Decision aiding model with entropy-based subjective utility

Abstract

An entropy-based method is presented to model a decision-maker’s (DM’s) subjective utility for a criterion value. The proposed method considers distribution of all the values that the criterion takes for the given set of alternatives. Based on the utility so modeled, and and the DM’s attitudinal character, a multi criteria decision aiding (MCDA) approach is developed to find the best alternative. The proposed method and the approach are applied in a real car selection case-study.

Introduction

In the real world, it is often difficult to find an alternative that is the best with respect to all the desirable criteria. Hence, the decision-maker (DM) makes a compromise among the conflicting criteria in order to determine his best choice. Most approaches explain this compromise through the concept of utility. It is averred in [37] that the utility maximizing behaviour is the key characteristic of human decision making. For instance, a consumer evaluates the utility of various alternatives against a set of desired properties (or criteria), and chooses the one giving him the maximum utility. Similarly, an organization determines the best production technology or the most suitable supplier; a student chooses a career stream among many; an individual chooses whether to be an employee or an entrepreneur. Any decision or choice by a DM is an outcome of a complex cognitive process that broadly involves:

• evaluation of utility for each of the criteria, and

• aggregation of such utility values to arrive at the net alternative utility.

Multi criteria decision aiding (MCDA) has become an important and pervasive subfield of operations research, which gives a systematic process to aid a decision-maker (DM) in choosing the best alternative. In the recent times, MCDA has received a lot of attention in the literature [32], [43]. The evaluation of subjective utility values is of crucial importance for a MCDA approach. In this regard, the works of Grable and Lytton [20], and Clemen and Reilly [15] are notable. Both are about the identification of the value of the utility function at a discrete set of points. The utility function is then obtained through a simple interpolation. The attempts in [1], [2], [39] to elicit the underlying utility function behind a DM’s choices have received attention. However, all these approaches require an extensive interaction with the DM, which makes them difficult to automate. Moreover, these methods need an ordinal preference for each of the alternatives. This restricts their usefulness in practical applications.

Some approaches have attempted to represent the utility function in terms of the parameters of aggregation operators, such as weighted averaging or Choquet integral [14]. In [40], the weight vector of weighting averaging operator is learned, while the parameter (fuzzy measure) of Choquet integral is resolved for a few datasets in supervised learning setup in [36]. Thus, these methods deduce parameter(s) that best represent(s) a given dataset. These methods are good at explaining the choices for a given dataset, but not so helpful in our goal of inferring the utility of a particular criterion value. Secondly, these approaches are based on the pairwise comparisions of the alternatives. This is achieved either by asking a DM directly about his preferences, or indirectly inducing through the DM’s pre-given ranking of the alternatives. In both the cases, it is quite a time-consuming and cumbersome task, when the number of alternatives exceeds 5 or 6.

In a similar vein, a few interesting approaches have appeared on optimization-based utility elicitation from a given dataset [8], [10]. Another set of studies considers that the knowledge of a DM is incomplete to precisely evaluate an alternative against a criterion [7], [29], [41], [45]. Different methods have been applied to deal with the incomplete information. In [12], [13], the authors make use of a probability distribution over different utility functions, and the final choice is determined by the expected utility averaged over such a distribution. However, these studies assume to be given with a database of partially elicited utility functions, which, in practice, are quite difficult to gather.

In  [11], a set of all those utility functions is constructed, which do not contradict the given information. The alternative that achieves minimum worst-case regret, in a mixed-integer programming approach, forms the best choice. Bayesian methods have been explored to quantify uncertainty about preferences in [23]. The studies [7], [16], [24] solve convex optimization problems of real-valued values to find the best choice. The utility and expected utility theory have been built upon for decision-making under uncertainty in [6] and  [17], respectively. The concept of probabilistic choice model has been propounded in [38].

These methods, despite their theoretical soundness, are not convenient to solve real MCDA problems, as they suffer from three fundamental limitations :

• they do not give the utility for a particular value. In practice, a criterion value is perceived differently by different individuals, and hence each DM evaluates a different utility (desirability) for a criterion value. For instance, length of a car (say 5000 mm), would be perceived differently by different individuals. While, it might be usual (normal) for one individual , it may be considered as too grand by another. With the extant methods, it is difficult to quantify this utility value that is specific to a DM.

• they are context-specific. A DM’s choices may completely differ with the availability of more alternatives that may be significantly better than the earlier alternatives, on offer. For instance, in the case of car selection, a new promising model often outshines the extant bestsellers. Also, quite often, a DM’s information regarding the number of criteria is often incomplete, which results in a modified utility evaluation in the face of new information. It is difficult to model such variations with most of the extant methods.

• they do not focus on the individualistic aspects of the aggregation process. Different individuals have different tolerance for an imperfect criterion. A tolerant DM displays a degree of compensation in the determination of the net alternative utility, in the sense that a poor utility score for a criterion can be compensated by a good utility score for another criterion. This degree of compensation varies from DM to DM. A perfectionist would have the expected utility as the minimum of the criteria utility values (maximin/ANDlike), displaying a nil compensation, whereas an extreme tolerant DM with maximum compensation has his expected utility towards the maximum of the criteria utility values (maximax/ORlike).

These limitations of the extant approaches significantly restrict their application in the real world problems. Hence, we are inspired to develop a more practical and systematic utility representation approach. The paper is organized as follows. Section 2 discusses the MCDA paradigm, and builds upon the motivation and contributions of the study. In Section 3, we look deeper at the concept of entropy to explore its usefulness to represent the subjective utility of a decision-maker. Section 4 gives a method to quantify subjective utility using the concept of entropy. In Section 5, we present a MCDA approach using the proposed entropy-based model. Section 6 illustrates the usefulness of the proposed concepts through a real application. Section 7 concludes the paper.