Stability analysis of tree structured decision functions

https://doi.org/10.1016/j.ejor.2003.10.007Get rights and content

Abstract

In multicriteria decision problems many values must be assigned, such as the importance of the different criteria and the values of the alternatives with respect to subjective criteria. Since these assignments are approximate, it is very important to analyze the sensitivity of results when small modifications of the assignments are made. When solving a multicriteria decision problem, it is desirable to choose a decision function that leads to a solution as stable as possible. We propose here a method based on genetic programming that produces better decision functions than the commonly used ones. The theoretical expectations are validated by case studies.

Introduction

In most real life problems, if we want to make a decision then generally we have to take into consideration more than one criterion. For example, if we want to buy a TV set, our criteria could be the cost, the quality of image, the physical aspect etc. In general, the criteria have different importance. In the case of TV set we would probably prefer the quality of image to the physical aspect. If a possible alternative has values not worse for all criteria and better value for at least one criterion than another alternative (i.e., the alternatives are Pareto ordered) then it is easy to make a decision. We would definitely choose the first alternative. Unfortunately, in most of the cases the criteria are conflicting. In particular, a TV set with better image could have higher cost.

In the economical world there are many multicriteria problems where a cardinal method is preferred to an ordinal one. For example, when deciding what product to buy, the director of a company could be interested not only in the importance order of the products. Sometimes they want to know some kind of efficiency numbers of the products. One would not be satisfied with theoretical explanations concerning the lack of exact mathematical solution for this problem. Since there is a high demand for solving problems of this type, we should think of as good numerical methods as possible. Our paper is an attempt to find such a method. We consider an infinite number of models described by multiparameter decision functions. We measure the quality of a model as the stability of the corresponding decision functions. This paper is only a first attempt to compare different models. Hopefully, inspired by this (probably far from perfect) approach a new trend will arise in multicriteria decision making. Throughout our paper we consider only the cardinal approach of multicriteria decision problems. This does not mean that we are not aware of the high importance of the ordinal methods, and that there are problems of ordinal nature where ordinal methods should be considered instead of cardinal ones. Of course, comparing the ordinal methods (beyond the scope of our paper) is also a sound issue worth studying. However, we think that it is no point in comparing cardinal methods with ordinal ones since they handle problems of different nature.

We are aware of the fact that our method has its limitations and may not be affordable for problems where the number of criteria and/or alternatives is very large. If the number of criteria is too large then a possibility could be to choose the most important criteria and neglect the others. If the number of alternatives is too large then they might be divided into groups and a two step analysis can be made: first the groups and then the individuals within the groups are ordered. The general global stability index described in the conclusion part of our paper can be adapted to this method, too. This kind of two step analysis is not unfamiliar in large real world applications. However, the nature of the problem will decide which decision method should be used. We believe that it is not possible to give a decision method, which works for every problem.

In real life problems we must make many assignments, such as importance of criteria and values of alternatives with respect to subjective criteria. For example, in the case of the TV set, quality of image is a subjective criterion, it is difficult to assign an exact value to it. When slightly perturbing the assigned values (i.e., weights and values of alternatives with respect to subjective criteria) we might obtain a different order of the alternatives.

By the sensitivity analysis of a multicriteria decision problem we mean the analysis of the results, when we make small perturbations to the weights and/or values of alternatives with respect to criteria. There are many different types of sensitivity analysis used by the known multicriteria methodologies [2], [6], [16], [17], [18]. A sensitivity analysis with respect to more than one parameter was given in the Promethee methodology [2]. The WINGDSS methodology [4] presents a sensitivity analysis with respect to all parameters (both alternative values and weights). Here we consider a similar sensitivity analysis (but either only with respect to alternative values or only with respect to weights). The originality of our approach is that we consider more complicated decision functions (represented in form of trees) rather than decision functions of linear fractional type only. Hence, instead of solving linear fractional programming problems we have to solve much more difficult global optimization problems. Also, the problem of searching for stable decision functions seems to be a new idea in the area of multicriteria decision making. As far as we can see, the classical methods of optimization are not appropriate in this case, this problem seems too complicated for classical methods, and therefore we apply a novel genetic programming based optimization method.

Multicriteria decision problems may have different goals. Usually the goal of a multicriteria problem is (1) to eliminate a number of worst alternatives, or (2) to choose a number of best alternatives, or (3) to rank the alternatives. In the problem of elimination or choice, the order between the eliminated or chosen alternatives could be also important. In this case we have a mixed problem of (4) choice and ranking.

Since the assignment of weights and alternative values with respect to subjective criteria is approximate, it would be always preferable either to get the same solution (i.e., to have a stable solution) or to obtain very close solutions (i.e., to have a solution with reasonable stability). For the beginning, we consider the problem (3) of full ranking of the alternatives. In order to rank the alternatives, we shall use some decision functions, which is a function of the values of an alternative with respect to the different criteria. The decision function must take into consideration the value on each criterion with the corresponding importance. In the future we shall consider the problems (1), (2) and (4), as well.

In this paper the stability index is given for the full ranking problem. It is possible to give stability indexes of similar nature for the problems (1), (2) and (4), too. First we consider the local stability of n−1 subproblems where n is the number of alternatives. These subproblems are the stability of the order of an alternative and the next alternative in the ranking. If all of these subproblems are stable then the ranking of the alternatives is stable. We shall introduce local stability indexes for these problems. The local stability index which takes values between 0 and 1 measures the stability of these n−1 orders (n−1 and not n since the last alternative has no follower).

We shall introduce a global index called the stability index, which takes values between 0 and 1 and measures the stability of a solution (here full ranking). If the stability index is 1 then the solution is stable. The closer is the index to 1, the more stable is a solution. The stability index is a function of the local stability indexes such that the global stability index is 1 if and only if all the local stability indexes are 1. This function can be a power mean with power α∈R∪{−∞,+∞} (or other kind of mean, see for example [7]) of the local stability indexes. Our idea was to penalize the solutions, which have a local stability index zero or very close to zero. We would like the stability index to be zero if one of the stability indexes is zero. There are two power means which satisfy this condition: the α-power mean with power α=−∞ (the minimum function) and the α-power mean with power α=0 (geometric mean). Although we want to penalize the solutions, which have a stability index close to zero we should give an average stability index to solutions, which have both small and large stability indexes. The minimum function does not satisfy this condition. So, we chose the geometric mean.

The goal of this paper is to find a decision function such that––for a fixed problem––the solution be as stable as possible. It is almost a hopeless task to search for a solution on the whole function space of possible decision functions. Therefore, we shall restrict the search to a subclass of these functions.

We call decision function a function, which satisfies some natural axioms. The most general class of functions satisfying these axioms that we could find can be represented in form of trees. The leaves of these trees are decision functions of weighted power mean type. On the upper level of the tree are power means. The value of the function is in the root of the tree (There exist other kinds of decision functions where the importance of criteria is given by capacities instead of weights [5], [6], [10], [11]. It would be interesting to see whether our method could be extended to that type of decision functions.). This value is calculated recursively, so that the value of each non-leaf node is the corresponding power mean of its children. We shall use genetic programming [12] for finding decision functions (of the above types) as stable as possible for certain multicriteria decision problems. The individuals in the genetic programming system are decision functions represented in the form of trees. Formally, the subtrees of such a tree are decision functions. However, no clear practical meaning can be given to a subtree. If the problem were a group decision problem we could think of the group decision function as the aggregation of the individual decision functions of the decision makers which are subtrees of the tree representing the group decision function. In that case the associativity of the aggregation would be an important issue [9]. In our framework we considered multicriteria decision problems with only one decision maker. In this case we could not think of a reason for the decision maker to consider more than one decision function and to aggregate them. Hence, we think that in our case the issue of the associativity is not important and we would not deal with it.

The paper is structured as follows: In Section 2 we shall describe the methodology of multicriteria decision problems: decision tables and decision functions. In 3 Sensitivity with respect to alternatives, 4 Sensitivity with respect to weights we shall make a sensitivity analysis with respect to alternatives and weights, respectively. In both cases we shall introduce a global stability index, which measures the stability of the order of alternatives with respect to small perturbations of alternative values and weights, respectively. In Section 5 we shall give experimental results and finally, in the last section we shall draw the conclusions and present the future perspectives.

Section snippets

Preliminaries

First let us define a multicriteria decision problem.

Definition 1

In a multicriteria decision problem n alternatives A1,…,An must be ranked by using m criteria C1,…,Cm (generally) of different importance expressed by using the positive numbers w1,…,wm called the weights of criteria C1,…,Cm, respectively. The values of the alternatives on each criterion can be organized in the following table:In this table aij>0 denotes the value of the jth alternative in the ith criterion and xj denotes the aggregated value

General setting

The decreasing order of the solution is not altered by the perturbations if the pairwise order of alternatives remains unchanged. The allowable perturbations may differ for the different pairs of consecutive alternatives. The stability of the order of two consecutive alternatives is a measure of the allowable perturbation. The notion of stability of order was introduced in the multicriteria group decision software WINGDSS 4.1 [3], [4], [15] developed at the Laboratory of Operations Research and

Sensitivity with respect to weights

In the previous section we supposed that the weights of the different criteria are constant. Here we suppose that the values of alternatives with respect to the criteria are constant (scaled into the interval [1,9] similar to the previous section) and the weights are subject to change. We perform a stability analysis with respect to weights.

Definition 6

For each alternative Ajxj(ε)=minf(w̃,a):w̃∈∏i=1,…,m[wi−εwi,wi+εwi]andxj+(ε)=maxf(w̃,a):w̃∈∏i=1,…,m[wi−εwi,wi+εwi]are the minimum and, respectively, the

Experimental results

In the following we present our findings regarding the stability analysis with respect to alternative values.

We conducted experiments on three multicriteria decision problems. In all cases we allowed variations of up to ε=10% of the alternative values. The experimental parameter setting including control parameters is summarized in Table 1.

Example 1

Let us consider the problem of buying a TV set. We would like to get the best TV set of certain dimensions for a given price, but there are 10 different

Conclusions

We presented here novel methods for the sensitivity and stability analysis of multicriteria decision problems with respect to alternatives and weights. In the case of alternative values, we created a genetic programming system for generating new decision functions that are more stable than the classical ones.

We think that in the future stability analysis should play a more important role in multicriteria decisions making. Small changes in the assignments of decision makers should not be allowed

Acknowledgements

The authors acknowledge the support of the School of Computer Science at the University of Birmingham, where a substantial part of this work was carried out. This work was supported by grant no. T029572 of the National Research Foundation of Hungary. S.Z. Németh was supported by the Bolyai János Research Fellowship. The authors are grateful to János Fülöp and Csaba Mészáros for many helpful discussions. Many thanks are due to the anonymous reviewers whose important comments helped improving the

References (18)

There are more references available in the full text version of this article.

Cited by (7)

  • Analysing and modelling runtime architectural stability for self-adaptive software

    2017, Journal of Systems and Software
    Citation Excerpt :

    Yet, the results are sensitive to the analysis step and accuracy of data used to build the model. Our method for reasoning about stability can make use of “sensitivity analysis” (Ekárt and Németh, 2005; Kjaerulff and Madsen, 2008), in order to test the extent to which small perturbations to the inputs of the model, i.e. entries of the conditional probability distributions, can affect the stability of the whole architecture. Two types of sensitivity analysis could be performed in probabilistic models: (i) evidence sensitivity analysis, in which how the result of an evidence is sensitive to the variations in the set of evidences, and (ii) parameter sensitivity analysis, in which how the result of an evidence is sensitive to the variations in a parameter of the model.

  • Decision-making management: A tutorial and applications

    2017, Decision-Making Management: A Tutorial and Applications
  • Cloud adoption: Prioritizing obstacles and obstacles resolution tactics using AHP

    2014, Proceedings of the ACM Symposium on Applied Computing
  • Gaining confidence on dependability benchmarks' conclusions through 'back-to-back' testing (practical experience report)

    2014, Proceedings - 2014 10th European Dependable Computing Conference, EDCC 2014
  • Decision support systems and environment: Role of MCDA

    2009, Decision Support Systems for Risk-Based Management of Contaminated Sites
View all citing articles on Scopus
View full text