Permutation polyhedra and minimisation of the variance of completion times on a single machine
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 2(n-3) where 2(n-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 less than or equal to 10, and whose domination number is Omega(2(n-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.
permutation polyhedra dominance completion time variance dominance value