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A Multi-CriterionDecision Making

Approach toProblem Solving

M. HERMAN, IrM. HERMAN, IrRoyal Defense College (Brussels - Belgium)Royal Defense College (Brussels - Belgium)

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MCDM, Quality and Productivity

• Actions : Alternative Strategies, Procedures for improvement

• Criteria : impact on– Productivity (% process time adding value)– Quality

• Customer satisfaction• Timeliness of the production/service• Accuracy of results• Efficiency of the process (reduce rework)

– Cost-effectiveness

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MCDM, Quality and Productivity

• Data : Assessment of Actions on Criteria– Measurements : numerical data– Ranking of qualitative assessments : ordinal

data• Problem : Rank or Select alternative

strategies or procedures for improvement

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Some Typical MCDM Applications

• Selection of high-tech industrial development zones

• A multi-attribute decision making approach for industrial prioritisation

• Selection of a thermal power plant location• An approach to industrial locations

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Some MCDM Applications (cont.)

• Selecting oil and gas wells for exploration • Multi-attribute decision modelling for tactical

and operations management planning in a batch processing environment

• New campus selection by an MCDM approach• Selection of an automated inspection system• Selection of an incident management procedure

in a computer center

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Some MCDM Applications (cont.)• Acquisition of equipment (vehicles, helicopters,

computers,...)• Personnel selection and ranking• Personnel assignment to jobs• Ranking and selection of investment plans• Ranking of loan requests by banks• Burden sharing allocation in international organisations

(EU, ASEAN,…)• …...

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Early Literature (1)

• B. Roy, “Méthodologie multicritère d’aide à la décision”, Economica, Paris, 423 p, 1985 - translated into English

• B. Roy and D. Bouyssou, “Aide multicritère à la Décision : Méthodes et Cas”, Economica, Paris, 700 p, 1993

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Early Literature (2)• J.P. Brans, B. Maréschal and Ph. Vincke,

“How to select and how to rank projects : the Prométhée Method”, EJOR (European Journal of O.R.), 24, pp. 228-238, 1986

• B. Maréschal and J.P. Brans, “Geometrical Representation for MCDM, the GAIA procedure”, EJOR (European Journal of O.R.), 34, pp. 69-77, 1988

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Early Literature (3)• M. Roubens, “Analyse et agrégation des

préférences : modélisation, ajustement et résumé de données relationnelles”, Revue Belge Stat. Inf. O.R. (JORBEL) 20(2), pp. 36-67, 1980

• M. Roubens, “Preference Relations on Actions and Criteria in Multicriteria Decision Making”, EJOR 10, pp. 51-55, 1982

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Early Literature (4)

• R. Van den Berghe and G. Van Velthoven, “Sélection multicritère en matière de rééquipement”, Revue X (Belgium), Vol. 4, pp. 1-8, 1982

• H. Pastijn and J. Leysen, “Constructing an Outranking Relation with Oreste”, Mathematical Computation and Modelling, Vol. 12, No. 10/11, pp. 1255-1268, 1989

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First approach to solve MCDM Problems

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Ranking of criteria

K

K1

K2Kn

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Combining criteria

xx

i

j jj

jj

.

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• Drawbacks of this method

* The problem of assigning weights

* The problem of compensation

Productivityp=0.65

Intelligencep=0.35

Arithmeticalmean

Conjunctivereasoning

Disjunctivereasoning

AB

0.450.60

0.900.55

0.60750.5825

0.450.55

0.900.60

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Concept of generalized average:

E p .Ei iR

i

1/R

where

p 1ii

R is any real number

It can be shown that for:

R = 1, the arithmetical mean is obtainedR = 0, the geometric mean is obtainedR = -, pure conjunctive reasoning is obtainedR = +, pure disjunctive reasoning is obtained

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Xg = (0.5 . 2R + 0.5 . 5R) 1/R

R D'R

-400 2.00-200 2.01-100 2.01-50 2,03-10 2,14-5 2,29-4 2,36-3 2,47-2 2,63-1 2,860 3,131 3,502 3,813 4,054 4,235 4,3610 4,6750 4,93100 4,97200 4,98400 4,99

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* The problem of incomparability

* The problem of indifference

• Interactive compromises

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Feature of MCDM ProblemsActions Quality Productivity

a b

d c

a b

d c

a b

d c

a 15 500

b 30 400

c 50 200

d 30 350

Majority Principle

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MCDM methods for richer dominance relations

• Aggregation by majority principles yields VERY POOR DOMINANCE RELATION: – A lot of Incomparabilities (R)– Some Indifferencies (I) and Preferences (P)

• MCDM methods should make the dominance relation richer (take into account more information than majority principles do)– Less R (making decisions easier)– More I and P

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Requirements for MCDM methods

Actions Criteria

a 100 100

b 30 20a P b

Actions Criteria

a 100 20

b 30 100a R b

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Requirements for MCDM methodsActions Criteria

a 100 99

b 20 100a P b

Actions Criteria

a 100 99

b 99 100a I b

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Requirements for MCDM methodsActions Criteria

a 100 100

b 99 99a I b

Actions Criteria

a 100 99

b 99 100a I b

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Scaling Effect on the Average

a 100 99 99.5

Criteria Average

b 20 100 60a P b

a 100 990 545b 20 1000 510

a P b

a 100 9900 5000 b 20 10,000 5010

b P a

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Requirements for an MCDM Method

• Deviations have to be considered• Elimination of scale effects• Pairwise comparison must lead to partial ranking

(incomparabilities) or to complete ranking• Methods must be transparant (“simple”)• Technical parameters must have an interpretation by the

decision maker• Weights allocated to criteria must have a clear interpretation• Conflict analysis of the criteria

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Some MCDM Methods

• Prométhée : numerical data• Oreste : ordinal data

• Electre : Pairwise comparisons - outranking with Incomparabilities

• AHP : Pairwise comparisons - No Incomparabilities

• ….

Complete & Partial Ranking

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The PROMETHEE METHOD

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- a number of Actions (strategies, candidates, etc.) to be ranked in orderof preference:

i.e. the set [a1, a2, a3, a4, ........... ak ] or "A"

- a number of Criteria of preference or of selection:

i.e. the set [ C1 , C2 , C3, ........ Cm ]

- the Weights assigned to each criterion:

i.e. the set [ w1 , w2 , w3, ........ wm ] or "W"such that:

Swi = 1

- a set of Preference functions expressing the way in which action" a1" is preferred over action " a2":

i.e. the set [P1, P2, P3, ............. Pm] or "P"

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There are six possible bids:

a1, a2, a3, a4, a5, a6

The following six criteria will be taken into account:

C1(a): mean maintenance time per day (in minutes)

C2(a): technical value of the equipment (score out of 100)

C3(a): cost (106 FB)

C4(a): estimated maintenance costs for the equipment's useful life (106 FB (discounted)

C5(a): estimated mean number of failures per year (based on standard use)

C6(a): safety rating of the equipment offered

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The data required to apply this method can be summarized as follow

Functions P1 P2 P3 P4 P5 P6Weight w1 w2 w3 w4 w5 w6Criteria C1 C2 C3 C4 C5 C6a1a2a3a4a5a6

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The foundations of the PROMETHEE method

• The three steps of the method

– (1) Selecting generalized criteria

– (2) Determining an outranking relationship

– (3) Evaluating preferences

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The concept of generalized criteria• Where Ci(a) is a criterion to be optimized• We consider a preference function

d = Ci(a1) - Ci(a2)

P(a1,a2)

d

1

0

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d

1

H

0

Préférence de a1 par rapport à a2Préférence de a2 par rapport à a1

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Choice of transformation functions

• Operational criteria : type III• Financial short term, acquisition cost, construction

cost : type V• Financial long term, maintenance cost, life cycle cost :

type IV• Discrete resources, manpower (roughly estimated) :

type II• Ecology, dramatic impact : type I• Security, Quality, Aesthetics : type VI

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Parameter settings• Indifference threshold : q

– high if uncertainty, low accuracy of data• Preference threshold : p

– close to maximum deviation if no loss of information is advisable (accurate data)

• Interactive choice in Promcalc

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• For each criterion Ci we will associate the preference function P.

(a1, a2) = S wi * Pi (a1, a2)(Different weights)

(a1, a2) = (1/m) *S Pi (a1, a2)(All weights are equal)

The outranking relationship

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• We have:

0 ( a1, a2) 1

• Furthermore,

– if ( a1, a2) 0 slight preference for "a1" over "a2"

– if ( a1, a2) 1 strong preference for "a1" over "a2"

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The outranking relationship

(a1,a2)

(a2,a1)

a1

a2

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Evaluating preferences

a . In p u t: + (a 1 ) =1

1k S ( a1 , a i)

b . O u tp u t: -(a1 ) =1

1k S ( a i , a1 )

c . N e t flo w : (a 1) = + (a 1 ) - -(a 1 )

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The PROMETHEE I method

a1 P+ a2 if +(a1) > +(a2) a1 I+ a2 if +(a1) = +( a2)

a1 P- a2 if -(a2) > -(a1) a1 I- a2 if -(a2) = -(a1)

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a1 P a2 "a1" outranks "a2" if: a1 P+ a2 and a1 P- a2 a1 P+ a2 and a1 I- a2 a1 I+ a2 and a1 P- a2

• a1 I a2 " a1" and " a2" are indifferent if:a1 I+ a2 and a1 I- a2

• a1 R a2 "a1" and "a2" are incomparable:in all other cases

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The PROMETHEE II method

• a1 PII a2 "a1" outranks "a2" if (a1) > (a2)

• a1 III a2 "a1" and "a2" are indifferent if (a1) = (a2)

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Example :There are six possible bids:

a1, a2, a3, a4, a5, a6

The following six criteria will be taken into account:

C1(a): mean maintenance time per day (in minutes)

C2(a): technical value of the equipment (score out of 100)

C3(a): cost (106 FB)

C4(a): estimated maintenance costs for the equipment's useful life (106 FB (discounted)

C5(a): estimated mean number of failures per year (based on standard use)

C6(a): safety rating of the equipment offered

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Selecting the generalized criteria

Criterion Type ParametersC1(a) II q = 10C2(a) III p = 30C3(a) V q = 50; p = 500C4(a) IV q = 10 ; p = 60C5(a) IC6(a) VI = 5

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The data

Criterion Min/max a1 a2 a3 a4 a5 a6C1(a) Min 80 65 83 40 52 94C2(a) Max 90 58 60 80 72 96C3(a) Min 600 200 400 1000 600 700C4(a) Min 54 97 72 75 20 36C5(a) Min 8 1 4 7 3 5C6(a) Max 5 1 7 10 8 6

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Devising the flow tableC 1(a):

1

- 10 0 + 10

H (d)

d

II

-cr iter io n to b e m in im ized-a 1 is th u s w orse th a n a 2 : C 1 (a 1) - C 1(a 2 ) = 15

a nd th erefore

P 1 (a1 , a2 ) = 0P 1 (a2 , a1 ) = 1

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Devising the flow tableC 2 (a ):

1

0

H (d)

d-30 +30

III

-c r ite r io n to b e m a x im iz e d

-a 1 is th u s b e tte r th a n a 2 : C 2 (a 1 ) - C 2 (a 2 ) = 3 2

a n d th e r e fo r e

P 2 (a 1 , a 2 ) = 1P 2 (a 2 , a 1 ) = 0

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Devising the flow table

C 3 (a ):

1

0

H (d)

d50 500- 500 - 50

V

- cr iter io n to b e m in im iz e d- a 1 is th u s w o r se th a n a : C 3 (a 1 ) - C 3 (a 2 ) = 4 0 0

a n d th e re fo re

P 3 (a 1 , a 2 ) = 0P 3 (a 2 , a 1 ) = 0 .7 7 8

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Devising the flow tableC 4 (a ):

1

0

H (d)

d10 60

IV

- 60 -10

1/2

- c r i t e r io n to b e m in im iz e d

- a 1 i s b e t te r th a n a 2 : C 4 (a 1 ) - C 4 (a 2 ) = - 4 3

a n d th e r e f o r e

P 4 (a 1 , a 2 ) = 0 .5P 4 (a 2 , a 1 ) = 0

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Devising the flow tableC 5 (a ):

1

0

H (d)

d

I

- c r i t e r io n to b e m in im iz e d

- a 1 i s th u s w o r s e th a n a 2 : C 5 (a 1 ) - C 5 (a 2 ) = 7

a n d th e r e fo r e

P 5 (a 1 , a 2 ) = 0P 5 (a 2 , a 1 ) = 1

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Devising the flow tableC6(a):

1

0

H (d)

d

VI

- criterion to be maximized

H6(d) = 1 - e

d

- a1 is better than a2:

and therefore

P6 (a1, a2) = 1 - e

4 2

22.5 = 0.274

P6(a2, a1) = 0

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Summary:

Average of Pi(a1, a2): (a1,a2) =16

(0+1+0+0,5+0+0.274) = 0.296

Average of Pi(a2, a1): (a2,a1) =16

(1+0+0,778+0+1+0) = 0.462

By applying the above reasoning to all the pairs (ai, aj), we obtain the followingtable:

a1 a2 a3 a4 a5 a6a1 0.296 0.250 0.268 0.100 0.185a2 0.462 0.389 0.333 0.296 0.500a3 0.236 0.180 0.333 0.056 0.429a4 0.399 0.505 0.305 0.223 0.212a5 0.444 0.515 0.487 0.380 0.448a6 0.286 0.399 0.250 0.432 0.133

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a1 a2 a3 a4 a5 a6 + = +--a1 0.296 0.250 0.268 0.100 0.185 0.220 -0.146a2 0.462 0.389 0.333 0.296 0.500 0.396 +0.017a3 0.236 0.180 0.333 0.056 0.429 0.247 -0.089a4 0.399 0.505 0.305 0.223 0.212 0.329 -0.020a5 0.444 0.515 0.487 0.380 0.448 0.455 +0.293a6 0.286 0.399 0.250 0.432 0.133 0.300 -0.055

- 0.366 0.379 0.336 0.349 0.162 0.355

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The ranking obtained using the Promethee I method

a2

a5 a4 a6

a1

a3

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a2 a6 a1

a5 a4 a3

The ranking obtained using the Promethee II method

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Flexibility of Prométhée• Weights

• Transformation functions = generalised criteria

• Parameter settings

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Thanks for your attention

MCDM

Questions ?

Suggestions ?

04/22/23 AREOPA MOBIUS RUG RMA H.Pastijn

58

Questions ?