Strengthening PROMETHEE with elements from value theory

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1 2η Συνάντηση Πολυκριτήριας Ανάλυσης Αποφάσεων Ελληνική Εταιρία Επιχειρησιακών Ερευνών Χανιά, 2-22 Οκτωβρίου 2004 Strengthening PROMETHEE with elements from value theory Εμπλουτισμός της μεθόδου PROMETHEE με στοιχεία της Πολυκριτήριας Θεωρίας Χρησιμότητας George Mavrotas*, Danae Diakoulaki Laboratory of Industrial & Energy Economics National Technical University of Athens

2 Contents PROMETHEE Value theory and partial value functions Combination of PROMETHEE with value theory Numerical example Conclusions

3 PROMETHEE Outranking method Pairwise comparison of the alternatives Results Promethee I, partial preorder (allow incomparabilities) Promethee II, complete preorder (no incomparabilities)

4 Brief description of PROMETHEE The difference between two alternatives in one criterion is transformed to [0,] using one of six functional forms (criteria types). Pairwise comparison Square matrix with pairwise multicriteria scores Leaving flow=row sum, Entering flow=column sum, Net flow=leaving flow-entering flow Complete preorder=rating of alternatives according to their net flow

5 Type of criteria 0 0 q 0 p Type Type 2 Type 3 0 q p 0 q p 0 s Type 4 Type 5 Type 6

6 PROMETHEE parameters q = indifference threshold (types 2, 4, 5) p = strict preference threshold (types 3, 4, 5) s = gaussian s standard deviation (type 6) Basic assumption: Parameters are constant throughout the entire range of the criterion

7 Implication The pairwise outranking score p j (a,b) depends only on the difference between the two alternatives (d ab ) and not on their position in the criterion scale. This implies a partial value function (p.v.f.) with a constant slope for the criteria (linear p.v.f.)

8 Example: Car selection Criterion: maximization of car s speed A 20 km/h C 220 km/h B 50 km/h D 250 km/h B is preferred against A the same as D is preferred against C

9 Remarks on the example Sometimes the same distance between alternatives is differently perceived, depending on the position of the alternatives in the criterion scale. Example of car s speed criterion: usually we are more sensitive in the differences at the beginning of the scale (normal speed figures) than at the end of the scale (Formula speed figures). In some cases the uniform degree of preference throughout the criterion range is not realistic Remedy: nonlinear partial value functions

10 Partial value functions (p.v.f.) Three usual types linear Uniform preference of the DM convex The DM is more sensitive in differences at the end of the scale concave The DM is more sensitive in differences at the beginning of the scale

11 Car s example revisited The DM is more sensitive in differences between low figures Car speed weak preference strong preference

12 Incorporation in PROMETHEE Enrichment of PROMETHEE with elements from value theory Flexible PROMETHEE The DM may choose the shape of the p.v.f. for a criterion, if he/she considers that it is different than the underlying linear.

13 Preference elicitation Estimate the shape of the partial value function Techniques Bisection (indicate the mid-value point) Difference (compare differences in values)

14 In practice The DM expresses his/her preferences as in normal PROMETHEE Criterion type ( 6) Required parameters (q, p, s) Weights (w, w 2,, w m ) One additional piece of information: A rough estimate of the partial value function for each criterion (not detailed estimation)

15 Categories of p.v.f. The DM chooses a representative partial value function for each criterion among five categories: concave more or less concave linear more or less convex convex concave more or less concave linear more or less convex convex 0

16 Measuring the position of the compared alternatives: the centrifugal coefficient According to the position of the compared alternatives A and B in the j-th criterion, the centrifugal coefficient is defined as: c jab xaj + xbj midrange j = 2 c [ 0.5, 0.5] range jab j If c > 0 then A and B are considered to be located in the upper half If c < 0 then A and B are considered to be located in the lower half A high c A and B are close to one edge of the criterion scale

17 Graphical representation range min max A B C D midrange (x A +x B )/2 (x C +x D )/2

18 Adjustable parameters Assume t is an initially assigned parameter from the DM for the specific criterion (q, p or s). The relative adjusted parameter (t adj ) is calculated using the following relationships: concave: t adj = t (+2c) More or less concave: t adj = t (+c) Linear: t adj = t More or less convex: t adj = t (- c) adjusting coefficents convex: t adj = t (- 2c)

19 Insight concave: t adj =t (+2c) If c>0 A, B in upper half parameter less sensitive If c<0 A, B in lower half parameter more sensitive Linear Uniform parameter convex: t adj =t(-2c) If c>0 A, B in upper half parameter more sensitive If c<0 A, B in lower half parameter less sensitive

20 Graphical example Assume a type 5 criterion with a concave partial value function thresholds for two alternatives in the lower half thresholds for two alternatives in the upper half 0 q L q q U p L p p U

21 Generalization General rule Value of parameter intensity of preference s s 2 0 q q 2 0 Type 2 Type 6 The smaller the parameter the easier we move from indifference to strict preference

22 Numerical example Problem: choose action plan in order to comply with ozon standards for year 200 in a Greek city 6 candidate action plans (scenarios) 3 criteria Cost (million ) Social Acceptance (qualitative scale) Expert opinion Environmental Safety Margin distance from the imposed limit (20 μg/m 3 ) in terms of concentration (μg/m 3 ).

23 Evaluation matrix and criteria characteristics AP AP2 Cost (million ) Social acceptance Safety margin (μg/m 3 ) 8 2 We are more sensitive in differences near to the imposed limit. AP3 AP4 AP5 AP The same difference counts more if we are close to the limit than far away from it direction Criteria characteristics min max max Why? type q p weight p.v.f. shape linear linear concave Motivation for the proposed approach

24 Example of calculations in third criterion Consider AP with AP2 and AP3 with AP4 Normal Promethee: p j (a,b) = (d ab q)/(p-q) p 3 (ΑP2, AP)=(4-2)/(5-2)=0.67 p 3 (ΑP4, AP3)=(4-2)/(5-2)=0.67

25 Promethee with adjusted parameters Centrifugal coefficient: c jab x Aj + = 2 x Bj midrange range j j =(8+2)/2 c(ap2,ap)=(0-6.5)/23= c=0.44 midrange = (28+5)/2 range = (28-5) c(ap4,ap3)=(26-6.5)/23= c=.82 =(24+28)/2

26 Graphical representation (=.82*5) (=0.44*5) 0.67 p 3 (AP2, AP)= p 3 (ΑP4, AP3)= The same difference results in diverse pairwise outranking scores (=0.44*2) (=.82*2)

27 Results The algorithm was coded in VBA for Excel Flexible Promethee (adjusted parameters) Normal Promethee (uniform parameters) Φ+ Φ- Φ rank Φ+ Φ- Φ rank AP AP AP AP AP AP AP AP AP AP AP AP

28 Comparison Cost (million ) Social acceptance Safety margin (μg/m 3 ) Rank (Normal Promethee) Rank (Flexible Promethee) AP AP AP AP AP AP

29 Conclusions The idea is to stengthen PROMETHEE with elements from value theory Adjustable parameters offer more flexibility Simple information required by the DM Easily implemented as an extra option in normal PROMETHEE

30 Thank you for your attention

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