Abstract
PROMETHEE is a powerful method, which can solve many multiple criteria decision making (MCDM) problems. It involves sophisticated preference modelling techniques but requires too much a priori precise information about parameter values (such as criterion weights and thresholds). In this paper, we consider a MCDM problem where alternatives are evaluated on several conflicting criteria, and the criterion weights and/or thresholds are imprecise or unknown to the decision maker (DM). We build robust outranking relations among the alternatives in order to help the DM to rank the alternatives and select the best alternative. We propose interactive approaches based on PROMETHEE method. We develop a decision aid tool called INTOUR, which implements the developed approaches.
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Appendices
Appendix A: INPRO-A Algorithm
- Step 1::
-
Take the f i (x k) ∀i and ∀k and convert them into r i k ∀i and ∀k.
- Step 2::
-
Ask the DM for q i , p i or σ i ∀i. If no thresholds are given, assign q i =p i =σ i =0 ∀i. Compute P i (x k,x j) ∀i and ∀k, j, j ≠ k and φ i (x k) ∀i and ∀k.
Ask the DM for initial partial information on criterion weights, W.
- Step 3::
-
In order to find φ̲(x k) and of each alternative, solve LPO for each alternative. (If the optimal objective function value of LPO is infeasible, then ask the DM to delete one or more of the constraints in order to resolve inconsistency.)
Present the following results to the DM with corresponding weight vectors:
• φ̲(x k), , alternatives ranked by descending order of lower and upper bounds.
Find and display the following information on the current solution to the DM:
• φ p, sets of OAs and NOAs, number of OAs and NOAs.
- Step 4::
-
If the DM wants to make a final decision using the results of the current iteration or if single alternative is left in the set of NOAs, STOP. Display φ̲(x k) and with corresponding weight vectors, and the sets of OAs and NOAs along with further ranks of OAs, if any. Otherwise, go to Step 5.
- Step 5::
-
(a) If the DM wants to put a threshold on φ p, update φ p and the sets of NOAs and OAs.
(b) If the DM wants to provide additional constraints on criterion weights, update W. Go to Step 3.
(c) If the DM wants to make a pairwise comparison, update W. Go to Step 3.
Appendix B: INPRO-B Algorithm
- Step 1::
-
Take f i (x k) ∀i and ∀k and convert them into r i k ∀i and ∀k.
- Step 2::
-
Ask the DM for q i , p i or σ i ∀ i . If no thresholds are given, assign q i =p i =σ i =0 ∀i. Compute P i (x k,x j) ∀i and ∀k,j,j≠k, and φ i +(x k) and φ i −(x k) ∀i and ∀k.
Ask the DM for initial partial information on criterion weights, W.
- Step 3::
-
In order to find φ̲ +(x k) and of each alternative, solve LPOL for each alternative (If LPOL is infeasible, then ask the DM to delete one or more of the constraints in order to resolve inconsistency.) In order to find φ̲ −(x k) and of each alternative, solve LPOE for each alternative. Present the following results to the DM with corresponding weight vectors:
• φ +(x k), , φ −(x k), , alternatives ranked by descending order of upper bound on leaving flows and lower bound on entering flows.
Find and display the following information on the current solution to the DM:
• φ p+, φ p−, sets of OAs and NOAs, number of OAs and NOAs.
- Step 4::
-
If the DM wants to make a final decision using the results of the current iteration, STOP.
Display φ +(x k), , φ −(x k) and with corresponding weight vectors, and sets of OAs and NOAs along with further ranks of OAs, if any. Otherwise, go to Step 5.
- Step 5::
-
(a) If the DM wants to put a threshold on φ p+ and/or φ p−, update φ p+ and/or φ p− and the sets of NOAs and OAs.
(b) If the DM wants to provide additional constraints on criterion weights, update W. Go to Step 3.
(c) If the DM wants to make a pairwise comparison, update W. Go to Step 3.
Appendix C: INPRO-C Algorithm
All steps except Step 2 and 3 are the same as those of INPRO-A. Steps 2 and 3 of INPRO-A algorithm are to be read as follows:
- Step 2::
-
Ask the DM for p, p̄, q and q̄
Compute P i (x k,x j) ∀i and ∀k, j,j≠k, first for p and q and then for p̄ and q̄.
Ask the DM for initial partial information on criterion weights, W.
- Step 3::
-
In order to find φ̲(x k) and of each alternative follow Steps 3.1 and 3.2.
- Step 3.1::
-
Set p=p and q=q, compute φ i +(x k) ∀i and ∀k, and then set p=p̄ and q=q̄ and compute φ i −(x k) ∀i and ∀k.
Solve LPO1 for each alternative in order to find the upper bound on the net flow of each alternative.
- Step 3.2::
-
Set p=p̄ and q=q̄, compute φ i +(x k) ∀i and ∀k, and then set p=p and q=q and compute φ i −(x k) ∀i and ∀k.
Solve LPO2 for each alternative in order to find the lower bound on the net flow of each alternative.
If the optimal objective function value of LPO1 is infeasible, then ask the DM to delete one or more of the constraints in order to resolve inconsistency. Otherwise continue as in INPRO-A algorithm.
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Özerol, G., Karasakal, E. Interactive outranking approaches for multicriteria decision-making problems with imprecise information. J Oper Res Soc 59, 1253–1268 (2008). https://doi.org/10.1057/palgrave.jors.2602458
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DOI: https://doi.org/10.1057/palgrave.jors.2602458