Permutation Polyhedra and Minimisation of the Variance of Completion Times on a Single Machine

Abstract

We consider the problem of minimising variance of completion times when n-jobs are to be processed on a single machine. This problem is known as the CTV problem. The problem has been shown to be difficult. In this paper we consider the polytope P _n whose vertices are in one-to-one correspondence with the n! permutations of the processing times [p ₁, p ₂, ..., p _n]. Thus each vertex of P _n represents a sequence in which the n-jobs can be processed. We define a V-shaped local optimal solution to the CTV problem to be the V-shaped sequence of jobs corresponding to which the variance of completion times is smaller than for all the sequences adjacent to it on P _n. We show that this local solution dominates V-shaped feasible solutions of the order of 2ⁿ⁻³ where 2ⁿ⁻² is the total number of V-shaped feasible solutions.

Using adjacency structure an P _n, we develop a heuristic for the CTV problem which has a running time of low polynomial order, which is exact for n ≤ 10, and whose domination number is Ω(2ⁿ⁻³). In the end we mention two other classes of scheduling problems for which also ADJACENT provides solutions with the same domination number as for the CTV problem.