A multiplicative version of Promethee II applied to multiobjective optimization problems

doi:10.1016/j.ejor.2006.10.002

European Journal of Operational Research

Volume 183, Issue 2, 1 December 2007, Pages 729-740

https://doi.org/10.1016/j.ejor.2006.10.002 Get rights and content

Abstract

The method Promethee II has produced attractive results in the choice of the most satisfactory optimal solution of convex multiobjective problems. However, according to the current literature, it may not work properly with nonconvex problems. A modified version of this method, called multiplicative Promethee, is proposed in this paper. Both versions are applied to some analytical problems, previously optimized by an evolutionary algorithm. The multiplicative Promethee got much better results than the original Promethee II, being capable of solving convex and nonconvex problems, with continuous and discontinuous Pareto fronts.

Introduction

The choice of the final solution of multiobjective optimization problems (MOP) can be stated as a typical multicriteria decision problem: determine which is the best among all possible efficient solutions, according to the decision-maker (DM) preferences, taking into account several criteria [1].

There are three manners of combining decision methods with optimization algorithms in order to determine the final solution of MOPs: the a priori, the progressive and the a posteriori decision-making [1], [2], [3]. In the a priori decision-making, the DM is consulted just one time, before the optimization process, and his preferences are used to guide the search toward the favorite solution from the Pareto front. In the progressive decision-making, the information about the DM preferences is repeatedly obtained in the course of the iterative optimization process in order to guide the search algorithm. The a posteriori decision-making starts with the execution of a multiobjective optimization algorithm in order to find a discrete approximation of the Pareto front. After that, the DM may use a decision method to compare the available alternatives and choose a unique final solution.

Promethee II is a popular decision method that has been successfully applied in the selection of the final solution of convex MOPs [4]. It generates a ranking of available points, according to the DM preferences, and the best ranked one is considered the favorite final solution. It is based on the concept of outranking relation, which is a binary relation defined between every pair (a, b) of alternatives, in such way that, if a is preferred to b (according to the DM interests), then it is said that a outranks b. When these relations are defined between all pairs of alternatives, they are exploited according to some rules in order to rank all solutions from best to worst.

In the current literature, there are several examples where Promethee II is used in the choice of the final solution of MOPs. For instance, in [5], [6], [7], [8], [9], [10], the a priori decision is performed by associating Promethee with Genetic Algorithms (GA). Promethee assigns a score to each point from the search region and the GA is used to meet the feasible solution with maximum score. Ref. [7] also shows that the progressive decision-making can be performed with Promethee linked to a GA. When the DM changes the Promethee input parameters, the optimization process is redirected toward another region of the Pareto front. Ref. [16] implements the a posteriori decision with the execution of Promethee II after the run of a multiobjective GA. The a posteriori decision is also performed in [11], [12], [13], [14], [15]: initially, a multiobjective GA is executed and, afterwards, a hybrid decision method, based on Promethee II and Electre (another decision method), selects the best efficient solution. Ref. [17] contains a graphical procedure which combines the Promethee ranking with the Pareto optimality concept, in order to determine the best solution of decision problems. Although it does not involve the execution of search algorithms, it can also be used in the a posteriori decision approach.

A modified version of this decision method, the multiplicative Promethee, is introduced in this paper. The a posteriori decision-making approach is performed with two decision programs: one based on Promethee II and the other on the multiplicative Promethee. Both of them are applied to some MOPs, previously optimized by a multiobjective search program based on the Nondominated Sorting Genetic Algorithm II (NSGA-II) [18]. In order to compare the quality of their solutions, nine analytical problems with very distinct characteristics are adopted. The results show that, in general, the multiplicative Promethee gets more attractive results than the original Promethee II, principally when nonconvex problems are considered. It represents an important contribution, as there are several real problems, from distinct areas, whose nondominated front delimits a nonconvex portion of the feasible region. Refs. [19], [20], [21], [22], [23], [24] contain examples of such problems.

Section snippets

The multicriteria decision problem definition

Consider a generic MOP stated as: $\begin{matrix} minimize [f_{1} (\vec{x}), f_{2} (\vec{x}), \dots, f_{m} (\vec{x})], \\ subjected to g_{i} (\vec{x}) ⩾ 0,for i = 1, 2, \dots, n . \end{matrix}$

In the a posteriori decision-making approach, the decision problem originated from this MOP involves the following elements [1]:

•
The set A of available alternatives or possible solutions. It corresponds to an approximation of the Pareto-optimal front met by the optimization algorithm. Hence, it is a discrete set of enumerable solutions that may have hundreds of elements.
•
The set of consequences of

Promethee II

Given that d_i = f_i(b) − f_i(a), when minimization problems are considered, the classical decision methods admit three possible situations:

•
if d_i = 0, then a is indifferent to b;
•
if d_i > 0, then a is preferred to b;
•
if d_i < 0, then b is preferred to a.

The method Promethee II extends this classical approach by modeling the DM preferences through a preference function p_i(d_i) for the ith criterion, in such way that it reflects the preference level of a over b, from 0 to 1. If p_i(d_i) = 0, both alternatives are

The multiplicative version of Promethee II

The original and the multiplicative versions of Promethee II are very similar. Both of them start with the specification of a preference function for each criterion. Their differences lie in the way each of them computes the net flow and defines the outranking relation. In the multiplicative version, the net flow of each alternative does not require the global preference index calculation as in Eq. (5).

Indeed, in the multiplicative version, a distinct directed valued graph is defined for each

Methodology

Both versions of Promethee are compared here based on the following aspects:

•
The processing time. In order to compare the computational effort required by each decision method, they are simulated in the same computer and the mean time taken by each of them is measured. These values are normalized, being divided by the mean time taken by the original Promethee II.
•
The stability of the results, when different discrete approximations of the Pareto front are considered. It was already observed by

Results

In the case of MOPs with two objective functions, both versions of Promethee are simulated with 1296 different configurations of their input parameters: 12 values of s₁ × 12 values of s₂ × 9 pairs of weights. In the case of MOPs with three objective functions, 364 different configurations of their input parameters are considered: 4 values for s_i and 7 sets of weights. All analytical problems considered in this study are described in Table 1.

Table 2 lists the mean processing time taken by the

Conclusions

This paper introduced a multiplicative version of Promethee II and compared it with its original version, by applying them in the selection of the final solution of multiobjective optimization problems. Some metrics originally proposed in Ref. [26] to test the quality of the nondominated fronts met by multiobjective algorithms were adapted and used here to verify whether these decision methods can get well-distributed solutions for different priority weights.

The obtained results confirm the

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Decision SupportA multiplicative version of Promethee II applied to multiobjective optimization problems

Abstract

Introduction

Section snippets

The multicriteria decision problem definition

Promethee II

The multiplicative version of Promethee II

Methodology

Results

Conclusions

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

Chemometrics and Intelligent Laboratory Systems

Electric Power Systems Research

Decision making in multiobjective optimization problems

Multicriterion decision making

Handling preferences in evolutionary multiobjective optimization: A survey

A multiple objective grouping genetic algorithm for assembly line design

Journal of Intelligent Manufacturing

Decision Support
A multiplicative version of Promethee II applied to multiobjective optimization problems