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J. OpI Res. Soc. Vol. 35, No. 8, pp. 759-767, Printed in Great Britain. All rights reserved
0160-5682/84 $3.00 + 0.00 1984 Operational Research Society Ltd
A Special Multi-Objective Assignment Problem
D. J. WHITE
University of Manchester The following assignment problem is considered. There are n activities to be assigned to n personnel. The cost of assigning activity i to person j is cii. It is required to find all the efficient assignments, i.e. those for which there exists no other assignment which has at least as small costs for each person and strictly smaller costs for at least one person. The main results are as follows. In Theorem 1 it is shown that whereas, for many integer problems, the standard scalar weighting factor approach will not produce all the efficient solutions, in this case it will. In Theorem 2 it is shown that when each efficient vector is determined by a single assignment solution, the efficient set is identical to the set of efficient vertices of the convex hull of the assignment solution set.
MANY REAL-LIFE PROBLEMS involve more than one objective. For example, cost, time, distance and so on may all appear as objectives in a single problem situation. If it is possible to aggregate the complete set of objectives into a single value function, or super objective, then one may optimize this value function in many cases. In practice, satisfactory ways of doing this may not exist for particular circumstances. Nevertheless, the number of options open to a decision maker may be very large, and the task then befalls the operational researcher of reducing the set of options to a manageable set using whatever information he may have about the decision maker's(s') preferences. The simplest form of information is usually that of the monotone property of preferences in terms of the levels of the objective functions. Thus with objectives such as cost, time and distance, the lower the levels, then the more preferable the option. Such considerations give rise to the need to determine the so-called efficient set of the options, viz. the subset of all options for which no other option exists whose associated levels of objective functions are all at least as good as those for an option in the subset, and better than the objective function level for at least one objective function. With the monotonicity assumption it is clear that, even if a value function existed in principle, we need not go outside the efficient solution set in seeking the best, or acceptable, option. Thus the determination of the efficient set, or even a knowledge of how the efficient set may be characterized in principle, would be of some value. It is, of course, true that the efficient set may itself be quite large. However, modern developments...