A Multi-Criterion Decision Making Approach to Problem Solving
description
Transcript of A Multi-Criterion Decision Making Approach to Problem Solving
![Page 1: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/1.jpg)
04/22/23 1
A Multi-CriterionDecision Making
Approach toProblem Solving
M. HERMAN, IrM. HERMAN, IrRoyal Defense College (Brussels - Belgium)Royal Defense College (Brussels - Belgium)
![Page 2: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/2.jpg)
04/22/23 2
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
![Page 3: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/3.jpg)
04/22/23 3
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
![Page 4: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/4.jpg)
04/22/23 4
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
![Page 5: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/5.jpg)
04/22/23 5
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
![Page 6: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/6.jpg)
04/22/23 6
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,…)• …...
![Page 7: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/7.jpg)
04/22/23 7
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
![Page 8: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/8.jpg)
04/22/23 8
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
![Page 9: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/9.jpg)
04/22/23 9
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
![Page 10: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/10.jpg)
04/22/23 10
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
![Page 11: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/11.jpg)
04/22/23 11
First approach to solve MCDM Problems
![Page 12: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/12.jpg)
04/22/23 12
Ranking of criteria
K
K1
K2Kn
![Page 13: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/13.jpg)
04/22/23 13
Combining criteria
xx
i
j jj
jj
.
![Page 14: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/14.jpg)
04/22/23 14
• 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
![Page 15: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/15.jpg)
04/22/23 15
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
![Page 16: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/16.jpg)
04/22/23 16
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
![Page 17: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/17.jpg)
04/22/23 17
* The problem of incomparability
* The problem of indifference
• Interactive compromises
![Page 18: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/18.jpg)
04/22/23 18
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
![Page 19: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/19.jpg)
04/22/23 19
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
![Page 20: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/20.jpg)
04/22/23 20
Requirements for MCDM methods
Actions Criteria
a 100 100
b 30 20a P b
Actions Criteria
a 100 20
b 30 100a R b
![Page 21: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/21.jpg)
04/22/23 21
Requirements for MCDM methodsActions Criteria
a 100 99
b 20 100a P b
Actions Criteria
a 100 99
b 99 100a I b
![Page 22: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/22.jpg)
04/22/23 22
Requirements for MCDM methodsActions Criteria
a 100 100
b 99 99a I b
Actions Criteria
a 100 99
b 99 100a I b
![Page 23: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/23.jpg)
04/22/23 23
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
![Page 24: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/24.jpg)
04/22/23 24
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
![Page 25: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/25.jpg)
04/22/23 25
Some MCDM Methods
• Prométhée : numerical data• Oreste : ordinal data
• Electre : Pairwise comparisons - outranking with Incomparabilities
• AHP : Pairwise comparisons - No Incomparabilities
• ….
Complete & Partial Ranking
![Page 26: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/26.jpg)
04/22/23 26
The PROMETHEE METHOD
![Page 27: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/27.jpg)
04/22/23 27
- 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"
![Page 28: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/28.jpg)
04/22/23 28
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
![Page 29: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/29.jpg)
04/22/23 29
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
![Page 30: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/30.jpg)
04/22/23 30
The foundations of the PROMETHEE method
• The three steps of the method
– (1) Selecting generalized criteria
– (2) Determining an outranking relationship
– (3) Evaluating preferences
![Page 31: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/31.jpg)
04/22/23 31
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
![Page 32: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/32.jpg)
04/22/23 32
d
1
H
0
Préférence de a1 par rapport à a2Préférence de a2 par rapport à a1
![Page 33: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/33.jpg)
04/22/23 33
![Page 34: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/34.jpg)
04/22/23 34
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
![Page 35: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/35.jpg)
04/22/23 35
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
![Page 36: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/36.jpg)
04/22/23 36
• 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
![Page 37: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/37.jpg)
04/22/23 37
• 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"
![Page 38: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/38.jpg)
04/22/23 38
The outranking relationship
(a1,a2)
(a2,a1)
a1
a2
![Page 39: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/39.jpg)
04/22/23 39
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 )
![Page 40: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/40.jpg)
04/22/23 40
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)
![Page 41: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/41.jpg)
04/22/23 41
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
![Page 42: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/42.jpg)
04/22/23 42
The PROMETHEE II method
• a1 PII a2 "a1" outranks "a2" if (a1) > (a2)
• a1 III a2 "a1" and "a2" are indifferent if (a1) = (a2)
![Page 43: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/43.jpg)
04/22/23 43
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
![Page 44: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/44.jpg)
04/22/23 44
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
![Page 45: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/45.jpg)
04/22/23 45
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
![Page 46: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/46.jpg)
04/22/23 46
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
![Page 47: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/47.jpg)
04/22/23 47
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
![Page 48: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/48.jpg)
04/22/23 48
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
![Page 49: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/49.jpg)
04/22/23 49
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
![Page 50: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/50.jpg)
04/22/23 50
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
![Page 51: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/51.jpg)
04/22/23 51
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
![Page 52: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/52.jpg)
04/22/23 52
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
![Page 53: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/53.jpg)
04/22/23 53
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
![Page 54: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/54.jpg)
04/22/23 54
The ranking obtained using the Promethee I method
a2
a5 a4 a6
a1
a3
![Page 55: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/55.jpg)
04/22/23 55
a2 a6 a1
a5 a4 a3
The ranking obtained using the Promethee II method
![Page 56: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/56.jpg)
04/22/23 56
Flexibility of Prométhée• Weights
• Transformation functions = generalised criteria
• Parameter settings
![Page 57: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/57.jpg)
04/22/23 57
Thanks for your attention
MCDM
Questions ?
Suggestions ?
![Page 58: A Multi-Criterion Decision Making Approach to Problem Solving](https://reader035.fdocuments.us/reader035/viewer/2022070500/56816836550346895dddf3f5/html5/thumbnails/58.jpg)
04/22/23 AREOPA MOBIUS RUG RMA H.Pastijn
58
Questions ?