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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

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