NAUTILUS method: An interactive technique in multiobjective optimization based on the nadir point

Abstract

Most interactive methods developed for solving multiobjective optimization problems sequentially generate Pareto optimal or nondominated vectors and the decision maker must always allow impairment in at least one objective function to get a new solution. The NAUTILUS method proposed is based on the assumptions that past experiences affect decision makers’ hopes and that people do not react symmetrically to gains and losses. Therefore, some decision makers may prefer to start from the worst possible objective values and to improve every objective step by step according to their preferences. In NAUTILUS, starting from the nadir point, a solution is obtained at each iteration which dominates the previous one. Although only the last solution will be Pareto optimal, the decision maker never looses sight of the Pareto optimal set, and the search is oriented so that (s)he progressively focusses on the preferred part of the Pareto optimal set. Each new solution is obtained by minimizing an achievement scalarizing function including preferences about desired improvements in objective function values. NAUTILUS is specially suitable for avoiding undesired anchoring effects, for example in negotiation support problems, or just as a means of finding an initial Pareto optimal solution for any interactive procedure. An illustrative example demonstrates how this new method iterates.

Introduction

Handling optimization problems in almost any application field typically leads to the need of considering several conflicting objectives simultaneously. This means that we must be able to solve multiobjective optimization problems and many methods have been developed for this need during the years (see, e.g. Branke et al., 2008, Chankong and Haimes, 1983, Miettinen, 1999, Sawaragi et al., 1985). Assuming the problem has been correctly specified, the methods usually concentrate on Pareto optimal solutions where none of the objective values can be improved without impairing at least one of the others. Thus, the concept of Pareto optimality necessitates trading off among the objectives.

Solving multiobjective optimization problems can be understood as finding the Pareto optimal solution that best satisfies the needs of the decision maker (DM). Thus, the final solution of a multiobjective optimization method is often referred to as the most preferred solution. The class of interactive methods (see, e.g. Luque et al., in press, Miettinen et al., 2008, and references therein) is a widely developed one consisting of a variety of approaches that iteratively proceed towards the most preferred solution and the DM can learn about the interdependencies among the objectives during the solution process and adjust one’s preferences accordingly. Interactive methods generate Pareto optimal solutions based on the DM’s preferences and when the DM changes one’s preferences, a new or some new Pareto optimal solutions are obtained. According to the definition of Pareto optimality, moving from one Pareto optimal solution to another necessitates that the DM is willing to impair the value of at least one objective function. This is not always easy and researches have reported surprisingly low numbers of iterations in real applications. Namely, for example, the median number of iterations has been between three and eight in the interactive solution processes reported in the literature according to the study by Gardiner and Vanderpooten (1997).

One would like to think that the low number of iterations taken in interactive solution processes would be explained by the fact that the methods used have been able to support the DM well and finding the most preferred solution has not required any more iterations for this reason. However, one possible explanation to the low number of iterations, as suggested by Korhonen and Wallenius (1996), is the fact that people do not react symmetrically to gains and losses. According to the prospect theory of Kahneman and Tversky (1979), our attitudes to losses loom larger than gains. For example, the pleasure of gaining a sum of money seems to be lower than the dissatisfaction of losing the same amount of money. Thus, requiring the DM to trade-off when changing consideration from one Pareto optimal solution to another may hinder her/his willingness to move from the current Pareto optimal solution. If, on the other hand, we start from a point where improvement in all objectives is possible, the DM can direct the solution process more freely and always attain only gains.

Some studies available in the scientific literature support our idea of allowing improvement in all objectives because iterating using Pareto optimal solutions only (and consequently, forcing the DM to trade-off among them) is not always advisable. For example, Janis and Mann (1977) state that “trade-off conflict is a major source of decisional stress”. Furthermore, Aloysius et al. (2006) found out in the behavioral literature that “choice sets that are high in trade-off conflict led to less accurate decision making”, and they carried out some experiments to conclude that “the explicit attribute tradeoffs required by a Multi-Criteria Decision Support System (MCDSS) can cause negative affect or decisional conflict in a decision maker. This decisional conflict is a significant determinant of user acceptance of decision aids, and thus can have influence on the adoption and continued utilization of these MCDSS by the end user. There remains much potential to extend this research by exploring features of MCDSS other that the preference elicitation technique that may effect user adoption”. Following the same line of reasoning, Dyer et al. (1992) encourage researchers to “bring behavioral and psychological insights to bear on the design of MCDSS”.

Another observation in the background of this study is the fact that our past experiences affect our hopes and the solutions considered may fix our expectations to a too limited range. This phenomenon can be defined as anchoring (Miettinen, 1999), where the DM fixes one’s thinking on some (possible irrelevant) information. For example, the starting point of the interactive method can be an anchor and the DM may fail to adjust one’s preferences and move away from the anchor. This means that the choice of the starting point may play a significant role in the success or failure in finding the most preferred solution. Anchoring has been studied, for example, in Buchanan and Corner (1997). Observations in real life confirm that if we first see a very unsatisfactory solution, a somewhat better solution is more satisfactory than otherwise. As formulated in Kahneman and Tversky (1979), “the past and present context of experience defines an adaptation level, or reference point, and stimuli are perceived in relation to this reference point.” A frequent reason of the anchoring effect is the fact that real, experienced decision makers usually do not have time to spend in a long interactive procedure. Therefore, they rather rely on their intuition, although the danger exists that this intuitions might result in anchoring and fixation of interests.

Based on the reasoning given, it is in order to question the setting that multiobjective optimization methods should generate Pareto optimal solutions throughout the solution process. Instead, it seems promising to start from the nadir point reflecting the worst possible values of each objective function and allow the DM to iteratively direct the search towards the most preferred solution. In this way, improvement is possible in each objective value and the DM can reach any Pareto optimal solution. This completely new methodological approach is the basic idea of the NAUTILUS method that we propose in this paper.

In some sense, NAUTILUS works as a submarine, if we regard the nadir solution as the point of maximum depth, and the Pareto optimal set in the objective space as the surface. From the deepest point, any Pareto optimal solution can be reached in principle. At every iteration, a step is taken towards the surface, in a direction determined by the preference information given by the DM. The specification of preferences allows the DM to focus on a part of the Pareto optimal set that will progressively shrink as we approach the surface. This implies that every time we take a step towards the surface, certain parts of the Pareto optimal set will not be reachable from that point if we keep ascending (and thus, it would be required to descend again if they were to be reconsidered).

At each step of NAUTILUS, the information shown to the DM is much more that just a solution dominating the previous one. The DM is informed both about the reachable part of the Pareto optimal set from the current iteration, and about the proximity to the surface. This information substitutes the trade-offs of a classical interactive method. It gives the DM an idea of what is lost and what can be achieved when approaching the Pareto optimal set in a certain direction, and thus, the DM can confirm the step or choose a different direction. Each iteration is based on the reference point scheme. But because the direct consideration of this scheme would take us to the Pareto optimal set in a single step, the DM is asked to specify the number of iterations (s)he wishes to perform, and the steplengths in the ascending directions are internally calculated by the method accordingly. This number of iterations is not a critical issue of NAUTILUS, because the DM can reconsider it at any moment, either by directly changing the number of remaining iterations (if the distance to the Pareto optimal set is regarded as inappropriate), or by taking steps backwards (if the current achievable part of the Pareto optimal set is not satisfactory). It must be pointed out that with NAUTILUS, we do not question the idea of finding a Pareto optimal solution as the final one. Moreover, the DM does not loose sight of the Pareto optimal set at any moment during the solution process. In this way, we wish to support the DM in free search for the most preferred solution.

Apart from its use in general multiobjective problems, a natural field of application of the NAUTILUS method is negotiation support. In fact, when several decision makers with conflictive objectives negotiate, it is not always a good policy to start from a Pareto optimal solution. This can result in anchoring for those negotiators who regard the current Pareto optimal solution as advantageous for their interests. Therefore, starting from the nadir point and producing solutions that dominate the previous ones at each step enables the decision makers to continue negotiating, because they all know that they can improve their situations. In addition, NAUTILUS can be used to find a non-anchored starting point for any other interactive method.

Although there exist some decision making models based on the prospect theory (see, for example, the decisional wealth approach in Karasakal and Michalowski (2003)), surprisingly few methods have been proposed in the literature so far that start the solution process from the nadir point. Among the few, we can mention the interior primal–dual multiobjective linear programming algorithm of Arbel and Korhonen (1996) (designed for linear problems only) and the IMGP (Interactive Multiple Goal Programming) method (Nijkamp and Spronk, 1980), which starts from the nadir point but changes goals through intervals and solves several ε-constraint problems.

The rest of this paper is arranged as follows. In Section 2 we describe the basic concepts and notations used. The NAUTILUS method is motivated and described in Section 3 and Section 4 illustrates the way it iterates with a simple example. We conclude with some additional comments in Section 5 and conclusions in Section 6.