Liu, Zhengliang and Ehrgott, Matthias (2016) Optimisation over the non-dominated set of a multi-objective optimisation problem. PhD thesis, Lancaster University.
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Abstract
In this thesis we are concerned with optimisation over the non-dominated set of a multiobjective optimisation problem. A multi-objective optimisation problem (MOP) involves multiple conflicting objective functions. The non-dominated set of this problem is of interest because it is composed of the “best” trade-off for a decision maker to choose according to his preference. We assume that this selection process can be modelled by maximising a function over the non-dominated set. We present two new algorithms for the optimisation of a linear function over the non-dominated set of a multi-objective linear programme (MOLP). A primal method is developed based on a revised version of Benson’s outer approximation algorithm. A dual method derived from the dual variant of the outer approximation algorithm is proposed. Taking advantage of some special properties of the problem, the new methods are designed to achieve better computational efficiency. We compare the two new algorithms with several algorithms from the literature on a set of randomly generated instances. The results show that the new algorithms are considerably faster than the competitors. We adapt the two new methods for the determination of the nadir point of (MOLP). The nadir point is characterized by the componentwise worst values of the non-dominated points of (MOP). This point is a prerequisite for many multi-criteria decision making (MCDM) procedures. Computational experiments against another exact method for this purpose from the literature reveal that the new methods are faster than the competitor. The last section of the thesis is devoted to optimising a linear function over the non-dominated set of a convex multi-objective problem. A convex multi-objective problem (CMOP) often involves nonlinear objective functions or constraints. We extend the primal and the dual methods to solve this problem. We compare the two algorithms with several existing algorithms from the literature on a set of randomly generated instances. The results reveal that the new methods are much faster than the others.