ZHANG JIANG
-
Upload
alexandr-arroyo -
Category
Documents
-
view
229 -
download
0
Transcript of ZHANG JIANG
-
8/8/2019 ZHANG JIANG
1/110
THREE ESSAYS ON INVENTORY MANAGEMENT
By
JIANG ZHANG
Submitted in partial fulfillment of the requirements
for the Degree of Doctor of Philosophy
Thesis Advisor: Dr. Matthew J. Sobel
Department of Operations
CASE WESTERN RESERVE UNIVERSITY
August 2004
-
8/8/2019 ZHANG JIANG
2/110
CASE WESTERN RESERVE UNIVERSITY
SCHOOL OF GRADUATE STUDIES
We hereby approve the dissertation of
______________________________________________________
candidate for the Ph.D. degree *.
(signed)_______________________________________________
(chair of the committee)
________________________________________________
________________________________________________
________________________________________________
________________________________________________
________________________________________________
(date) _______________________
*We also certify that written approval has been obtained for any
proprietary material contained therein.
JIANG ZHANG
Lisa M. Maillart
Yunzeng Wang
Peter H. Ritchken
06/16/2004
Matthew J. Sobel
-
8/8/2019 ZHANG JIANG
3/110
-
8/8/2019 ZHANG JIANG
4/110
To my mother: Huishu Jiang
my father: Shuqing Zhang
my wife: Dr. Yan Cao
-
8/8/2019 ZHANG JIANG
5/110
Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1 Inventory Replenishment with a Financial Criterion . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Dynamic Programming Analysis . . . . . . . . . . . . . . . . . . . 11
1.4 Optimality of (sn, Sn) Replenishment Policies . . . . . . . . . . . 14
1.5 Infinite Horizon Convergence . . . . . . . . . . . . . . . . . . . . . 18
1.6 Models with Smoothing Costs . . . . . . . . . . . . . . . . . . . . 19
1.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 23
v
-
8/8/2019 ZHANG JIANG
6/110
2 Fill Rate of General Review Supply Systems. . . . . . . . . . . . . . . . . . . 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 General Periodic Review System . . . . . . . . . . . . . . . . . . . 29
2.3 Uncapacitated Single-stage Systems . . . . . . . . . . . . . . . . . 32
2.4 Gamma and Normal Demand in Single-stage Systems . . . . . . . 35
2.4.1 Gamma Demand Distribution . . . . . . . . . . . . . . . . 35
2.4.2 Normal Demand Distribution . . . . . . . . . . . . . . . . 36
2.4.3 Fill Rate Approximation for Normal Demand Distribution 39
2.5 Multi-Stage General Review Systems . . . . . . . . . . . . . . . . 40
2.5.1 Fill Rate in Two-Stage Systems . . . . . . . . . . . . . . . 42
2.5.2 Fill Rate in Two-Stage Systems with General Leadtime . . 48
2.5.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . 51
2.6 Fill Rate in a Three-Stage System . . . . . . . . . . . . . . . . . . 51
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Interchangeability of Fill Rate Constraints and Backorder Costs
in Inventory Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2 Model and Problem Formulations . . . . . . . . . . . . . . . . . . 66
3.3 Continuous Demand . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4 Discrete Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Interchangeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
vi
-
8/8/2019 ZHANG JIANG
7/110
3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.6.1 Strictly Positive Demand Density . . . . . . . . . . . . . . 83
3.6.2 Non Strictly Positive Demand Density . . . . . . . . . . . 84
3.6.3 Discrete Demand . . . . . . . . . . . . . . . . . . . . . . . 86
3.7 Generalizations and Summary . . . . . . . . . . . . . . . . . . . . 89
3.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
vii
-
8/8/2019 ZHANG JIANG
8/110
List of Tables
2.1 Fill Rate and its Approximation for Normal Demand . . . . . . . . . 57
2.2 Fill Rate of Two-stage Systems for Normal Demand (a) . . . . . . . 58
2.3 Fill Rate of Two-stage Systems for Normal Demand (b) . . . . . . . 59
3.4 Distribution Function and Expected Number of Backorders . . . . . . 87
3.5 Values ofG1() and B1() . . . . . . . . . . . . . . . . . . . . . . 88
3.6 S-optimal Base-Stock Levels and Fill Rates at which they are F-optimal 88
3.7 F-optimal Base-Stock Levels and Unit Stockout Costs at which they are
S-optimal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
viii
-
8/8/2019 ZHANG JIANG
9/110
List of Figures
2.1 The standard N-stage serial inventory system . . . . . . . . . . . 30
2.2 The Fill Rate Integral for a system with Normal Demand . . . . 38
3.3 Dependence of S-Optimal Base-Stock Level on Stockout Cost: Non-
negative Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4 Locus of{(b, f)} with the Same Optimal Base-Stock Level: Non-
negative Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
ix
-
8/8/2019 ZHANG JIANG
10/110
Acknowledgements
I would like to express my sincere gratitude to my mentor, Professor Matthew
J. Sobel, who encouraged and guided me through various phases of my doctoral
studies with patience. I would also like to thank him for his incredible effort and
willingness to help me at any time and any where.
I would specially like to thank my dissertation committee members, Profes-
sors Lisa Maillart, Peter Ritchken, and Yunzeng Wang for their generous insight,
comments, and support on this work. In addition, my thanks are also owed to Pro-
fessors Apostolos Burnetas, Hamilton Emmons, Kamlesh Mathur, Daniel Solow,
and George Vairaktarakis for their help throughout my doctoral studies.
I would like to express my appreciation to the Department of Operations, Case
Western Reserve University, for their generous financial support. Special thanks
to departments staff, Elaine Iannicelli, Sue Rischar, and Emily Anderson for their
help throughout my study in the department.
I had a fabulous time at Case which would not have been possible without the
company of friends like Junze Lin, Zhiqiang Sun, Yuanjie He, Huichen Chiang,
x
-
8/8/2019 ZHANG JIANG
11/110
Wei Wei, Xiang Fang, Qiaohai Hu, Will Millhiser, Ant Printezis, Halim Hans, and
Kang-hua Li who have always helped and cheered me up in every possible way.
Finally, I would like to thank my family for their unconditional love, support
and encouragement. My special thanks are for my wife Yan who is always there
for me through everything.
xi
-
8/8/2019 ZHANG JIANG
12/110
Three Essays on Inventory Management
Abstract
by
JIANG ZHANG
This dissertation consists of three essays that are related to inventory man-
agement.
The first essay models a single-product equity-owned firm which orders prod-
ucts from an outside supplier, borrows short-term capital for solvency, and issues
dividends to its shareholders while facing financial risks and demand uncertainty.
The firm maximizes the expected present value of the time stream of dividends. If
there is a setup cost in this model, we show that an (s, S) replenishment policy is
optimal by jointly optimizing the firms operational and financial decisions. The
analysis is not a straightforward copy of Scarfs argument. The second part of
this essay studies the same model with a smoothing cost (instead of a setup cost)
and shows that the optimal policy has the same form as the traditional smoothing
xii
-
8/8/2019 ZHANG JIANG
13/110
cost model. Although operational decisions and financial decisions interact with
each other in these models, the optimal inventory policies have standard forms.
The second essay obtains fill rate formulas for general review inventory models
with base-stock-level policies. Ordering decisions in a general review model are
made every R (R 1) periods but demand arises every period. We provide
exact fill rate formulas for single-stage model with a general demand distribution.
A simple fill rate expression is derived for the model with normally distributed
demand. For multi-stage models, we first discuss a general review procedure at
each stage and then provide exact fill rate formulas for two-stage and three-stage
models.
There are parallel streams of literature which analyze identical models except
that one stream has backorder costs and the other has fill rate constraints. The
third essay clarifies redundancy in the two streams of dynamic inventory models
with linear purchase costs. We show that optimal policies for either kind of
model can be inferred from the other. That is, inventory fill rate constraints
and backorder costs are interchangeable in dynamic newsvendor models.
xiii
-
8/8/2019 ZHANG JIANG
14/110
Chapter 1
Inventory Replenishment with a
Financial Criterion
1.1 Introduction
Nearly all the literature on optimal inventory management uses criteria of cost
minimization or profit maximization. An inventory managers goal for example,
is modeled as minimizing cost or maximizing profit while satisfying customers
demands. If inventory decisions do not affect the revenue stream, these two crite-
ria result in the same optimal replenishment policy. Most of this literature treats
firms inventory decisions and financial decisions separately. This dichotomy is
perhaps due to the perception that inventory managers in a large firm cannot in-
fluence the firms financial policy and that financial officers are usually detached
1
-
8/8/2019 ZHANG JIANG
15/110
2
from the inventory decisions. This separate consideration of financial and opera-
tional decisions simplifies management and has its foundation in corporate finance.
The pathbreaking papers, Modigliani and Miller (1958) and Modigliani and Miller
(1963) (hereafter referred to as M-M), show that the firms capital structure and
its financial decisions should be independent of the firms investment and opera-
tional decisions if capital markets are perfect.
However, when market imperfections such as taxes and transaction fees are in-
troduced, the results characterized from these separate treatments may no longer
hold. Treating real and financial decisions of the firm as independent may not
be justified.(Dammon and Senbet 1988). Other sources of market imperfections
include asymmetric information between supplier and retailer, asymmetric infor-
mation between shareholders and managers, and differential access to financial
resources by different firms. For example, many small and medium-sized firms
are cash constrained, and their operational decisions are heavily dependent on
their financial decisions (such as short-term borrowing). Although the assumed
independence of operations and finance has led to the development of intuitively
appealing and insightful results, there remains the question of whether joint opti-
mization of both the operational and financial decisions of a firm will generate new
insights regarding firm behavior and perhaps overturn or modify existing results.
The M-M theorem not only allows separation of operations and finance, but
also establishes that the firms optimal decisions are the same no matter if it
-
8/8/2019 ZHANG JIANG
16/110
3
optimizes the value of the firm, dividends, or retained earnings. However, when
capital markets are imperfect, the equivalence of different objectives is no longer
valid. The literature on agency theory finds that corporate managers, the agents
of shareholders, have conflicting interests with the shareholders. Those conflicts
are primarily caused by dividends paid to shareholders. Payouts to shareholders
reduce the resources under managers control, thereby reducing managers control,
and making it more likely they will incur the monitoring of the capital markets
which occurs when firm must obtain new capital(Jensen 1986, p. 1). So managers
have to seek external funds to finance their projects. Since the external funds are
usually unavailable for certain firms or available only at high prices, it somewhat
reduces the profitability (or increases the operating costs) of the projects and may
affect the performance evaluation of the managers. This conflict may discourage
managers from disgorging the cash to shareholders and cause organizational inef-
ficiencies.
There is a literature on finance that recognizes the interdependence issues
among a firms decisions. Most of this literature focuses on the effects of mar-
ket imperfections on financial structures and decisions. Miller and Rock (1985)
extend the standard finance model of the firms dividend/investment/financing
decisions by allowing asymmetric information between the firms managers and
outside investors and they show that there exists an equilibrium investment pol-
icy which leads to lower levels of investment than the optimum achievable under
-
8/8/2019 ZHANG JIANG
17/110
4
full information. Myers (1974) shows how investment decisions, i.e., acceptance
or rejection of projects, affect the optimal financial structure of a firm and why
investment in turn, should be affected by leverage. Long and Racette (1974)
shows that the cost of capital of a competitive firm facing stochastic demand is
affected by the level of production. Hite (1977) examines the impact of leverage
on the optimal stock of capital by the firm and its capital-to-labor ratio. Datan
and Ravid (1958) analyze the interaction between the optimal level investment
and debt financing. In their model, a firm faces an uncertain price and has to
decide on its optimal level investment and debt simultaneously. They show that
a negative relationship exists between investment and debt. Dammon and Senbet
(1988) extend capital structure model in DeAngelo and Masulis (1980) and study
the effect of corporate and personal taxes on the firms optimal investment and
financing decisions under uncertainty. However, this literature does not address
how a firm operates (quantitatively) by considering the interrelationships.
This paper considers interactions between operational and financial decisions
and uses a dividend criterion. The primary purpose of this paper is to study how
operational decisions are affected by jointly considering financial and operational
decisions and using a non-traditional operations objective. We consider a sin-
gle product equity-owned retail firm which periodically reviews its inventory and
retained earnings. Every period the firm faces random demand and replenishes
-
8/8/2019 ZHANG JIANG
18/110
5
stocks to satisfy customer demand. In addition, every R (a positive integer) peri-
ods, the firm issues a dividend to its shareholders. The firm seeks to maximize the
expected present value of the time stream of dividends (also called shareholders
wealth).
Shareholders wealth maximization is a widely accepted objective of the firm in
the literature (cf. Hojgaard and Taksar 2000, Milne and Robertson 1996, Moyer,
McGuigan, and Kretlow 1990, Sethi 1996, and Taksar and Zhou 1998). This goal
states that management should seek to maximize the present value of the expected
future returns to the shareholders of the firm. These returns can take the form of
periodic dividend payments or proceeds from the sale of stock.
If the objective of the firm is to minimize the expected present value of or-
dering, holding, and shortage costs and the capital market is perfect, the optimal
replenishment policies have been characterized for a broad range of conditions. See
Porteus (1990), Graves et al. (1993), and Zipkin (2000) for details and references
to the literature.
Some recent research addresses the coordination of financial and operational
decisions. Li, Shubik, and Sobel (2003) examine the relationship between de-
cisions on production, dividends, and short-term loans in dynamic newsvendor
inventory models. They show that there are myopic optimal base-stock policies
associated with production decisions and dividend decisions. The present paper
proceeds directly from Li, Shubik, and Sobel (2003) and augments their model
-
8/8/2019 ZHANG JIANG
19/110
6
with a setup cost and smoothing costs. Buzacott and Zhang (1998) look at the
interface of finance and production for small firms with limited borrowing. They
maximize profit over a finite horizon using a mathematical programming model
to optimize inventory and borrowing decisions, and they assume that the demand
for the product is known. Archibald, Thomas, Betts, and Johnston (2002) as-
sert that start-up firms are more concerned with the probability of survival than
with profitability. They present a sequential decision model of a firm which faces
an uncertain bounded demand and whose inventory replenishment decisions are
constrained by working capital.
Buzacott and Zhang (2003) incorporate financial capacity into production de-
cisions using an asset-based constraint on the available working capital in a single-
period newsvendor model. They model the available cash as a function of assets
and liabilities that will be updated according to the dynamics of the production
activities. They analyze a leader-follower game between the bank and the retailer,
and illustrate the importance of jointly considering production and financial deci-
sions. Babich and Sobel (2002) consider capacity expansion and financial decisions
to maximize the expected present value of a firms IPO. They treat the IPO event
as a stopping time in an infinite-horizon Markov decision process, characterize an
optimal capacity-expansion policy, and provide sufficient conditions for a mono-
tone threshold rule to yield an optimal IPO decision.
-
8/8/2019 ZHANG JIANG
20/110
7
The rest of the paper is organized as follows. Section 1.2 formulates the finan-
cial inventory model and 1.3 analyzes the corresponding dynamic program. The
structure of the finite-horizon optimal replenishment policy is explored in 1.4
and extended to the infinite horizon optimal policy is discussed in 1.5. Section
1.6 studies the financial inventory model with smoothing costs and characterizes
optimal replenishment policies. Section 1.7 concludes the paper.
1.2 Model Formulation
We consider an equity-owned retail firm that sells a single product to meet uncer-
tain periodic demand and orders the product from a supplier with an ample supply.
The firm can make short-term loans, if necessary, to obtain working capital. As
discussed in the introduction, every R periods, dividends are issued to the share-
holders and the objective of the firm is to maximize the expected present value
of the time stream of dividends. Negative dividends are interpreted as capital
subscriptions, a common phenomenon for young firms. The following chronology
occurs in each period. The firm observes the level of retained earnings, wn, and
the current physical inventory level, xn. A default penalty (or bankruptcy) p(wn)
is assessed if wn < 0, but it is convenient to define p() as a function on . Weassume that p() is convex nonincreasing on . Then the firm chooses the level
of its short-term loan, bn, and the order quantity, zn. The restriction R = 1 sim-
plifies the presentation and Section 6 substantiates that R = 1 is without loss of
-
8/8/2019 ZHANG JIANG
21/110
8
generality.
At the beginning of each period, the firm also decides on the amount of divi-
dend to declare, vn. if vn 0 then it is a dividend issued to the shareholders; if
vn < 0 then it is a capital subscription. Also at the beginning of the period, the
loan interest (bn) (where we assume () is a convex increasing function on +) is
paid, and the ordering decision is implemented at a cost of K (zn) + czn, where K
is an ordering setup cost, (zn) = 1 ifzn > 0, and (zn) = 0 otherwise. Then de-
mand Dn in period n is realized, and sales revenue net of inventory cost, denoted
g(yn, Dn), is received. For specifity, let g(y, d) = r min{y, d}h(yd)+(dy)+
where yn, r, h and denote available goods in period n after delivery, unit sale
price, holding cost and shortage penalty cost, respectively. However, we only use
convexity of g(, d) for each d 0. Finally, the loan principal bn is repaid. We as-
sume that the demands D1, D2, are independent nonnegative random variablesand unmet demands are backlogged.
For convenience but without loss of generality, we assume that the order lead-
time is 0. Therefore, the amount of goods that is available to satisfy demand in
period n is
yn = xn + zn (1.1)
Let In be the internally generated working capital in period n:
In = wn p(wn) vn czn (bn) K(zn) (1.2)
-
8/8/2019 ZHANG JIANG
22/110
9
That is, In is working capital after the dividend is issued, loan interest and replen-
ishment costs are paid, and before the loan is made and revenue and inventory
costs are realized. The total working capital available at the beginning of period n
is bn + wn and the residual cash left in the firm before sales is bn + In. We assume
that there is an interest rate associated with In, that is, if working capital is
positive, the firm will gain In if In 0 or pay a penalty In if In < 0.
Since excess demand is backlogged, the dynamics are as follows:
xn+1 = xn + zn Dn
wn+1 = (1 + )[wn p(wn) vn czn (bn) K(zn)] + g(yn, Dn)
The first equation balances the flow of physical goods and the second equation
balances the cash flow. Using (1.1) and (1.2), the balance equations become
xn+1 = yn Dn (1.3)
wn+1 = (1 + )In + g(yn, Dn) (1.4)
We assume that the loan and replenishment quantities are nonnegative:
bn 0 and zn 0 (1.5)
The following liquidity constraint prevents the expenditures in period n from
exceeding the sum of retained earnings and the loan proceeds:
wn + bn (bn) p(wn) + vn + czn + K (zn) (1.6)
-
8/8/2019 ZHANG JIANG
23/110
10
Given xn and wn, from (1.1) and (1.2) the decision variables in period n can be
specified as yn, In and bn instead of zn, vn and bn.
Let B denote the present value of the time stream of dividends and let
denote the single period discount factor :
B =
n=1
n1vn (1.7)
Remark Discount factor can be regarded as the risk neutral discount rate for
the shareholders and need not to be a constant every period. Letting (xn, wn)
be an endogenous random variable which depends on the levels of inventory and
retained earnings will not alter the results.
For n = 1, 2, , let Hn denote the history up to the beginning of period n,
namely,
Hn = (x1, w1, b1, I1, y1, D1, , xn1, wn1, bn1, In1, yn1, Dn1, xn, wn)
Let n be the set of all possible Hn sequences. A policyis a nonanticipative rule for
choosing y1, I1, b1, y2, I2, b2, . . . . That is, a policy is a rule that, for each n, chooses
yn, In, and bn as a function of Hn. An optimal policy maximizes E(B|Hn = )
for each
n, for all n = 1, 2, . . .. Since a policy specifies the three decisions
each period (the amount of supply level, yn, the amount of internally generated
working capital, In, and the short-term loan, bn), the firm can easily determine
the order size and the amount of dividends to issue to shareholders by using (1.1)
-
8/8/2019 ZHANG JIANG
24/110
11
and (1.2). The theme of 3 and 4 is the characterization of an optimal policy,
i.e., one that maximizes the expected value of (1.7) subject to (1.3)-(1.6).
1.3 Dynamic Programming Analysis
This section gives the dynamic programming equations which correspond to the
dynamics in 2. We then give a proposition that reduces the dimensionality of
the decision space from three to two. From (1.2) and (1.3), vn = wn
p(wn)
In cyn + cxn (bn) K (yn xn) and xn = yn1 Dn1 (n > 1), and using
standard procedures (Veinott and Wagner 1965), substituting vn and xn in (1.7),
then inserting (1.4) and rearranging terms yields
B = cx1
n=1
ncDn
+
n=1
n1wn p(wn) In (1 )cyn (bn) K(yn xn)
= cx1 + w1 p(w1)
n=1
ncDn
n=1
n1K(yn xn)
+
n=1
n1(1 )cyn In +
(1 + )In + g(yn, Dn)
p(1 + )In + g(yn, Dn)
(bn)
For (b , I , y) 3 let
L(b , I , y) = (1 )(I+ cy) + I + E
g(y, D) p(1 + )I+ g(y, D)
(b)
(1.8)
-
8/8/2019 ZHANG JIANG
25/110
12
Then the expected present value of the dividends can be stated as
E(B) = cx1+w1p(w1)E
n=1
n
cDn
+E
n=1
n1L(bn, In, yn)K(ynxn)
(1.9)
As in Veinott and Wagner (1965) and in many other references since then, we
interpret L(, , ) as a generalized inventory reward function (revenue minus costs
and penalties). Because the first four terms in (1.9) do not depend on the decision
variables, a policy maximizes (1.9) if and only if it maximizes the last term of (1.9).
So we utilize (1.8) and (1.9) and pursue the following objective:
sup EH1
n=1
n1[L(bn, In, yn) K(yn xn)]
subject to yn xn, bn + In 0, and bn 0. (1.10)
where the supremum is over the set of all policies, H1 = (x1, w1) is the initial
state, and the constraints in (1.10) follow from (1.1), (1.2), (1.5), and (1.6).
It is convenient to analyze the following finite horizon counterpart of (1.10)
and then let N :
sup EH1 N
n=1
n1[L(bn, In, yn) K (yn xn)]
subject to yn xn, bn + In 0, and bn 0 (1.11)
A dynamic recursion that corresponds to (1.11) is 0() 0 and for each x
and n = 1, 2, . . .,
n(x) = sup{Jn(b , I , y) K(y x) : y x, I+ b 0, b 0} (1.12)
-
8/8/2019 ZHANG JIANG
26/110
13
Jn(b , I , y) = L(b , I , y) + E[n1(y D)]. (1.13)
This dynamic program has one state variable, x, and three decision variables (b,
I, y), but the original problem has two state variables, x and w. This reduction
occurs because equation (1.4) allows embedding state variable w into the decision
variable I, which is also a surrogate decision variable for v.
Let bn(x), In(x) and yn(x) be optimal values of b, I and y in (1.12). The
following proposition states that borrowing should not exceed the amount needed
to cover current expenses. That is, bn(x) = [In(x)]+. The proof is similar to
that of Proposition 3.2 in (Li, Shubik, and Sobel 2003).
Proposition 1.3.1 For all n = 1, 2, and x , if the supremum in (1.12) is
achieved, then b = (I)+ without loss of optimality in (1.12).
Proof From (1.8), (1.12) and (1.13),
n(x) = supb,I,y
(1 )(I+ cy) + I + E
g(y, D) p[(1 + )I+ g(y, D)]
(b) + E[n1(y D)] K(y x) : y x, I+ b 0, b 0
= supy
c(1 )y + E[g(y, D)] + E[n1(y D)] K(y x)
+supI
(1 )I+ I E
p[(1 + )I + g(y, D)]
+sup
b
(b) : I + b 0, b 0
: I
: y x
The last supremum is achieved by b = 0 ifI 0 and by b = I if I < 0 because
-
8/8/2019 ZHANG JIANG
27/110
14
() is monotone increasing. Therefore,
n(x) = supy
c(1
)y + Eg(y, D) + En1(y
D) K (y
x)
+ supI
(1 )I+ I E
p((1 + )I+ g(y, D))
[(I)+] : I
: y x
.
The next section uses Proposition 1.3.1, (1.12), and (1.13) to analyze the
dynamic problem and establish conditions that guarantee the optimality of (s, S)
policies.
1.4 Optimality of (sn, Sn) Replenishment Poli-
cies
This section establishes conditions under which the optimal ordering policy turns
out to be an (s, S)-type policy for the dividend criterion inventory model. An (s,
S) policy brings the level of inventory after ordering up to S if the initial inventory
level x is below s (where s S), and orders nothing otherwise. For a finite horizon
dynamic inventory problem in which the ordering cost is linear plus a fixed setup
cost and the other one-period costs are convex, Scarf (1959) and Zabel (1962)
show that the optimal ordering policy is (sn, Sn). Iglehart (1963) shows that
the limiting (s, S) policy characterizes the optimal policy for the infinite horizon
problem. Scarfs proof uses the important concept of K-convexity. In this paper,
we use K-concavity.
-
8/8/2019 ZHANG JIANG
28/110
15
Definition A real-valued function f() on is K concave (K 0) if for all
x , 0, and > 0,
f(x + ) f(x) K+
[f(x) f(x )] (1.14)
Properties of K-concave functions are analogous to those of K-convex func-
tions.
Lemma 1.4.1 (a) f() is 0-concave f() is concave on ;
(b) fi() is K-concave, i=1,2,. . . 1f1 + 2f2 is 1K1 + 2K2 concave (1 >
0, 2 > 0);
(c) f() is K-concave f() is V-concave for all V K;
(d) f() is K-concave f() is continuous on .
Proof f() is K-concave if and only if f() is K convex. So (a) through (d)
follow from properties of K-convex functions (Scarf 1959).
For dynamic program (1.12) and (1.13), we use the following results to show
that there is an optimal (s, S) policy.
Let and Q() be the mean and distribution function of D. Let G(y) =
E[g(y, D)].
Lemma 1.4.2 (a) p[(1 + )I + g(y, d)] is convex with respect to (I, y) 2 (for
each d 0);
(b) L(, , ) is a concave function on its domain3.
-
8/8/2019 ZHANG JIANG
29/110
16
Proof (a) Since g(, d) is a convex on for each d 0, (1+ )I+ g(y, d) is jointly
convex in (I, d). The conclusion in (a) follows from monotonicity and convexity
of p(). p[(1 + )I + g(y, d)] is convex on I , so is E{p[(1 + )I+ g(y, D)]}.
(b) Concavity of L(, , ) follows from definition (1.8) and (a).
Theorem 1.4.3 If L(b , I , y) as |y| for all b 0, and b + I 0,
then there is an optimal (s, S) policy.
Note that the hypothesis uses concavity of L(
,
,
) and ensures the existence of
maxima of L(b,I, ). The proof of this theorem is not a paraphrasing of Scarf
(1959). Indeed, it exploits Proposition 1.3.1 to reduce the decision space from 3
to 2, and establishes the K-concave properties of embedded functions.
Proof The concavity of 0() 0 initiates an inductive proof of K-concavity for
each n.
For any n 1, if n1() is K-concave, then
Jn(b ,I,y) = L(b , I , y) + E[n1(y D)]
is K-concave ( 0 < < 1 ) in y because L(, , ) is concave and E[n1(y D)]
is K-concave due to Lemma 1.4.1 (b).
To prove that n() is K-concave, we show that Tn() is K-concave whereTn(y) = sup{Jn(b , I , y) : b 0, b+I 0}; Proposition 1.3.1 asserts that b = (I)+
is optimal. So
Tn(y) = sup{L(b ,I,y) + E[n1(y D)] : b 0, b + I 0}
-
8/8/2019 ZHANG JIANG
30/110
17
= sup{L(0, I , y) + E[n1(y D)] (b) : b 0, b + I 0}
= E[n1(y
D)] + sup
{L(0, I , y)
[(
I)+] : I
}.
Let (y) = supI{L(0, I , y) [(I)+]}. Since () : C and C = {(I, y) :
I , y x} is a convex set, a slight modification of Heyman and Sobel (1984)
(Proposition B-4) shows that () is concave on by proving that supI{L(0, I , y)
[(I)+]} is concave in y.
So Tn(
) is K-concave because E[n1(y
D)] is K-concave by applying
Lemma 1.4.1 (b).
Therefore, n() is K-concave; hence it is K-concave.
The proof that an (s, S) policy is optimal for period n follows the next lemma
which is proved in lemma 7-3 (Heyman and Sobel 1984) (p.314).
Lemma 1.4.4 Suppose T() is K-concave, attains its global maximum at S, and
there is a smallest number s S such that
T(s) V + T(S) (1.15)
where V K. Then () is V-concave where
(x) = sup
{T(y)
V (y
x) : y
x
}x
(1.16)
Since Tn(y) as |y| and Tn() is continuous due to K-concavity,
the global maximum of Tn() is attained, say at Sn. Also, let sn be the smallest
number x such that x Sn and Tn(x) K + Tn(Sn); so sn is well defined.
-
8/8/2019 ZHANG JIANG
31/110
18
Finally, K > K implies K-concavity of
n(x) = sup{Tn(y) K(y x) : y x} x (1.17)
Therefore, a policy that utilizes (sn, Sn) policy for each n = 1, 2, , N is
optimal.
1.5 Infinite Horizon Convergence
The analysis in 1.3 replaces the infinite planning horizon in (1.10) with a finite
planning horizon (n < ) in (1.11). This section shows that the earlier conclu-
sions regarding the qualitative properties of an optimal policy remain valid for
(1.10). We draw on Iglehart (1963) and Heyman and Sobel (1984, sections 8-5,
8-6), and proves that (a) there are upper and lower bounds for the sequences
{sn} and {Sn} of the finite horizon optimal policy, (b) the value function of the
finite horizon dynamic program converges as n , (c) the limit value function
satisfies the functional equation of dynamic programming, (d) as n , the fi-
nite horizon optimal policy converges to a policy that is optimal in the functional
equation, and (e) the limit policy inherits the qualitative properties of the finite
optimal policies.
We rewrite function Tn():
Tn(y) = E[n1(y D)] + (y). (1.18)
-
8/8/2019 ZHANG JIANG
32/110
-
8/8/2019 ZHANG JIANG
33/110
20
The present value of the time stream of dividends can be expressed as
B = cx1
n=1
n
cDn +
n=1
n1
[wn p(wn) In (1 )cyn (bn) e+(zn zn1)+ e(zn1 zn)+] (1.20)
Define e = (e+ + e)/2 and observe that
e+ (zn zn1)+ + e (zn1 zn)+ = e|zn zn1| + e+ e
2(zn zn1)
Proceeding as in Sobel (1969), rearranging and collecting terms in (1.20) yields
B =
n=1
n1(1 )(In + cyn) + In + g(yn, Dn)
p(1 + )In + g(yn, Dn)
(bn) e|zn zn1| e
+ e2
(1 )2yn
+
cx1 + w1 p(w1)
n=1
ncDn
+e+ e
2
(1 )x1
n=1
n(1 )Dn z0
Let
M(b , I , y) = (1 )(I+ cy) + I + E
g(y, D) p[(1 + )I+ g(y, D)]
(b) e+ e
2(1 )2y
Therefore,
E(B) =
n=1
n1E
M(bn, In, yn) e|zn zn1|
+E
cx1 + w1 p(w1)
n=1
ncDn (1.21)
+e+ e
2
(1 )x1
Nn=1
n(1 )Dn z0
-
8/8/2019 ZHANG JIANG
34/110
21
Since the last two rows of (1.21) depend only on the distribution of demand and
the initial state, we proceed to optimize the first row of (1.21).
As in 1.4, we analyze a finite-horizon counterpart of the infinite-horizon prob-
lem and the former converges to the latter. Henceforth, we optimize the following
objective:
sup EH1{N
n=1
n1[M(bn, In, yn) e|zn zn1|]}
subject to yn xn, bn + In 0, and bn 0 (1.22)
where the supremum is over the set of all policies.
A dynamic programming recursion that corresponds to (1.22) is 0(, ) 0
and for each n = 1, 2, , N, x , and z 0,
n(x, z) = sup[Jn(b , I , y) e|y x z| : y x, I+ b 0, b 0] (1.23)
Jn(b , I , y) = M(b , I , y) + E[n1(y D, y x)] (1.24)
Let bn(x), In(x) and yn(x) be optimal values of b, I and y, respectively, in
(1.23). It can be shown that bn(x) = [In(x)]+ is optimal as in Proposition 1 in
the setup cost model.
Corollary 1.6.1 For all n = 1, 2, and x , if the supremum in (1.23) is
achieved, then b = (I)+ is without loss of optimality.
Using Corollary 1.6.1, (1.23) and (1.24) can be written as follows:
n(x, z) = supyx
e|y x z| + sup{Jn(b , I , y) : I+ b 0, b 0}
-
8/8/2019 ZHANG JIANG
35/110
22
= supyx
e|y x z| + hn(y)
(1.25)
where hn(y) = sup
{Jn(b , I , y) : I+ b
0, b
0
}.
The assumption that M(, , ) is a concave function on its domain 3 leads to the
concavity of hn() on y.
Theorem 1.6.2 If M(b , I , y) as |y| for all (b, y) such that b 0,
and b + I 0, then for each n and x, there are numbers un(x) and Un(x) with
un(x) Un(x) for each x , such that an optimal policy in (1.25) is
y =
un(x) if x + z < un(x)
x + z if un(x) x + z < Un(x)
Un(x) if x Un(x) x + z
x if Un(x) < x.
(1.26)
Proof Sketch: It is straightforward to prove inductively that n(
,
), Jn(
,
)
and hn() are concave functions on their respective domains because M(, , ) is
concave. Since concave functions have one-sided derivatives (except possibly at
their boundaries), let hn() denote the left-hand derivative of hn(). Concavity
and a modification of Sobel (1969) leads to the structural result in Theorem 1.6.2.
Moreover,
un(x) = sup{y : hn(y) e}
Un(x) = sup{y : hn(y) e}
-
8/8/2019 ZHANG JIANG
36/110
23
and (1.26) corresponds to the following optimal replenishment quantity
zn =
un(xn) xn, if xn + zn1 < un(xn)
zn1, if un(xn) xn + zn1 < Un(xn)
Un(xn) xn, if xn Un(xn) xn + zn1
0, if Un(xn) < xn.
Sobel (1971) shows in great detail that finite horizon nonstationary policies
converges to infinite horizon stationary policies in the smoothing costs model.
His method is based on the convexity of the value functions, and can be directly
applied in our model. So the infinite horizon counterparts of (1.22), (1.23), (1.24),
and (1.26) are valid and the following stationary policy is optimal:
y =
u(x) if x + z < u(x)
x + z if u(x) x + z < U(x)
U(x) if x U(x) x + z
x if U(x) < x
(1.27)
where u(x) = sup{y : h(y) e}, U(x) = sup{y : h(y) e}, and h(y) is the
limit of hn(y) as n .
1.7 Concluding Remarks
In this paper, we consider periodic review inventory systems where the objective
is to maximize the expected present value of the time stream of dividends. We
consider joint financial and replenishment decisions and examine models with a
setup cost and with smoothing costs by embedding the state variable for working
-
8/8/2019 ZHANG JIANG
37/110
24
capital in the replenishment decision variable. This embedding simplifies the
analysis and allows the use of existing stochastic optimization techniques to obtain
qualitative results. In the setup cost case, the proof is not straightforward. In
both cases, the same form of replenishment policy is optimal as when the criterion
is cost minimization.
A generalization that permits dividends to be issued every R periods, where
R is a positive integer, would lead to similar results. The only change would be to
constrain vn = 0 for non-dividend-paying periods n. With these constraints added
to the model, Proposition 1.3.1 and Corollary 1.6.1 remain valid. So Theorems
1.4.3 and 1.6.2 remain essentially unchanged with only minor changes in their
proofs.
-
8/8/2019 ZHANG JIANG
38/110
Chapter 2
Fill Rate of General Review
Supply Systems
2.1 Introduction
All inventory systems face a difficult tradeoff between inventory costs and cus-
tomer service. The fill rate, the long-run average fraction of demand which is sat-
isfied immediately from on-hand inventory, is perhaps the most important measure
of customer service in professional practice.
There is a literature on formulas for the fill rate under different inventory re-
plenishment policies. Most of it concerns the fill rate in a single-stage system with
a demand process consisting of independent and identically distributed normal
random variables. Johnson et al. (1995) review the literature on approximations
25
-
8/8/2019 ZHANG JIANG
39/110
26
for the item fill rate in an uncapacitated single-stage system with normally dis-
tributed demand, develop a new approximation, and evaluate the accuracy via
simulations of several approximations to estimate the exact fill rate.
Sobel (2004), Glasserman and Tayur (1994), and Glasserman and Liu (1997)
consider the fill rate of capacitated periodic review multi-stage supply systems
in which each stage reviews its inventory periodically, and there is a constant
transportation leadtime between stages. Glasserman et al. develop asymptotic
bound and approximations, including diffusion approximations with higher order
correction terms, for fill rate and optimal base-stock levels of multi-stage systems,
whereas the fill rate formulas in Sobel (2004) are exact and the bounds are valid
without asymptotics.
Research on fill rate of periodic review inventory systems usually assumes that
the system reviews its inventory every period. In practice, although customer
demand may arise every period, a firm may not review its inventory and make
ordering (replenishment) decisions every period. For example, consider a retailer
who is supplied by a wholesaler who ships products to the retailer by truck once
a week. Although the retailer might prefer to replenish her inventory daily, she
should reorder goods only shortly before the truck leaves. One may argue that in
this situation, if we define the unit period as one week, the system becomes the
usual periodic review system. Indeed, this is a widely held perception. However,
our results show that the fill rate computed via a rescaled periodic review system
-
8/8/2019 ZHANG JIANG
40/110
27
differs from the actual fill rate of the system.
Fill rate expressions are sometimes used to optimize the parameters of an
inventory policy subject to a lower bound on the fill rate induced by the policy.
Many authors consider optimization problems with service level constraints and
most of this literature consists of heuristics and approximations.
Tijms and Groenevelt (1984) consider both periodic review and continuous re-
view (s, S) inventory systems and present a practical approximation for the reorder
point s subject to a fill rate constraint and find that the normal approximation
gives good results for required service levels when the coefficient of variation of
the demand during lead time and review periods does not exceed 0.5.
Silver (1970), Yano (1985), and Platt, Robinson, and Freund (1997) propose
heuristic solutions to fill-rate constrained models using (R, Q) policies. Axsater
(2003) considers a continuous-review fill-rate constrained serial system with batch
ordering. The system faces a discrete compound Poisson demand process in which
the leadtime demand has a negative binomial distribution. He shows that an
optimal policy consists of a mixed multistage echelon stock (R, nQ) policy with
one of the reorder points varying over time.
Schneider (1978) and Schneider and Ringuest (1990) study service-constrained
models with setup costs, focus on (s, S) policies where the order quantities are
predetermined, and present several approximations to estimate the reorder point
s such that the required service level is achieved. Schneider and Ringuest consider
-
8/8/2019 ZHANG JIANG
41/110
28
a periodic review system with a fixed leadtime.
Boyaci and Gallego (2001) and Shang and Song (2003) study a periodic re-
view service-constrained serial inventory system where the leadtime demand for
the end product is Poisson distributed. Their service measure, the limiting prob-
ability of having positive on-hand inventory at the last stage, differs from the fill
rate. Boyaci and Gallego focus on base-stock policies, develop heuristic solutions,
and discuss the relationship between stockout cost and service-constrained models.
Shang and Song study the same model, and develop closed-form heuristics to ap-
proximate optimal base-stock policies for serial service-constrained systems. Our
paper concerns exactly optimal policies for periodic review single-stage models
with general demand distributions.
An important purpose of modeling is to analyze the sensitivity of system per-
formance to various parameters. So fill rate equations are used to analyze the
sensitivity of inventory levels to alternative fill rate goals. In this sense, the role
of fill rate targets is similar to that of stockout costs, but practitioners seem to
prefer fill rate targets. Van Houtum and Zijm (2000) discuss the possible relations
between backorder cost and several types of service contraints. In particular, they
establish the one to one correspondence between backorder cost and modified fill
rate (one minus the ratio of the average backlog at the end of a period and the
mean demand per period) constraint. Chapter 3 of this dissertation show that fill
rate constraints and backorder costs are interchangeable in dynamic newsvendor
-
8/8/2019 ZHANG JIANG
42/110
29
models and establish monotone mappings between the set of optimal polices with
backorder costs and the set of optimal policies with fill rate constraints.
The rest of this paper is organized as follows. Section 2.2 introduces the general
review inventory model. A single-stage general review model is considered in 2.3
and 2.4. Section 2.3 provides fill rate formulas for general demand distribution.
Specific fill rate formulas are developed for Gamma and Normal distributions
of demand in 2.4. Section 2.5 discusses the review mechanisms of a multi-stage
system and has the fill rate formulas for general review two-stage systems. Section
2.6 has a formula for the fill rate of a three-stage system. We conclude the paper
in 2.7.
2.2 General Periodic Review System
The following model describes a periodic review N-stage serial system which is
displayed in Figure 2.1. Materials, parts or products can be ordered from any
stage and are then shipped to the next downstream stage. The inventory level
at each stage n is reviewed every Rn periods at which time an order is placed
for additional items, if any. An order for material placed at the beginning of a
review period t with destination stage n arrives at that stage at the beginning of
period t+Ln (Ln and Rn are positive integers.), if sufficient materials are available
at stage n + 1. The outside supplier preceding stage N has ample supplies and
can deliver any order that is placed by stage N. Customer demand for the end
-
8/8/2019 ZHANG JIANG
43/110
30
Figure 2.1: The standard N-stage serial inventory system
product arises solely at stage 1 and any excess demand is backlogged. If Rn = 1
(1 n N), this general review model is a standard serial multi-stage model.
At the beginning of period t, let xnt denote the number of items that are in
storage at stage n (n = 2, , N; t = 1, 2, ). Let x1t be the analogous quantity
at stage 1 minus the number of items backlogged, if any, at the beginning of period
t. That is, x1t is the on-hand physical inventory ifx1t 0, and x1t is the amount
of backordered demand ifx1t < 0. Let zNt be the number of items purchased from
an outside supplier in period t, and for n < N, let znt be the number of items
that are ordered by stage n and removed from stage n + 1 in period t.
Let Dt be the demand in period t, and let D1, D2, be independent, identi-
cally distributed, and nonnegative random variables with distribution function G
and finite mean . To avoid trivialities, it is assumed that G(0) < 1. Let G(k)()
denote the k-fold convolution of G(), i.e., the distribution function ofkj=1 Dj,and let G0(a) = 1 (0) if a
(
-
8/8/2019 ZHANG JIANG
44/110
31
stages;
The order size vector (z1t,
, zNt) is chosen;
Finally, demand at stage 1 occurs.
Let ynt be the inventory at stage n in period t after deliveries of previously ordered
goods but before demand occurs:
y1t = x1t + ztL1; ynt = xnt + zn,tLn if n > 1. (2.28)
Because an order cannot exceed the upstream inventory,
0 znt yn+1,t if 1 n N; 0 zNt. (2.29)
Notice that znt = 0 if period t is not a review period for stage n. For expository
convenience, let period t with t|R = 0 (where a|b denotes a modulo b when a and
b are integers) be a review period at stage 1. Then an order decision is made at
stage 1 with order size zt units which will be delivered at period t + L1. Ift|R = 0,
z1t = 0. The on-hand inventory that is available to satisfy demand in period t is
(y1t)+. Because excess demand (if any) is backlogged, the inventory dynamic are
as follows:
x1,t+1 = y1t Dt; xn,t+1 = ynt zn1,t (1 < n N). (2.30)
The fill rate, , is the long run average fraction of demand that can be satisfied
immediately from on-hand inventory. So,
= limT
E
Tt=1 min{(y1t)+, Dt}T
t=1 Dt
(2.31)
-
8/8/2019 ZHANG JIANG
45/110
32
where (u)+ denote max{u, 0}. The expectation and limit exist in (2.31) for the
base-stock policies that are analyzed in subsequent sections.
2.3 Uncapacitated Single-stage Systems
This section considers an uncapacitated single stage general review system in
which products are ordered from an outside supplier every fixed R periods and
are available to satisfy demand in period t + L (where both R and L are positive
integers). We assume that the system uses a base-stock-level policy. Base-stock
policies have been proved to be optimal for periodic review single-stage system
under general conditions and are very easy to implement in practice (cf. Zipkin
2000 and Porteus 2002). Let be the base-stock level, so
z1t = ( y1t)+ if t|R = 0 and z1t = 0 otherwise for t = 1, 2, .
It follows from Lemma 1 in Sobel (2004) that there is no loss of generality
in assuming that initial inventory is never higher than , i.e., x11 . As a
consequence, for every t L,
y1t = L+[(tL)|R]
k=1
Dtk, for (t L)|R = 0, 1, , R 1 (2.32)
In Sobel (2004), there is an inconsistency between the fill rate definition and
the proof of Theorem 1 . He uses y1t in the fill rate definition [(3) on page 43], but
uses x1t to derive the fill rate formula in the proof of Theorem 1 (line 9, page 44).
However, because of the chronology differences between his model and the present
-
8/8/2019 ZHANG JIANG
46/110
33
model, substituting x1t in the proof by y1t of our model results in the same fill
rate formulas. Formulas (4) and (5) in Sobel (2004) remain valid here if R = 1.
The following theorem gives an exact fill rate formula for a general review
single-stage inventory system.
Theorem 2.3.1
=1
R
0
[G(L)(b) G(L+R)(b)]db (2.33)
Proof From (2.31), the fill rate is the long run average fraction of demand that
is met directly from on-hand inventory. So,
= limT
E Tt=1
min{(y1t)+, Dt}/T
t=1
Dt
= limT
E
[T
t=1
min{(y1t)+, Dt}/T]/[T
t=1
Dt/T]
=1
lim
TE
T
t=1min{(y1t)+, Dt}
/T
=1
Rlim
TE Tt=1
min{(y1t)+, Dt}
/(T /R)
=1
Rlim
TE
R1i=0
t{t:(tL)|R=i}
min{(y1t)+, Dt}
/(T /R)
=1
Rlim
TE
R1i=0
t{t:(tL)|R=i}
min{(y1t)+, Dt}/(T /R)
Let Hj =
t{t:(tL)|R=j} min{(y1t)+, Dt}, j = 0, 1, , R 1. Using (2.32), for all
j {0, 1, , R 1}, it can be shown that
limT
E[Hj/(T /R)] = E
min
(L+jk=1
Dk)+, DL+j+1
= E
DL+j+1 [DL+j+1 ( L+jk=1
Dk)+]+
-
8/8/2019 ZHANG JIANG
47/110
34
= E
[DL+j+1 (L+jk=1
Dk)+]+
Let () make explicit the dependence of on . Then
() =1
R
R1j=0
Hj =1
R
R1j=0
E
[DL+j+1 (
L+jk=1
Dk)+]+
= 1 1R
R1j=0
E
[DL+j+1 ( L+jk=1
Dk)+]+
and let K() = [1 ( ,L,R)]. So,
K() = 1R
R1j=0
E
[DL+j+1 (L+jk=1
Dk)+]+
=1
R
R1j=0
0
a
(a + b )dG(L+j)(b)dG(a)
+0
a
dG(L+j)(b)dG(a)
=1
R
R1j=0
1 G(L+j)()
+0
a
(a + b )dG(L+j)(b)dG(a)
Leibnitz Rule yields
K() =1
R
R1j=0
0
a
dG(L+j)(b)dG(a)
=1
R
R1j=0
0
b
dG(a)dG(L+j)(b)
=1
R
R1j=0
0
G( b) 1
dG(L+j)(b)
=1
R
R1
j=0G(L+j+1)() G(L+j)()
= 1R
G(L+R)() G(L)()
Since (0) = 0, K(0) = [1 (0)]. Therefore,
() = 1 K()/
-
8/8/2019 ZHANG JIANG
48/110
35
= 1 [K(0) +0
K(a)da]/
=1
R
0
[G(L)(b)
G(L+R)(b)]db.
Theorem 2.3.1 characterizes the dependence of the system fill rate on the base-
stock level (), demand distribution (G), review period (R), and leadtime (L).
2.4 Gamma and Normal Demand in Single-stage
Systems
This subsection specializes (2.33) for the gamma distribution and the normal
distribution.
2.4.1 Gamma Demand Distribution
Let (j,) denote the sum of j independent, identically distributed random vari-
ables, each one exponential with parameter , i.e., (j,) is a gamma random
variable with parameters j and . If D is (, ) where is a positive integer,
then = E(D) = /, V ar(D) = /2, the probability density function of D is
ea(a)1
()
and the distribution function of D is
G(a) = P(D a) = 1 1j=0
ea(a)j/j!
-
8/8/2019 ZHANG JIANG
49/110
36
Consequently, G(L) and G(L+R) are (L,) and [(L + R), ], respectively. So,
G
(L)
(a) = 1 L1
j=0 e
a
(a)
j
/j!
G(L+R)(a) = 1 (L+R)1
j=0
ea(a)j/j!
It follows from (2.33) that
=1
R
0
(L+R)1j=0
ea(a)j/j! L1j=0
ea(a)j/j!
da
= R0
(L+R)1j=L
ea(a)j/j!
da
=1
R
(L+R)1j=L
0
ea(a)j/j!
da
Therefore,
=1
R
(L+R)j=L+1
P{(j,) } (2.34)
2.4.2 Normal Demand Distribution
Many researchers investigate the fill rate of an inventory system with normally
distributed demands. In order to compare our results with others, we analyze the
general review inventory systems when demand (D) is normally distributed with
mean and variance 2 > 0.
Let () and () denote the distribution and density function, respectively,
of a standard normal random variable (with mean 0 and variance 1), and let
b(a, j) = (a j)/(j). The normality and independence of demand imply
-
8/8/2019 ZHANG JIANG
50/110
37
that sums of demands are normally distributed; that is, G(S)(a) = [b(a, S)] for
S I+. So, formula (2.33) yields
=1
R
0
[b(a, L)] [b(a, L + R)]
da
=1
R
Lb(,L)b(0,L)
(x)dx
L + Rb(,L+R)b(0,L+R)
(x)dx
(2.35)
The evaluation of the integral in (2.35) exploits the following equation (Hadley
and Whitin 1963; Sobel 2004; Zipkin 2000).
t
[1 (x)]dx = (t) + t(t) t
This equation implies
st
(x)dx = (s) (t) + s(s) t(t). (2.36)
Using (2.36) in (2.35) yields the following equation for the fill rate that uses only
the standard normal density and tabulated standard normal distribution function:
=1
R
L
[b(, L)] [b(0, L)] + b(, L)[b(, L)] b(0, L)[b(0, L)]
L + R
[b(, L+ R)] [b(0, L+ R)]
+ b(, L+ R)[b(, L+ R)] b(0, L+ R)[b(0, L+ R)]
=1
R
L[b(, L)] [b(0, L)]
L + R
[b(, L+ R)] [b(0, L+ R)]
+ ( L)[b(, L)] [b(, L+ R)]+ R
[b(, L+ R)]
+ L
[b(0, L)]
(L + R)[b(0, L+ R)]
. (2.37)
Another deviation of 2.37 uses the following formula:
ba
x(x)dx = (a) (b). (2.38)
-
8/8/2019 ZHANG JIANG
51/110
38
Figure 2.2: The Fill Rate Integral for a system with Normal Demand
Write the integral in (2.35) as
=1
R
0
b(a,L)
(x)dx b(a,L+R)
(x)dxda
whose integrand in the a x plane covers the area in the northeast and southeast
quadrants depicted in Figure 2.2 and bounded by the lines a = 0, a = , x =
b(a, L), and x = b(a, L + R). As shown in Figure 2.2 , this area is the union of
three sets in the a x plane. Interchanging the order of integrations, employingformula (2.38), and integrating A1, A2, and A3 individually leads to the same fill
rate expression as (2.37).
-
8/8/2019 ZHANG JIANG
52/110
39
2.4.3 Fill Rate Approximation for Normal Demand Dis-
tribution
Although (2.37) can be calculated easily, the following simple approximation yields
valuable insights:
A =1
R
L + R[b(, L + R)]
+ ( L)
1 [b(, L + R)]
+ R[b(, L + R)]
(2.39)
Approximation (2.39) is derived from (2.37) by observing from Figure 2.2 that
[b(0, L)] 0, [b(, L)] 0, [b(0, L+R)] 0, [b(, L)] 1, [b(0, L+R)] 0,
and [b(0, L)] 0. Our numerical results show that the fill rate approximation
(2.39) is very accurate when (L + R). The numerical comparisons are shown
in Table 2.1.
Table 2.1 use the same normal demand distribution as Table 1 in Sobel (2004),
compute the exact value and approximation of the fill rate in a general review
single-stage system with = 2000 and L + R = 5. The last columns of the tables
report the absolute error (100%) of fill rate and its approximation.
In summary, we have the following observations:
The fill rate increases as the variance (2) goes down or base-stock level ()
goes up.
In general, shorter leadtime yields higher fill rate if L + R is fixed.
-
8/8/2019 ZHANG JIANG
53/110
40
The approximation performs better when the demand coefficient of variation
(/) decreases. There are three cases (with high variance, long leadtime) in
our example that cause large errors, but the errors diminish when base-stock
level increases.
2.5 Multi-Stage General Review Systems
An echelon base-stock policy is optimal for a periodic-review multi-stage systems
with linear inventory holding costs at all stages and linear backorder costs at stage
one (Clark and Scarf 1960, Federgruen and Zipkin 1984). It is quite natural to use
echelon base-stock policies in our general review systems. To clearly understand an
echelon base-stock policy, it is useful to define the following echelon variables(cf.
Clark and Scarf 1960):
echelon inventory level of a stage is the inventory on hand at this stage plus
inventories at or in transit to all its downstream successor stages minus total
customer backorder at the lowest stage.
echelon inventory position of stage is the sum of echelon inventory level at
this stage and inventory in transit to the stage.
sbnt = the (beginning) echelon inventory level at stage n before any order is
received
-
8/8/2019 ZHANG JIANG
54/110
41
abnt = the (beginning) echelon inventory position at stage n before any order
is placed
snt = echelon inventory level at stage n
snt = the echelon inventory position at stage n before demand occurs.
The evolution of the system and dynamics of these echelon variables can be
specified as:
abnt = sbnt +
Ln
k=1
zn,tk ant = abnt + znt snt = s
bn,t + zn,tLn (2.40)
and
abn,t+1 = ant Dt sbn,t+1 = snt Dt (2.41)
The first expression in (2.40) can be written
ant = snt +Ln1
k=0zn,tk
The formulation in installation variables xnt, ynt, and znt is equivalent to a
formulation in echelon variables because xnt = ynt znt, ynt = snt an1,t (let
s0t = a0t = 0), and znt = ant Ln1k=1 zn,tk snt. Because s1t = y1t, (2.31) canbe written
= limT
E
Tt=1 min{(s1t)+, Dt}
Tt=1
Dt
(2.42)
The constraints on the order quantities (2.29) correspond to
s1t a1t s2t sNt aNt. (2.43)
-
8/8/2019 ZHANG JIANG
55/110
42
An echelon base-stock policy depends on echelon base-stock levels 1, , N
and can be expressed as: If period t is a review period, aNt = max{abNt, N} and
for n < N,
ant =
abnt if n abntmin{n, sn+1,t} otherwise,
(2.44)
and ant = abnt if period t is not a review period for all n N. The order quantity
specified by (2.44) is znt = min{(nabnt)+, (sn+1,t abnt)+} ift is a review period,
and znt = 0 otherwise.
The subsequent sections discuss the fill rate of general review two-stage and
three-stage supply systems.
2.5.1 Fill Rate in Two-Stage Systems
This subsection considers a two-stage general review system in which both stages
have the same review intervals (R1 = R2 = R) and order leadtimes are L1 = L
and L2 = 1. Based on the review procedure, if t is a review period, then t 1 is a
review period for stage two. Both stages have the same length of review intervals.
We are interested in this system because it is very similar to the single-stage
system; in essence, the only difference is that the order quantity in stage one at
a review period is constrained by the inventory at stage two. We shall explicitly
characterize the relationship between the echelon inventory level at stage one and
base-stock levels 1 and 2 for stage one and 2, respectively. It is without loss
of generality to assume that the initial echelon inventory positions are no higher
-
8/8/2019 ZHANG JIANG
56/110
-
8/8/2019 ZHANG JIANG
57/110
44
= a1t L1k=0
a1,tk (a1,tk1 Dtk1)
= a1,tL L
k=1 Dtk
Lemma 2.5.1 states that the echelon inventory level at stage one in period t is
equal to its echelon inventory position in period t L minus the preceding L
periods demands.
As a consequence [similar to the derivation of (2.32)], s1t for any period t can
be expressed explicitly. It is without loss of generality to assume that stage one
reviews its inventory at period t when t|R = 0. Let = 2 1.
Lemma 2.5.2 For any t L,
s1t = 1 + min{0, Dt[L+(tL)|R]1} L+(tL)|R
k=1
Dtk (2.45)
Proof At t = L, then (t L)|R = 0, and a2,L1 = 2 because stage two reviewsits inventory one period ahead of stage one. Lemma 2.5.1 assures
s1L = a1,L L
k=1
Dtk
= min{1, 2 DL1} L
k=1
Dtk
= 1 + min{0, DL01} L+0
k=1Dtk
So (2.45) is valid at t = L and initiates an inductive proof of (2.45).
Assume that (2.45) is valid at t and note that
s1,t+1 = s1,t Dt + zt+1L
-
8/8/2019 ZHANG JIANG
58/110
-
8/8/2019 ZHANG JIANG
59/110
46
= limT
E
[T
t=1
min{(s1t)+, Dt}/T]/[T
t=1
Dt/T]
=
1
limTE
T
t=1 min{(s1t)
+
, Dt}/T=
1
Rlim
TE Tt=1
min{(s1t)+, Dt}/(T /R)
=1
Rlim
TE
R1i=0
t{t:(tL)|R=i}
min{(s1t)+, Dt}
/(T /R)
=1
Rlim
TE
R1i=0
t{t:(tL)|R=i}
min{(s1t)+, Dt}/(T /R)
Let Hj = t{t:(tL)|R=j}min{(s1t)+, Dt}, j = 0, 1, , R1. Using (2.45) and
noticing that the demand variables are independent,
limT
E[Hj/(T /R)] = E
min
1 + min{0, D1} L+j+1k=2
Dk+
, DL+j+2
For all j {0, 1, , R 1}, it can be shown that
(Hj) = E
min
1 + min{0, D1} L+j+1k=2
Dk+
, DL+j+2
= E
DL+j+1
DL+j+2 [1 + min{0, D1} L+j+1k=2
Dk]++
= 1 E
DL+j+2 [1 (D1 )+ L+j+1k=2
Dk]++
Let (1, ) make explicit the dependence of on 1 and . Then
(1, ) =1
R
R1j=0
(Hj)
= 1R
R1j=0
1 EDL+j+2 [1 (D1 )+
L+j+1k=2
Dk]++
Let K(1, ) = [1 (1, )]. So,
K(1, ) =1
R
R1j=0
E
DL+j+2 [1 (D1 )+ L+j+1k=2
Dk]++
-
8/8/2019 ZHANG JIANG
60/110
47
=1
R
R1j=0
1+c0
1+bc
(a + b + c
1
)dG(L+j+2)(a)dG
(L+j)(b)dG(c)
+
1+c
0
a dG(L+j+2)(a)dG(L+j)(b)dG(c)
+0
10
1b
(a + b 1) dG(L+j+2)(a)dG(L+j)(b)dG(c)
+0
1
0
a dG(L+j+2)(a)dG(L+j)(b)dG(c)
Leibnitz Rule yields
K(1, )1
= 1R
R1j=0
1
+c
0
1+bc dG(L+j+2)(a)dG(L+j)(b)dG(c)
+
0
a dG(L+j+2)(a)g(L+j)(1 + c)dG(c)
+
0
a dG(L+j+2)(a)g(L+j)(1 + c)dG(c)
+0
10
1b
dG(L+j+2)(a)dG(L+j)(b)dG(c)
+0
0
a dG(L+j+2)(a)g(L+j)(1)dG(c)
+00 a dG(L+j+2)(a)g
(L+j)
(1)dG(c)
(2.47)
Notice that the sum of the second and third lines in (2.47) is zero and the sum of
the fifth and sixth lines is zero. Further simplifying (2.47),
K(1, )
1=
1
R
R1j=0
1+c0
1+bc
dG(L+j+2)(a)dG(L+j)(b)dG(c)
+
0
1
0
1b dG(L+j+2)(a)dG
(L+j)(b)dG(c)
=1
R
R1j=0
1+c0
G(L+j+2)(1 + b c) 1
dG(L+j)(b)dG(c)
+0
10
G(L+j+2)(1 b) 1
dG(L+j)(b)dG(c)
=1
R
R1j=0
G(L+j+1)(1 + c) G(L+j)(1 + c)
dG(c)
-
8/8/2019 ZHANG JIANG
61/110
48
+0
G(L+j+1)(1) G(L+j)(1)
dG(c)
=1
R
R1
j=0
G(L+j+1)(1 + c) G
(L+j)(1 +
c)dG(c)+G(L+j+1)(1) G(L+j)(1)
G()
=1
R
G(L+R)(1 + c) G(L)(1 + c)
dG(c)
+G(L+R)(1) G(L)(1)
G()
(2.48)
Similarly, it is straightforward to show that for all 0, K(0, ) = .
Therefore,
(1, ) = 1 K(1, )/
= 1 K(0, ) +
10
K(a, )
ada/
= 10
K(a, )
ada/
=1
R
G()
10
G(L)(a) G(L+R)(a)
da
+10
G(L)(a + c) G(L+R)(a + c)dG(c)da
2.5.2 Fill Rate in Two-Stage Systems with General Lead-
time
The formula for a two-stage general review system with a single period leadtime
in stage two can be easily extended to a two-stage general review system in which
the review intervals are R1 = R2 = R and order leadtimes are L1 and L2. We use
the following review procedure which is bases on complete information-sharing
between stages 1 and 2: if t is a review period, then t L2 is a review period for
-
8/8/2019 ZHANG JIANG
62/110
49
stage two. Both stages have the same length of review intervals.
Lemma 2.5.1 still holds for this general two-stage system. We give the following
lemma to express the echelon inventory level at stage one.
Lemma 2.5.4 For any t L1 + L2,
s1t = 1 + min
0, L2k=1
Dt[L1+(tL1)|R]k
L1+(tL1)|Rk=1
Dtk (2.49)
The proof of Lemma 2.5.4 is similar to that of Lemma 2.5.2 except that the order
placed by stage two has L2 delay before it can be used to satisfy stage ones order.
Replacing D1 and G(c) byL2
k=1 Dk and G(L2)(c), respectively, in the proof of
Theorem 2.5.3 yields the next result.
Theorem 2.5.5
=1
R10
G(L2)()G(L1)(a) G(L1+R)(a)
+
G(L1)(a + c) G(L1+R)(a + c)
dG(L2)(c)
da (2.50)
In what follows, we present an alterative formula for (2.50) using the single
stage fill rate formulas. Let 1(, L) be the fill rate of a single stage system
with review period R, base-stock level , and order leadtime L. We introduce
incomplete convolutions (cf. van Houtum et al. 1996). Let G(k)
be the distribution
functions ofk
j=1. Let > 0 and define
G(k) (x) =
G(k)(x + ) if x 0
0 if x < 0,
-
8/8/2019 ZHANG JIANG
63/110
50
The incomplete convolution, denoted G[m,n] , is
G
[m,n]
(x) =0 G
(m)
( + x u)dG(n)
(u).
note that G[m,n]0 (x) = G
(m+n)(x)
Corollary 2.5.6
=1
R
G(L2)()1(1, L1) + 1(1 + , L1 + L2) 1(, L1 + L2)
1
0G[L1,L2] (a)
G[L1+R,L2] (a)da (2.51)
Proof Rewriting (2.50) and using (2.33),
=1
R
10
G(L1)(a + c) G(L1+R)(a + c)
dG(L2)(c)da
+G(L2)()10
G(L1)(a) G(L1+R)(a)
da
=1
R
10
a+
G(L1)(a + c) G(L1+R)(a + c)
dG(L2)(c)da
+G(L2)()1(1, L1)
=1
R
10
a+0
G(L1)(a + c) G(L1+R)(a + c)
dG(L2)(c)da
10
0
G(L1)(a + c) G(L1+R)(a + c)
dG(L2)(c)da
+G(L2)()1(1, L1)
=1
R
10
G(L1+L2)(a + ) G(L1+L2+R)(a + )
da
1
0
[G[L1,L2]
(a) G[L1+R,L2]
(a)
da + G(L2)
()1(1, L1)
=1
R
1+
G(L1+L2)(u) G(L1+L2+R)(u)
du
10
[G
[L1,L2] (a) G[L1+R,L2] (a)
da + G(L2)()1(1, L1)
=1
R
1+0
G(L1+L2)(u) G(L1+L2+R)(u)
du
-
8/8/2019 ZHANG JIANG
64/110
51
0
G(L1+L2)(u) G(L1+L2+R)(u)
du
1
0[G
[L1,L2] (a)
G
[L1+R,L2] (a)da + G(L2)()1(1, L1)
=1
R
G(L2)()1(1, L1) + 1(1 + , L1 + L2) 1(, L1 + L2)
10
G
[L1,L2] (a) G[L1+R,L2] (a)
da
(2.52)
2.5.3 Numerical Example
It is clear from (2.50) that the fill rate of a two-stage general review system depends
on R, L, 1, (2), and G(). However, it is not easy to reveal the dependences.
Therefore, we perform another set of numerical study on (2.50) to illustrate their
relationships. In this study, demand is normally distributed with = 10.
We make the following observations from Tables 2.2 and 2.3:
The fill rate increases as variance goes down, and as echelon base-stock level
at stage one goes up.
The fill rate decreases as leadtime at stage one and stage two increase.
The fill rate increases as echelon base-stock level at stage two increases.
2.6 Fill Rate in a Three-Stage System
This section considers a three-stage general review system in which the review
interval is the same at each stage (R1 = R2 = R3 = R) and order leadtimes are
-
8/8/2019 ZHANG JIANG
65/110
-
8/8/2019 ZHANG JIANG
66/110
53
Theorem 2.6.2
=
1
R10
2
1+2e
G
(L1)
(a + 1 + 2 c e)G(L1+R)(a + 1 + 2 c e)
dG(L2)(c)dG(L3)(e)
+G(L1)(a) G(L1+R)(a)
2
G(L2)(1 + 2 e)dG(L3)(e)
+G(L3)(2)1
G(L1)(a + 1 c) G(L1+R)(a + 1 c)
dG(L2)(c)
+G(L1)(a) G(L1+R)(a)
G(L2)(1)G
(L3)(2)
da. (2.54)
Proof From (2.31),
= limT
E Tt=1
min{(s1t)+, Dt}/T
t=1
Dt
=1
Rlim
TE
R1i=0
t{t:(tL)|R=i}
min{(s1t)+, Dt}/(T /R)
Let Hj =
t{t:(tL)|R=j} min{(s1t)+, Dt}, j = 0, 1, , R 1, use (2.53), and
notice that the demand random variables are independent. Let D(j) have the
distribution ofj
k=1 Dk. Then limT E[Hj/(T /R)] is equal to
E
min
1 + min
0, 1 + min {0, 2 D(L3)} D(L2)
D(L1+j)
+, D
For all j {0, 1, , R 1}, it can be shown that
(Hj) = 1
ED
1
[D(L3)
2]
+ + D(L2)
1
+
D(L1+j)
+
+
Let (1, 1, 2) make explicit the dependence of on 1, 1, and 2. Then
(1, 1, 2) =1
R
R1j=0
(Hj)
-
8/8/2019 ZHANG JIANG
67/110
54
Let K(1, 1, 2) = [1 (1, 1, 2)]. So,
K(1,1,2)
=1
R
R1j=0
E
D
1
[D(L3) 2]+ + D(L2) 1
+ D(L1+j)++
=1
R
R1j=0
2
1+2e
1+1+2ec0
1+1+2ecb
(a + b + c + e 1 1 2)dG(a)dG(L+j)(b)dG(L2)(c)dG(L3)(e)
=
2
1+2e
1+1+2ec
0
adG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)
+
2
1+2e0
10
1b
(a+ b 1)dG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)
+2
1+2e0
1
0
adG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)
+
20
1
1+1c0
1+1cb
(a + b + c 1 1)dG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)
+
20
1
1+1c
0
adG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)
+
20
10
10
1b
(a + b 1)dG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)
+ 2
0
1
0
1
0
adG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)Leibnitz Rule yields
K(1, 1, 2)
1
=1
R
R1j=0
2
1+2e
1+1+2ec0
1+1+2ecb
dG(a)dG(L+j)(b)dG(L2)(c)dG(L3)(e)
21+2e0
101b dG(a)dG
(L1+j)
(b)dG
(L2)
(c)dG
(L3)
(e)
20
1
1+1c0
1+1cb
dG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)
20
10
10
1b
dG(a)dG(L1+j)(b)dG(L2)(c)dG(L3)(e)
=1
R
2
1+2e
G(L1+R)(1 + 1 + 2 c e)
-
8/8/2019 ZHANG JIANG
68/110
-
8/8/2019 ZHANG JIANG
69/110
56
2.7 Conclusion
This paper develops formulas for the fill rates of single-stage and multi-stage
supply systems that use base-stock-level policies and have general review intervals.
We provide fill rate formulas for a single-stage general review system and general
distributions of demand. When demand is normally distributed, an exact fill
rate expression uses only the standard normal distribution function and density
function. For the general review multi-stage systems, we first discuss how each
stage reviews its inventory and provide a general approach to compute the system
fill rate.
-
8/8/2019 ZHANG JIANG
70/110
57
/ R L A %Error
200 0.1 11000 1 4 0.999014 0.999014 0200 0.1 11000 2 3 0.999507 0.999507 0200 0.1 11000 3 2 0.999671 0.999671 0200 0.1 11000 4 1 0.999754 0.999754 0
1000 0.5 11000 1 4 0.789395 0.760095 2.931000 0.5 11000 2 3 0.880264 0.880047 0.02171000 0.5 11000 3 2 0.919869 0.920032 0.01631000 0.5 11000 4 1 0.938963 0.940024 0.10612000 1 11000 1 4 0.590938 0.335729 25.52092000 1 11000 2 3 0.686992 0.667864 1.91282000 1 11000 3 2 0.766131 0.778576 1.24452000 1 11000 4 1 0.815568 0.833932 1.8364200 0.1 12000 1 4 1 1 0200 0.1 12000 2 3 1 1 0200 0.1 12000 3 2 1 1 0200 0.1 12000 4 1 1 1 0
1000 0.5 12000 1 4 0.895048 0.886563 0.84851000 0.5 12000 2 3 0.943282 0.943282 01000 0.5 12000 3 2 0.962025 0.962188 0.01631000 0.5 12000 4 1 0.97058 0.971641 0.10612000 1 12000 1 4 0.679695 0.520189 15.9506
2000 1 12000 2 3 0.765023 0.760095 0.49282000 1 12000 3 2 0.826923 0.840063 1.3142000 1 12000 4 1 0.861683 0.880047 1.8364200 0.1 13000 1 4 1 1 0200 0.1 13000 2 3 1 1 0200 0.1 13000 3 2 1 1 0200 0.1 13000 4 1 1 1 0
1000 0.5 13000 1 4 0.95544 0.953442 0.19981000 0.5 13000 2 3 0.976695 0.976721 0.00261000 0.5 13000 3 2 0.984318 0.984481 0.01631000 0.5 13000 4 1 0.987299 0.98836 0.10612000 1 13000 1 4 0.758474 0.664425 9.40492000 1 13000 2 3 0.829454 0.832213 0.27592000 1 13000 3 2 0.874769 0.888142 1.33732000 1 13000 4 1 0.897742 0.916106 1.8364
Table 2.1: Fill Rate and its Approximation for Normal Demand
-
8/8/2019 ZHANG JIANG
71/110
58
/ 1 1 R L1 L2 2 0.2 50 0 1 4 1 0.17472 0.2 50 0 1 4 2 0.00372 0.2 50 0 2 3 1 0.49852 0.2 50 0 2 3 2 0.08922 0.2 50 0 3 2 1 0.6654
2 0.2 50 0 3 2 2 0.33362 0.2 50 0 4 1 1 0.74912 0.2 50 0 4 1 2 0.55 0.5 50 0 1 4 1 0.30305 0.5 50 0 1 4 2 0.10525 0.5 50 0 2 3 1 0.47025 0.5 50 0 2 3 2 0.20415 0.5 50 0 3 2 1 0.62005 0.5 50 0 3 2 2 0.34855 0.5 50 0 4 1 1 0.7141
5 0.5 50 0 4 1 2 0.49325 0.5 50 20 1 4 1 0.63575 0.5 50 20 1 4 2 0.53955 0.5 50 20 2 3 1 0.77735 0.5 50 20 2 3 2 0.69485 0.5 50 20 3 2 1 0.85035 0.5 50 20 3 2 2 0.79065 0.5 50 20 4 1 1 0.88685 0.5 50 20 4 1 2 0.8419
/ 1 1 R L1 L2 2 0.2 80 0 1 4 1 12 0.2 80 0 1 4 2 0.99402 0.2 80 0 2 3 1 12 0.2 80 0 2 3 2 0.99702 0.2 80 0 3 2 1 1
2 0.2 80 0 3 2 2 0.99802 0.2 80 0 4 1 1 12 0.2 80 0 4 1 2 0.99855 0.5 80 0 1 4 1 0.97495 0.5 80 0 1 4 2 0.85465 0.5 80 0 2 3 1 0.98685 0.5 80 0 2 3 2 0.91475 0.5 80 0 3 2 1 0.99125 0.5 80 0 3 2 2 0.94275 0.5 80 0 4 1 1 0.9933
5 0.5 80 0 4 1 2 0.95715 0.5 80 20 1 4 1 0.99875 0.5 80 20 1 4 2 0.99445 0.5 80 20 2 3 1 0.99935 0.5 80 20 2 3 2 0.99715 0.5 80 20 3 2 1 0.99945 0.5 80 20 3 2 2 0.99795 0.5 80 20 4 1 1 0.99865 0.5 80 20 4 1 2 0.9975
Table 2.2: Fill Rate of Two-stage Systems for Normal Demand (a)
-
8/8/2019 ZHANG JIANG
72/110
59
/ 1 1 R L1 L2 5 0.5 50 30 1 4 1 0.6373
5 0.5 50 30 1 4 2 0.62845 0.5 50 30 2 3 1 0.77855 0.5 50 30 2 3 2 0.77145 0.5 50 30 3 2 1 0.85125 0.5 50 30 3 2 2 0.84625 0.5 50 30 4 1 1 0.88745 0.5 50 30 4 1 2 0.88375 0.5 80 30 1 4 1 0.99875 0.5 80 30 1 4 2 0.99855 0.5 80 30 2 3 1 0.9993
5 0.5 80 30 2 3 2 0.99915 0.5 80 30 3 2 1 0.99945 0.5 80 30 3 2 2 0.99935 0.5 80 30 4 1 1 0.99865 0.5 80 30 4 1 2 0.9985
/ 1 1 R L1 L2 5 0.5 50 50 1 4 1 0.6373
5 0.5 50 50 1 4 2 0.63735 0.5 50 50 2 3 1 0.77855 0.5 50 50 2 3 2 0.77855 0.5 50 50 3 2 1 0.85125 0.5 50 50 3 2 2 0.85125 0.5 50 50 4 1 1 0.88745 0.5 50 50 4 1 2 0.88745 0.5 80 50 1 4 1 0.99875 0.5 80 50 1 4 2 0.99875 0.5 80 50 2 3 1 0.9993
5 0.5 80 50 2 3 2 0.99935 0.5 80 50 3 2 1 0.99945 0.5 80 50 3 2 2 0.99945 0.5 80 50 4 1 1 0.99865 0.5 80 50 4 1 2 0.9986
Table 2.3: Fill Rate of Two-stage Systems for Normal Demand (b)
-
8/8/2019 ZHANG JIANG
73/110
-
8/8/2019 ZHANG JIANG
74/110
61
the fraction of demand which is immediately met from on-hand inventory. During
this latter period, the relative importance of the two risks has often been param-
eterized with holdings costs and a lower bound on the fill rate. As a result, there
are parallel streams of literature which analyze identical models except that one
stream has stockout costs and the other has fill rate constraints.
These streams of literature correspond in mathematical programming to op-
timization subject to constraints and to optimization of an unconstrained La-
grangean. As in nonlinear deterministic optimization, in stochastic optimization
the two approaches do not always yield the same results. This paper investigates
whether there is redundancy in the two streams of dynamic inventory models with
linear purchase costs, namely dynamic newsvendor models. We show the extent
to which optimal policies for either kind of model can be inferred from the other.
Here, an inventory replenishment policy is called Stockout-optimal, or S-
optimalfor short, if it minimizes the long-run average sum of holding and stockout
costs per unit time. So, S-optimality corresponds to an unconstrained Lagrangean
formulation. A policy is called Fill-Rate-Optimal, or F-optimal for short, if it
minimizes the long-run average holding cost per unit time subject to a fill-rate
constraint. If demand is continuous, i.e., if the distribution function of demand
has a density function, then S-optimality and F-optimality are shown to be equiv-
alent in the following sense:
(a) Corresponding to any unit stockout cost b, there is a base-stock level y and a
-
8/8/2019 ZHANG JIANG
75/110
62
fill-rate f such that a base-stock policy with parameter y is both S-optimal with
stockout cost b and F-optimal with constraint parameter f.
(b) Corresponding to any fill-rate constraint parameter f, there is a base-stock
level y and a stockout cost b such that a base-stock policy with parameter y is
both S-optimal with stockout cost b and F-optimal with constraint parameter f.
If demand is an integer-valued random variable, the situation is more com-
plicated. Although a deterministic base-stock level policy is S-optimal for every
stockout cost b, a randomized base-stock level policy is F-optimal for most con-
straint parameters f. Nevertheless, a parametric analysis of either kind of opti-
mality can be accomplished via the other kind in the following sense:
(c) Corresponding to any unit stockout cost b, there is a base-stock level y and a
fill-rate f such that a base-stock policy with parameter y is both S-optimal with
stockout cost b and F-optimal with constraint parameter f.
(d) There are sequences of fill-rate constraint parameters 0 < f1 < f2 < < 1,
base-stock level parameters y1 < y2 < , and stockout cost parameters b1 1, rewrite
(3.63) as
inf
E N
n=1
H(yn)
: E N
n=1
B(yn) N(1 f)
(3.66)
For any N > 1, yn 0 for n = 1, , N. Then (3.66) is a convex nonlinear
program for which the following Karush-Kuhn-Tucker conditions are necessary
and sufficient when demand has a density:
E[H(yn)] + E[B(yn)] n = 0 for n = 1, 2, , N(3.67)
N
n=1
E[B(yn)] N(1 f)
= nyn = 0 for n = 1, 2, , N(3.68)N
n=1
E[B(yn)] N(1 f), yn 0, 0, and n 0. for n = 1, 2, , N(3.69)
-
8/8/2019 ZHANG JIANG
87/110
74
Since H() > 0 and B() < 0 on (m, M), if = 0, then (3.67) implies that
n = E[H(yn)] > 0 for n = 1, , N. Consequently, from (3.68), yn = 0 for
n = 1, , N. But then,
Nn=1
E[B(yn)] = N = N(L + 1) > N(1 f) for all f > 0
which contradicts (3.69).
So = 0, and (3.68) yields
Nn=1
E[B(yn)] = N(1 f). (3.70)
Also, from (3.67),
E[H(y1)]
E[B(y1)]=
E[H(y2)]
E[B(y2)]= = E[H
(yN)]
E[B(yN)]
So, y1 = y2 = = yN because H() and B() are convex and monotone. There-
fore, with (3.70),
E[B(y1)] = E[B(y2)] = = E[B(yN)] = (1 f)
Since B() is an injection, it follows that
y1 = y2 = = yN = y = B1[(1 f)]
Since x1 y, xn+1 = yn Dn and P{Dn 0} = 1 ensure that yn = y xn for
all n. Therefore, adding the constraints yn xn for all n reduces the feasibility set
of (3.66), but does not affect the optimality ofyn = y for all n (given x1 y).
-
8/8/2019 ZHANG JIANG
88/110
75
The transition from (3.63) to (3.62) is consistent with the large literature which
connects finite horizon and infinite horizon inventory models. We exploit the fact
that eventually the inventory level is at least as low as the back-stock level y
(regardless of the initial inventory level). The proof is brief, straightforward, and
omitted.
Lemma 3.3.3 With probability one there is a period n < , such that xn y
for all n n.
Proposition 3.3.4 If demand has a density, then the base stock policy yn =
max{y, xn} for all n, with y specified in (3.65), is F-optimal for (3.62), the
infinite horizon problem with a fill rate constraint.
Proof For all N and for all such that E|x1[N
n=1 B(yn)] N(1 f), if
x1 y,NH(y) E|x1
Nn=1
H(yn)
(Lemma 3.3.2)
H(y) E|x1 1
N
Nn=1
H(yn)
Therefore, if is feasible in (3.62),
H(y
) limN infE|x11
N
N
n=1 H(yn)
Therefore, is optimal in (3.62) because it is feasible due to Lemma 3.3.3 and
limN
infE|x1 N
n=1
H(yn)
= limN
infE|x1