Decision making with multiple conflicting objectives

Multiobjective decision making problems occur in most disciplines. Despite the considerable va- riety of techniques that have been developed in the field of operations research since the 1950s, solving such problems has presented a non-trivial challenge to researchers. Indeed, the earli- est theoretical work on multiobjective problems dates back to 1895, when the mathematicians Cantor and Hausdorff laid the foundations of infinitely dimensional ordered spaces [72]. Cantor also introduced equivalence classes and utility functions a few years later [29], while Hausdorff published the first example of a complete ordering set [29].

In most decision making situations, the decision maker is required to consider more than one 2_{A set of objectives is decomposable if the decision maker is able to think about each objective easily without}

criterion simultaneously in order to build a complete, accurate decision making model. These criteria, however, typically conflict with one another as informed by the values of the decision maker. Objectives are said to be conflicting if trading an alternative with a higher achievement measure in terms of a certain criterion comes at a cost (i.e. a decrease in the levels of achievement of some of the other criteria for that alternative). A crucial problem in multiobjective decision making therefore lies in analysing how to best perform trade-offs between the values projected by these conflicting objectives.

There exist multiple techniques for determining the quality of an alternative in a decision making process [27]. An appropriate technique should be selected based on factors such as the complexity of the problem, the nature of the decision maker, the time frame for solving the problem or the minimum quality level of the best alternative. The additive preference model, for example, is a popular method in the literature [43, 117], in which the decision maker is required to construct a so-called additive utility function for comparing the attribute levels of available alternatives, as well as the fundamental objectives, in terms of their relative importance, using weight ratios. This decision making model approach is, however, incomplete, as it ignores certain fundamental characteristics of choice theory amongst multiatribute alternatives [27]. In order to resolve this discrepancy, slightly more complex methods such as multiattribute utility models were designed [104]. Here, the decision maker considers attribute interactions by establishing sub-functions for all pairwise combinations of individual utility functions, in contrast to utilising an additive combination of preferences for individual attributes.

In general, the approach adopted toward solving a multiobjective decision making problem may be classified into two paradigms [27, 81]. The first involves combining the individual objective functions into a single composite function, or to move all except one objective to the set of constraints. This is typically the case in utility theory and weighted sum methods. The additive utility function and multiattribute utility models mentioned above, are examples of decision making techniques in this paradigm. Optimisation methods in this paradigm therefore return a single “optimal” solution to a multiobjective decision making problem. This procedure of handling multiobjective optimisation problems is a relatively simple one. Due to the subjective nature of the decision making problem in terms that depend purely on the decision maker’s pref- erences, however, it may often be very difficult to determine all the necessary utility functions and weight ratio parameters accurately, particularly when the number of attributes to be consid- ered is large. Perhaps more importantly, parts of the front are inaccessible when adopting fixed weights in the case of non-convex problems [42]. This optimisation procedure may be enhanced to some extent by considering multiple a priori weight vectors; this is particularly useful in cases where choosing a single appropriate set of weight ratio parameters is not obvious. In most non-linear multiobjective problems, however, it has been shown that a uniformly distributed set of weight vectors need not necessarily result in a uniformly distributed set of Pareto-optimal solutions [42]. Once again, since the nature of this mapping is not usually known, it is often difficult to select weight vectors which are expected to result in a Pareto-optimal solution located within some desired region of the objective space.

Alternatively, the second paradigm, referred to as multiobjective optimisation, aims to enumer- ate and filter the set of all alternatives into a suggested decision set in such a way that the decision maker is indifferent between any two alternatives within the set and so that there exists no alternative outside the set which is preferred to any alternatives within the set. Due to the conflicting nature of the objectives, no single solution typically exists that minimises or max- imises all objectives simultaneously. In other words, each alternative in the suggested decision set yields objective achievement measures at an acceptable level without being outperformed by any other alternatives [81]. Decision making techniques within this paradigm therefore aim to

2.1. Decision theory 17

identify a set of alternative solutions which represent acceptable inter-attribute compromises. This approach is elucidated in the following section. One of the principal advantages of multiob- jective optimisation is that the relative importance of the objectives can be decided a posteriori with the Pareto front on hand.