Chapter 10 Multicriteria Decision-Marking Models
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Transcript of Chapter 10 Multicriteria Decision-Marking Models
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Chapter 10
Multicriteria Decision-Marking Models
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Application Context
multiple objectives that cannot be put under a single measure; e.g.,
distribution: cost and time
as a single objective function problem if time can be converted into cost
supply chain: customer service and inventory cost 多目標而目標沒有
共同的衡量方式。
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Chapter Summary
10.0 Scoring Model 評點法 10.1 Weighting Method 權重法 10.2 Goal Programming 目標規劃 10.3 AHP (Analytical Hierarchy Process) 層
級分析法我們只學每個方法
基本的概念。
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Motivation Problem
dinner, two factors to consider: distance and cost
three restaurants A: (2, 3) B: (7, 1) C: (4, 2)
which one to choose?
home
(distance from home, cost)
(2, $$$)
(7, $)
(4, $$)
A
B
C
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Scoring Model 評點法
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Scoring Model 評點法 a subjective method
assign weights to each criterion assign a rating for each decision alternative on each
criterion
Restaurant Selection Example: Version 1 min w1 (distance) + w2 (cost)
home
(distance from home, cost)
(2, $$$)
(7, $)
(4, $$)
AB
C
w1 w2 A (2, $$$) B (7, $) C (4, $$) choice
1 1 5 8 6 A
1 3 11 10 10 B, C
Version 1 只需要決定各目標的權重。
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Scoring Model 評點法 Restaurant Selection Example: Version 2
weights of criteria, and ratings on criteria for alternatives
Example: Tom dislikes walking and likes good food (from expensive restaurants)
每個選擇在每一項目標中有點數( ratings, scores ),而目標
有各自的權重( weight )。
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Scoring Model 評點法 Restaurant Selection Example, Version 2
weights of walking and price by Tom: 1 to 1
ratings (scores) of each restaurant for walking and price:
56C (4, $$)
23B (7, $)
810A (2, $$$)
w2 = 1w1 = 1
pricedistance
restaurantcriterion
objective of Tom: max w1 (rating of distance)
+ w2 (rating of price)
dislikes walking and likes expensive, good food
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Scoring Model 評點法 a subjective method on assigning
weights ratings
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Example 10-5 Product Selection
to expand the product line by adding one of the following: microwave ovens, refrigerators, and stoves
decision criteria manufacturing capability/cost market demand profit margin long-term profitability/growth transportation costs useful life
assigning weights to the criteria and ratings to the three alternatives for each criterion
maximizing the total score
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Example 10-5 Product Selection
Scoremicro = 4(4)+5(8)+3(6)+5(3)+2(9)+1(1) = 108
Scorerefer = 4(3)+5(4)+3(9)+5(6)+2(2)+1(5) = 98
Scorestove = 4(8)+5(2)+3(5)+5(7)+2(4)+1(6) = 106
weight microwave refers stoves
manuf. cap./cost 4 4 3 8market demand 5 8 4 2profit margin 3 6 9 5(long-term) prof./growth 5 3 6 7
Transp. costs 2 9 2 4useful life 1 1 5 6
any comments on the relative
values?
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Weighting Method 權重法
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Weighting Method 權重法 a form of scoring method transforming a multi- to a single-
criterion objective function by finding the weights of the criteria
以目標的權重( weight )將多目標的問題轉化為單目標的問
題。
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Weighting Method 權重法 max Z(x) = [z1(x), z2(x), …, zP(x)]
s.t. x S turning into a single-criterion objective
function by weighting (with weights)
max Z(x) = w1z1(x)+w2z2(x)+… +wpzP(x) s.t. x S
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Weighting Method 權重法 criteria (i.e., objectives)
max z1(x) = 2x1+3x2x3
min z2(x) = 6x1x2
max z3(x) = 2x1+x3
constraints x1+x2+x3 15
x1+2x2+x3 20
x3 2 x1, x2, x3 0
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Weighting Method 權重法 somehow got: w1 = 1, w2 = 2, w3 = 4
max z1(x)2z2(x)+4z3(x) = (2x1+3x2x3) 2(6x1x2) + 4(2x1+x3) = 18x1+5x2+3x3,
s.t. x1+x2+x3 15 ; x1+2x2+x3 20; x3 2; x1, x2, x3 0.
max z1(x) = 2x1+3x2x3, min z2(x) = 6x1x2
,, max z3(x) = 2x1+x3,
s.t. x1+x2+x3 15 ; x1+2x2+x3 20; x3 2; x1, x2, x3 0.
negative sign
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Goal Programming 目標規劃
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1/4: Introducing the Ideas of Goal Programming
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Goal Programming
GP: priority + goal
priority of the goals (i.e., of the criteria)
(saving) money is most important: B
(shortest) distance is most important: A
(best) food is the most important: A
home
(2, $$$)
(7, $)
(4, $$)
A
B
C
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Goal Programming
a goal an objective with a desirable quantity no good to be over and under this quantity
short, not enough exercise,distance
long, too tiring.
goal
v(
)over u(
)under
low, not tasty,money
high, expensive.
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General Idea of Goal Programming
suppose the goals are: 3 units for distance, and 2 units (i.e., $$) for price
C(4, 2)
B(7, 1)
A(2, 3)
v(p,)u(p,)v(d,)u(d,)
pricedistance
home
(2, $$$)
(7, $)
(4, $$)
AB
C
0010
0140
1001
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General Idea of Goal Programming
priority P1 > P2 > P3 > …
0010C
0140B
1001A
v(p,)u(p,)v(d,)u(d,)
pricedistanceP1up, P2vd, P3ud, P4vp
A
P1up, P2ud, P3vd, P4vp
C
P1vp, P2ud, P3vd, P4up
C
B is dominated by C, i.e., C is optimal for any priority that B is optimal.
P1up > P2vd > P3ud > P4vp
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2/4 : A More General Goal Programming Approach
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General Idea of Goal Programming
a goal program parts like a linear program
with decisions variables with hard constraints
parts unlike a linear program with soft constraints
expressed as goals to be achieved co-existence of constraints such as x1 10 and x1 7 in a
GP if they are soft constraints with the objective function in LP replaced by the
priorities of goals in GP
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Deviation Variables for a Soft Constraints
example: a soft constraint on labor hour x1 units of product 1, each for 4 labor hours
x2 units of product 2, each for 2 labor hours
goal: 100 labor hours
a soft constraint: 4x1+2x2 100
2 deviation variables u and v: 4x1+2x2 + u v = 100
u: under utilization of labor v: over utilization of labor
人世間有不少 soft constraints (可以
斟酌的限制式)
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Example 10-1: Formulation of a GP
three products, quantities to produce x1, x2, and x3
objectives in order of priority min overtime in assembly min undertime in assembly min sum of undertime and overtime in packaging
product x1 x2 x3 availabilitymaterial (lb/unit) 2 4 3 600 pounds
assembly (min. unit) 9 8 7 900 minutes
packaging (min/unit) 1 2 3 300 minutes
Suppose that the material availability is a hard constraint, i.e., there is no
way to get more material.
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Example 10-1: Formulation of a GP
GP min P1v1, P2u1, P3(u2+v2), s.t. 2x1 + 4x2 + 3x3 600 (lb., hard
const.) 9x1 + 8x2 + 7x3 + u1 v1 = 900 (min., soft
const.) 1x1 + 2x2 + 3x3 + u2 v2 = 300 (min., soft
const.) all variables 0
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3/4 : Solution of a Goal Program
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Example: Solution of a GP
min P1u1, P2u2, P3u3,
s.t.
5x1 + 3x2 150 (hard const.)(A)
2x1 + 5x2 + u1 v1 = 100 (soft const.) (1)
3x1 + 3x2 + u2 v2 = 180 (soft const.) (2)
x1 + u3 v3 = 40 (soft const.)(3)
all variables 0
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u1 = 0, v1 > 0
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Example: Solution of a GP
min P1u1, P2u2, P3u3, s.t.5x1 + 3x2 150 (A) 2x1 + 5x2 + u1 v1 = 100 (1)3x1 + 3x2 + u2 v2 = 180 (2) x1 + u3 v3 = 40 (3) all variables 0
x1
x2
50
30
feasible solution
space
5x1 + 3x2 = 150
x1
x2
50
20 2x1 + 5x2 = 100
u1 > 0, v1 = 0
direction of improvement in
P1
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Example: Solution of a GP
x1
x2
50
30
20
50
Hard (A
)
Soft (1)
P1
region with u1 = 0
x1
x2
50
30
20
50
Hard (A
)
Soft (1)
P1
optimal with (A), (1), and (2)60
60
Soft (2)
P2
x1
x2
50
30
20
50
Hard (A
)
Soft (1)
P1
optimal with (A), (1), (2),
and (3)
60
60
Soft (2)
P2 P3
Soft (3)
Actually at this point we know that the point is optimal even
with the third constraint added and the third goal considered.
Why?
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Example 10-2
min P1v1, P2u2, P3v3, s.t.5x1 + 4x2 + u1 v1 = 200 (1)2x1 + x2 + u2 v2 = 40 (2)2x1 + 2x2 + u3 v3 = 30 (3) all variables 0x1
x2
50
40
P1
region with v1 = 0
40
20 x1
x2
50
40
P1
P2
40
20 x1
x2
50
40
P1
P215
15
P3
optimal, with v1 = u2 = v3 = 0
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4/4 : Another Form of Goal Programming
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Another Form of GP: Weighted Goals
goals with weights u1 = 30, u2 = 20, v2 = 20, u3 = 20, v3 = 10
the GP expressed as LP
min P1u1, P2u2, P3u3, s.t.5x1 + 3x2 150 (A) 2x1 + 5x2 + u1 v1 = 100 (1)3x1 + 3x2 + u2 v2 = 180 (2) x1 + u3 v3 = 40 (3) all variables 0
min 30u1+20u2+20v3 +20u3 + 10v3 s.t.5x1 + 3x2 150 (A) 2x1 + 5x2 + u1 v1 = 100 (1)3x1 + 3x2 + u2 v2 = 180 (2) x1 + u3 v3 = 40 (3) all variables 0
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Assignment #4
#1. Chapter 8, Problem 16 (a). Find the maximal flow for this network.
Show all the steps. (b). Formulate this problem as a linear
program.
#2. Chapter 10, Problem 1
#2. Chapter 10, Problem 4