 ZZ − for the first

constraint and a fraction of maximum tolerance for others. The solution is a unique

optimal solution.

Forth approach (Zimmermann’s approach): A decision maker may want to solve a FLP

problem with a fuzzy objective and fuzzy constraints, when the goal 0b of the fuzzy

objective and its minimum tolerance are given. That is,

subject to

,0

,,~

,~max

1

≥

∀+≤

=

∑=

X

ipbXa

XcZ

ii

n

jjij

T

θ (5.17)

where iij bpbac ,,,, 00 and ipi ∀, are given. The problem given by (5.17) is actually

equivalent to:

Find X,

subject to

,0

,,

,~max

1

≥

∀+≤

=

∑=

X

ipbXa

XcZ

ii

m

jjij

T

θ (5.18)

with the membership function of the fuzzy constraints as previously described in (5.6)

and the membership function of the fuzzy objective 0μ as follows:

⎪⎪

⎩

⎪⎪

⎨

⎧

−<

≤≤−−−

−

>

=

.if0

,if1

,if1

00

000010

0

0

pbXc

bXcpbZZ

Xcb

bXc

T

TT

T

μ (5.19)

Thus by use of the maximum concept, (5.18) is actually equivalent to:

λmax

subject to

,0],1,0[and,,,and

0

i0

≥∀∈

∀≥

Xi

i

iμμλλμμ

(5.20)

75

where iij bpbac ,,,, 00 and ipi ∀, are given. Let θλ −=1 . Then (5.15) will be

equivalent to:

subject to

,0and]1,0[

,,

,,max

1

00

≥∈

∀+≤

+≥

∑=

X

ipbXa

pbXc

ii

m

jjij

iT

θ

θ

θ

θ

(5.21)

where iij bpbac ,,,, 00 and ipi ∀, are given and θ is a fraction of the maximum

tolerances. The optimal solution (5.21) is unique.

When a fuzzy objective is assumed, what Zimmermann and Werner’s approaches are

asumming is essentially a performance function on the objective,

]1,0[)()( ∈= XcFXf T .

Thn, in all cases, if ]1,0[),(* ∈θθZ , is the fuzzy solution to the problem, the

corresponding point solution for each fuzzy objective (performance function associated to

Zimmermann’s or Werner’s approach) to be considered, can be obtained by solving the

point-fix equation. It is showed that in the application.

Fifth approach: A decision maker may want to solve a FLP problem with a fuzzy

objective and fuzzy constraints, while only the goal 0b of the fuzzy objective is given, but

its tolerance 0p is not given. That is,

subject to

,0

,,~

,x~ma~max

1

≥

∀≤

=

∑=

X

ibXa

XcZ

i

n

jjij

T

(5.22)

where iij bbac ,,,, 0 and ipi ∀, are given, but 0p is not given. While 0p is not given, we

do know that 0p should be in between 0 and 00 Zb − . For each ],0[ 0

00 Zbp −∈ , we

can obtain the membership function of the fuzzy objective as (5.19). Sice in a high-

productivity system the objective value should be larger then 0Z at 0=θ , there is no

meaning to given a positive grade of membership for those which are less than 0Z .

76

The difference between this problem and Zimmermann is that 0p is not initially given in

this problem. Therefore, we may assume a set of sp0 , where ],0[ 000 Zbp −∈ . Then, the

problem of each 0p given is a Zimmermann problem.

The decision maker may choose a refined 0p among the solution for this given set of

sp0 . Then a Zimmermann problem with the decision maker’s refined 0p is solved. This

solution will be the final optimal solution for (5.22).

5.3 Interactive fuzzy linear programming

Decision processes are better described and solved using fuzzy sets theory, rather

than precise approaches. However, the decision maker himself always plays the

most important role in using fuzzy sets theory. Therefore, an interactive process

between the decision maker and the decision process is necessary to solve our

problem. That is actually a user-dependent fuzzy LP technique.

Furthermore, a problem-oriented concept is also a vitally important concept in

solving practical problems, as noted by Simon.

By use of fuzzy sets theory, and user dependent (interactive) and problem,

oriented concepts, the flexibility and robustness of LP techniques are improved.

An IFLP approach which is a symmetric integration of Zimmermann’s, Werner’s,

Verdegay’s and Chanas’s FLP approaches is developed and additionally it

provides a decision support system for solving a specific domain of a rel-world

LP system Lai and Hwang [92]. Lai and Hwang suggested “expert decision

support system” that give an aggregate solution to all possible cases.

The system determines fuzzy-efficient extreme solution and a fuzzy efficient

compromise solution. They are judged by the decision maker and he decides

whether modification are necessary. In the latter case the decision maker change

membership functions assisted by the system, Werners [202].

The application of FLP implies that the problems will be solved in an interactive

way. In the first step, the fuzzy system is modeled by using only the information

77

which the decision maker can provide without any expensive additional

information acquisition. Knowing a first “compromise solution” the decision

maker can perceive which further information should be obtained and he is able to

justify the decision by comparing carefully additional advantages and arising

costs. In doing so, step by step the compromise solution are improved. The

procedure obviously offer the possibility to limit the acquisition and processing

information to the relevant components and therefore information costs will be

distinctly reduced, Rommefanger [154].

The most important element that affects solutions of FLP problems is

parameters which are used reflecting fuzziness of model. How these parameters

define fuzzy geometry is the most sensitive point. Because the success of solution

depends on the success of reflecting the system of model.

Moreover, the interactive concept provides a learning process about the

system and makes allowance for psychological convergence for the decision

maker, whereby, (s)he learns recognize good solutions, the relative importance of

factors in the system and then design a high-productivity system, instead of

optimizing a given system.

This IFLP system provides integration-oriented, adaption and learning

features by considering all possibilities of specific domain of LP problems which

are integrated in logical order using an IF-THEN rule.

IFLP methods have been studied, since 1980. Typical works are Baptistella and

Ollero, Fabian, Cibiobanu and Stoica, Ollero, Aracil and Camacho, Sea and

Sakawa, Slowinski, Werners and Zimmermann, Zimmermann described some

general concepts and modeling methods of decision support system and expert

system in a fuzzy enviroment. Others developed interactive approaches to solve

multiple criteria decision making problem, Lai and Hwang [92].

With the aim of solutions for the models like these, there are many studies

on LP models. However the studies of Zimmermann, Chanas, Werners and

78

Vedegay have become quite efficient for improving LP models with decision

support to solve real-world problem.

5.3.1 Interactive fuzzy linear programming algorithm

Step 1 Solve a traditional LP problem of (5.3) by use of the simplex method.

The unique optimal solution with its corresponding consumed resources is

presented to the decision maker.

Step 2 Does this solution satisfy the decision maker ? Consider the following

cases.

1.If solution is satisfied the print out results an top

2.If resource i, for some i are idle then reduce available ib and go to Step 1.

3. If available resources are not precise and some tolerances are possible then

make a parametric analysis with and go to Step 3.

Step 3 Solve a parametric LP problem of (5.4). Then results are depicted in a

table. At the same time, let us identify )0(*0 == θZZ and )1(*1 == θZZ .

Step 4 Do any of these solutions shown in table satisfy the decision maker?

Consider the following cases:

1. If solution is satisfied then print out results ad stop.

2. If resource i, for some i are idle then decrease ib (and change ip ) and

then go to Step 1.

3. If tolerance i, for same i are not acceptable then change ip as desired and

go to Step 3.

4. If the objective should be considered as imprecise then to Step 5.

Step 5 After reffering to first table, the decision maker is then asked for his

subjective goal 0b and its tolerance 0p for solving a symmetric FLP problem.

If the decision maker does not like to give his goal for the fuzzy objective, to to

Step 6. If 0b is given, go to Step 8.

79

Step 5 Solve problem of (15). A unique Werners’s solution is the provided.

Step 7 Is the solution of (5.16) satisfying ? Consider the following cases:

1. If the solution is satisfied then print out results and stop.

2. If the user has realized his/her goal then give the goal 0b and go to Step 8.

3. If resource i, for some i are idle then decrease ib (and change ip ) and then

go to Step 1.

4. If tolerance i, for same i are not acceptable the change ip as desired and go

to Step 3.

Step 8 Is 0p determined by the decision maker ? If the decision maker would like

to specify 0p , we should provide a table to help the decision maker. Then go to

Step 9. If 0p is not given, then go to Step 11.

Step 9 Solve problem of (5.21). A unique Zimmermann’s solution is obtained.

Step 10 Is the solution of (5.21) satisfying ?

1. If solution is satisfied then print out results and stop.

2. If the user has realized beter his/her goal (and its tolerance) then give the

goal 0b ( and 0p ) and go to Step 8.

3. If resource i, for some i are idle then decrease ib (and change ip ) and then

go to Step 1.

4. If tolerance i, for same i are not acceptable the change ip as desired and go

to Step 3.

Step 11 Solve last problem. That is, call Step 9 to solve problem of (5.21) for a set

of sp0 . Then the solutions are depicted in a table.

Step 12 Are the solution satisfying ? If yes, print out the solution and then

terminate the solution procedure. Otherwise, go to Step 13.

80

Step 13 Ask the decision maker to specify the refined 0p , and then go to Step 0.

It is rather reasonable to ask the decision maker 0p at this step, because he has a

good idea about 0p now figure 5.1.

For implementing the above IFLP, we need only two solution-finding techniques,

the simplex method and parametric method. Therefore, the IFLP approach

proposed here can be easily programmed in a PC system for its simplicity, Lai and

Hwang [92].

5.4 Portfolio problem

The process of selecting a portfolio may be divided into two stages. The first stage starts

with observation and experience and end with beliefs about the future performance of

available securities. The second stage starts with the relevant beliefs about future

performances and ends with the choice of portfolio by Markowitz [114]. This chapter is

concerned with the second stage.

The problem of standard portfolio selection is as follows. Assume

(a) n securities,

(b) an initial sum of money to be invested,

(c) the beginning of a holding period,

(d) the end of the holding period,

and let x1,…,xn be the investors investment proportion weights. These are the proportions

of the initial sum to be invested in the n securities at the beginning of the holding period

that define the portfolio to be held fixed until the end of the holding period. The standard

view is that there is only one purpose in portfolio selection, and that is to maximize

portfolio return, the percent return earned by the portfolio over the course of the holding

period.

Now we consider the problem (4.11)-(4.12) given in Chapter 4, for 1=υ . Thus we have

)(max xf (5.23)

subject to YxR SSDf)( , (5.24)

X∈x . (5.25)

81

Here R→X:f is a concave continuous functional. Also in particular, we may

use

)]([)( xRxf E=

and this will still lead to nontrivial solutions, due to the presence of the dominance

constraint.

Yes

No

Yes

No

Figure 5.1 Flow chart decision support system (Werner’s, 1987)

Using the Chapter 4, for 1=υ , we get the following proposition.

Proposition 5.1 Assume that Y has a discrete distribution with realizations miyi ,1, = .

Then relation (5.24) is equivalent to

])[(]))([( ++ −≤− YyxRy ii EE , mi ,1=∀ . (5.26)

Model formulation

Efficient Extremesolution

Compromise Solution Local Information

Solution Acceptable ?“Best”

Compromise STOP

Modification of membership functionstion

Local consequences ?

82

Let us assume now that the returns have a discrete joint distribution with realizations jtr ,

Tt ,1= , nj ,1= , attained with probabilities tp , Tt ,1= . The formulation of the

stochastic dominance relation (5.24) respectively (5.26) simplifies event further.

Introducing variables its representing shortfall of R(x) below yi in realization t, mi ,1=

and Tt ,1= , we obtain the following proposition.

Proposition 5.2 The problem (5.23)-(5.25) is equivalent to the problem

)(max xf (5.27)

subject to iitj

n

jjt ysxr −≤−−∑

=1, mi ,1= , Tt ,1= , (5.28)

);(21

iit

T

tt yYFsp ≤∑

=

, mi ,1= (5.29)

0≥its mi ,1= , Tt ,1= , (5.30)

∑=

≤n

jjx

11, (5.31)

∑=

−≤−n

jjx

11 , (5.32)

0≥jx , nj ,1= , (5.33)

and problem (5.27)-(5.33) can be written as

)(max Xϕ = ∑=

n

jjj Xc

1, (5.34)

subject to: i

mTn

jjij bXa ≤∑

+

=1, 2,1 ++= mmTi , (5.35)

0≥jX , mTnj += ,1 , (5.36)

where, ),...,,...,,...,,,...,,,...,( 12211111 mTmTTn ssssssxxX = .

83

⎪⎩

⎪⎨

⎧

+−++=−=++=−−=++==−

=otherwise

iTnnjandTKmKKmiTKmKKminjr

aij

ij

,01)1(,1,)1(,0,)1(,1,1

)1(,0,)1(,1,,1,

⎩⎨⎧ +==

=otherwise

mTinjaij ,01,,1,1

⎩⎨⎧ +==−

=otherwise

mTinjaij ,02,,1,1

⎩⎨⎧ +++==++−+=

= −−−

otherwisemmTmTimKTKnKTnjpa KTnj

ij ,02,3,,1,,1)1(,)1(

In the next section we extended this result to fuzzy decisions theory.

5.5 Case of fuzzy technological coefficients and fuzzy right-hand side numbers

5.5.1 Case of fuzzy technological coefficients

In this section presents an approach to portfolio selection using fuzzy decisions theory.

We consider the problem (5.34) – (5.36) with fuzzy technological coefficients Gasimov

[57].

)(max Xϕ =∑=

n

jjj Xc

1 (5.37)

subject to i

mTn

jjij bXa ≤∑

+

=1

~ , 2,1 ++= mmTi , (5.38)

0≥jX , mTnj += ,1 . (5.39)

Assumption 5.1. ija~ is a fuzzy number for any i and j.

In this case we consider the following membership functions:

(i) 1. For )1(,0,)1(,1 −=++= TKmKKmi and nj ,1=

⎪⎩

⎪⎨

⎧

+−≥+−<≤−−+−

−<=

.0,/)(

,1)(

ijij

ijijijijijij

ji

a

drtifdrtrifdtdr

rtift

ijμ

84

2. For )1(,0,)1(,1 −=++= TKmKKmi and j=n+T(i-Km-1)+K+1

⎪⎩

⎪⎨

⎧

+−≥+−<≤−−+−

−<=

,10,11/)1(

11)(

ij

ijijija

dtifdtifdtd

tift

ijμ

(ii) For 2,3 +++= mmTmTi , mK ,1= and TKnKTnj ++−+= ,1)1(

⎪⎩

⎪⎨

⎧

+≥+<≤−+

<=

−−−

−−−−−−

−−−

−−−

,

)1(

)1(

)1()1(

)1(

0,/)(

,1)(

ij

ijijijKTnja

dptifdptpifdtdp

ptift

KTnj

KTnjKTnj

KTnj

ijμ

where Rt ∈ and 0>ijd for all 2,1 ++= mmTi , )1(,0 −= TK and mTnj += ,1 .

For defuzzification of this problem, we first fuzzify the objective function. This is done

by calculating the lower and upper bound of the optimal values first. The bounds of the

optimal values lz and uz are obtained by solving the standard linear programming

problems

)(max1 Xz ϕ= (5.40)

subject to ij

mTn

jij bXa ≤∑

+

=1, 2,1 ++= mmTi , (5.41)

0≥jX , mTnj += ,1 , (5.42)

and

)(max2 Xz ϕ= (5.43)

subject to ij

mTn

jij bXa ≤∑

+

=1

ˆ , 2,1 ++= mmTi , (5.44)

0≥jX , mTnj += ,1 , (5.45)

where

85

⎪⎩

⎪⎨

⎧

−=+−++=++=+−−=++==+−

=otherwised

TKandiTnnjmKmKmidTKandmKKminjdr

a

ij

ij

ijij

ij

,)1(,0,1)1(,1,)1(,1,1

)1(,0)1(,1,,1,ˆ

⎪⎩

⎪⎨⎧ +==+

=otherwised

mTinjda

ij

ijij ,

1,,,1,1ˆ

⎪⎩

⎪⎨⎧ +==+−

=otherwised

mTinjda

ij

ijij ,

2,,1,1ˆ

⎪⎩

⎪⎨⎧ +++==++−+=+

= −−−

otherwisedmmTmTiandmKTKnKTnjdp

aij

ijKTnjij ,

2,3,,1,,1)1(,ˆ )1(

The objective function takes values between 1z and 2z while technological

coefficients vary between ija and ijij da + . Let ),min( 21 zzz =l and ),max( 21 zzz =u .

Then lz and uz are called the lower and upper bounds of the optimal values,

respectively.

Assumption 5.2. The linear crisp problems (5.40)-( 5..42) and (5.43)-(5.45) have finite

optimal values.

In this case the fuzzy set of optimal values, G, which is subset of mTnR + , is defined as

Klir and Yuan [84 ]

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≥

≤≤−−

<

=

∑

∑∑

∑

=

==

=

n

jjj

n

jjj

n

jjj

n

jjj

G

zXcif

zXczifzzzXc

zXcif

X

1

11

1

1

)/()(

0

)(

u

ullul

l

μ (5.46)

The fuzzy set of the ith constraint, iC , which is a subset of mTnR + , is defined by

(i) 1. For mKKmi )1(,1 ++= and )1(,0 −= TK

86

)(XiCμ =

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+−≥

+−<≤−+

−<

∑

∑ ∑∑ ∑

∑

=

= == =

=

n

jjijjii

n

j

n

jjijijijij

n

j

n

jjijjiji

n

jjjii

Xdrb

XdrbXrXdXrb

Xrb

1

1 11 1

1

)(,1

)(,/)(

,0

(5.47)

2. For mKKmi )1(,1 ++= and )1(,0 −= TK

=)(XiCμ

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+−≥

+−<≤−+

−<

∑

∑ ∑∑ ∑

∑

+=

+= +=+= =

+=

),(

1

),(

1

),(

1

),(

1 1

),(

1

,)1(,1

,)1(,/)(

,,0

Kin

njjiji

Kin

nj

Kin

njjijij

Kin

nj

n

jjijji

Kin

njji

Xdb

XdbXXdXb

Xb

(5.48)

where n(i,K)=n+T(i-Km-1)+K+1

(ii) For ,2,3 +++= mmTmTi and mK ,1=

)(XiCμ =

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+≥

+<≤−

<

∑

∑ ∑∑ ∑

∑

+

−+=−−−

+

−+=

+

−+=−−−−−−

+

−+=

+

−+=−−−

+

−+=−−−

TKn

KTnjjijKTnji

TKn

KTnj

TKn

KTnjjijKTnjijKTnj

TKn

KTnj

TKn

KTnjjijjKTnji

TKn

KTnjjKTnji

Xdpb

XdpbXpXdXpb

Xpb

)1()1(

)1( )1()1()1(

)1( )1()1(

)1()1(

.)(,1

,)(,/)(

,,0

(5.49)

By using the definition of the fuzzy decisions proposed by Bellman and Zadeh

[9], we have

))((min),(min()( XXXjCjGD μμμ = .

i.e.

))((min),(min(max))((max00

XXXjCjGXDX

μμμ≥≥

=

87

Consequently, the problem (5.37)-(5.39) can be written as

λmax (5.50)

,)( λμ ≥XG (5.51)

2,1,)( ++=≥ mmTiXiC λμ , (5.52)

0≥jX , 10 ≤≤ λ , mTj ,1= . (5.53)

By using (5.46) and (5.47)-(5.53), we obtain the following theorem.

Theorem 5.1 The portfolio problem with fuzzy technological coefficient can be reduced

to the following problem

λmax (5.54)

0)( 21

21 ≤+−− ∑=

zXczzn

jjjλ , (5.55)

∑+

=

≤−mTn

jijij bXa

10)(ˆ λ , 2,1 ++= mmTi , (5.56)

0≥jX , 10 ≤≤ λ , mTnj += ,1 . (4.57)

where

⎪⎩

⎪⎨

⎧

+−+=−=++=+−−=++==+−

=otherwise,,

,1)1(,1)1(,0,)1(,1,1,)1(,0,)1(,1,,1,

)(ˆ

ij

ij

ijij

ij

diTnjandTKmKKmid

TKandmKKminjdra

λλλ

λ

⎪⎩

⎪⎨⎧ +==+

=,otherwise,

,1,,1,1)(ˆ

ij

ijij d

mTinjda

λλ

λ

⎪⎩

⎪⎨⎧ +==+−

=otherwise,,

,2,,1,1)(ˆ

ij

ijij d

mTinjda

λλ

λ

⎪⎩

⎪⎨⎧ +++=++−+=+

= −−−

. otherwise,2,3,,1)1(,

)(ˆ )1(

ij

ijKTnjij d

mmTmTiTKnKTnjdpa

λλ

λ

Notice that, the constraints in problem (5.54)-(5.57) containing the cross product term

jXλ are not convex. Therefore the solution of this problem requires the special approach

88

adopted for solving general nonconvex optimization problem (Rockafellar and Wets

[153] and White [209]).

5.5.2 Portfolio problems with fuzzy technological coefficients and fuzzy right-

hand-side numbers

We consider the linear programming problem (5.34)-(5.36) with fuzzy technological

coefficients and fuzzy right-hand-side numbers

)(max Xϕ =∑=

n

jjj Xc

1 (5.58)

subject to i

mTn

jjij bXa ~~

1≤∑

+

=

, 2,1 ++= mmTi , (5.59)

0≥jX , mTnj += ,1 . (5.60)

Assumption 5.3. ija~ and ib~ are fuzzy numbers for any i and j.

In this case we consider the following linear membership functions:

i) 1. For )1(,0,)1(,1 −=++= TKmKKmi and nj ,1= ,

⎪⎩

⎪⎨

⎧

+−≥+−<≤−−+−

−<=

+

,

)1(

0,/)(

,1)(

ijij

ijijijijijij

jk

a

drtifdrtrifdtdr

rtift

ijμ

2. For )1(,0,)1(,1 −=++= TKmKKmi and j=n+T(i-Km-1)+K+1,

⎪⎩

⎪⎨

⎧

+−≥+−<≤−−+−

−<=

,10,11/)1(

11)(

ij

ijijija

dtifdtifdtd

tift

ijμ

(ii) For mKmmTmTi ,1,2,3 =+++= and TKnKTnj +−+= ),1( ,

⎪⎩

⎪⎨

⎧

+≥+<≤−+

<=

,0,/)(

,1)(

ijij

ijijijijijij

ij

a

dptifdptpifdtdp

ptift

ijμ

and

89

⎪⎩

⎪⎨

⎧

+≥+<≤−+

<=

ii

iiiiii

i

b

pbtifpbtbifptpb

btift

i

0,/)(

,1)(μ

where Rt ∈ and 0>ijd for all 2,1,,1 ++== mmTinj . For defuzzification of this

problem (5.58)-(5.60), we first calculate the lower and upper bounds of the optimal

values. The optimal values lz and uz can be defined by solving the following standard

linear programming problems, for which we assume that all they have the finite optimal

values.

Now defuzzification of this problem (5.58)-(5.60). first we fuzzify the objective function.

This is done by calculating the lower and upper bound of the optimal values first. The

bounds of the optimal values lz and uz are obtained by solving the standard linear

programming problems

)(max1 Xz ϕ= (5.61)

subject to ijij

n

jij bXdr ≤+−∑

=

)(1

, 2,1 ++= mmTi (5.62)

0≥jX , nj ,1= , (5.63)

and

)(max2 Xz ϕ= (5.64)

subject to iij

n

jij pbXr +≤−∑

=1, (5.65)

0≥jX , nj ,1= , (5.66)

and

)(max3 Xz ϕ= (5.67)

subject to iij

n

jijij pbXdr +≤+−∑

=1)( , (5.68)

0≥jX , nj ,1= , (5.69)

and

90

)(max4 Xz ϕ= (5.70)

subject to: ij

n

jij bXr ≤−∑

=1, (5.71)

0≥jX , nj ,1= . (5.72)

Let ),,,min( 4321 zzzzz =l and ),,,max( 4321 zzzzzu = . The objective function takes

values between lz and uz while technological coefficients take values between ijr− and

ijij dr +− and the right-hand side numbers take values between ib and ii pb + .

Then, the fuzzy set of optimal values, G, which is a subset of mTnR + , is defined by

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≥

≤≤−−

<

=

∑

∑∑

∑

=

==

=

n

jjj

n

jjj

n

jjj

n

jjj

G

zXcif

zXczifzzzXc

zXcif

X

1

11

1

.,1

,,)/()(

,,0

)(

u

ullul

l

μ (5.73)

The fuzzy set of the ith constraint, iC , which is a subset of mTnR + , is defined by:

(i) 1. For mKKmi )1(,1 ++= and )1(,0 −= TK

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

++−≥

++−<≤−++

−<

=

∑

∑ ∑∑ ∑

∑

=

= == =

=

n

jijijiji

n

j

n

jijijijijij

n

j

n

jijijjiji

n

jjiji

C

pXdrb

pXdrbXrpXdXrb

Xrb

Xi

1

1 11 1

1

.)(,1

,)(,)(/()(

,,0

)(μ

(5.74)

2. For mKKmi )1(,1 ++= and )1(,0 −= TK

91

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

++−≥

++−<≤−++

−<

=

∑

∑ ∑∑ ∑

∑

+=

+= +=+= =

+=

),(

1

),(

1

),(

1

),((

1 1

),(

1

.)1(,1

,)1(,)(/()(

,,0

)(

Kin

njijiji

Kin

nj

Kin

njijijij

Kin

nj

n

jijijji

Kin

njji

C

pXdb

pXdbXpXdXb

Xb

Xi

μ

(5.75)

where 1)1(),( ++−−+= KKmiTnKin .

(ii) For 2,3 +++= mmTmTi and mK ,1=

)(XiCμ =

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

++≥

++<≤+−

<

∑

∑ ∑∑ ∑

∑

+

−+=

+

−+=

+

−+=

+

−+=

+

−+=

+

−+=

TKn

KTnjijijiji

TKn

KTnj

TKn

KTnjijijijijij

TKn

KTnj

TKn

KTnjijijjiji

TKn

KTnjjiji

pXdpb

pXdpbXppXdXpb

Xpb

)1(

)1( )1(,

)1( )1(

)1(

.)(,1

)(,)(/)(

,,0

(5.76)

By using the definition of the fuzzy decisions proposed by Bellman and Zadeh [9], we

have

)))((min),(min()( XXXjCjGD μμμ = .

In this case the an optimal fuzzy decision is a solution of the problem

))((min),(min(max))((max00

XXXjCjGXDX

μμμ≥≥

= ).

Consequently, the problem (5.58)-(5.60) can be written as to the following optimization

problem

λmax (5.77)

λμ ≥)(XG (5.78)

2,1,)( ++=≥ mmTiXiC λμ (5.79)

0≥X , 10 ≤≤ λ . (5.80)

92

By using the method of defuzzification as for the problem (5.50)-(5.53), we get the

following theorem.

Theorem 5.2 The problem (5.58)-(4.60) is reduced to one of the following crisp

problems :

λmax (5.81)

0)( 11

12 ≤+−− ∑=

zXczzn

jjjλ , (5.82)

∑=

≤−++−n

jiijijij bpXdr

10)( λλ , mKKmi )1(,1 ++= and )1(,0 −= TK (5.83)

0≥jX , nj ,1= , 10 ≤≤ λ ; (5.84)

λmax (5.85)

0)( 11

12 ≤+−− ∑=

zXczzn

jjjλ , (5.86)

∑+=

≤−++−),(

10)1(

Kin

njiijij bpXd λλ , mKKmi )1(,1 ++= and )1(,0 −= TK (5.87)

0≥jX , ),(,1 Kinnj += , 10 ≤≤ λ , (5.88)

where n(i,K)=n+T(i-Km-1)+K+1;

λmax (5.89)

0)( 11

12 ≤+−− ∑=

zXczzn

jjjλ , (5.90)

∑+

−+=

≤−++TKn

KTnjiijijij bpXdp

)1(0)( λ , 2,3 +++= mmTmTi and mK ,1= (5.91)

0≥jX , TKnKTnj +++= ),1( , 10 ≤≤ λ . (5.92)

Notice that, the problem given in this theorem are also nonconvex programming

problems, similar for the problem (5.77)-(5.80).

93

5.6 The modified subgradient method

In this section, we briefly present an algorithm of the modified subgradient method

suggested by Gasimov [57] which can be applied for solving a large class of nonconvex

and nonsmooth constrained optimization problems. This method is based on the

construction of dual problems by using sharp Lagrangian functions and has some

advantages Azimov and Gasimov[6], Gasimov [58], Rockafellar [150]. Some of them are

the following:

- The zero duality gap property is proved for suffciently large class of problems;

- The value of dual function strongly increases at each iteration;

- The method does not use any penalty parameters;

- The presented method has a natural stopping criterion.

Now, we give the general principles of the modified subgradient method. Let 0X be any

topological linear space, 0XS ⊂ be a certain subset of 00 ,YX be a real normed space

and *0Y be its dual. Consider the primal mathematical programming problem defined as

(P) 0)(

)(infinf

=

=∈

xgtosubject

xfPSx

where f is a real-valued function defined on S and g is a mapping of S into 0Y :

For every 0Xx∈ and 0Yy∈ let

⎩⎨⎧

∞+=∈

=Φ.,

)(),(),(

otherwiseyxgandSxifxf

yx (5.93)

We define the augmented Lagrange function associated with problem (P) in the following

form: (see Azimov and Gasimov [6] and Rockafellar and Wets [153),

),(),(inf),,( uyycyxcuxLYy

−+Φ=∈

for 0Xx∈ , *0Yu∈ and 0≥c . By using (5.93) we concretize the augmented Lagrangian

associated with (P):

))),((()()(),,( uxgxgcxfcuxL −+= , (5.94)

where 0Xx∈ , *0Yu∈ and 0≥c .

94

It is easy to show that,

),,(supinfinf*),(

cuxLPYcuSx

+×∈∈=

R

The dual function H is defined as

),,(inf),( cuxLcuHSx∈

= (5.95)

for *0Yu∈ and 0≥c . Then, a dual problem of (P) is given by

),(,sup*

0),(

* cuHPSupYcu +×∈

=R

.

Any element +×∈ R*

0),( Ycu with ),(* cuHPSup = is termed a solution of *P .

Proofs of the following three theorems can be found in Gasimov [58].

Theorem 5.3. Suppose in (P) that f and g are continuous, S is compact and a feasible

solution exists. Then *PSupInfP = and there exists a solution to (P). Furthermore, in

this case, the function H in ( *P ) is concave and finite everywhere on +×R*

0Y , so this

maximization problem is efficiently unconstrained.

Theorem 5.4. Let *supinf PP = and for some +×∈ R*

0),( Ycu ,

)),(()()(),,(inf uxgxgcxfcuxLSx

−+=∈

. (5.96)

Then x is a solution to (P) and ),( cu is a solution to ( *P ) if and only if

g(x) = 0. (5.97)

When the assumptions of the theorems, mentioned above, are satisfied, the maximization

of the dual function H by using the subgradient method will give us the optimal value of

the primal problem.

It will be convenient to introduce the following set :

)()()(minimizes),( SxoverxguxgcxfxxcuS ∈−+=

95

Theorem 5.5 Let S be a nonempty compact set in nR and let f and g be continuous so

that for any ),(,),( cuScu mkk+×∈ RR is not empty. If ),( cuSx ∈ , then

))(),(( xgxg− is a subgradient of H at ),( cu .

Now we are able to present the algorithm of the modified subgradient method.

Algorithm

Initialization Step. Choose a vector ),( 11 cu with 01 ≥c let k = 1, and go to main step.

Main Step.

Step 1. Given ),( kk cu . Solve the following subproblem :

( )

.)),(()()(min

Sxtosubjectuxgxgcxf kk

∈

−+

Let kx be any solution. If 0)( =kxg , then stop; ),( kk cu is a solution to dual problem

( *P ), kx is a solution to primal problem (P). Otherwise, go to Step 2.

Step 2. Let

)()(

)(1

1

kkkkk

kkkk

xgscc

xgsuu

ε++=

−=+

+

(5.76)

where ks and kε are positive scalar stepsizes, replace k by k + 1; and repeat Step 1.

One of the stepsize formulas which can be used is

3)(5)),((

k

kkkkk

xgcuHH

s−

=α

where kH is an approximation to the optimal dual value, 20 << kα and kk s<< ε0 .

The following theorem shows that in contrast with the subgradient methods developedfor

dual problems formulated by using ordinary Lagrangians, the new iterate improves the

cost for all values of the stepsizes ks and kε .

Theorem 5.6. Suppose that the pair +×∈ RR mkk cu ),( is not a solution to the dual

problem and ),( kkk cuSx ∈ . Then for a new iterate ),( 11 ++ kk cu calculated from (5.76)

for all positive scalar stepsizes ks and kε we have

96

211 )()2(),(),(0 k

kkkkkk xgscuHcuH ε+≤−< ++ .

5.7 Defuzzification and solution of defuzzificated problem

In this section, we present the modified subgradient method (Gasimov [57]) and

use it for solving the defuzzificated problems (5.55)-(5.57) for nonconvex constrained

problems and can be applied for solving a large class of such problems.

Notice that, the constraints in problem (5.55)-(5.57) is generally not convex. These

problems may be solved either by the fuzzy decisive set method, which is presented by

Sakawa and Yana [165], or by the linearization method of Kettani and Oral [83].

5.7.1 A modified subgradient method to fuzzy linear programming

For applying the subgradient method ( Gasimov [57]) to the problem (5.54)-(5.57),

we first formulate it with equality constraints by using slack variables 0y and iy ,

2,1 ++= mmTi . Then, we can be written as

λmax , (5.99)

0)(),,( 021

210 =++−−= ∑=

yzXczzyXgn

jjjλλ , (5.100)

=),,( λyXgi ∑+

=

=+−mTn

jiijij ybXa

10)(ˆ λ , 2,1 ++= mmTi , (5.101)

0≥jX , 0,0 ≥iyy , 10 ≤≤ λ , mTnj += ,1 , 2,1 ++= mmTi . (5.102)

where ),...,( 0 nyyy =

For this problem we define the set S as

10,0,0),,( ≤≤≥≥= λλ yXyXS .

Since )min(max λλ −−= and ),...,( 20 ++= mmTggg the augmented Lagrangian

associated with the problem (5.99)-(5.102) can be written in the form

97

.2

1)(ˆ021

)21(0

21

22

1)(ˆ

2

021

)21(),,(

1

1

∑++

=⎟⎠⎞

⎜⎝⎛ +−∑−⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+∑

=+−−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∑++

=⎟⎠⎞

⎜⎝⎛ +−∑+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

++∑=

−−+−=

+

=

+

=

mmT

iybXauyz

n

j jXjczz

mmT

iybXayz

n

j jXjczzccuXL

iij

mTn

jiji

iij

mTn

jij

λλμ

λλλ

The modified subgradient method may be applied to the problem (5.99)-(5.102) in the

following way:

Initialization Step. Choose a vector ),,...,,( 112

11

10 cuuu mmT ++ with 01 ≥c . Let 1=k , and

go to main step.

Main Step.

Step 1 . Given ),,...,,( 210kk

mmTkk cuuu ++ ; solve the following subproblem :

,2

1)(

1ˆ021

)21(0

21

22

1 1)(ˆ

2

021

)21(min

∑++

= ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−∑

+

=−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+∑

=+−−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∑++

= ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−∑

+

=+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

++∑=

−−+−

mmT

i iyibjXmTn

j ijaiuyzn

j jXjczzu

mmT

i iyibmTn

j jXijayzn

j jXjczzc

λλ

λλλ

.),,( SyX ∈λ

Let ),,( kkk yX λ be a solution. If 0),,( =kkk yXg λ , then stop; ),,...,,( 210kk

mmTkk cuuu ++

is a solution to dual problem, ),,( kkk yX λ is a solution to problem (5.54)-(5.57).

Otherwise, go to Step 2.

Step 2 . Let

⎟⎟⎠

⎞⎜⎜⎝

⎛++−−−= ∑

=

+02

1210

10 )( yzXczzhuu

n

jjj

kkk λ

⎟⎟⎠

⎞⎜⎜⎝

⎛+−−= ∑

+

=

+mTn

jiijij

kki

ki ybXahuu

1

1 )(ˆ λ , 2,1 ++= mmTi

),,()(1 kkkkkkk yXghcc λε++=+

98

where kh and kε are positive scalar stepsizes and 0>> kkh ε , replace k by k + 1; and

repeat Step 1.

5.7.2 Fuzzy decisive set method

For a fixed value of λ , the problem (5.54)-(5.57) is a linear programming

problem. Thus obtaining the optimal solution *λ to the problem (5.54)-(5.57) is

equivalent to determining the maximum value of λ so that the feasible set is nonempty.

Bellow is presented the algorithm (Gasimov [57]), of this method for the problem (5.54)-

(5.57).

Algorithm

Step 1. Set λ = 1 and test whether a feasible set satisfying the constraints of the problem

(5.54)-(5.57) exists or not using phase one of the simplex method. If a feasible set exists,

set λ = 1: Otherwise, set 0=Lλ and 1=Rλ and go to the next step.

Step 2. For the value of 2/)( RL λλλ += ; update the value of Lλ and Rλ using the

bisection method as follows :

λλ =L if feasible set is nonempty for λ ;

λλ =R if feasible set is empty for λ .

Hence, for each λ , we test whether a feasible set of the problem (5.54)-(5.57)

exists or not using phase one of the Simplex method and determine the maximum value *λ satisfying the constraints of the problem (5.54)-(5.57).

Example 5.1.

Solve the optimization problem, see Gasimov [40]

,0,61~3~42~1~

32max

21

21

21

21

≥≤+

≤+

+

xxxx

xx

xx

(5.103)

which take fuzzy parameters as )2,3(3~),3,2(2~),1,1(1~ LLL === and )3,1(1~ L= as

used by Shaocheng [171]. That is,

99

⎥⎦

⎤⎢⎣

⎡=

1321

)( ija , ⎥⎦

⎤⎢⎣

⎡=

3231

)( ijd ⇒ ⎥⎦

⎤⎢⎣

⎡=+

4552

)( ijij da

For solving this problem we must solve the folowing two subproblems:

,0,6342

32max

21

21

21

211

≥≤+≤+

+=

xxxxxx

xxz

and

,0,645452

32max

21

21

21

212

≥≤+≤+

+=

xxxxxx

xxz

Optimal solutions of these subproblems are

2.16.18.6

2

1

1

===

xxz

and

,82.047.006.3

2

1

2

===

xxz

respectively. By using these optimal values, problem (5.103) can be reduced to the

following equivalent non-linear programming problem:

0,10

3236

32406.38.6

06.332max

21

21

21

21

21

21

≥≤≤

≥+−−

≥+−−

≥−

−+

xx

xxxx

xxxx

xx

λ

λ

λ

λ

λ

that is

100

0,10

6)31()23(4)32()1(

74.306.332max

21

21

21

21

≥≤≤

≤+++≤+++

+≥+

xx

xxxx

xx

λλλλλ

λλ

(5.104)

Let's solve problem (5.104) by using the fuzzy decisive set method.

For 1=λ , the problem can be written as

0,6454528.632

21

21

21

21

≥≤+≤+≥+

xxxxxx

xx

and since the feasible set is empty, by taking 0=Lλ and Rλ =1; the new value of

21)

210( =+=λ is tried.

For 21

=λ , the problem can be written as

,0,

6254

427

23

9294.432

21

21

21

21

≥

≤+

≤+

≥+

xx

xx

xx

xx

and since the feasible set is empty, by taking 0=Lλ and 21

=Rλ , the new value of

412/)

210( =+=λ is tried.

For 41

=λ , the problem can be written as

101

,0,

647

27

44

1145

9941.332

21

21

21

21

≥

≤+

≤+

≥+

xx

xx

xx

xx

and since the feasible set is empty, by taking 41

=Lλ and 21

=Rλ , the new value of

832/)

21

41( =+=λ is tried.

For 83

=λ , the problem can be written as

,0,

68

174

15

4825

811

4618.432

21

21

21

21

≥

≤+

≤+

≥+

xx

xx

xx

xx

and since the feasible set is nonempty, by taking 83

=Lλ and 21

=Rλ , the new value

of 1672/)

21

83( =+=λ is tried.

For 4375.0167==λ , the problem can be written as

,0,

61637

831

41653

1623

6956.432

21

21

21

21

≥

≤+

≤+

≥+

xx

xx

xx

xx

and since the feasible set is nonempty, by taking 83

=Lλ and 167

=Rλ , the new value

102

of 32132/)

167

83( =+=λ is tried.

For 40625.03213

==λ , the problem can be written as

,0,

63271

32122

432

1033245

5787.432

21

21

21

21

≥

≤+

≤+

≥+

xx

xx

xx

xx

and since the feasible set is empty, by taking 83

=Lλ and 3213

=Rλ , the new value of

64252/)

3213

83( =+=λ is tried.

For 390625.06425

==λ , the problem can be written as

,0,

664

13964242

464203

6489

5202.432

21

21

21

21

≥

≤+

≤+

≥+

xx

xx

xx

xx

and since the feasible set is empty, by taking 6425

=Lλ and 3213

=Rλ , the new value of

128512/)

3213

6425( =+=λ is tried.

For 398475.012851

==λ , the problem can be written as

103

,0,

6128281

128486

4128409

128179

5494.432

21

21

21

21

≥

≤+

≤+

≥+

xx

xx

xx

xx

and since the feasible set is empty, by taking 6425

=Lλ and 12851

=Rλ , the new value of

2561012/)

12851

6425( =+=λ is tried.

The following value of λ are obtained in the next thirteen iterations:

256/101=λ = 0.39453125

512/1203=λ = 0.396484325

1024/407=λ = 0.396972656

4096/1627=λ = 0.397216796

8192/3255=λ = 0.397338867

16384/6511=λ = 0.397399902

32768/13021=λ = 0.397369384

65536/26043=λ = 0.397384643

131072/52085=λ = 0.397377014

262144/104169=λ = 0.3973731

524288/208337=λ = 0.3973733

1048576/416675* =λ = 0.3973723

Consequently, we obtain the optimal value of λ at the twenty first iteration by using the

fuzzy decisive set method.

Now, let's solve the same problem by using the modified subgradient method. Before

solving the problem, we first formulate it in the form

)min(max λλ −−=

104

,0,,0,10

06)31()23(04)32()1(006.33274.3

210

21

221

121

021

≥≥≤≤

=+−+++=+−+++=++−−

pppxx

pxxpxx

pxx

λλλλλ

λ

where 10 , pp and 2p are slack variables. The augmented Lagrangian function for this

problem is

).6)31()23(()4)32()1(()06.33274.3(])6)31()23((

)4)32()1(()06.33274.3[(),,(

22121211

021021

2221

2121

2021

pxxupxxupxxupxx

pxxpxxccuxL

+−+++−+−+++−++−−−+−++++

+−++++++−−+−=

λλλλλλλ

λλλλ

Let the initial vector is ),,,( 112

11

10 cuuu = (0; 0; 0; 0) and let's solve the following

subproblem

.2.174.06.182.0

10)0,0,(min

2

1

≤≤≤≤

≤≤

xx

xLλ

The optimal solutions of subproblem are obtained as

.1),,(

2),,(8.4),,(

101

1113

1112

1111

2

1

−=

−=

=

===

λ

λ

λ

λ

pxg

pxgpxg

xx

Since 0),,( 111 ≠λpxg , we calculate the new values of Lagrangemultipliers

),,( 222

21

20 cuuu by using Step 2 of the modified subgradient method. The solutions of the

second iteration are obtained as

105

62223

62222

62221

*2

1

1031.2),,(108.3),,(

109),,(3973723.0

75147.01475877,1

−

−

−

×=

×−=

×=

=

==

λ

λ

λ

λ

pxgpxg

pxg

xx

Since )(xg is quite small, by Theorem 5.4 75147.0,1475877.1 *2

*1 == xx and =*λ

3973723.0 are optimal solutions to the problem (5.100). This means that, the vector

),( *2

*1 xx is a solution to the problem (5.99) which has the best membership grade *λ .

Note that, the optimal value of λ found at the second iteration of the modified

subgradient method is approximately equal to the optimal value of λ calculated at the

twenty first iteration of the fuzzy decisive set method.

Example 5.2.

Solve the optimization problem, see Gasimov [40]

,0,4~3~2~3~2~1~

max

21

21

21

21

≥≤+

≤+

+

xxxx

xx

xx

(5.105)

which take fuzzy parameters as

)2,3(3~),2,3(3~),2,2(2~),1,2(2~),1,1(1~ LbLLLL ====== and

)3,4(4~2 Lb == as used by Shaocheng [171]. That is,

⎥⎦

⎤⎢⎣

⎡=

3221

)( ija , ⎥⎦

⎤⎢⎣

⎡=

2211

)( ijd ⇒ ⎥⎦

⎤⎢⎣

⎡=+

5432

)( ijij da

⎥⎦

⎤⎢⎣

⎡=

43

)( ib , ⎥⎦

⎤⎢⎣

⎡=

32

)( ip ⇒ ⎥⎦

⎤⎢⎣

⎡=+

75

)( ii pb .

To solving this problem, first, we must solve the folowing two subproblems:

106

,0,454332

max

21

21

21

211

≥≤+≤++=

xxxxxx

xxz

and

,0,732

52max

21

21

21

212

≥≤+≤+

+=

xxxx

xxxxz

Optimal solutions of these subproblems are

011

2

1

1

===

xxz

and

,05.35.3

2

1

2

===

xxz

respectively. By using these optimal values, problem (5.105) can be reduced to the

following equivalent non-linear programming problem:

0,10

22324

2315.3

1max

21

21

21

21

21

21

≥≤≤

≥+−−

≥+−−

≥−−+

xx

xxxx

xxxx

xx

λ

λ

λ

λ

λ

that is

0,10

6)31()23(4)32()1(

74.306.332max

21

21

21

21

≥≤≤

≤+++≤+++

+≥+

xx

xxxx

xx

λλλλλ

λλ

(5.106)

Let's solve problem (5.106) by using the fuzzy decisive set method.

107

For 1=λ , the problem can be written as

0,154132

5.3

21

21

21

21

≥≤+≤+

≥+

xxxxxx

xx

and since the feasible set is empty, by taking 0=Lλ and Rλ =1; the new value of

212/)10( =+=λ is tried.

For 21

=λ , the problem can be written as

,0,2543

225

23

25.2

21

21

21

21

≥

≤+

≤+

≥+

xx

xx

xx

xx

and since the feasible set is empty, by taking 0=Lλ and 21

=Rλ , the new value of

412/)

210( =+=λ is tried.

For 41

=λ , the problem can be written as

,0,4

1327

25

25

49

45

625.1

21

21

21

21

≥

≤+

≤+

≥+

xx

xx

xx

xx

and since the feasible set is empty, by taking 41

=Lλ and 41

=Rλ , the new value of

812/)

410( =+=λ is tried.

108

For 81

=λ , the problem can be written as

,0,829

413

49

822

817

89

3125.1

21

21

21

21

≥

≤+

≤+

≥+

xx

xx

xx

xx

and since the feasible set is nonempty, by taking 81

=Lλ and 41

=Rλ , the new value

of 1632/)

41

81( =+=λ is tried.

The following value of λ are obtained in the next thirteen iterations:

16/3=λ = 0.1875

32/5=λ = 0.15625

64/11=λ = 0.171875

128/23=λ = 0.1796875

256/47=λ = 0.18359375

512/93=λ = 0.181640625

1024/187=λ = 0.182617187

2048/375=λ = 0.183349609

4096/751=λ = 0.183166503

8192/1501=λ = 0.397377014

16384/3001=λ = 0.183166503

32768/6003=λ = 0.183197021

65536/12007=λ = 0.18321228

131072/24015=λ = 0.183219909

62144/48029=λ = 0.183216095

524288/96057=λ = 0.183214187

1048576/192115=λ = 0.183215141

2097152/383231=λ = 0.183215618

109

4194304/786463=λ = 0.183215856

8388608/1536927=λ = 0.183215975

16777216/3073853* =λ =0.183215916

Hence,, we obtain the optimal value of λ at the twenty fifth iteration by using the fuzzy

decisive set method.

By using the modified subgradient we solve the same problem method. Before solving

the problem, we first formulate it in the form

)min(max λλ −−=

,0,,0,10

04)23()22(03)2()1(

015.2

210

21

221

121

021

≥≥≤≤

=+−+++=+−+++

=++−−

pppxx

pxxpxx

pxx

λλλ

λλλ

where 10 , pp and 2p are slack variables. The augmented Lagrangian function for this

problem is

).4)23()22(()3)2()1(()15.2(])4)23()22((

)3)2()1(()15,2[(),,(

22121211

021021

2221

2121

2021

pxxupxxupxxupxx

pxxpxxccuxL

+−+++−+−+++−++−−−+−++++

+−++++++−−+−=

λλλλλλλ

λλλλ

Let the initial vector is ),,,( 112

11

10 cuuu = (0; 0; 0; 0) and let's solve the following

subproblem

.005.31

10)0,0,(min

2

1

≤≤≤≤≤≤

xx

xLλ

The optimal solutions of subproblem are obtained as

110

3),,(

1),,(5.2),,(

101

1113

1112

1111

2

1

=

=

=

===

λ

λ

λ

λ

pxg

pxgpxg

xx

Since 0),,( 111 ≠λpxg , we calculate the new values of Lagrangemultipliers

),,( 222

21

20 cuuu by using Step 2 of the modified subgradient method. The solutions of the

second iteration are obtained as

82223

82222

72221

*

82

1

1083.7),,(102.8),,(

1028.3),,(1832159.0

108.7

45804,1

−

−

−

−

×−=

×=

×=

=

×=

=

λ

λ

λ

λ

pxgpxg

pxg

x

x

Since )(xg is quite small, by Theorem 5.4, 0108.7,45804.1 8*2

*1 ≅×== xx and

=*λ 0.1832159 are optimal solutions to the problem (5.106). This means that, the vector

),( *2

*1 xx is a solution to the problem (5.105) which has the best membership grade *λ .

Note that, the optimal value of λ found at the second iteration of the modified

subgradient method is approximately equal to the optimal value of λ calculated at the

twenty first iteration of the fuzzy decisive set method.

5.8 Portfolio problem with fuzzy multi-objective

The Fuzzy Multiple Objective Decision Model (FMODM) studied by Lai and Hwang

[93] states that the effectiveness of a decision makers’ performance in a decision process

can be improved as a result of the high quality of analytic information supplied by a

computer. They propose an Interactive Fuzzy Multiple Objective Decision Model

(IFMODM) to solve a specific domain of Multiple Objective Decision Model (MODM).

111

In this section we consider this approach for (4.2)-(4.4) portfolio model. Thus we have

the following problem

))(),...,(( xfxfMax qi (5.107)

subject to

itjt

n

jji srxy ≤−∑

=1, Ttmi ,1,,1 == , (5.108)

);(21

iit

T

tt yYFsp ≤∑

=

, ,,1 mi = (5.109)

0≥its , Ttmi ,1,,1 == , (5.110)

0≥x , X∈x . (5.111)

where qkxcxfn

iiikk ,1,)(

1

==∑=

.

Let us now consider the case of a decision-maker who has a fuzzy goal such as “the

objective function )(xf k should be much greater than minkp ”. Further, let us assume that

the degree of satisfaction of the decision-maker with respect to achieving the objective

does not change beyond the level maxkp . Then the corresponding linear membership

function that characterises the fuzzy goal of the decision-maker is given by:

⎪⎪⎩

⎪⎪⎨

⎧

≤

<<−−

≤

=

.)(;1

,;)(,;0

)]([max

maxminminmax

min

minmin

kk

kkkkk

kk

kk

kk

pxf

pfppppxf

pf

xfμ (5.112)

Given the membership functions for the various objectives of the decision-maker, the

maximizing decision can be computed by solving the following optimization problem:

)]([minmaximize,1

xfkkqkμ

= (5.113)

subject to itjt

n

jji srxy ≤−∑

=1, mi ,1= , (5.114)

);(21

iit

T

tt yYFsp ≤∑

=

, mi ,1= (5.115)

112

0≥its , Ttmi ,1,,1 == (5.116)

0≥x , X∈x . (5.117)

By introducing the auxiliary variable λ , the above optimization problem can be reduced

to the following conventional linear programming problem :

λMaximize (5.118)

subject to qkxfkk ,1,)]([ =≥ λμ , (5.119)

itjt

n

jji srxy ≤−∑

=1, Ttmi ,1,,1 == , (5.120)

);(21

iit

T

tt yYFsp ≤∑

=

, mi ,1= (5.121)

0≥its , Ttmi ,1,,1 == (5.122)

0≥x , X∈x , R∈λ , 10 ≤≤ λ . (5.123)

Let us consider the case of a fund manager who has to choose a structured portfolio from

an investment universe of n assets with jl and jl , nj ,1= being the minimum and

maximum weight of the ith asset in the portfolio. In order to select the structured

portfolio, the fund manager may examine k potential market scenarios, and for each of

these scenarios the decision maker may wish to maximize the portfolio return. To achieve

the return objective the fund manager could formulate the following optimization

problem:

maximize ( )(),...,(1 xRxR q ) (5.124)

subject to itjt

n

jji srxy ≤−∑

=1, Ttmi ,1,,1 == , (5.125)

);(21

iit

T

tt yYFsp ≤∑

=

, ,,1 mi = (5.126)

11

=∑=

m

iix , (5.127)

jjj lxl ≤≤ , nj ,1= , (5.128)

113

0≥its , Ttmi ,1,,1 == , (5.129)

0≥x , X∈x . (5.130)

where qkxRxRn

jj

kj

k ,1,)(1

==∑=

In equation (5.76), kjr denotes the return from the jth asset for the kth market scenario at

the end of the investment period and )(xRk the portfolio return for the kth scenario.

Since the above optimization problem has multiple objective functions, one has to

compute a Pareto optimal solution for the problem (see Sakawa [164]). Also we can use

the model of Chapter 2 for instance, one could characterize the set of Pareto optimal

solutions using the weighted minimax method and select one solution from this set. The

set of Pareto optimal solutions to the above optimization problem is characterized by:

Maximize λ (5.131)

subject to λ≥)(xRw kk , pk ,1= , (5.132)

itjt

n

jji srxy ≤−∑

=1, Ttmi ,1,,1 == , (5.133)

);(21

iit

T

tt yYFsp ≤∑

=

, mi ,1= (5.134)

11

=∑=

n

jjx , (5.135)

jjj lxl ≤≤ , ,,1 nj = (5.136)

0≥its , Ttmi ,1,,1 == (5.137)

0≥x , X∈x , R∈λ . (5.138)

In above relation, λ is an auxiliary variable and qkwk ,1, = are any arbitrarily chosen

nonnegative weights. Given any suitable weighting vector, one can determine the Pareto

optimal solution. Here, we assume without loss of generality that ,0)( >xRk

jjj lxl ≤≤ , nj ,1= . If this is not the case, the objective functions can be rewritten as

qkCRxR kk ,1),()(ˆ == , (5.139)

114

where C is a suitable constant that ensures kxRk ∀> ,0)(ˆ . Incorporating this change in

equation (5.132), one can compute the Pareto optimal solution.

The optimization problem formulated above is a linear programming problem and can be

easily solved using standard algorithms. However, finding a satisfactory Pareto optimal

solution requires one to define the a priori probabilities of various scenarios that

incorporate the market views. In the face of uncertainty these a priori probabilities are not

computable, and hence it is difficult to compute a Pareto optimal solution that can be

characterised as being satisfactory. Moreover, the fund manager may like to structure the

portfolio such that the return targets are different for each market scenario, for instance

with those scenarios that he/she considers more likely to occur (although no experimental

evidence is available) being targeted to achieve greater return. Transforming such goals

into suitable weights qkwk ,1,0 => for the various scenarios is not obvious from the

fund manager’s perspective.

Let us now consider a fund manager structuring a portfolio based on p potential

market scenarios. For each such scenario, the fund manager may have a target range for

the expected return over the investment period. We will denote by minkp and max

kp the

minimum and maximum expected return for the jth market scenario. Note that it is quite

easy for the fund manager to provide information on the expected target range of return

for various scenarios rather than to define the a priori probabilities for different scenarios.

Using the linear membership function given in equation (5.112) it is possible to compute

the degree of satisfaction ))(( xRkkμ for any given portfolio x for the kth market

scenario. Given that the degree of satisfaction to the fund manager for the kth market

scenario is ))(( xRkkμ , the structured portfolio can be computed by solving the

following optimization problems, for qk ,1=

λMaximize (5.140)

subject to λμ ≥)(xRkk , (5.141)

itjt

n

jji srxy ≤−∑

=1, Ttmi ,1,,1 == , (5.142)

115

);(21

iit

n

jt yYFsp ≤∑

=

, mi ,1= (5.143)

11

=∑=

n

jjx , (5.144)

jjj lxl ≤≤ , ,,1 mi = (5.145)

0≥its , Ttmi ,1,,1 == (5.146)

0≥x , X∈x , R∈λ . (5.147)

It is easy to show that the solution to the above optimization problem (if one exists) will

be Pareto optimal, Sakawa [164]. It is again useful to remind that we can interpret the

membership function ))(( xRkkμ for the kth market scenario in (5.136)) as modelling the

fuzzy utility of the investor for the given scenario. In this case, the structured portfolio

computed by solving the above optimization problem maximizes the fuzzy utility of the

investor.

5.9. Multiobjective fractional programming problems under fuzziness

Fractional programming has attracted the attention of many researchers in the

past. The main reason for interest in fractional programming stems from the fact that

linear fractional objective functions occur frequently as measures of performance in a

variety of circumstances such as when satisfying objectives under uncertainty. In some

real world decision-making situations, when formulating fractional objectives, some or all

of the parameters of the optimization problem are described by fuzzy or stochastic

variables.

Saad [160 ] presented a solution procedure for solving linear fraction programs

having fuzzy parameters in the right-hand side of the constraints. These parameters have

been characteristized by fuzzy numbers and the concept of α -optimality has been

introduced. On the other hand, Bicriterion integer nonlinear fractional programs

(BINOLFP) involving fuzzy parameters in the objective functions have been studied by

Saad and Abdelkader [161] .

116

Moreover, a solution algorithm has been described to solve the (BINOLFP). Furthermore,

a solution algorithm has been proposed by Saad and Abd-Rabo [162] was based upon the

chance-constrained programming technique Seppala [168] along with the branch-and-

bound method Ammar (1988). Recently, Saad and Sharif developed a solution method to

solve integer linear fractional program with chance-constraints and having statistically

independent random parameters Dutta (1992). Pareto-optimality for multiobjective linear

fractional programming problems with fuzzy parameter has been discussed by Sakawa

and Yano [165]. Programming with linear fractional functions was introduced into the

literature by Charnes and Cooper [30]. Since we can use a fuzzy multiobjective fractional

portfolio models, in this section we give some recently results on fuzzy multiobjective

fractional programming problem.

5.9.1 Problem formulation and the solution concept

The problem to be considered in this paper is the following fuzzy multiobjective

fractional programming problem:

(FMOFP) ,)()(

,...,)(

()()(max

1

1

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡xgxf

xgxf

xgxf

p

p

subject to 0,~)~,( ≥≤∈=∈ xbAxRxbAXx n ,

where A is an )( nm× -matrix b~ is an m-vector of fuzzy parameters and we suppose that

they are given by fuzzy numbers, estimated from the information provided by the

decision maker. Moreover ),1(,0)( pixgi => for all x in the feasible region of problem

(FMOFP).

Definition 5.1 (Dubois and Prade [44]) It is apropiate to recall that a real fuzzy number

a~ is a continuous fuzzy subset from the real R whose membership function )(~ aaμ is

defined by:

(1) A continuous mapping function R to the closed interval [0,1],

(2) 0)(~ =aaμ for all ],( 1aa −∞∈ ,

(3) )(~ aaμ is strictly increasing on ],[ 21 aa ,

117

(4) 1)(~ =aaμ for all ],( 32 aaa∈ ,

(5) )(~ aaμ is strictly decreasing on ],( 43 aaa∈ ,

(6) 0)(~ =aaμ for all ),[ 4 ∞∈ aa ,

Figure 5.2 Illustrates the graph of possible shape of a membership function of a fuzzy

number a~ .

Here, the vector of fuzzy parameters b~ involved in problem (FMOFP) is a vector of

fuzzy numbers whose membership function is )(~ bbμ .

)(~ aaμ 1 0 1a 2a 3a 4a a

Fig. 5.2 Membership function of a fuzzy number a~ .

In what follows, we give the definition of the α -level set or α -cut of the fuzzy vector

]~,...,~[~1 mbbb = .

Definition 5.2 [44] The α -level set of the vector of fuzzy parameters b~ in problem

(FMOFP) is defined as the ordinary set )()~( ~ αμα ≥∈= bRbbL bm .

For a certain degree *αα = in [0,1], estimated by the decision maker, the (FMOFP) can

be understood as the following nonfuzzy α -multiobjective fractional programming

problem (α -MOFP):

(α -MOFP):

).~(,0,)~,(

,)()(

,...,)(

()()(max

1

1

bLbxbAXRxbAXxtosubject

xgxf

xgxf

xgxf

n

p

p

α∈≥≤∈=∈

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

It should be emphasized here in the (α -MOFP) above that the vector of parameters

b is treated as a vector of decision variables rather than constants.

Problem (α -MOFP) can be reformulated in the following form:

118

(P) ))(),...,(()(max 1 xFxFxF p= ,

subject to ),,( bAXx∈

where )()()()( * xgxxfxF iiii θ−= and ),1(,)()(,0 *

*** pi

xgxf

r

rri ==≥ θθ are fixed

parameters and for their specification, Singh and Hanson [174].

Based on Definition 2 of the α -level set of the vector of fuzzy numbers b~ , we introduce

the concept of α -efficient solution of problem (P) above as follows:

Definition 5.3 (Sakawa and Yano [165]) A point ),(* bAXx ∈ is said to be an α -

efficient solution of problem (P) if only if there exists no other )~(),,(* bLbbAXx α∈∈

such that ),1();()( * pixFxF ii =≤ with strictly inequality holding for at least one i,

where the corresponding values of parameters ),1(* mrbr = are called the α -level

optimal parameters.

Now, consider λ is a p-dimensional strictly positive fixed vector, then problem (P) can

be written again in a problem of scalar single-objective function )( λP in the following

form:

)( λP :

).,(

),(max1

bAXxtosubject

xFp

iii

∈

∑=

λ

Let ),( bAX denote the set of feasible solutions of problem (α -MOFP) or (P) or )( λP .

We assume that 0)( ≥xf , 0)( >xg , for all ),( bAXx∈ . We further assume that f,-g

are concave functions and ),( bAX is a convex set. It follows that F is concave (Singh

and Hanson [174]).

Problem )( λP can be solved at 1* == ii λλ with the corresponding fixed parameters

),1(,* piii ==θθ using any available nonlinear programming package, for example,

GINO (Lieberman, et al [102], to find the α -optimal solution *x together with the

optimal parameters ),1(* mrbr = .

119

It should be noted from Singh and Hanson [174] that *x is an α -efficient solution to

problem (α -MOFP) or problem (P) with the corresponding α -level optimal parameters

),1(* mrbr = if there exists 0* ≥λ such that *x solves problem )( λP and either one of

the following conditions holds:

i. 0* >= ii λλ for all ),1( pi = .

ii. *x is the unique maximizer of problem )( λP .

Definition 5.4 (Geofrion [60]) Consider the multiobjective programming problem

))(),...,(()(max 1 xxx kφφφ = ,

subject to nRSx ⊆∈ .

We say that Sx ∈0 is efficient if ond only if there exists no Sx∈ such that

)()( 0 xx φφ ≤ .

Definition 5.5 (Geofrion [60]) For the multiobjective programming problem in

Definition 4, we say that an efficient solution 0x is properly efficient if only if for each i

and Sx∈ , there exists a positive real number M and a j such that

0)()( 0 >− xx jj φφ and )))()((()()( 00 xxMxx jjii φφφφ −≤− ,

whenever 0)()( 0 >− xx ii φφ .

Before we go any further, the reader is reminded that for multiobjective linear

fractional programming, when the emphasis is on finding efficient solution, there is no

general method for finding all the efficient solutions but Choo and Atkins [34] have

developed an algorithm, using row parameters, for solving the bicriterion linear fractional

programming problem (BLFP). Choo [33] has also shown that if 0x is an efficient

solution to (BLFP) then 0x is properly efficient Geofrion [60].

The nonnegativity of *iθ is needed to establish part (b) of Theorem 5.7 bellow.

Theorem 5.7

(a) If *x is an α -optimal solution of )( λP , then *x is properly an α -efficient for )(P .

(b) If f and -g are concave and *x is properly an α -efficient for (P), then it is an α -

optimal for )( λP .

120

To prove Theorem 5.7 above, the reader is referred to Geofrion [60].

Theorem 5.8 The point ),(* bAXx ∈ is an α -efficient solution of (α -MOFP) if it is an

α -efficient of (P) with 0)( * =xF .

Proof Suppose ),(* bAXx ∈ is an α -efficient solution of (α -MOFP). Then by

Definition4, there is no ),(* bAXx ∈ such that

pixgxf

xgxf

i

i

i

i ,1,)()(

)()(

*

*

=∀≤ .

Letting )()(

*

**

xgxf

i

ii =θ for pi ,1= , we see from the above inequality that there does not

exists an ),( bAXx∈ such that

pixFxgxf iiii ,1),()()(0 * =∀=−≤ θ

Since pixFxgxf iiii ,1),()()(0 ** ==−= θ , we see that there exists no x in ),( bAX

such that pixFxF ii ,1),()( * =≤ . Therefore, *x is an α -efficient of (P) with

0)( * =xF .

Conversely, suppose that *x is an α -efficient solution of (P) with

)()(0)( **** xgxfxF θ−== . That means, by Definition 5.4, there exists no

),( bAXx∈ such that

pixgxfxFxF iii ,1),()()()(0 ** =∀−=≤= θ .

That is, there exists no ),( bAXx∈ such that

pixgxf

xgxf

i

ii

i

i ,1,)()(

)()( *

*

*

=∀≤= θ

Hence, *x is an α -efficient solution of (α -MOFP).

For the development that follows, we assume that there exists real numbers 0,0 >> Kk

such that Kxgk i << )( for all i. Applying Definition 5.5 of the proper efficiency to

121

problem (α -MOFP), we note that an α -efficient solution *x of problem (α -MOFP) is

properly α -efficient if there exists a real number 0>M such that for each i, we have

)](/)()(/)([)(/)()(/)( *** xgxfxgxfMxgxfxgxf jjjjiiii −≤−

for some j such that )(/)()(/)( ** xgxfxgxf jjjj < whenever ),( bAXx∈ and

)(/)()(/)( ** xgxfxgxf iiii > . Or, rewriting these inequalities slightly differently, we

say an α -efficient solution *x of problem (α -MOFP) is properly α -efficient if there

exists a real number 0>M such that for each i, we have

)(/)](/)()()([)(/)]()()()([ ****** xgxgxfxgxfMxgxgxfxgxf jjjjjiiiii −≤− ,

(5.148)

where kMKM /= for some j such that

)()(/)()( ** xfxgxgxf jjj < (5.149)

Whenever ),( bAXx∈ and

0)]()()()([ ** >− xgxfxgxf iiii . (5.150)

To link proper α -efficiently of problem (α -MOFP) and (P), we prove the following

theorem.

Theorem 5.9 The point ),(* bAXx ∈ is a properly α - efficiently solution of problem

(α -MOFP) if and only if it is a properly α - efficiently solution on (P) with 0)( * =xF .

Proof. Supose *x is a properly α - efficiently solution of problem (α -MOFP). Then by

Theorem 5.8, we know its an α - efficiently solution on (P) with 0)( * =xF . Now *x is

a properly α - efficiently solution of problem (P) if there exists a positive real number M

such that for each i,

))()(()()( ** xFxFMxFxF jjii −≤− . (5.151)

for some j such that

. 0)()( * <− xFxF jj (5.152)

whenever ),( bAXx∈ and

0)()( * >− xFxF ii . (5.153)

122

Or [in view of the fact that 0)( * =xF for all i and )()()( * xgxfxF iiii θ−= with

)(/)( *** xgxf iii =θ for pi ,1= ], the result holds if and only if there exists an 0>M

such for each i,

)(/)](/)()()([

)](/)]()()(/)([***

***

xgxgxfxgxfM

xgxgxfxgxf

jjjjj

iiiii

−

≤− (5.154)

for some j such that

0)()()()( ** <− xgxfxgxf jjjj (5.155)

whenever ),( bAXx∈ and

0)()()()( ** >− xgxfxgxf iiii (5.156)

Relation (5.154)-(5.156) hold by (5.148)-(5.150) with MM = . Conversely, suppose *x

is a properly α -efficient solution of (P) with 0)( * =xF . Then by Definition 5.5,

relation (5.151)-(5.153) hold for some M and eacj i and ),( bAXx∈ . From this it follows

that (5.154)-(5.156) hold which are (5.148)-(5.150) with MM = .

5.9.2. Solution algorithm

A solution algorithm to solve fuzzy multiobjective fractional programming problem

(FMOFP) is described in a series of steps. The suggested algorithm can be summarized in

the following manner : Saad [159]

Step 1. Start with an initial level set 0* ==αα .

Step 2. Determine point ),,,( 4321 bbbb for the vector of fuzzy parameters b~ in problem

(FMOFP) to elicit a membership function )(~ bbμ satisfying assumptions (5.148)-(5.153)

in Definition 5.1.

Step 3. Convert problem (FMOFP) into its nonfuzzy version )( MOFP−α .

Step 4. Rewrite problem )( MOFP−α in the form of problem )( λP of single-objective

function.

123

Step 5. Choose 0* >= ii λλ and 11

* =∑=

p

iiλ with fixed values of ),1(,* piii ==θθ and

use GINO software package Lieberman, et al [102] to find the α -optimal solution *x of

prolem )( λP .

Step 6. Set ]1,0[)( * ∈+= stepαα and go to step 1.

Step 7. Repeat again the above procedure until the interval [0,1] is fully exhausted. Then,

stop.

Example. (Saad [160]) In what follows we provide a numerical example to clarify the

solution algorithm suggested above.

Let

222211

211 2)(,2)(,21)(,1)( xxgxfxxgxxf −==+=−=

So

.2

2)()()(,

211

)()()(

22

222

1

21

1

11 xxg

xfxFx

xxgxfxF

−==

+−

==

Consider the followings fuzzy bicriterion practionl programming problem (FBFP)

)),(),(()(max 21 xFxFxF =

subject to .0,

~

21

22

21

≥≤+

xxbxx

where b~ is a fuzzy parameter and is characterized by the following fuzzy

numbers:

)5,3,1,0(~=b

Assume that the membership function of these fuzzy numbers in the following

form:

124

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

≥

≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−

≤≤

≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−

≤

=

4

43

2

34

3

32

21

2

21

2

1

~

,0

,1

,1

,1

,0

)(

bb

bbbbbbb

bbb

bbbbbbb

bb

bbμ

Let 19.0=α , for example, then we get:

8.41.0 ≤≤ b

Choosing 1=b , then non-fuzzy α -bicriterion fractional programming problem

(α -BFP)becomes:

)),(),(()(max 21 xFxFxF =

subject to .0,1

21

22

21

≥≤+

xxxx

observe that point )0,0(* =x is an α -efficient solution of problem (α -BFP) since,

for each feasible x and then we have

02121

211)()( 2

2

22

21

22

21*

11 ≤++

−=−+−

=−xxx

xxxFxF ,

and

02

12

2)()(2

2

2

*22 ≥

−=−

−=−

xx

xxFxF ,

and there is no other feasible point for which

)1,1())(),(()( 21 ≥= xFxFxF .

We now consider the case when 1,2 == ji in the definition of a properly efficient

solution and therefore it can be seen that )0,0(* =x is also properly α -efficient solution.

When

)()( *22 xFxF − we have 0

2 2

2 >− xx

; that is, 02 >x .

125

Then

0212)()( 2

2

22

21

1*

1 >++

=−xxxxFxF and 0)()( *

22 >− xFxF .

0)2)(2(

)21(22

212

222 >+−

+=

xxxxxM .

We have

))()(()()( 1*

1*

22 xFxFMxFxF −≤− .

So that point )0,0(* =x is properly α -efficient solution for problem (α -BFP) with the

corresponding α -level set equals 0.19.

5.9.3. Basic stability notions for problem (FMOFP)

Based on definition of the set of feasible parameters; the solvability set and the stability

set of the first kind (SSK1) of problem (FMOFP) via problem (α -MOFP).

Let

⎭⎬⎫

⎩⎨⎧

=∈= ∑∑=

∈=

p

iiibAXx

p

iii

n xFxFRxE1),(1

**** )(max)()( λλλ

be the set of α -optimal solutions of problem )( *λP .

Definition 5.6 The set of feasible parameter of problem (α -MOFP), which is denoted b,

which is denoted by U, is defined by:

φα ≠=∈∈= ),(and,1),~( bAXmrbLbbU rrmR .

Definition 5.7 The solvability set of problem (α -MOFP), which is denoted by V, is

defined by:

)(where,solutionefficientanhas)(Problem *** λαα ExxMOFPUbV ∈−−∈= .

Definition 5.8 (The stability set of the first kind). Supose that Vb ∈* with the

corresponding α -efficient solution *x of problem (α -MOFP) such that )( ** λEx ∈ ,

then the stability set of the first kind(SSK1) of problem (α -MOFP), which is denoted by

)( *xS , is defined by:

126

)( problem ofsolution efficientn-an is)( ** MOFPxVbxS −∈= αα .

5.9.4. Utilization of Kuhn-Tucker conditions corresponding to problem )( λP

Problem )( λP can be written in the followings form:

)( λP : ,)(max1∑=

p

iii xFλ

subject to

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=≥

=≤≤

=≤=∑=

.,1,0

,,1,

,,1,),(1

njx

mrHbh

mrbxabx

j

rrr

n

jrjrjrrψ

It is clear that the constraint )~(bLb α∈ in the problem (P) has been replaced by the

equivalent constraint mrHbh rrr ,1, =≤≤ in problem )( λP , where rh and rH , are

lower and upper bounds on rb , respectively.

Therefor, the Kuhn-Tucker necessary optimality conditions corresponding to the

maximization problem )( λP we have the following form:

∑ ∑ ∑∑

∑ ∑ ∑

= = ==

= = =

=+−+∂

∂

=+∂

∂−

∂∂

m

r

m

r

m

rrr

m

rr

j

rrr

p

i

m

r

n

jj

j

rrr

j

ii

xbx

xbx

xxF

1 1 11

1 1 1

,0),(

,0),()(

ηγξψξ

βψξλ

rrr bbx ≤),(ψ ,

,rr bh ≤

,rr Hb ≤

,0≥jx

,0]),([ =− rrrr bbxψξ

,0)( =− rrr bhγ

127

,0)( =− rrr Hbη

0=jj xβ

0≥rξ ,

,0≥jβ

,0≥rγ

,0≥rη

where ,...,1 mIr =∈ and ,...,1 nJj =∈ . In addition, all the expressions of Kuhn-

Tucker conditions are evaluated at the α -optimal solution *x of problem )( *λP .

Furthermore, rrjr ηγβξ ,,, are the Lagrange multipliers.

The first two together with the last four relations of the above system of the Kuhn-Tucker

conditions represent a Polytope in ξβγη -space for which its vertices can be determined

using any algorithm based upon the simplex method, for example, Balinski (1961).

According to whether any of the variables ,...,1,,,, mIrrrjr =∈ηγβξ and

,...,1, nJjj =∈β are zero positive, then the set of parameters mrbr ,1, = for which

the α -efficient solution *x for one vector of parameters Vb ∈* rests efficient for all

parameters Vb∈ .

128

CHAPTER 6

A POSSIBILISTIC APPROACH FOR A PORTFOLIO SELECTION PROBLEM

6.1. Introduction

A half century ago, H. Markowitz pioneered the modern finance theory by his

meanvariance portfolio selection model. Although his work is perhaps technically simple

in today’s view, his idea still inspires the work in finance.

Zhou and Li [218] explored Markowitz’s work in a complete continuous-time financial

market. Variance has been commonly taken as a measure of risk. Meanwhile there are a

lot of researches on how to measure the risk of an investment, such as semivariance

advised by Markowitz. In the mean-risk framework, only mean-variance was widely

accepted in the discrete-time market.

The portfolio selection model of Markowitz [112, 113] consists of two interrelated

modules:

- a nonlinear programming problem where risk-averse investors solve a utility

maximization problem involving the risk and the expected rate of return of any

portfolio, subject to the constraint of an efficiency frontier. The latter is defined

pointwise, as a sequence of solutions to a quadratic programming problem which

minimizes the risk associated with each possible portfolios expected rate of return

subject to the constraint that the elements of the portfolio be non-negative and sum to

unity, and

- a parametric stochastic returns-generating process by which, in each period, the

investors determine the requisite vector of expectations and the variance-covariance

matrix of the investors anticipated rates of return on all risky assets.

The managers are constantly faced with the dilemma of guessing the direction of

market moves in order to meet the return target for assets under management. Given the

uncertainty inherent in financial markets, the managers are very cautious in expressing

their market views. The information content in such cautious views can be best

129

described as being “fuzzy” or vague, in terms of both the direction and the size of

market moves. Nevertheless, such fuzzy views are the ones needed to structure

portfolios so that the target return, which is assumed to be higher than the risk-free

theory to select optimal portfolios that target returns above the risk-free rate by taking

only market risk.

6.2 A Mean VaR portfolio selection multi-objective model

with transaction costs

A value-at-risk (VaR) model measures market risk by determining how much the value of

a portfolio could decline over a given period of time with a given probability as a result of

changes in market prices or rates. The two most important components of VaR models are

the length of time over which market risk is to be measured and the confidence level at

which market risk is measured. The choice of these components by risk managers greatly

affects the nature of the value-at-risk model.

We begin by using the rates of return of the risky securities in the economic have a

multivariate normal distribution. In practice, this is a popular assumption when

computing a portfolio’s VaR ( see Hull and White [74]).

In this section, we will study a possibilistic mean VaR multi-objective model with

transaction costs.

6.2.1 Case of Mean downside-risk

In this section we extended Chen et al [31], Inuiguchi and Ramik [75] for n

assets. In practice investors are concerned about the risk that their portfolio value falls

below a certain level. That is the reason why different measures of downside-risk are

considered in the multi asset allocation problem. Denoted the random variable iν ,

qi ,1= the future portfolio value, i.e., the value of the portfolio by the end of the

planning period, then the probability

))(( ii VaRP <ν , qi ,1=

130

that iν the portfolio value falls below the iVaR)( level, is called the shortfall probability.

The conditional mean value of the portfolio given that the portfolio value has fallen

below (VaR)i , called the expected shortfall, is defined as

))(( iii VaRE <νν .

Other risk measures used in practice are the mean absolute deviation

)())(( iiii EEE νννν <− ,

and the semi-variance

))())((( 2iiii EEE νννν <− ,

where we consider only the negative deviations from the mean.

Let ),1( njx j = represents the proportion of the total amount of money devoted to

security j and jM1 and jM 2 represent the minimum and maximum proportion of the

total amount of money devoted to security j , respectively. For nj ,1= , qi ,1= let jir

be a random variable which is the rate of the i return of security j. Then we have

∑=

=n

jjjii xr

1ν .

Assume that an investor wants to allocate his/her wealth among n risky securities. If the

risk profile of the investor is determined in terms of (VaR)i, qi ,1= , a mean-VaR

efficient portfolio will be a solution of the following .

Multi-objective optimization problem

[ ])(,),(max 1 qRx

EEn

νν L∈

(6.1)

qiVaRtosubject iii ,1,)(Pr =≤≤ βν , (6.2)

∑=

=n

jjx

11, (6.3)

njMxM jjj ,1,21 =≤≤ . (6.4)

In this model, the investor is trying to maximize the future value of portfolio,

which requires the probability that the future value of his portfolio falls below (VaR)i not

to be greater than iβ , qi ,1= .

131

6.2.2. Case of the proportional transaction costs model

The introduction of transaction costs adds considerable complexity to the optimal

portfolio selection problem. The problem is simplified if one assumes that the transaction

costs are proportional to the amount of the risky asset traded, and there are no transaction

costs on trades in the riskless asset. Transaction cost is one of the main sources of concern

to managers (see Arnott and Wagner [1], Zhou and Li [218] are found that ignoring

transaction costs would result in efficient portfolio and some conclusion.

Assume the rate of transaction cost of security j ( nj ,1= ) and allocation of i, qi ,1=

assets is jic , thus the transaction cost of security j and allocation of i assets is jji xc . The

transaction cost of portfolio ),...,( 1 nxxx = is qixcn

jjji ,1,

1=∑

=

. Considering the

proportional transaction cost and the shortfall probability constraint, we purpose the

following mean VaR portfolio selection model with transaction costs:

⎥⎦

⎤⎢⎣

⎡−− ∑∑

==∈

n

jjjkk

n

jjj

RxxcvExcvEMax

n11

11 )(....,,)( (6.5)

iii VaRvtosubject β≤< )(Pr , qi ,1= , (6.6)

∑=

=n

jjx

11, (6.7)

njMxM jjj ,1,21 =≤≤ . (6.8)

6.3 Possibilistic mean Var portfolio selection model.

In this section we introduce the concepts of possibilistic mean VaR portfolio selection

model and with assume that the rates of return on securities are modeled by possibility

distributions rather than probability distributions. Applying possibilistic distribution may

have two advantages (Hull and White [74]): (1) the knowledge of the expert can be easily

introduced to the estimation of the return rates and (2) the reduced problem is more

tractable than that of the stochastic programming approach. Possibility theory may be

132

quantitative or qualitative (Dubois and Prade, [44]) according to the range of these

measures which may be the real interval [0, 1], or a finite linearly ordered scale as well.

6.3.1 Possibilistic theory. Some preliminari

We consider the possibilistic theory proposed by Zadeh [216]. Let a~ and b~ be

two fuzzy numbers with membership functions a~μ and b~μ respectively. The possibility

operator (Pos) is defined as follows (Dubois and Prade [44]).

⎪⎩

⎪⎨

⎧

∈==<∈=<≤∈=≤

.))(),(supmin()~~(,,))(),(supmin()~~(

,,))(),(supmin()~~(

~~

~~

~~

RR,R,

xxxbaPosyxyxyxbaPosyxyxyxbaPos

ba

ba

ba

μμμμμμ

(6.9)

In particular, when b~ is a crisp number b, we have

( )

⎪⎩

⎪⎨

⎧

==<∈=<≤∈=≤

).()~(,,)(sup)~(

,,)(sup~

~

~

~

bbaPosbxxxbaPosbxxxbaPos

a

a

a

μμμ

RR

(6.10)

Let RRR →×:f be a binary operation over real numbers. Then it can be

extended to the operation over the set of fuzzy numbers. If we denoted for the fuzzy

numbers ba ~,~ the numbers )~,~(~ bafc = , then the membership function c~μ is obtained

from the membership function a~μ and b~μ by

),(,,))(),(supmin()( ~~~ yxfzyxyxz bac =∈= Rμμμ (6.11)

for R∈z . That is, the possibility that the fuzzy number )~,~(~ bafc = achives value

R∈z is as great as the most possibility combination of real numbers x,y such that z =

f(x,y), where the value of a~ and b~ are x and y respectively.

133

6.3.2 Triangular and trapezoidal fuzzy numbers

Let the rate of return on security given by a trapezoidal fuzzy number ),,,(~4321 rrrrr =

where 4321 rrrr <≤< . Then the membership function of the fuzzy number r~ can be

denoted by:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≤≤−−

≤≤

≤≤−−

=

.,0

,

,,1

,,

)(,43

43

4

32

2112

1

otherwise

rxrrrrx

rxr

rxrrrrx

xμ (6.12)

We mention that trapezoidal fuzzy number is triangular fuzzy number if 32 rr = .

)(~ xbμ )(~ xrμ 1 δ 0 b1 b2 r1 b3 xδ r2 r3 b4 r4

Figura 6.1: Two trapezoidal fuzzy number r~ and b~ .

Let us consider two trapezoidal fuzzy numbers ),,,(~

4321 rrrrr = and =b~

),,,( 4321 bbbb , as shown in Figure 6.1.

If 32 br ≤ , then we have

( ) yxyxbrPos br ≤=≤ )(),(minsup~~~~ μμ

,11,1min)(),(min 3~2~ ==≥ br br μμ

which implies that 1)~~( =≤ brPos . If 32 br ≥ and 41 br ≤ then the supremum is

achieved at point of intersection xδ of the two membership function )(~ xrμ and )(~ xbμ .

A simple computation shows that

134

( ))()(

~~1234

14

rrbbrbbrPos

−+−−

==≤ δ

and

δδ )( 121 rrrx −+= .

If 41 br > , then for any yx < , at least one of the equalities

0)(,0)( ~~ == yx br μμ

hold. Thus we have ( ) 0~~ =≤ brPos . Now we summarize the above results as

( )⎪⎩

⎪⎨

⎧

≥≤≥

≤=≤

.,0,,,

,,1~~

41

4132

32

brbrbr

brbrPos δ (6.13)

Especially, when b~ is the crisp number 0, then we have

( )⎪⎩

⎪⎨

⎧

≥≤≤

≤=≤

0,00,

0,10~

1

21

2

rrr

rrPos δ (6.14)

where

21

1

rrr−

=δ . (6.15)

We now turn our attention the following lemma.

Lemma 6.1 (Dobois and Prade [42]) Let ( )4321 ,,,~ rrrrr = be a trapezoidal fuzzy

number. Then for any given confidence level α with ( ) αα ≥≤≤≤ 0~,10 rPos if and

only if 1)1( rα− + 02 ≤rα .

The λ level set of a fuzzy number ( )4321 ,,,~ rrrrr = is a crisp subset of R and denoted

by ,)(]~[ Rxxxr ∈≥= λμλ , then according to Carlsson et al [26], we have

)](),([,)(]~[ 344121 rrrrrrRxxxr −−−+=∈≥= λλλμλ .

Given )](),([]~[ 21 λλλ aar = , the crisp possibilistic mean value of ( )4321 ,,,~ rrrrr =

is

∫ +=1

0 21 ))()(()~(~ λλλλ daarE .

135

where E~ denotes fuzzy mean operator.

We can see that if ( )4321 ,,,~ rrrrr = is a trapezoidal fuzzy number then

63

))()(()~(~ 41321

0 344121rrrrdrrrrrrrE +

++

=−−+−+= ∫ λλλλ . (6.16)

6.3.3 Construction efficient portfolios

Let jx the proportional of the total amount of money devoted to security j,

jM1 and jM 2 represent the minimum and maximum proportion respectively of the total

amount of money devoted to security j . The trapezoidal fuzzy number of jir is

( )4)(3)(2)(1)( ,,,~jijijijiji rrrrr = where 4)(3)(2)(1)( jijijiji rrrr <≤< . In addition, we denote the

(VaR)i level by the fuzzy number trapezoidal ( )4321 ,,,~iiiii bbbbb = , qi ,1= .

Using this approach we see that the model given by (6.5)-(6.8) reduces itself to the form

from the following theorem.

Theorem 6.1 The possibilistic mean VaR portfolio selection for the vector mean VaR

efficient portfolio model (6.5)-(6.8) is

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛∑∑∑∑====∈

n

jjjq

n

jjjq

n

jjj

n

jjj

RxxcxrExcxrE

n111

11

1~~,...,~~max (6.17)

i

n

jijji bxrPostosubject β≤⎟⎟⎠

⎞⎜⎜⎝

⎛<∑

=1

~~ , qi ,1= , (6.18)

∑=

=n

jjx

11, (6.19)

njMxM jjj ,1,21 =≤≤ . (6.20)

In the following using Section 6.3 we obtain the efficient portfolios given by the Theorem

6.1.

Theorem 6.2. If qii ,1,0 =>λ , then an efficient portfolio for possibilistic model is an

optimal solution of the following problem:

136

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛∑∑∑===∈

n

jjji

n

jjji

q

ii

RxxcxrE

n111

~~max λ (6.21)

i

n

jijji bxrPostosubject β≤⎟⎟⎠

⎞⎜⎜⎝

⎛<∑

=1

~~ , qi ,1= , (6.22)

∑=

=n

jjx

11, (6.23)

njMxM jjj ,1,21 =≤≤ . (6.24)

Using the fact that rate of return on security ),1( njj = is given by trapezoidal fuzzy

number, then we get the following results.

Theorem 6.3 Let rate of return on security ),1( njj = by the trapezoidal fuzzy number

( )4)(3)(2)(1)( ,,,~jijijijiji rrrrr = where 4)(3)(2)(1)( jijijiji rrrr <≤< and addition

( )4321 ,,,~iiiii bbbbb = is trapezoidal fuzzy number for VaR level and 0>iλ ,with qi ,1= .

Then using the possibilistic mean VaR portfolio selection model an efficient portfolio is

an optimal solution for the following problem:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−+

++

∑∑∑∑∑

∑=

====

=∈

n

jjji

n

jjji

n

jjji

n

jjji

n

jjjiq

ii

Rxxc

xrxrxrxr

n1

14)(

11)(

13)(

12)(

1 63max λ (6.25)

:. toubjects

( ) 011

32)(1

41)( ≥⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−− ∑∑

==

n

jijjii

n

jijjii bxrbxr ββ , qi ,1= , (6.26)

∑=

=n

jjx

11, (6.27)

njMxM jjj ,1,21 =≤≤ . (6.28)

Proof : Really, from the equation (6.16), we have

137

63

~~ 14)(

11)(

13)(

12)(

1

∑∑∑∑∑ ====

=

++

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛

n

jjji

n

jjji

n

jjji

n

jjjin

jjji

xrxrxrxrxrE , pi ,1= .

From Lemma 6.1, we have that, for any qi ,1= ,

i

n

jijji bxrPos β≤⎟⎟⎠

⎞⎜⎜⎝

⎛<∑

=1

~~ , is equivalent with

( ) 011

32)(1

41)( ≥⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−− ∑∑

==

n

jijjii

n

jijjii bxrbxr ββ .

Furthermore, from (6.25)-(6.28) given by Theorem 6.2, we get that this problem is of the

following form :

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−+

++

∑∑∑∑∑

∑=

====

=∈

n

jjji

n

jjji

n

jjji

n

jjji

n

jjjiq

ii

Rxxc

xrxrxrxr

n1

14)(

11)(

13)(

12)(

1 63max λ (6.29)

tosubject

( ) 011

32)(1

41)( ≥⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−− ∑∑

==

n

jijjii

n

jijjii bxrbxr ββ , qi ,1= (6.30)

∑=

=n

jjx

11, (6.31)

njMxM jjj ,1,21 =≤≤ . (6.32)

This completes the proof.

Problem (6.29)-(6.32) is a standard multi-objective linear programming problem. For

optimal solution we can used some algorithms of multiobjective programming (Slowinski

and Teghem [175] and White [209]).

138

6.4 A Weighted possibilistic mean value approach

In this section introducing a weighting function measuring the importance of λ -

level sets of fuzzy numbers we shall define the weighted lower possibilistic and upper

possibilistic mean values, crisp possibilistic mean value of fuzzy numbers, which is

consistent with the extension principle and with the well-known definitions of expectation

in probability theory. We shall also show that the weighted interval-valued possibilistic

mean is always a subset (moreover a proper subset excluding some special cases) of the

interval-valued probabilistic mean for any weighting function.

A trapezoidal fuzzy number ),,,(~4321 rrrrr = is a fuzzy set of the real line R with a

normal, fuzzy convex and continuous membership function of bounded support. The

family of fuzzy numbers will be denoted by F. A λ -level set of a fuzzy number

),,,(~4321 rrrrr = is defined by ,)(]~[ R∈≥= xxxr λμλ , then

)](),([,)(]~[ 414121 rrrrrrxxxr −−−+=∈≥= λλλμλ R ,

if 0>λ and 0)(]~[ ≥∈= xxclr μλ R (the closure of the support of r~ ) if 0=λ . It is

well-known that if r~ is a fuzzy number then λ]~[r is a compact subset of R for all

]1,0[∈λ .

Definition 6.1 (Majlender [110]) Let F∈r~ be fuzzy number with =λ]~[r

)],(),([ 21 λλ aa ]1,0[∈λ . A function ]1,0[:w → R is said to be a weighting function if

wis non-negative, monoton increasing and satisfies the following normalization

condition

∫ =1

01)( λλ dw . (6.33)

We define the w-weighted possibilistic mean (or expected) value of fuzzy number r~ as

∫+

=1

021 )(

2)()()~(~ λλλλ dwaarEw . (6.34)

It should be noted that if ]1,0[,2)( ∈= λλλw then

∫ +=1

0 21 .)]()([)~(~ λλλλ daarEw

139

That is the w-weighted possibilistic mean value defined by (6.24) can be

considered as a generalization of possibilistic mean value in Chen [32]. From the

definition of a weighting function it can be seen that )(λw might be zero for certain

(unimportant) λ -level sets of r~ . So by introducing different weighting functions we can

give different (case-dependent) important to λ -levels sets of fuzzy numbers.

Let ),,,(~21 βαrrr = be a fuzzy number of trapezoidal form and with peak ],[ 21 rr , left-

with 0>α and right-with 0>β and let

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−−=

−

1)1()12()( 21γλγλw ,

where 1≥γ . It’s clear that w is weighting function with 0)0( =w and

∞=−→

)(lim01

λλ

w .

Then the w-weighted lower and upper possibilistic mean values of r~ are computed by

( )[ ]∫ −−−−−= −− 1

0

2/11 11)12]()1([)~(~ λλγαλ γ drrEw

)14(2)12(

1 −−

−=γγαr ,

and

( )[ ]∫ −−−−−= −+ 1

0

2/11 11)12]()1([)~(~ λλγβλ γ drrEw

)14(2)12(

2 −−

+=γγβr

and therefore

⎥⎦

⎤⎢⎣

⎡−−

+−−

−=)14(2)12(,

)14(2)12()~(~

21 γγβ

γγα rrrEw

)14(4

))(12(2

)~(~ 21

−−−

++

=γ

αβγrrrEw . (6.35)

This observation along with Theorem 6.1 as in Section 6.3.3 leads to the

following theorem.

140

Theorem 6.4 The mean VaR efficient portfolio model is

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛∑∑∑===∈

n

jjji

n

jjjiw

q

ii

RxxcxrE

n111

~~max λ (6.36)

i

n

jijji bxrPostosubject β≤⎟⎟⎠

⎞⎜⎜⎝

⎛<∑

=1

~~ , qi ,1= , (6.37)

∑=

=n

jjx

11, (6.38)

njMxM jjj ,1,21 =≤≤ . (6.39)

In the next theorem we extend Theorem 6.3 to the case weighted possibility mean value

approach with a special weighted )(λw .

Theorem 6.5 Let ⎥⎥⎦

⎤

⎢⎢⎣

⎡−−−=

−

1)1()12()( 21γλγλw , 1≥γ the weighted possibility mean

of the trapezoidal fuzzy number ( )4)(3)(2)(1)( ,,,~jijijijiji rrrrr = where ≤< 2)(1)( jiji rr

4)(3)( jiji rr < and addition ( )4321 ,,,~iiiii bbbbb = is a trapezoidal fuzzy number for (VaR)i

level, qi ,1= . Then the possibilistic mean VaR portfolio selection model is

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

++

∑∑∑∑∑

∑=

====

=∈

n

jjji

n

jjji

n

jjji

n

jjji

n

jjjiq

ii

Rxxc

xrxrxrxr

n1

13)(

14)(

12)(

11)(

1 )14(4

)12(

2max

γ

γλ (6.40)

tosubject

( ) 011

32)(1

41)( ≥⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−− ∑∑

==

n

jijjii

n

jijjii bxrbxr ββ , qi ,1= , (6.41)

∑=

=n

jjx

11, (6.42)

njMxM jjj ,1,21 =≤≤ . (6.43)

141

Proof : From the equation (6.16), we have

)14(4

)12(

2~~ 1

3)(1

4)(1

2)(1

1)(

1 −

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

++

=⎟⎟⎠

⎞⎜⎜⎝

⎛ ∑∑∑∑∑ ====

= γ

γn

jjji

n

jjji

n

jjji

n

jjjin

jjjiw

xrxrxrxrxrE , 1≥γ .

From Lemma 6.1, we have that

i

n

jijji bxrPos β≤⎟⎟⎠

⎞⎜⎜⎝

⎛<∑

=1

~~ , qi ,1= is equivalent with

( ) 011

32)(1

41)( ≥⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−− ∑∑

==

n

jijjii

n

jijjii bxrbxr ββ , qi ,1= .

Furthermore, from (6.40)-(6.43) given by Theorem 6.4, we get the following form :

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

++

∑∑∑∑∑

∑=

====

=∈

n

jjji

n

jjji

n

jjji

n

jjji

n

jjjiq

ii

Rxxc

xrxrxrxr

n1

11)(

14)(

12)(

11)(

1 )14(4

)12(

2max

γ

γλ (6.44)

tosubject

( ) 011

32)(1

41)( ≥⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−− ∑∑

==

n

jijjii

n

jijjii bxrbxr ββ , qi ,1= (6.45)

∑=

=n

jjx

11, (6.46)

njMxM jjj ,1,21 =≤≤ . (6.47)

This completes the proof.

Problem (6.44)-(6.47) is a standard multi-objective linear programming problem. Also we

can obtain an optimal solution by using some algorithms of multi-objective programming

(Kacprzyk and Yager [76] and Stanley and Li [177]).

For ∞→γ we see that =∞→

)~(~lim rEwγ 824321 rrrr −

++

. Thus we get

Corollary 6.1 For +∞→γ , the weighted possibilistic mean VaR efficient portfolio

selection model can be reduce to the following linear programming problem:

142

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

++

∑∑∑∑∑

∑=

====

=∈

n

jjji

n

jjji

n

jjji

n

jjji

n

jjjiq

ii

Rxxc

xrxrxrxr

n1

11)(

14)(

12)(

11)(

1 82max λ

subject to

( ) 011

32)(1

41)( ≥⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−− ∑∑

==

n

jijjii

n

jijjii bxrbxr ββ , qi ,1= ,

∑=

=n

jjx

11,

njMxMjj j ,1,21 =≤≤ .

6.5 A weighted possibilistic mean variance and covariance of

fuzzy numbers The classical mean-variance portfolio selection problem uses the variance as the

measure for risk, which puts the same weight on the down side and upside of the return.

In this section, we study the “weighted” possibilistic mean-variance and covariance

portfolio selection model.

Definition 6.2 (Fuller and Majlender, [53]) Let F∈r~ be a fuzzy number with

)](),([]~[ 21 λλλ rrr = , ]1,0[∈λ . The w -weighted possibilistic variance of r~ is

∫ ⎟⎠⎞

⎜⎝⎛ −

=1

0

212 )(

2)()()~( λλλλ dwrrrVarw

λλλλ

λλλλ dwrrrrrr )(

2)()()()(

2)()(

211

0

221

2

2

121∫ ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡ +

−+⎥⎦⎤

⎢⎣⎡ −

+=

where weighting function is non-decreasing and satisfies

∫ =1

01)( λλ dw . (6.48)

We note that the weighted possibilistic variance of r~ is defined as the expected value of

the squared deviations between the arithmetic mean and the endpoints of its level sets, i.e.

the lower possibility-weighted average of the squared distance between the left-hand

endpoint and the arithmetic mean of the endpoints of its level sets plus the upper

143

possibility weighted average of the squared distance between the right-hand endpoint and

the arithmetic mean of the endpoints their of its level sets.

The standard deviation of r~ is defined by

)~(~ rVarr =σ (6.49)

Let r~ fuzzy number and w be a weighting function, we define the weighted possibilistic

variance of r~ by

∫ ⎟⎠⎞

⎜⎝⎛ −

=1

0

212 )(

2)()(

)~( λλλλ

dwrr

rVarw

and the weighted covariance of r~ and b~ is defined as

∫ ⎟⎠⎞

⎜⎝⎛ −−

=1

01212 )(

2)()(.

2)()(),~( λλ

λλλλ dwbbrrbrCovw .

If ]1,0[,2)( ∈= λλλw

∫ −=1

0 12 ))()((21)~( λλλλ drrrVarw , (6.50)

and

∫ −−=1

0 1212 2))()()(()((21),~( λλλλλλ dbbrrbrCovw . (6.51)

Let ),,,(~4321 rrrrr = and ),,,(~

4321 bbbbb = be fuzzy numbers of trapezoidal form.

Let

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=

−

1)1()12()( 21γλγλw ,

where 1≥γ , be a weighting function then the power-weighted variance and covariance

r~ and b~ are computed by

λλλλ

γ γ drrrVarw ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟

⎠⎞

⎜⎝⎛ −

−=−1

0

212

12 1)1(2

)()()12()~(

⎥⎦

⎤⎢⎣

⎡−

++−

+−+−

−−

=318

1)(28

1))((212

1)(4

)12( 2434312

212 γγγ

γ rrrrrrrr

144

⎥⎦

⎤⎢⎣

⎡−

++

−+−

+−

−−=

)16(3)(

)14(2))((2

12)(

4)12( 2

4343122

12

γγγγ rrrrrrrr

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+−−+−−−=

−1

0

21

43124312 1)1(2

))(1)((.

2))(1)((

)12()~,~( λλλλ

γ γ dbbbbrrrr

brCovw

= ⎥⎦

⎤⎢⎣

⎡−++

+−

++−+

−−−−

)16(3))((

)14(2))()((

12))((

4)12( 43434343121212

γγγγ bbrrrrbbrrbbrr

Theorem 6.6 The mean-variance efficient portfolio model is

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛∑∑∑===∈

n

jjji

n

jjjiw

q

ii

RxxcxrE

n111

~~max λ (6.52)

tosubject i

n

jijji bxrPos β≤⎟⎟⎠

⎞⎜⎜⎝

⎛<∑

=1

~~ , qi ,1= , (6.53)

∑=

=n

jjx

11, (6.54)

njMxM jjj ,1,21 =≤≤ . (6.55)

In the next theorem we extend Theorem 6.3 to the case weighted possibility mean-

variance approach with a special weighted )(λw .

Theorem 6.7 Let ⎥⎥⎦

⎤

⎢⎢⎣

⎡−−−=

−

1)1()12()( 21γλγλw , 1≥γ the weighted possibility mean

variance of the trapezoidal number ( )4)(3)(2)(1)( ,,,~jijijijiji rrrrr = where

4)(3)(2)(1)( jijijiji rrrr <≤< and addition ( )4321 ,,,~iiiii bbbbb = is a trapezoidal number for

(VaR)i level, qi ,1= . For qii ,1,0 =>λ , then the possibilistic mean variance portfolio

selection model is

145

⎢⎢⎢⎢⎢

⎣

⎡

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

+−

⎟⎟⎠

⎞⎜⎜⎝

⎛−− ∑∑∑∑ ∑

∑ ==== =

=∈ )14(4

)12(

)12(4

)12(max 1

4)(1

3)(1)(1

2)(

2

1 11)(2)(

1 γ

γ

γ

γλ

n

jjjij

n

jjijji

n

jjji

n

j

n

jjjijjiq

ii

Rx

xrxrxrxrxrxr

n

+

⎥⎥⎥⎥⎥

⎦

⎤

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

∑∑∑

=

==n

jjji

n

jjji

n

jjji

xcxrxr

1

2

14)(

13)(

)16(12

)12(

γ

γ (6.56)

tosubject

( ) 011

32)(1

41)( ≥⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−− ∑∑

==

n

jijjii

n

jijjii bxrbxr ββ , qi ,1= , (6.57)

∑=

=n

jjx

1

1, (6.58)

njMxM jjj ,1,21 =≤≤ . (6.59)

Proof: The proof is the on the line of Theorem 6.5.

146

CHAPTER 7

ATZBERGER’S EXTENSION OF MARKOWITZ PORTFOLIO SELECTION

7.1 Introduction

Recently Atzberger [3] we represent one basic manner by which Markowitz’s theory for

portfolio selection can be extended to account for non-gaussian distributed returns. Thus

we discuss how a model incorporating information about the performance of the assets in

different market regimes over the holding period can be developed. This basic extension

follows the work of Buckley, Comezana, Djerrond, and Seco [20].

When attempting to apply Markowitz’s theory for portfolio selection in practice a

number of well known challenges arise. First, in the theory the mean returns μ and

covariance V must somehow be estimated. One approach is to use historic data, but this

has its obvious limitations in predicting the future performance of assets. In practice, the

historic data is often used in combination with securities analysis to construct forecasts

incorporating information about the fundamentals of the security and the opinions of

financial experts. Second, in the theory the variance is used as the primary indicator of

risk. However, the variance does not capture important information about the risk of an

investment if the returns are non-Gaussian with a multimodal or strongly skewed

distribution, see figures 7.1–7.3. Third, the covariance structure and realized returns for

assets may be drastically altered if the market conditions suddenly change, such as a

rally with investors having bullish expectations changing to a sell-out with bearish

expectations because of the announcement of important news, such as the results of an

election, oil production adjustments, or interest rate changes. In particular, the character

of the asset returns may change drastically depending on which important economic

conditions prevail over the holding period of the asset, leading to a break-down of the

assumptions usually justifying a Gaussian distribution. To better enable the key insights

147

of the Markowitz theory to be adapted in this case, we shall discuss a basic extension

with the following features.

- Returns will be modeled by Gaussian Mixture Distributions, discussed in more

detail below.

- We shall have a separate mean and covariance structure to model returns for

each of the market scenarios anticipated over the holding period (for example:

election results, oil production adjustments, interest rate changes) and

probabilities assigned for these different scenarios to occur.

- Since the variance may no longer be a good indication of the uncertainty (risk)

of the asset returns, we shall introduce a new quantification of risk. In particular,

we shall consider the probability that the returns fall short of a desired return.

As we mentioned above we seek to model the asset returns for different market

conditions which may prevail over the holding period of the portfolio. One useful

way to extend the model, while retaining much of the computational convenience of

the Markowitz Theory, is to describe the asset returns across the different market

scenarios using Gaussian Mixture Distributions, which are distributions derived from

linear combinations of Gaussian distributions. One basic motivation for this

approach, is that the returns of the assets for each market scenario may still retain

their Gaussian behavior, while averaging over the different scenarios leads to

multimodal or skewed distributions. The mixture coefficients then correspond to the

probability of each scenario. We now give the mathematical details of this extension.

Figure 7.1: Asset return having a Gaussian distribution, well characterized by the mean and variance.

148

As we mentioned above we seek to model the asset returns for different market

conditions which may prevail over the holding period of the portfolio. One useful way to

extend the model, while retaining much of the computational convenience of the

Markowitz Theory, is to describe the asset returns across the different market scenarios

using Gaussian Mixture Distributions, which are distributions derived from linear

combinations of Gaussian distributions. One basic motivation for this approach, is that

the returns of the assets for each market scenario may still retain their Gaussian behavior,

while averaging over the different scenarios leads to multimodal or skewed distributions.

The mixture coefficients then correspond to the probability of each scenario. We now

give the mathematical details of this extension.

Figure 7.2: Asset return having a non-Gaussian distribution, not well characterized by the mean and variance as consequence of skew.

7.2 Gaussian Mixture Distributions

Definition 7.3 [3] A scalar random variable Z has the univariate Gaussian Mixture

(GM) distribution if its probability density has the form:

∑ ∑= =

⎟⎟⎠

⎞⎜⎜⎝

⎛ −==

n

i

n

i i

iiXi

zzfzf

i1 1

)()(σμ

φαα

where ),(~ 2iiiX σμη are Gaussian distributed random variables with mean iμ and

2iσ variance. The density of the standard Gaussian is denoted by

149

2

2

21)(

x

ex−

=π

φ .

The additional condition 11

=∑ =

n

i iα is imposed to ensure the resulting density

describes a probability distribution. We now extend this definition to the vector-valued

case.

Definition 7.3: [3] A vector-valued random variable Z has the multivariate GM

distribution if its probability density is of the form:

)(,)()(1

)(

1)()( zvzfzf

n

i

in

iXiz ii ∑∑

==

==μ

φα

Figure 7.3: Asset return having a non-Gaussian distribution, not well characterized by the mean and variance as a consequence of biomodality.

where )(iX are multivariate Gaussian random variables with component means given by

the vector )(iμ and covariances given by the matrix )(..( )()()( ia

ia

i XEeiV =μ and

)),cov( )()()(,

jb

ia

iba XXV = . The multivariate Gaussian random variables with the specified

means and variances is given by the density:

)()()(21

21

)(2/,

)(1)()(

)()(

)()2(

1)(iiTi

ii

zVz

iV e

DetVz

μμ

πμ

πφ

−−− −

= ,

We remark that by definition we shall assume 0,cov( )()( =jb

ia XX We now show how

moments can be computed for these random variables.

150

Proposition 7.1 [3] Let Z be a vector-valued GM random variable with mean )(iμ ,

covariance )(iV , and mixture weights iα , then for any function g we have

∑=

=n

i

ii XgEZgE

1

)( ))(())(( α .

where )(iX are the Gaussian random variables defined above.

Proof: This follows immediately from the definition of the GM random variable with a

density which is a linear combination of the densities of )(iX . We leave the details as an

exercise .

A particular consequence of this proposition is that ∑ ==

n

ii

iZE1

)()( μα .

Proposition 7.2 [3] Let Z be as in the previous proposition, then the ath component Za =

[Z]a has a variance which can be expressed as

( )

2

1

)()(2)(

1

1

)()(

1

)(

)()(

)()(var()var(

∑∑∑

∑∑∑

= <=

= <=

−+=

−+=

n

i

ja

ia

ijji

ia

n

ii

n

i

ia

ia

ijji

n

i

iaia XEXEXZ

μμαασα

ααα

Proof:

( )

( )∑ ∫

∫ ∑

∑ ∑∫

=

=

= =

−+−=

−==

===

n

iXa

ia

iai

n

iXiaa

ia

ia

i

n

iiXiaa

dzzz

dzzzZ

dzzzZE

ia

ia

ia

1

2)()(

1

22)(

)(1

1 1

)(

)()()var(

)()(

)(

)(

)(

φμμμα

φαμσ

μαφαμ

( ) ( )

( ) ( )∑ ∫

∑ ∫∑ ∫

=

==

−−+

−+−=

n

iXa

ia

iai

n

iXa

iai

n

iX

iai

dzzz

dzzdzzz

ia

ia

ia

1

)(2)(

1

2)(

1

2)(

)(2

)()(

)(

)()(

φμμμα

φμμαφμα

( ) ( )∑∑==

−+=n

i

iaai

n

i

iai X

1

2)(

1

)(var μμαα .

Now using ∑ ==

n

ii

aia 1)(μαμ and )(2)(22)( 2)()( i

aai

aai

aa μμμμμμ ++=− , we have

151

( ) ( )

( ) ( )( )

( )∑∑

∑

∑ ∑ ∑∑

= <

=

= = = =

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

−=−

n

i

ia

ia

ijji

n

ji

ja

ia

ja

iaji

n

i

n

i

ja

n

i

n

j

iaji

iai

iaai

1

2)()(

1,

)()(2)(2)(

1 1

)(

1 1

)(2)(2)(

221

μμαα

μμμμαα

μμααμαμμα

and substituting this above proves the proposition .

Proposition 7.3 [3] Let Z be defined as above then the covariance between the two

components of a and b is given by

( )( )∑∑∑− <=

−−+=n

i

jb

ib

ja

ia

ijji

iba

n

iiba VZZ

1

)()()()()(,

1),cov( μμμμααα

Proof: The proof is similar to the last proposition.

Remark 7.1: Let V denote the covariance matrix of Z then )),cov(( , baba ZZV =

( )( )Tn

i

jiji

ijji

in

iiVV ∑∑∑

− <=

−−+=1

)()()()()(

1μμμμααα .

7.3 An Extension of the Markowitz Portfolio Theory

We now discuss some results for the linear combinations of GM distributed random

variables, which will be useful in describing the return of a portfolio of assets.

Proposition 7.4 [3] Let Z be a vector-valued GM distributed random variable with m

components and let

∑=

=m

aaaZwY

1,

then

∑=

=n

iiY yyf

ii1

,)()( 2σμ

φα .

In the notation, ∑ ==

m

ai

aai w1

)(μμ and ∑ ∑= ==

m

aibab

m

b ai Vww1

)(,1

σ , where φ is the

density of a Gaussian as defined above.

Proof: The proof is similar to the previous propositions by making use of the fact that

the density is a linear combination of Gaussian densities.

152

From these propositions we see that the statistics of the GM distribution random

variables can be computed as readily as for Gaussian random variables, yet we can

construct a richer set of distributions. In principle, any distribution can be approximated

by a Gaussian mixture model provided a sufficient number are Gaussians are used.

As we discussed in the introduction, the variance may be a poor indicator of the risk of

an asset return. Instead, we shall consider a different approach to characterizing the risk

of a portfolio. An alternative criteria which can be used to model risk is to consider the

probability that a portfolio falls short of an intended (desired) return Bμ . We shall refer

to this measure of risk as the Probability of Shortfall (PoS). We remark that this is

similar in spirit, but distinct, from other measures used to evaluate risk, such as Value at

Risk (VaR).

Definition 7.4 [3] Probability of Short-fall (PoS) is defined for a desired return Bμ by

Pr BYPoS μ≤=

where Y is the portfolio return defined for a GM model of the returns Z given by

∑=

=m

aaaZwY

1

From the propositions above we have the following.

Proposition 7.5 [3] For asset returns modeled by a GM distribution the Probability of

Shortfall (PoS) is given by

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −Φ=

n

iiT

iTB

iBwVw

wwF1

)(

)(

),( μμαμ

Proof: Follows from similar calculations as the proofs above .

7.4 Portfolio Selection Problem (GM-PoS)

Using the PoS to quantify the risk associated with a portfolio and using the GM model

for the asset returns, the portfolio selection problem for obtaining a target return Pμ but

not falling below the return Bμ is:

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −Φ

n

iiT

iTB

iwVw

w1

)(

)(

min μμα

153

tosubject

PT

m

i

T

w

w

μμ ≥

=∑=

11

where ∑==

n

ii

i1)(μαμ and dyex

xy

∫ ∞−

−=Φ 2

2

21)(π

.

The objective function to be minimized is no longer quadratic as in Markowitz’s Theory

so in general a numerical optimization will have to be performed. Similarly, the lack of

analytic expressions for the solution requires that the efficient frontier be analyzed

numerically.

154

BIBLIOGRAPHY

[1] Arnott, R.D. and Wanger, W.H., The measurement and control of trading costs,

Financial Analysts Journal, 46(6), 73-80, 1990.

[2] Arnold B. C., Majorization and the Lorenz Order: A Brief Introduction, Lecture

Notes in Statistics 43, Springer-Verlag, Berlin, 1980.

[3] Atzberger P. J., An extension of Markowitz portfolio selection, http://www.math.

ucsb. edu/~atzberg/spring2006

[4] Artzner P., Delbaen F., Eber J.-M. and Heath D., Coherent measures of risk ,

Mathematical Finance, 9, pp. 203–228,1999.

[5] Atzberger Paul J., An Extension of Markowitz Portfolio Selection,

[6] Azimov, A,Y., Gasimov, R, N., On weak conjgary, weak subdifferentials and

duality with zero-gap in non-convex optimization, Iternational Journal of Applied

Mathematics, Vol. 1 pp. 171-192, 1999.

[7] Bawa V. S., Stochastic dominance: a research bibliography, Management

Science, 28,pp. 698–712,1982.

[8] Beale, E.M.L., On minimizing a convex function subject to linear inequalities," J.

of the Royal Statistical Society, Series B, 17, pp. 173-184. 1955.

[9] Bellman, R. and Zadeh, L.A., Decision making in a fuzzy environment,

Management Science,17, 141-164, 1970.

[10] Benders, J.F., Partitioning procedures for solving mixed variables programming

problems, Numerische Mathematik, 4, pp. 238-252, 1962.

[11] Ben Abdelaziz, F., Lang, P., and Nadeau, R., Pointwise efficiency in multi-

objective stochastic linear programming, Journal of Operational Research

Society, Vol. 45, pp. 1324-1224, 1994.

[12] Ben Abdelaziz, F., Lang, P., and Nadeau, R., Distribution efficiency in multi-

objective programming, Journal Operational Research, Vol. 85, pp. 399-415,

1995

[13] Bertsekas, D. and Tsitsiklis, J., Neuro-Dynamic Programming, Athena Scienti_c,

Belmont, MA., 1996.

155

[14] Birge, J.R., Decomposition and partitioning methods for multi-stage stochastic

linear programs," Operations Research, 33, pp. 989-1007, 1985.

[15] Birge, J.R. Stochastic Programming Computation and Applications," INFORMS

J. on Computing, 9, pp. 111-133. 1997.

[16] Birge, J.R. and Louveaux, F.V., Introduction to Stochastic Programming,

Springer, New York, 1997.

[17] Bloomfield P. and Steiger W. L., Least Absolute Deviations, Birkh¨auser, Boston

1983.

[18] Bonnans J.F. and Shapiro A., Perturbation Analysis of Optimization Problems,

Springer-Verlag, New York, 2000.

[19] Bowman, J., On the relationship of the Tchebychef norm and the efficient frontier

of multiple-criteria objective, in Tieriez, H. and Zionts, S. (eds), Multiple Criteria

Decision Making, Berlin: Springer, 1977.

[20] Buckley, I., Comenzana, G., Djerrond, B. and Seco, L., Portfolio optimization

when asset returns have the gaussian mixture distribution,

htt://ww.riskab.ca/seco/GM_AR, 2003

[21] Caballero, R., Cerda, E., Munoz, M. M., Rey, L., I. M. Stancu Minasian,

Efficient solution concepts and Theit relations in stochastic multi-objective

programming, JOTA, Vol. 110, No. 1, pp. 53-74, July 2001.

[22] Caballero, R., Cerda, E., Munoz, M. M., and Rey, L., Relations among several

efficiency concepts in stochastic multi-objective programming, Research and

Practice in Multi-criteria decision making, edited by Y. Y. Haimes and R.

Steuer, Lecture Notes in Econimics and Mathematical System, Springer Verlag,

Berlin, Germany, Vol. 484, pp. 57-68, 2000.

[23] Carino, D.R., T. Kent, D.H. Meyers, C. Stacy, M. Sylvanus, A.L. Turner, K.

Watanabe, and W.T. Ziemba, The Russell-Yasuda Kasai Model: An asset/liability

model for a Japanese insurance company using multistage stochastic

programming," Interfaces, 24, pp.29-49, 1994.

[24] Carrillo, M. J., A relaxtion algorithm for minimization of a quasi concave

function on concave polyhedron, Math. Prog., 13, 69-80, 1977.

[25] Carlsson, C. and Fuller, R., On possibilistic mean value and variance of fuzzy

numbers, Fuzzy sets and systems, 122, 315-326, 2001.

156

[26] Carlsson, C., Fuller, R. and Majlender, P., A possibilistic approach to selecting

portfolios with highest utilty score, Fuzzy sets and systems, 131, 13-21, 2002.

[27] Carlsson, C. and Korhonen, P., A Parametric Approach to Fuzzy Linear

Programming, Fuzzy Sets and Systems 20, 17-33, 1986.

[28] Carlsson, C., Approximate Reasoning for solving fuzzy MCDM problems,

Cybernetics and Systems: An International Journal, 18, 35-48, 1987.

[29] Chankong, V., and Haimes, Y. Y., Multiobjective Decision Making Theory and

methodology, Nort-Holand, New York, NY, 1983.

[30] Charnes, A. Cooper, W.W. and G.H. Symonds, Cost horizons and certainty

equivalents: An approach to stochastic programming of heating oil, Management

Sci., 4, pp.235–263, 1958.

[31] Charnes, A. and Cooper, W.W. Chance-constrained programming,"

Management Science, 5, pp. 73-79, 1959.

[32] Chen, G., et al, A possibilistic mean VaR model for portfolio selection, AMO-

Advanced Modeling and Optimization, Vol. 8, No. 1, 2006.

[33] Choo, E.U., Proper efficiency and the theory of vector maximization, Journal of

Mathematical Analysis and Applications, 22, 618-630, 1982.

[34] Choo, E. U. and Atkins, D. R., Bicriterion linear fractional vector maximum

problem, Operational Research, 32(1), 216-220, 1984.

[35] Chow, G., Portfolio selection based on return, risk, and relative performance,

Financial Analysts Journal, pages 5460, March-April 1995.

[36] Chen, S., X. J. Li, X. Y. Zhou. Stochastic linear quadratic regulators with

indefinite control weight costs. SIAM J. Control Optim. 36 1685–1702, 1998.

[37] Cox, J. C., C. F. Huang.. Optimal consumption and portfolio policies when asset

prices follow a diffusion process. J. Econom. Theory, 49 33–83, 1989.

[38] Cvitanic, J. and Karatzas, I., Convex duality in constrained portfolio

optimization, Annals of Applied Probability, 2, pp. 767-818, 1992.

[39] Dantzig, G.B., Linear programming under uncertainty," Management Science, 1,

pp. 197-206, 1955.

[40] Deb K., Multi-Objective Optimization using Evolutionary Algorithms. John

Wiley, New York, 2001.

157

[41] Dentcheva, D. and Ruszczynski, A., Optimization with stochastic dominance

constraints, Siam J. Optim.Society For Industrial And Applied Mathematics Vol.

14, No. 2, Pp. 548–566, 2003.

[42] Dentcheva, D. and Ruszczynski, A, Optimization under linear stochastic

dominance, Comptes Rendus de l’Academie Bulgare des Sciences 56, No. 6, pp.

6–11, 2003.

[43] Dentcheva, D. and Ruszczynski A., Optimization under nonlinear stochastic

dominance, Comptes Rendus de l’Academie Bulgare des Sciences 56, No. 7, pp.

19–25, 2003.

[44] Dubois, D. and Prade, H., Possibility Theory, Plenum Press, New York 1998.

[45] Duffie, D., Jackson, M., Optimal hedging and equilibrium in a dynamic futures

market. J. Econom. Dynam. Control 14 21–33, 1990.

[46] Duffie, D., Richardson H. M., Mean-variance hedging in continuous time. Ann.

Appl. Probab. 1 1–15. Follmer, H., D. Sondermann. 1986. Hedging of non-

redundant contingent claims. A. Mas-Colell, W. Hildenbrand, eds. Contributions

to Mathematical Economics. North-Holland, Amsterdam, The Netherlands, 205–

233, 1991.

[47] Edirisinghe, N.C.P. and W.T., Ziemba, Implementing bounds-based

approximations in convex-concave two stage programming," Mathematical

Programming, 19, pp. 314-340, 1996.

[48] Elliott, R.J. and Kopp P.E., Mathematics of Financial Markets, Springer-

Verlag,New York, 1999.

[49] Falk, J. E., and Hoffman, K. R., A successive underestimation method for

concave minimization problems, Math. Oper. Res., 1, 251-257, 1976.

[50] Fenchel W., Convex Cones, Sets, and Functions, lecture notes, Princeton

University, 1953.

[51] Fisher, M., J. Hammond, W. Obermeyer, and A. Raman, Configuring a supply

chain to reduce the cost of demand uncertainty," Production and Operations

Management, 6, pp.211-225 , 1997.

[52] Fishburn, P. C., Decision and Value Theory, John Wiley & Sons, New York,

1964.

158

[53] Frauendorfer, K., Stochastic Two-Stage Programming," Lecture Notes in

Economics and Mathematical Systems, 392, Springer-Verlag, Berlin. , 1992.

[54] Frauendorfer, K., Multistage stochastic programming: error analysis for the

convex case," Zeitschrift fur Operations Research, 39, pp. 93-122, 1994.

[55] Fuller, R. and Majlender, P., On weighted possibilistic mean value and variance

of fuzzy numbers, Turku Centre for Computer Science, 2002.

[56] Gassmann, H.I. and A.M. Ireland, On the formulation of stochastic linear

programs using algebraic modelling languages," Annals of Operations Research,

64, pp. 83-112, 1996.

[57] Gasimov, R. N., Yenilmez, K., Solving fuzzy linear programming problems with

linear membership functions, Turk J Math. 26 , 375 -396, 2002.

[58] Gasimov, R. N., Augmeted Laggrangian duality and nondifferentiable

optimization methods in nonconvex programming, ournal of Global

Optimization, Accepted, 2002.

[59] Gastwirth J. L., A general definition of the Lorenz curve, Econometrica, 39, pp.

1037–1039, 1971.

[60] Geofrion, A. M., Proper efficiency and the theory of vector maximization, Journal

of Mathematical Analysis and Applications, 22, 618-630, 1968.

[61] Jorion, P., Value at Risk: The New Benchmark for Managing Financial Risk,

McGraw-Hill, New York, 2001

[62] Glover, F. and Klingman, D., Concave programming applied to a special class of

0-1 integer programmer, Math. Res., 21, 135-140, 1973.

[63] Goicoechea, A., Hansen, D.R., and Duckstein, L., Multiobjective Decision

Analysis with engineering and Business application, John Wiley and Sons, New

York, New York 1982.

[64] Gotoh J. and Konno H., Third degree stochastic dominance and mean–risk

analysis, Management Science, 46, pp. 289–301,2000.

[65] Hadley, G. H., Linear Programming, Reading, MA: Addison-Wesley, 1973.

[66] Hadar, J. and Russell, W., Rules for ordering uncertain prospects, The

Amaerican Economic Review, 59, pp. 25–34, 1969.

[67] Hakansson, N.H., Capital growth and the mean–variance approach to portfolio

management, Journal of Financial Quantitative Analysis, 6, pp. 517-557, 1971.

159

[68] Hanoch, G. and Levy, H., The efficiency analysis of choices involving risk, Rev.

Econom.Stud., 36 , pp. 335–346, 1969.

[69] Hanqin J., Continuous-time portfolio optimization, Thesis, The Chinese

University of Hong Kong, 2004

[70] Hardy, G. H., Littlewood, J. E. and G. Polya, Inequalities, Cambridge University

Press, Cambridge, MA, 1934.

[71] Higle, J.L. and S. Sen, Stochastic Decomposition: An algorithm for two-stage

linear programs with recourse," Mathematics of Operations Research, 16, pp.

650-669, 1991.

[72] Higle, J.L. and Sen, S., On the convergence of algorithms with implications for

stochastic and nondifferentiable optimization," Math. of Operations Research,

17, pp. 112-131, 1992.

[73] Higle, J.L. and Sen S., Epigraphical Nesting: a unifying theory for the

convergence of algorithms," Journal of Optimization Theory and Applications,

84, pp. 339-360, 1995.

[74] Hull, J. C., White, D. J., Value-at-risk when daily changes in market variable

are not normally distributed, Journal of Derivatives 5, 9-19, 1998.

[75] Inuiguchi, M. and Ramik, J., Possibilistic linear programming: a brief review of

fuzzy mathematical programming and a comparison with stochastic

programming in portfolio selection problem, Fuzzy Sets and Systems, 111, 3-28,

2000.

[76] Kacprzyk J. and Yager, R.R., Using fuzzy logic withlinguistic quantifiers in

multiobjective decision making and optimization: A step towards more

humanconsistent models, in: R.Slowinski and J.Teghem eds., Stochastic versus

Fuzzy Approaches to Multiobjective Mathematical Programming under

Uncertainty, Kluwer Academic Publishers, Dordrecht, 331-350, 1990.

[77] Kall, P. and Wallace S.W., Stochastic Programming, John Wiley & Sons,

Chichester, England 1994.

[78] Kall, P. and Mayer J., SLP-IOR: an interactive model management system for

stochastic linear programs," Mathematical Programming, Series B, 75, 221-240,

1996.

160

[79] Karatzas, I, S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer-

Verlag, New York, 1988.

[80] Karatzas , I. and Shreve, S.E., Methods of Mathematical Finance, Springer-

Verlag, New York, 1998.

[81] Karatzas, I., J. P. Lehoczky, Shreve S. E., Optimal portfolio and consumption

decisions for a “small investor”on a finite horizon. SIAM J. Control Optim. 25

1557–1586, 1987.

[82] Karlin, S., Mathematical Methods and Theory in Games, Programming and

Economics, Reading, MA: Addison-Wesley, 1959.

[83] Kettani, O., Oral, M., Equivalent formulations of nonlinear integer problems for

eficient optimization, Management Science Vol. 36 No. 1 115-119, 1990.

[84] Klir, G.J., Yuan, B., Fuzzy Sets and Fuzzy Logic-Theory and Applications,

Prentice-Hall Inc., 574p., 1995.

[85] Kohlmann, M., Tang S., Optimal control of linear stochastic systems with

singular costs, and the meanvariance hedging problem with stochastic market

conditions. Working paper, Faculty for Mathematics and Information, University

of Konstanz, Konstanz, Germany, 2000.

[86] Konno, H. and Yamazak, H.i, Mean–absolute deviation portfolio optimization

model and its application to Tokyo stock market, Management Science, 37, pp.

519–531, 1991.

[87] Korner, R., On the variance of fuzzy random variables, Fuzzy Sets and Systems

92: 83-3, 1997.

[88] Korn, R., Optimal Portfolios: Stochastic Models for Optimal Investment and Risk

Management in Continuous Time. World Scientific, Singapore, 1997.

[89] Kramkov, D. and W. Schachermayer, Necessary and sufficient conditions in the

problem of optimal investment in incomplete markets, preprint, 2001.

[90] Kruse, R. and Meyer, K.D., Statistics with Vague Data, D. Reidel publishing

company, 1987.

[91] Kwakernaak, H., Fuzzy random variables: Definition and theorems, Information

Sciences, Vol. 15, 1-29, 1978.

[92] Lai, Y. J. and Hwang, C. L., Interactive fuzzy linear programming, Fuzzy Sets

and Systems, 45, 169-183, 1992.

161

[93] Lai, Y. J. and Hwang, C. L., Fuzzy Multiple Objective Decision Making: Methods

and Applications, Springer-Verlag, New York, 1994.

[94] Lamberto, C., Optimization Theory and Applications, Springer-Verlag, 1985

[95] Laporte, G. and Louveaux, F.V., The integer L-shaped method for stochastic

integer programs with complete recourse," Operations Research Letters, 13,

pp.133-142, 1993.

[96] Lehmann E., Ordered families of distributions, Annals of Mathematical Statistics,

26, pp. 399–419, 1955.

[97] Leibowitz, M.L. and Kogelman, S., Asset Allocation under Shortfall Constraints.

Journal of Portfolio Management, Winter, 18-23, 1991.

[98] Leon, T., Liern, V. and Vercher, E., Viability of infeasible portfolio selection

problem: a fuzzy approach, European Journal of Operational Research, 139, 178-

189, 2002.

[99] Levy, H., Stochastic dominance and expected utility: survey and analysis,

Management Science, 38, pp. 555–593, 1992.

[100] Levy, H. and Sarnat, M., Investment and Portfolio Analysis, New York, John

Wiley and Sons, 1972.

[101] Li, D. and Ng, W.L., Optimal dynamic portfolio selection: Multiperiod mean–

variance formulation, Math. Finance, 10, pp. 387-406, 2000.

[102] Lieberman, L., Lasdon, L., Schrage, L. and Waren, A., Modelling and

Optimization with GINO, The Scientific Press, Redwood City, 1986.

[103] Lim, A.E.B. and Zhou, X.Y., Mean–variance portfolio selection with random

parameters, Mathematics of Operations Research, 27, pp. 101-120, 2002.

[104] Liu, B. and Iwamura, K., Chance constrained programming with fuzzy

parameters, Fuzzy setsand systems, 94, 227-237, 1998.

[105] Li-Yong, Yu Xiao, Dong JI Shou and Yang Wang, Stochastic programming

models in financial optimization: A Survey, AMO – Advanced Modeling and

Optimization, Volume 5, Number 1, 2003.

[106] Lorenz M. O., Methods of measuring concentration of wealth, Journal of the

American Statistical Association, 9 (, pp. 209–219,1905.

162

[107] Li, X.. Zhou X.Y and Lim, A.E.B., Dynamic mean–variance portfolio selection

with no-shorting constraints, SIAM Journal on Control and Optimization, 40, pp.

1540-1555, 2001.

[108] Ma, J., Yong, J., Forward-Backward Stochastic Differential Equations and Their

Applications, Lect. Notes Math. 1702. Springer, New York, 1999.

[109] Marshall, A. W. and Olkin, I. , Inequalities: Theory of Majorization and Its

Applications, Academic Press, San Diego, 1979.

[110] Majlender, P., Strategic investment planning by using dynamic decision trees,

Proceedings of the 36th Hawaii Internasional Conference on System Science,

2003.

[111] Manfred Gilli and Evis K¨ellezi., A Global Optimization Heuristic for Portfolio

Choice with VaR and Expected Shortfall, This draft: January 2001.

[112] Markowitz, H. M., Portfolio Selection, Journal of Finance, 7, 77-91, 1952.

[113] Markowitz, H. M., Portfolio Selection, John Wiley & Sons, New York, 1959.

[114] Markowitz ,H. M., Mean–Variance Analysis in Portfolio Choice and Capital

Markets, Blackwell, Oxford, 1987.

[115] Mathieu-Nicot, B., Determination and Interpretation of the Fuzzy Utility of an

Act in an Uncertain Environment”, in J. Kacprzyk and M. Fedrizzi, (eds.).,

Multiperson Decision Making Using Fuzzy Sets and Possibility Theory, Kluwer

Academic Publishers, pp. 90-7, 1990.

[116] Matheiss, T. H. and Rubin, D. S., A survey and comparison of methods for

finding all vertices of concave polyhedral sets, Math. Oper., 5, 167-183, 1980.

[117] Merton, R. C., An analytic derivation of the efficient frontier. J. Finance Quant.

Anal. 7 1851–1872, 1972.

[118] Merton, R. C., Optimum consumption and portfolio rules in a continuous time

model, Journal of Economic Theory, 3, pp. 373-413; Erratum 6 (1973), pp. 213-

214, 1971.

[119] Merton, R. C., Continuous-Time Finance, Blackwell Publishers, pp. 213-214,

1992.

[120] Miller, L.B. and Wagner, H., Chance-constrained programming with joint

constraints, Oper. Res., 13, pp. 930–945, 1965.

163

[121] Mosler K. and Scarsini M. (Eds.), Stochastic Orders and Decision Under Risk,

Institute of Mathematical Statistics, Hayward, California, 1991.

[122] Muliere P. and M. Scarsini, A note on stochastic dominance and inequality

measures, Journal of Economic Theory, 49, pp. 314–323,1989.

[123] Mulvey, J.M. and A. Ruszczynski, A new scenario decomposition method for

large scale stochastic optimization," Operations Research, 43, pp. 477-490, 1995.

[124] Murphy, K. G., Solving the fixed charge problem by ranking the extreme points,

Oper. Res., 16, 268-283, 1968.

[125] Negoita, C.V.: Fuzziness in management, OPSA/TIMS, Miami, 1970.

[126] Neumann von, J. and Morgenstern, O., Theory of Games and Economic

Behavior, Princeton University Press, Princeton, NJ, 1947.

[127] Ogryczak ,W. and Ruszczynski, A., Dual stochastic dominance and related

mean–risk models, SIAM Journal on Optimization, 13, pp. 60–78, 2002.

[128] Ogryczak ,W. and Ruszczynski, A, From stochastic dominance to mean–risk

models: semideviations as risk measures, European Journal of Operational

Research, 116, pp. 33–50, 1999.

[129] Ogryczak ,W. and Ruszczynski, A, On consistency of stochastic dominance and

mean–semideviation models, Mathematical Programming, 89, pp. 217–232, 2001.

[130] Ostermask, R., A fuzzy control model (FCM) for dynamic portfolio management,

Fuzzy sets and Systems, 78, 243-254, 1998.

[131] Pardoux, E., S. Peng. Adapted solution of backward stochastic equation. Systems

Control Lett. 14 55–61, 1990.

[132] Perold, A. F., Large-scale portfolio optimization. Management Sci. 30 1143–

1160, 1984.

[133] Pliska, S.R., A discrete time stochastic decision model, Advances in Filtering and

Optimal Stochastic Control, edited by W.H. Fleming and L.G. Gorostiza, Lecture

Notes in Control and Information Sciences, 42, Springer-Verlag, New York, 290-

304, 1982.

[134] Pliska, S.R., A stochastic calculus model of continuous trading: Optimal

portfolios, Mathematics of Operations Research, 11, pp. 371-384, 1986.

[135] Popescu, C. and Sudradjat, S., An application of minsadbed regression, JAQM,

vol 1 No. 2 Winter 2006.

164

[136] Popescu, C. and Sudradjat, S., Parameter estimation for fuzzy sets, IJPAM,

accepted Novembers 6, 2006.

[137] Popescu, C., Sudradjat, S. and M. Ghica, On least squares approach in a fuzzy

setting, Conferinţă a Societăţii Probabilităţii şi Statistică din România, 13-14

Aprilie 2007.

[138] Prekopa, A., On probabilistic constrained programming, in Proceedings of the

Princeton Symposium on Mathematical Programming, Princeton University

Press, Princeton, NJ, pp. 113–138, 1970.

[139] Prekopa, A., Logarithmic concave measures with application to stochastic

programming," Acta Scientiarium Mathematiarum (Szeged), 32, pp. 301-316,

1971.

[140] Prekopa, A., Stochastic Programming, Kluwer Academic Publishers, Dordrecht,

Nederlands, 1995.

[141] Preda, V., Some optimality conditions for multiobjective programming problems

with set functions, Rev. Roum. Math. Pures Appl. 1993.

[142] Preda, V., Teoria Decizilor Statistice, Editura Academie Româna, 1992.

[143] Preda, V. , Sudradjat,S., On stochastic comparisons dispersive of order statistics

in two-sample problem with left truncated exponential, MATH. REPORTS 8(58),

4, 453-458, 2006.

[144] Preda, V, and Sudradjat, S., Conectings among some different classes of

solutions in multi objective programming with application to stochastic

programming, Proceeding of Romanian Academy, accepted, 2007.

[145] Puri, M.L., Ralescu, D.A., Fuzzy random variables, J. Math. Anal. Appl. 114,

151-158 Sciences, 15, 1-29, 1986.

[146] Quirk, J.P and Saposnik, R., Admissibility and measurable utility functions,

Review of Economic Studies, 29, pp. 140–146, 1962.

[147] Ralph, E., Steuer, Yue Qi and Markus Hirschberger, Multiple Objectives in

Portfolio Selection, June 16, 2005.

[148] Ramaswamy, S., Portfolio selection using fuzzy decision theory, working paper

of bank for international settlements, No. 59, 1998.

[149] Roy, B., Decision-Aid and Decision-Making, in Bana e Costa ed., Readings in

Multiple Criteria Decision Aid, Springer Verlag, 17-35, 1990.

165

[150] Rockafellar, R. T., Convex Analysis, Priceton, NJ: Princeton University Press,

1970.

[151] Rockafellar R.T. and Wets, R.J.-B., Stochastic convex programming: Basic

duality , Pacific J.Math., 62, pp. 173-195, 1976.

[152] Rockafellar, R.T. and Uryasev S., Conditional value-at-risk for general loss

distributions,J. Banking and Finance, 26, pp. 1443–1471, 2002.

[153] Rockafellar, R. T.,Wets, R. J-B., Variational Analysis, Springer-Verlag, Berlin,

1988.

[154] Rommelfanger, H., Fuzzy linear programming and applications, Europan Journal

of Operational Research, 92, 512-527, 1996.

[155] Rothschild, M. and Stiglitz, J. E., Increasing risk: I. A definition, Journal of

Economic Theory, 2, pp. 225–243, 1969.

[156] Ruszczynski, A. and Vanderbei R. J., “Frontiers of stochastically nondominated

portfolio” Operations research and financial engineering, Princeton University,

ORFE -0-01,2002.

[157] Ruszczynski, A. and Vanderbei, R.J., Frontiers of stochastically nondominated

portfolios, Econometrica, to appear in 2003.

[158] Ruszczynski, A., A regularized decomposition method for minimizing a sum of

polyhedral functions, Mathematical Programming 35, 309-333, 1986.

[159] Saad, O. M., On Stability of proper efficient solutions in multiobjective fractional

programming problems under fuzziness, Mathematical and Computer Modelling,

45, 221-231, 2007.

[160] Saad, O. M., On the solution of fuzzy linear fraction programs, in: The 30th

Annual Conference, ISSR, Vol. 30, Part (I V), Cairo University, Egypt, pp. 1-9,

1995.

[161] Saad, O. M. and Abdulkader, M. F., On the solution of bicriterion integer

nonlinear fractional programs with fuzzy parameters in the objective functions,

The Journal of Fuzzy Mathematics 10 (1), 1-7, 2002.

[162] Saad, O. M. and Adb-Rabo, K., On the solution of chance-constrained integer

linear fraction programs, in: The 32nd Annual Conference, ISSR, vol. 32, Part

(VI), Cairo University, Egypt, pp. 134-140, 1997.

166

[163] Saad, O. M, and Sharif, W. H., On the solution of integer linear fractional

programs with uncertain data, Institute of Mathematics & Computer Sciences

Journal 12 (2), 169-173,2001.

[164] Sakawa, M., Fuzzy Sets and Interactive Multiobjective Optimisation, Plenum

Press, London, 1993.

[165] Sakawa, M., Yana, H., Interactive decision making for multi-objective linear

fractional programming problems with fuzy parameters, Cybernetics Systems 16

377-397, 1985.

[166] Salukavadze, M. E. and Topchishvili, A. L., Weakly-efficient solutions limiting

multicriteria optimization problems, JOTA: vol. 77, No. 2, May 1993.

[167] Samuelson, P.A., Lifetime portfolio selection by dynamic stochastic

programming, Review on Economic Statistics, 51, pp. 371-382, 1986.

[168] Seppala, Y., On accurate linear approximations for chance-constrained

programming. Journal of operational Resarch Society, 39(7), 693-694, 1988.

[169] Sen, S. and Higle, J.L., An Introductory Tutorial on Stochastic Linear

Programming Models," Interfaces, 29, pp. 33-61, 1999.

[170] Sen, S. R.D., Doverspike and Cosares S.,, Network Planning with Random

Demand," Telecommunication Systems, 3, pp. 11-30, 1994.

[171] Shaocheng, T.: Interval number and Fuzzy number linear programming, Fuzzy

Sets and Systems 66 (1994) 301-306.

[172] Sharpe, W. F., A linear programming approximation for the general portfolio

analysis problem, Journal of Financial and Quantitative Analysis, 6, pp. 1263–

1275, 1971.

[173] Sharpe, W. F., Capital asset prices: A theory of market equilibrium under

conditions of risk. J. Finance 19 425–442, 1964.

[174] Singh, C. and Hanson, M. A., Multiobjective fractional programming duality

theory, Naval Research Logistics, 38, 925-933, 1991.

[175] Slowinski, R., and Teghem, J., Editors, Stochastic versus Fuzzy Approaches to

Multionjective Mathematical Programming under Uncertainty, Kluwer

Academic Publishers, Dordrecht, Nederlands, 1990.

[176] Sortino, F.A. and van der Meer R., Downside Risk, Journal of Portfolio

Management, 17, pp. 27-31, 1991.

167

[177] Stanley Lee, E. and Li, R.J., Fuzzy multiple objective programming and

compromise programming withP areto optimum, Fuzzy Sets and Systems, 53,

275-288, 1993.

[178] Stancu-Minasian, I. M., Stochastic Programming with Multiple-Objective

Function, D. Reidel Publishing Company, Dordrecht, Nederlands, 1984.

[179] Stancu-Minasian, I.M., The Stochastic programming with multiple fractile

criteria, Revue Roumaine de Mthematiques Pures et Appliquees, Vol 37, pp.

939-941, 1992.

[180] Stancu-Minasian, I.M., Tigan, S., The vetorial minimum risk problem,

Proceedings of the Colloquium on Approximation and Optimization, Cluj-

Napoca, Romania, pp. 321-328, 1984..

[181] Steuer, R. E., Qi, Y. and Hirschberger, M., Suitable-portfolio investors,

nondominated frontier sensitivity, and the effect of multiple objectives on

standard portfolio selection, Annals of Operations Research, 2005.

accepted/forthcoming.

[182] Steuer, R. E., Silverman, J,. and A. W. Whisman. A combined Tcheby-

cheff/aspiration criterion vector interactive multiobjective programming

procedure, Management Science, 39(10):12551260, 1993.

[183] Sudradjat, S., Popescu, C., On stochastic comparisons of order statistics in two-

sample problem for exponential distribution, Analele universităţii Bucureşti,

Matematică-Informatică, anul XLV, 2006.

[184] Sudradjat, S., On possibilistic approach for a portfolio selection, Mathematical

Reports, accepted, 2007.

[185] Sudradjat, S., The weighted possibilistic mean variance and covariance of fuzzy

numbers, accepted, inclusion in JAQM Fall, 2007.

[186] Sudradjat, S., On stochastic ordering for sample range of order statistic in two-

sample problem with left truncated exponential, the International symposium,

high and very high strength concrete, Bucharest, May 23-25, 2007, submitted.

[187] Sudradjat ,S., Popescu, C. and Ghica, M., .A portfolio selection problem with a

possibilistic approach, 22ND European Conference on operational research,

Prague July 2007, accepted.

168

[188] Sudradjat S., and Preda, V., On portfolio optimization using fuzzy decisions,

ICIAM, Elvetia, Zurich, accepted:IC/CT/010/j/3551, December 2006.

[189] Sudradjat, S. and Preda, V., Results relative to some classes of solutions in multi-

objective programming with application to stochastic programming, Submited,

2007.

[190] Suvrajeet Sen, Stochastic Programming: Computational Issues and Challenges,

From Encyclopedia of OR/MS, S. Gass and C. Harris (eds.), SIE Department

University of Arizona, Tucson, AZ 85721,-

[191] Takeda, E. and Nishida, T., Multiple criteria decision problem with fuzzy

dominance structures, Fuzzy Sets and System, 3, 123-136, 1980.

[192] Tanaka, H., Guo, P. and Türksen, I.B., Portfolio selection based on fuzzy

probabilities and possibility distributions, Fuzzy Sets and Systems, Vol. 11, pp.

387-397, 2000.

[193] Tanaka, H., Asai, K.: Fuzzy linear programming problems with fuzzy numbers,

Fuzzy Sets and Systems 13, pp. 1-10, 1984.

[194] Tanaka, H., Okuda, T., Asai, K.: On fuzzy mathematical programming, J.

Cybernetics 3, 37-46, 1984.

[195] Teghem, J., Dufrane, D., Thauvoye, M., and Kunsch, STRANE P., An interactive

method for multi-objective linear programming under uncertainty, Eouropean

Journal of Operational Research, Vol. 26, pp. 65-82, 1986.

[196] Urli, B., Nadeau, R., Stochastic MOLP with incomplete information: An

interactive Approach with recourse, Journal of Operational Research Society,

Vol. 41, pp. 1143-1152, 1990.

[197] Vanderbei R.J., Linear Programming: Foundations and Extensions. Kluwer

Academic Publishers, 2nd edition, 2001.

[198] Văduva Ion, Fiabilitatea Programelor, Editura Universităţii din Bu-cureşti, 2003.

[199] Văduva Ion, Modele de Simulare, Editura Universităţii din Bucureşti, 2004.

[200] Văduva Ion, Simulation of systems reliability, International Confer-ence on

Industrial Engineering and Production Management. Glasgow, July 12-15,

Book 1, p.201-210., Proc. IEPM., 1999.

169

[201] Watada, J., Fuzzy portfolio model for decision making in investment, In Y.

Yoshida editror, Dynamical Aspects in fuzzy decision making, Physica Verlag,

Heidelberg, 141-162, 2001.

[202] Werners, B., An interactive fuzzy programming system, Fuzzy Sets and Systems,

23, 131-147, 1987.

[203] Wets, R.J.-B., Programming under uncertainty: The equivalent convex program,

Siam J.Appl. Math., 14, pp. 89–105, 1966.

[204] Wets, R.J.-B,. Stochastic programs with fixed recourse: The equivalent

deterministic program,SIAM Rev., 16, pp. 309–339, 1974.

[205] Wets, R. J. B. and Zeimba, W. T., Stochastic programming: State of the Art,

1998, Anaks of Operations Research, Vol. 85, pp. 2-11, 1999.

[206] White, D. J., Optimality and Efficiency, Chichester: Willey, 1982.

[207] White, D. J., Weighting factor extentions for finite multiple objective vector

minimization problem, Eur. J. Oper, res., 36, 256-265, 1988.

[208] White, D. J., Finite horizon Markove decision processes with uncertain terminal

payoffs, University Manchester, 1991.

[209] White, D. J., Multiple objective weighting factor auxiliary optimization

problems, J. Multi-Criteria Decision Analysis, 4, 122-132, 1995.

[210] Whitmore, G. A. and Findlay, M.C., eds., Stochastic Dominance: An Approach

to Decision–Making Under Risk , D.C.Heath, Lexington, MA, 1978.

[211] Yitzhaki S., Stochastic dominance, mean variance, and Gini’s mean difference,

American Economic Review, 72, pp. 178–185, 1982.

[212] Young M. R., A minimax portfolio selection rule with linear programming

solution, Management Science, 44, pp. 673–683, 1998.

[213] Yoshimoto, A., The mean-variance approach to portfolio optimization subject to

transaction costs, Journal of the Operational Research Society of Japan, 39, 99-

117, 1996.

[214] Yong, J., X. Y. Zhou. Stochastic Controls: Hamiltonian Systems and HJB

Equations. Springer, New York, 1999.

[215] Yu, P. L., Multiple-Criteria Decision making, New York: Plenum, 1985.

[216] Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and

systems, 1, 3-28, 1978.

170

[217] Zeleny, M., Linear multiobjective programming, Springer Verlag, New York,

NY, 2(1991), pp. 461-471, 1974.

[218] Zhou ,X. Y. and Li, D., Continouos time mean- variance portfolio selection : A

stochastic LQ framework, Applied Mathematics and Optimization, 42, pp. 19-33,

2000.

[219] Zhao Y. and Ziemba, W.T., Mean-variance versus expected utility in dynamic

investment analysis, Working Paper, Faculty of Commerce and Business

Administration, University of British Columbia, 2000.

[220] Zhou, X.Y., Markowitz’s world in continuous time, and beyond, Stochastic

Modeling and Optimization, edited by D.D. Yao et al., Springer, New York, pp.

279-310, 2003.

[221] Zimmermann, H. J.: Fuzzy mathematical programming, Comput. & Ops. Res.

Vol. 10 No 4, 291-298, 1983.

[222] Zimmermann, H. J., Decision making in Ill-structured environments and with

multiple criteria, in Bana e Costa ed., Readings in Multiple Criteria Decision Aid,

Springer Verlag, 119-151, 1990.

171

NOTATIONS

0\+R : positive real number sets +R : the set of nonnegative real numbers −R the set of nonpositive real numbers m+R the set of m-dimension real vector with

nonnegative components m−R the set of m-dimension real vector with

nonpositive components ))(,,,(1 ⋅θμλqAP : auxilitiy optimization problem 1

11 ),,,( EqM ⊆θμλ : set of optimal solutions nD R⊆ : a compact set of feasible actions

FSDf : First-order stochastic dominance

SSDf : second-order stochastic dominance (Ω,F, P) : probability space Z : A decision vector in an appropriate space Z

][XE : mean X

n,1 : 1,2,…,n

nRR ,...,1 : random returns of assets n,1 )(xμ : mean return

[ ])()( xRarx V=ρ : the variance of the return

))()((max xxXx

λρμ −∈

: the mean–risk portfolio optimization problem

),( kt

kt uD θ : a function of the slope vector

+− )( vη : )0,max( v−η

ija : technology coefficient

B : right-hand side vector of the constraint

ija~ : fuzzy number

)(XGμ : the fuzzy set of optimal values, G

iC : the fuzzy set of the ith constrain

),,( pXg λ : augmented Lagrangian

),,,(~4321 rrrrr = : Trapezoidal

172

minkp : the minimum expected return for the kth

market scenario maxkp : the maximum expected return for the kth

market scenario il : the minimum weight of the ith asset in the

portfolio il : the maximum weight of the ith asset in the

portfolio; r~σ : the standard deviation of r~

)~(rVarw : weighted possibilistic variance of r~ u : υuu ,...,1

)ˆ,...,ˆ(ˆ 1)( υυ uuu = : )ˆ,...,ˆ( 1 υuu ),,,(~

4321 rrrrr = : E

F : the family of fuzzy numbers E : set of efficient solutions

wE : sets of weakly efficient solutions GE : Set Geoffrion/proper efficient solutions

PKE : the set of efficient solutions

173

ACRONYMS & ABBREVIATIONS

ALM : assets liability management a.s : almost surely BFP bicriterion fractional programming BINOLFP bicriterion integer nonlinear fraction programs BSDE backward stochastic differential equation Covr : covariance dom : domain F : he family of fuzzy numbers FLP : fuzzy linear programming FMODM : fuzzy multiple objective decision model FMOFP Fuzzy multiobjective fractional programming IFLP : interactive fuzzy linear programming IFMODM : interactive fuzzy multiple objective decision

model LQ linear quadratic MOFP multiobjective fractional programming MODM : multiple objective decision model ODE ordinary differential equation Pos : possibilistic; Pr : probability resp : respectively SRE stochastic riccati equation SSK1 the stability set of the first kind VaR : value at risk Var : variance

174

INDEX

Arnott,131 Wagner, 131

auxiliary, 27 optimization, 27 algorithm, 32, 34, 78, 95, 137, 141 asset return, 147 Bellman-Zaded, 3 BINOLFP, 151 concave, 28,32,36

convex, 36 Nondecreasing 40 continuous, 41, 54, 60, 94 functional, 81 function, 38

concavity, 28,32 convex, 28

function, 17,,18,19, 25,34 convexity, 32, 36 concavity, 26 cone 70 nonconvex, 87, 93 programming, 92 combination, 83 polyhedral 73 programming, 36, 93 set 17, 32,57

continuous, 116 fuzzy, 116 mapping, 116

classes, 9, 21

nonconvex, 9,28 auxiliary, 32 multiobjective, 9 solution, 21 covariance, 97, 98, 142 continuous,41, 44 cumulative, 42 functional, 41, 54 variable, 10 structur, 146 matrix, 143 defuzzification, 84, 95 Dentcheva, 3, 4, 41, 42, 66

Ruszcynski, 3, 4, 41, 42,54 66,79

dual, 45 Risk, 45 functions, 93, 94 problems, 60, 92, 94 ,110

solution, 80 duality, 56, relation, 60 function, 60 gap, 93 distributed, 146 return, 146 efficient 21

portfolio, 131,135 solution, 10, 21 sets, 10, 20 SSD/FSD, 26

175

stochastic,27 expected value, 4, 10, 11

efficient, 10 standard deviation, 10, 11 minimum, 13

fuzzy fuzzy numbers, 60, 88, 123, 140, , 134, 86, 88, 91 number trapezoidal,135 approach, 5, 68, 81 fuzzy decision, 2, 5,103 constraints, 70 efficient, 76 compromise, 76 bicriterion, 123 emviroment, 3,77 decision, 5, 91 decesive, 98 function, 82 geometry, 77 goal, 111 linear, 5, 69, 70 mean-operator, 135 parameter, 98, 115, 122, 123 multiobjective, 110, 122 fractional, 116 objective, 72, 75 Resources, 70, 71 system, 77 utility, 2, 114 technological coefficient, 88 right-hand, 88 convex, 138 Fuzzy mean operator, 135 multicriteria, 3 random, 5

nonfuzzy,117, 122 Geoffrion, 13, 119 Gaussian, 119, distribution, 146 mixture distribution, 147 investment, 7, 112, 113, 128 investor, 80, 115 interactive approach, 77 interactive fuzzy, 76, 110, linear, 78 multiobjective, 110 Karush-Kuhn-Tucker, 58 multipliers, 58 linear programming, 76 linear fractional, 115 nonlinear fractional, 115 nonlinear programming, 115 nonconvex, 28 auxiliary 32, 108 programming, 92 markowitz, 4, 60, 83

model, 4 mathieu-Nicot, 2 mean-variance, 142 portfolio, 142 return, 55, 113, 128

assets, 44 objective, 112 total return, 3, 44 security, 130, 136 rate, 131 target, 114, 128 scurity, 136

pareto-optimal, 9 portfolio retun, 112, 113 possibilistic, 7, 8 , 131

constraint, 8 distribution, 131 theory, 132

176

model, 135 mean VaR, 131 mean variance, 131, 142 mean covariance, 143

proper efficient, 21 random return, 55 splitting, 76 right-hand-side, 88 shortfall, 130

probability, 82, 130 stochastic contol, 8

dominance, 40, 48 multi-objective, 1, 11, 30, problem, 21, 40 programming, 7 optimization, 53 return, 128 variable, 115

sakawa-Yana method, 116 SSD/FSD efficient, 26, slater-optimal, 20 subgradient, 93, 94, 1044 method, 93

transaction, 128, cost, 129

security, 131 triangular, 133, fuzzy, 136 trapezoidal, 133, 143 fuzzy, 134 ,140 value

at Risk, 11, 129, portfolio, 140, 141 random, 149

variance, 128 covariance, 128,142 matrix, 128

semivariance, 128 of return, 45

weakly, 19, 37 efficient, 19,21 dominant, 22 weighted, 17

problem, 17 possibilistic, 142, 138

variance, 142 covariance, 143

177