Problem Formulation

Contents

In this chapter we discuss another class of discrete optimization problems, namely sequencing problems, which belong to a wide class of scheduling problems. Theory of scheduling is an important part of discrete optimization. It deals with a huge number of problems involving different types of resources. In this chapter we discuss only the simplest problems in which the set of resources consists of one machine and a schedule is a sequence (permutation) of a given set of jobs. We seek a feasible sequence to achieve a given goal. Every job has a nonnegative processing time and, typically, we wish to find an order of jobs in which the jobs are completed as quickly as possible (however, in some applications we should also avoid jobs completion being too early). Even this restricted class includes a large number of different problems. Unfortunately, most of them turned out to be NP-hard and their minmax regret versions are NP-hard as well.

In this chapter we adopt the maximal regret criterion to solve a sequencing problem under uncertainty. As in the previous part of this monograph, we will model the imprecise parameters by closed intervals. The first difficulty that arises is that there may be now several parameters associated with a job instead of a single weight as for the minmax regret combinatorial optimization problems considered in the previous part. Hence the characterization of the worst case scenario and the computation of the value of the maximal regret of a given solution become more difficult. In consequence, solving the minmax regret problem is also more challenging.

We start this chapter by recalling the formulation of the deterministic sequencing problem together with some examples. We introduce then the minmax regret sequencing problem, that is the one in which the imprecise parameters are specified as closed intervals and we seek a solution that minimizes the maximal regret. Later, we consider three particular minmax regret sequencing problems whose deterministic counterparts are polynomially solvable.