module - 9 lecture notes 3 multi-criteria...

Water Resources Systems Planning and Management: Advanced Topics – Multi-criteria Decision Making

D Nagesh Kumar, IISc, Bangalore

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MODULE - 9 LECTURE NOTES – 3

MULTI-CRITERIA DECISION MAKING

INTRODUCTION

Multicriterion decision making (MCDM) is a process of evaluating real world situations,

based on various qualitative / quantitative criteria in certain / uncertain / risky environments

to suggest a course of action / choice/ strategy / policy among the available options. In this

lecture we will discuss about the structured decision making and various MCDM methods.

STRUCTURED DECISION MAKING

The evaluation of multicriterion situations using conventional approaches may be difficult. A

structured decision making (SDM) is necessary to visualize the decision making. It also helps

in replicating the same steps if the given decision is proved to be right. The steps in SDM are

shown in figure 1. These steps are to be followed to arrive at an effective decision.

Fig. 1 Steps in SDM

Purpose and necessity of decision making

Identification of decision makers

Identification of criteria that will influence decision making

Assume suitable priority to each criterion

Formulate various strategies on which criteria can be evaluated

Estimate each strategy using SDM

Determine the suitable strategy

Implement the strategy

Evaluate the outcome of the decision

Document the lessons that have been learnt in this process for

further improvement of decision making skills



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STEPS IN MCDM METHODOLOGY

The steps for the selection of best alternative from a set of available alternatives are

(Duckstein et al., 1989):

Defining the problem and fixing the criteria

Data collection

Establishment of feasible alternatives

Formulation of payoff matrix i.e., matrix comprising evaluation of alternatives with

reference to criteria

Selection of appropriate method

Incorporation of decision maker’s preferences

Choosing one or more best alternatives for further analysis

MCDM METHODS

MCDM methods can be classified into four groups:

1. Distance

(a) Compromise Programming (CP)

(b) Cooperative Game Theory (CGT)

(c) Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS)

(d) Composite Programming (COP)

2. Outranking

(a) Preference Ranking Organisation METHod of Enrichment Evaluation

(PROMETHEE)

(b) ELimination Et Choix Traduisant la REalite (ELECTRE)

3. Priority / Utility and

(a) Weighted Average Method

(b) Multi Attribute Utility Theory

(c) Analytic Hierarchy Process

4. Mixed category

(a) Multicriterion Q- Analysis -2

(b) EXPROM-2

(c) STOPROM-2

In this lecture we will discuss about compromise programming, PROMETHEE and weighted

average method.



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COMPROMISE PROGRAMMING (CP)

The objective in CP is to obtain a solution that is as ‘close’ as possible to some ‘ideal’

solution in terms of distance. The distance measure used in Compromise Programming is the

family of Lp - metrics and expressed as

pJ

j

p

jj

p

jp affwaL

1

1

)()( * (1)

Normalizing between the range [0, 1], eqn. (1) becomes,

pJ

j

p

jj

jjp

jpmM

affwaL

1

1

)()(

*

(2)

where )(aLp = pL - metric for alternative a, )(af j = Value of criterion j for alternative a,

jM = Maximum value of criterion j in set N , jm = Minimum value of criterion j in set N ,

*

jf = Ideal value of criterion j, jw = Weight assigned to the criterion j, p =

Parameter/balancing factor reflecting the attitude of the decision maker with respect to

compensation between deviations. For p = 1, all deviations from *

jf are taken into account in

direct proportion to their magnitudes. For p , the largest deviation is the only one taken

into account corresponding to zero compensation between deviations. The flow chart of CP

methodology is given in figure 2.

Fig. 2 Flow chart of Compromise Programming methodology

Enter the number of alternatives, criteria, payoff matrix, and weight

of each criterion

Specify the parameter p ; Compute pL - metric value

Print results

Start

Stop

Rank alternatives based on minimum pL - metric value



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Example

Compute pL - metric values of alternatives and corresponding ranking pattern for the payoff

matrix presented in Table 1 [Brans et al. (1986)] using Compromise Programming method

for p = 1, 2, . Assume equal weights for each criterion. Alternatives A1 to A6 in payoff

matrix represent hydropower projects and criteria C1 to C6 correspond to man power,

Hydropower (MW), construction cost (109 $), maintenance cost (10

6 $), number of villages

to be evacuated and security level respectively.

Table 1. Payoff Matrix

Crit.

Alt.

C1 C2 C3 C4 C5 C6

A1 80 90 6 5.4 8.0 5

A2 65 58 2 9.7 1.0 1

A3 83 60 4 7.2 4.0 7

A4 40 80 10 7.5 7.0 10

A5 52 72 6 2.0 3.0 8

A6 94 96 7 3.6 5.0 6

Max/Min Min Max Min Min Min Max

Solution:

Negative sign is assigned to the criterion of minimization in nature to enable to analyze the

problem uniformly in maximization perspective i.e., (-min) = max. Table 2 presents payoff

matrix after this transformation, where all criteria are made to be of maximization in nature.

Table 2. Transformed payoff matrix

Crit.

Alt.

C1 C2 C3 C4 C5 C6

A1 -80 90 -6 -5.4 -8.0 5

A2 -65 58 -2 -9.7 -1.0 1

A3 -83 60 -4 -7.2 -4.0 7

A4 -40 80 -10 -7.5 -7.0 10

A5 -52 72 -6 -2.0 -3.0 8

A6 -94 96 -7 -3.6 -5.0 6



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pL - metric value can be computed as follows:

pJ

j

p

jj

jjp

jpmM

affwaL

1

1

)()(

*

Parameters that are required for the computation of pL - metric value are: maximum,

minimum and ideal value for each criterion, weight of each criterion and parameter p. These

parameters are presented in Table 3.

Table 3. Parameters required for computation of pL - metric value

Parameters

required for

each criterion

Notation C1 C2 C3 C4 C5 C6

Maximum

value jM -40.00 96.00 -2.00 -2.00 -1.00 10.00

Minimum

value jm -94.00 58.00 -10.00 -9.70 -8.00 1.00

Ideal value *

jf -40.00 96.00 -2.00 -2.00 -1.00 10.00

Weights jw 1 1 1 1 1 1

Normalized*

weights jw 0.1666 0.1666 0.1666 0.1666 0.1666 0.1666

* Since there are six criteria of equal importance, normalized weight of each criterion is 1/6

i.e., 0.1666 each.

pL - metric value of alternative A1 i.e., pL ( A1) is computed [on the basis of equation (2),

Table 2 and Table 3] as below:

Criterion C1 =

p

p

))00.94(00.40(

))00.80(00.40(1666.0 =

p1234074.0

Criterion C2 =

p

p

)00.5800.96(

)00.9000.96(1666.0 =

p02630526.0

Criterion C3 =

p

p

))00.10(00.2(

))00.6(00.2(1666.0 =

p0833.0

Criterion C4 =

p

p

))70.9(00.2(

))4.5(00.2(1666.0 =

p07356364.0



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Criterion C5 =

p

p

))00.8(00.1(

))00.8(00.1(1666.0 = p1666.0

Criterion C6 =

p

p

)00.100.10(

)00.500.10(1666.0 = p09255556.0

For alternative A1, pL - metric value for given p is

ppppppp

1

09255556.01666.007356364.00833.002630526.01234074.0

For p = 1 , 1L (A1) is as follows:

1

1

09255556.01666.007356364.00833.002630526.01234074.0 = 0.56573

For p = 2 , 2L (A1) is as follows:

2

1222222 09255556.01666.007356364.00833.002630526.01234074.0 =0.25415

For p = 10 (approximating for ), 10L (A1) is as follows:

10

1101010101010 09255556.01666.007356364.00833.002630526.01234074.0 =

0.16748

Similarly pL - metric values for other alternatives are computed. Best alternative is the one

which is having minimum pL - metric value from ideal solution. Accordingly ranking of

alternatives is done. Table 4 presents pL - metric values of alternatives A1 to A6 and

corresponding ranking pattern (values in parenthesis).

Table 4. pL - metric values and corresponding ranking pattern

Alternative p = 1 p = 2 p =

A1 0.56573 (4) 0.25415 (3) 0.16748 (4)

A2 0.57693 (6) 0.29869 (6) 0.18595 (6)

A3 0.57159 (5) 0.25512 (4) 0.16087 (2)

A4 0.49855 (3) 0.25929 (5) 0.17034 (5)

A5 0.31017 (1) 0.15171 (1) 0.10620 (1)

A6 0.47459 (2) 0.23311 (2) 0.16682 (3)



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It is observed that for p = 1, alternatives A5, A6,A4, A1 , A2, A3 occupied ranks 1 to 6. For p =

2, these are A5, A6, A1, A3 , A4, A2 whereas for p = , these are A5, A3, A6, A1 , A4, A2. It

is observed that first position is occupied by A5 for all the three scenarios of p = 1, 2, .

PROMETHEE-2

PROMETHEE-2 (Preference Ranking Organisation METHod of Enrichment

Evaluation) is a MCDM method of outranking nature. The method is based on preference

function approach. Mathematically, preference function ),( baPj depends on the pairwise

difference jd between the evaluations )(af j and )(bf j of alternatives a and b for criterion

j, chosen criterion function and corresponding parameters (here criterion and criterion

function are different). Six types of criterion functions are listed in Table 5. Similarly

parameter jq represents indifference threshold that represents the largest difference that is

considered negligible by the decision maker when comparing two alternatives on that

criterion; parameter jp represents the smallest difference that justifies a strict preference for

one of the two alternatives. Mathematical analysis of preference function and its relationship

with jd for various criterion functions are also presented in Table 6. Multicriterion

Preference Index, ),( ba , a weighted average of the preference functions ),( baPj for all the

criterion is defined as:

J

jj

j

J

jj

w

baPw

ba

1

1

),(

),( (3)

)1(

),(

)(N

ba

a A (4)

)1(

),(

)(N

ab

a A

(5)

)()()( aaa (6)



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Table 5. Types of various criterion functions and relevant preference function values in

PROMETHEE-2

Types of generalized criterion functions Preference function values for various types

of criterion functions

1 Usual

criterion

01

00)(

j

j

jdif

difdH

2 Quasi

criterion

jj

jj

jqdif

qdifdH

1

0)(

3

Criterion with

linear

preference

and no

indifference

area

jj

jj

j

j

j

pdif

pdifp

d

dH

1

)(

4 Level

criterion

jj

jjj

jj

j

pdif

pdqif

qdif

dH

1

5.0

0

)(

5

Criterion with

linear

preference

and

indifference

area

jj

jjj

jj

jj

jj

j

pdif

pdqifqp

qd

qdif

dH

1

)(

)(

0

)(

6 Gaussian

criterion

]1[)(2'

2

2 j

jd

j edH

The value of '

j is the distance between

the origin and the point of inflexion of the

considered preference function.

where jw = Weight assigned to the criterion j; )(a = Outranking index of a in the

alternatives set N ; )(a = Outranked index of a in the alternatives set N ; )(a = Net

ranking of a in the alternatives set N ; J = Number of criteria. The alternative having the

H (d j)

d j

1

H (d j)

1

q j d j

H (d j)

1

p j d j

H (d j)

1

d j

0.5

-p j -q j q j p j

H (d j)

1

d j -p j -q j q j p j

1

d j - ‘j

‘j

H (d j)



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Yes

highest )(a value is considered to be the best/suitable. Figure 3 presents flow chart of the

PROMETHEE-2 methodology.

Figure 3 Flow chart of PROMETHEE-2 methodology

Start

Enter number of alternatives, criteria, payoff matrix, weight of each criterion

Compute pairwise difference between values of alternatives for each criterion ( )jd

Choose type of criterion function, indifference and/or preference threshold values for

each criterion.

Compute preference function matrix for each criterion based on jd and type of

chosen criterion function

Compute Multicriterion Preference Index

Compute net value for each alternative and corresponding rank

Select the best/suitable alternative having highest net value

Any need to specify another set of parameters

Print results

Stop



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Example

Compute ranking pattern of alternatives for the payoff matrix presented in Table 1 using

PROMETHEE-2 method. Parameters required for the method are presented in Table 6.

Assume equal weights for each criterion.

Table 6 Parameters required for PROMETHEE-2 method

Criteria Type Criterion function Parameters (thresholds) Normalized

weights

of the criterion

C1 2 Quasi criterion Indifference jq = 10 0.1666

C2 3 Linear preference

and no indifference

area

Preference jp = 30 0.1666

C3 5 Linear preference

and indifference

area

Indifference jq = 0.50

Preference jp = 5.00

0.1666

C4 4 Level criterion Indifference jq = 1.00

Preference jp = 6.00

0.1666

C5 1 Usual criterion _______ 0.1666

C6 6 Gaussian criterion '

j = 5.00 0.1666

Solution:

Table 2 is reproduced here as Table 7.

Table 7 Transformed payoff matrix

Crit.

Alt.

C1 C2 C3 C4 C5 C6

A1 -80 90 -6 -5.4 -8.0 5

A2 -65 58 -2 -9.7 -1.0 1

A3 -83 60 -4 -7.2 -4.0 7

A4 -40 80 -10 -7.5 -7.0 10

A5 -52 72 -6 -2.0 -3.0 8

A6 -94 96 -7 -3.6 -5.0 6

Solution methodology of PROMETHEE-2 is divided into various steps for the computation

of net and ranking pattern as explained below (Analysis is performed for maximization and

accordingly right hand side portion of Table 5 i.e., after H(dj) function mark is considered).



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Step 1: Pairwise difference between values of alternatives for criteria

Pairwise difference between values of alternatives for each criterion (six in this case) is to be

performed. For example, for criterion C1, pairwise difference between alternatives A1 and A2

are -80-(-65) = -15. Similarly pairwise difference between alternatives A2 and A1 for C1 are -

65- (-80) = 15. Pairwise difference between alternative A1 versus A1 is zero as they are

comparing the same i.e., alternative A1 versus A1. Table 8 (a) to Table 8 (f) present pairwise

differences between alternatives for criterion C1 to C6.

Table 8 (a) Pairwise difference matrix and preference function values for C1: Man power

(Quasi criterion function; jq = 10)

Alt. A1 A2 A3 A4 A5 A6

A1 0

(0)

-15

(0)

3

(0)

-40

(0)

-28

(0)

14

(1)

A2 15

(1)

0

(0)

18

(1)

-25

(0)

-13

(0)

29

(1)

A3 -3

(0)

-18

(0)

0

(0)

-43

(0)

-31

(0)

11

(1)

A4 40

(1)

25

(1)

43

(1)

0

(0)

12

(1)

54

(1)

A5 28

(1)

13

(1)

31

(1)

-12

(0)

0

(0)

42

(1)

A6 -14

(0)

-29

(0)

-11

(0)

-54

(0)

-42

(0)

0

(0)

Table 8 (b) Pairwise difference matrix and preference function values for C2: Power (Linear

preference and no indifference area; jp = 30)


A1

0

(0)

32

(1)

30

(1)

10

(0.3333)

18

(0.6)

-6

(0)

A2

-32

(0)

0

(0)

-2

(0)

-22

(0)

-14

(0)

-38

(0)

A3 -30

(0)

2

(0.0667)

0

(0)

-20

(0)

-12

(0)

-36

(0)

A4 -10

(0)

22

(0.7333)

20

(0.6667)

0

(0)

8

(0.2667)

-16

(0)

A5

-18

(0)

14

(0.4667)

12

(0.4)

-8

(0)

0

(0)

-24

(0)

A6 6

(0.2)

38

(1)

36

(1)

16

(0.5333)

24

(0.8)

0

(0)



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Table 8 (c) Pairwise difference matrix and preference function values for C3: Construction

cost (109 $) (Linear preference and indifference area jq = 0.50; jp = 5.00)


A1

0

(0)

-4

(0)

-2

(0)

4

(0.7778)

0

(0)

1

(0.1111)

A2

4

(0.7778)

0

(0)

2

(0.3333)

8

(1)

4

(0.7778)

5

(1)

A3 2

(0.3333)

-2

(0)

0

(0)

6

(1)

2

(0.3333)

3

(0.5556)

A4 -4

(0)

-8

(0)

-6

(0)

0

(0)

-4

(0)

-3

(0)

A5

0

(0)

-4

(0)

-2

(0

4

(0.7778)

0

(0)

1

(0.1111)

A6 -1

(0)

-5

(0)

-3

(0)

3

(0.5556)

-1

(0)

0

(0)

Table 8 (d) Pairwise difference matrix and preference function values for C4: Maintenance

cost (109 $) (Level criterion jq = 1.00; jp = 6.00)


A1

0

(0)

4.3

(0.5)

1.8

(0.5)

2.1

(0.5)

-3.4

(0)

-1.8

(0)

A2

-4.3

(0)

0

(0)

-2.5

(0)

-2.2

(0)

-7.7

(0)

-6.1

(0)

A3 -1.8

(0)

2.5

(0.5)

0

(0)

0.3

(0)

-5.2

(0)

-3.6

(0)

A4 -2.1

(0)

2.2

(0.5)

-0.3

(0)

0

(0)

-5.5

(0)

-3.9

(0)

A5

3.4

(0.5)

7.7

(1)

5.2

(0.5)

5.5

(0.5)

0

(0)

1.6

(0.5)

A6 1.8

(0.5)

6.1

(1)

3.6

(0.5)

3.9

(0.5)

-1.6

(0)

0

(0)



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Table 8 (e) Pairwise difference matrix and preference function values for C5: Number of

villages to be evacuated (Usual criterion)


A1

0

(0)

-7

(0)

-4

(0)

-1

(0)

-5

(0)

-3

(0)

A2

7

(1)

0

(0)

3

(1)

6

(1)

2

(1)

4

(1)

A3 4

(1)

-3

(0)

0

(0)

3

(1)

-1

(0)

1

(1)

A4 1

(1)

-6

(0)

-3

(0)

0

(0)

-4

(0)

-2

(0)

A5

5

(1)

-2

(0)

1

(1)

4

(1)

0

(0)

2

(1)

A6 3

(1)

-4

(0)

-1

(0)

2

(1)

-2

(0)

0

(0)

Table 8 (f) Pairwise difference matrix and preference function values for C6: Security level

(Gaussian criterion: '

j = 5.00)


A1

0

(0)

4

(0.2739)

-2

(0)

-5

(0)

-3

(0)

-1

(0)

A2

-4

(0)

0

(0)

-6

(0)

-9

(0)

-7

(0)

-5

(0)

A3 2

(0.0769)

6

(0.5132)

0

(0)

-3

(0)

-1

(0)

1

(0.0198)

A4 5

(0.3935)

9

(0.8021)

3

(0.1647)

0

(0)

2

(0.0769)

4

(0.2739)

A5

3

(0.1647)

7

(0.6247)

1

(0.0198)

-2

(0)

0

(0)

2

(0.0769)

A6 1

(0.0198)

5

(0.3935)

-1

(0)

-4

(0)

-2

(0)

0

(0)

Step 2: Exploring preference function values for criteria C1 to C6 (on the basis of Figure 3,

Tables 6 and 7) : Sample calculations are presented below with reference to A1 and A2:

Criterion C1: Type 2: Quasi criterion function; Indifference threshold jq = 10

Pairwise difference between alternatives A1 and A2 for criterion C1 are -80-(-65) = -15. For

quasi criterion function, with indifference threshold value of 10, preference function value

P1(A1, A2) = 0 (as -15 ).10 Here P1 () represents for criterion C1 . Similarly difference for

alternatives A2 and A1 are -65-(-80) = 15 and corresponding preference function value P1(A2,



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A1) = 1 (15 ).10 In case of Quasi criterion function, elements of preference function matrix

are either 0 or 1.

Criterion C2: Type 3: Linear preference and no indifference area; preference threshold jp =

30

Pairwise difference between alternatives A1 and A2 for criterion C2 is 90-58 = 32 and

corresponding preference function value P2(A1, A2) is 1 (as 32 ).30 Similarly for A2 to A1

pairwise difference is -32 and corresponding P2(A2,A1) = 0. In case of Linear preference and

no indifference area criterion function, elements of preference function matrix vary between

0 and 1.

Criterion C3: Type 5: Linear preference and indifference area; Indifference threshold jq =

0.50; Preference threshold jp = 5.00

Pairwise difference between alternatives A1 and A2 for criterion C3 is -6-(-2) = - 4 and

corresponding preference function value P3(A1, A2) is 0 (as -4 ).5.0 Similarly for A2 to A1

pairwise difference is 4 (greater than jq value of 0.5 and less than jp value of 5) and

corresponding P3(A2,A1) is computed as (4-0.5)/(5-0.5) = 0.7778. In case of Linear

preference and indifference area criterion function, elements of preference function matrix

vary between 0 and 1.

Criterion C4: Type 4: Level criterion; Indifference threshold jq = 1.00; Preference threshold

jp = 6.00

Pairwise difference between alternatives A1 and A2 for criterion C4 is -5.4-(-9.7) = 4.3. As

pairwise difference is 4.3 (greater than jq value of 1.00 and less than jp value of 6), and

corresponding preference function value P4(A1, A2) = 0.5. In case of Level criterion, elements

of preference function matrix take values either 0, 0.5 or 1.

Criterion C5: Type 1: Usual criterion

Pairwise difference between alternatives A1 and A2 for criterion C5 is -8-(-1) = -7 and

corresponding preference function value P5(A1, A2) is 0 (as -7 ).0 Similarly for A2 to A1

pairwise difference is 7 and corresponding preference function value P5(A2, A1) is taken as 1.

In case of Usual criterion function, elements of preference function matrix are either 0 or 1.

Criterion C6: Type 6: Gaussian criterion '

j = 5.00

Pairwise difference between alternatives A1 and A2 for criterion C6 is 5-1 = 4 and

corresponding preference function value P6(A1, A2) is 0.2739. Similarly pairwise difference



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between A2 and A1 is -4 and preference function value P6(A2, A1) = 0.0. In case of Gaussian

criterion function, elements of preference function matrix vary between 0 and 1.

Table 9 (a) to 9 (f) present the preference function values for criteria C1 to C6 (values in

parenthesis) corresponding to pairwise difference matrices of C1 to C6.

Step 3: Computation of Multicriterion Preference Index (based on equation 3)

Multicriterion Preference Index, ),( 21 AA for pairwise alternatives (A1,A2) is computed as

follows:

Preference function values for A1 and A2 for criterion C1 to C6 are 0, 1.0, 0, 0.5, 0, and

0.2739. Corresponding weights of criteria are 1, 1, 1, 1, 1, 1 (or normalized weights of the

criteria are 0.1666 each). Substitution in equation (3) yields,

),( 21 AA = 2956.06

7739.1

]111111[

12739.01015.0101110

Similarly, ),( 12 AA 4630.06

7778.2

]111111[

10111017778.01011

Table 9 presents Multicriterion Preference Index values.

Table 9 Multicriterion Preference Index values


A1 0 0.2956 0.2500 0.2685 0.1000 0.1852

A2 0.4630 0 0.3889 0.3333 0.2963 0.5000

A3 0.2350 0.1800 0 0.3333 0.0556 0.4292

A4 0.3989 0.5059 0.3052 0 0.2239 0.2123

A5 0.4441 0.5152 0.4866 0.3796 0 0.4480

A6 0.2866 0.3989 0.2500 0.4315 0.1333 0

Step 4: Computation of (as per equation 4)

(A1) = 2199.05

1852.01000.02685.02500.02956.00

Step 5: Computation of (as per equation 5)

(A1) = 3655.05

2866.04441.03989.02350.04630.00

Step 6: Computation of net (as per equation 6)

Net (A1) = (A1) - (A1) = 0.2199 - 0.3655 = -0.1456



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Similarly , ,net values for other alternatives are computed. The best alternative is

the one which is having the highest net value. Accordingly, ranking of alternatives is done.

Table 10 presents , , net and corresponding ranking pattern of each alternative.

Table 10. , , and ranking pattern for alternatives

Alt. Rank

A1 0.2199 0.3655 -0.1456 6

A2 0.3963 0.3791 0.0172 2

A3 0.2466 0.3362 -0.0896 5

A4 0.3293 0.3493 -0.0200 3

A5 0.4547 0.1618 0.2929 1

A6 0.3001 0.3549 -0.0548 4

It is observed from Table 10 that ranking pattern in the order of alternatives A1 to A6 is 6, 2,

5, 3, 1, 4 and alternative A5 with the highest net value of 0.2929 is considered as the best.

WEIGHTED AVERAGE METHOD

Weighted Average method is a utility type MCDM method. It is expressed as the average of

the weighted sum of criterion values i.e., jja uwuwuwU .......2211 where aU is the

overall utility value for alternative a; jwww ......,, 21 are the weights assigned to the criterion

and juuu ,.....,, 21 are the corresponding criteria values. The alternative having the highest

overall utility is considered the best. Suitable normalization approach can be used wherever

necessary.

Example

Compute priority of alternatives for the payoff matrix presented in Table 11 using Weighted

Average method. Assume weights for each criterion are 0.1, 0.12, 0.15, 0.20, 0.33, 0.1. Use

normalization method 4 for analysis.



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Table 11 Payoff matrix

Crit.

Alt. C1 C2 C3 C4 C5 C6

A1 8 9 6 5 2 4

A2 6.5 5.8 2 9 6 3

A3 8.3 6 4 7 4 4

A4 4 8 8 7 3 8

A5 5.2 7.2 4 2 1 6

A6 9.4 9 6 3.2 2 4

Solution:

Normalization of payoff matrix presented in Table 11 is performed using the formula

N

aj

j

af

af

1

2)(

)( and presented in Table 12.

Normalized weights of the criterion = 0.1, 0.12, 0.15, 0.20, 0.33, 0.1

Table 12. Normalized payoff matrix and weighted average values

Alt.

Normalized values of criterion Weighted

average

value

Rank

C1 C2 C3 C4 C5 C6

A1 0.45693 0.48281 0.45750 0.33846 0.23905 0.31923 0.35075 4

A2 0.37125 0.31115 0.15250 0.60922 0.71714 0.23943 0.47978 1

A3 0.47406 0.32187 0.30500 0.47384 0.47809 0.31923 0.41624 3

A4 0.22846 0.42917 0.60999 0.47384 0.35857 0.63847 0.44279 2

A5 0.29700 0.38625 0.30500 0.13538 0.11952 0.47885 0.23620 6

A6 0.53689 0.48281 0.45750 0.21661 0.23905 0.31923 0.33438 5

Weighted average of alternative A1 is computed as follows:

jjA uwuwuwU .......22111 i.e.,

35075.031923.01.023905.033.0

33846.02.045750.015.048281.012.045693.01.0(1AU



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Similarly weighted average values for other alternatives are computed. The best alternative is

the one which is having the highest weighted average value. Accordingly ranking of

alternatives is done. Table 12 presents the weighted average values of alternatives and

corresponding ranking pattern. It is observed that the ranking pattern in the order of

alternatives A1 to A6 is 4, 1, 3, 2, 6, 5. It is observed that A2 and A4 occupied first and second

positions due to their higher utility values of 0.47978 and 0.44279.

module - 9 lecture notes 3 multi-criteria...

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