Exact algorithm over an arc-time-indexed formulation for parallel machine scheduling problems

Abstract

This paper presents an exact algorithm for the identical parallel machine scheduling problem over a formulation where each variable is indexed by a pair of jobs and a completion time. We show that such a formulation can be handled, in spite of its huge number of variables, through a branch cut and price algorithm enhanced by a number of practical techniques, including a dynamic programming procedure to fix variables by Lagrangean bounds and dual stabilization. The resulting method permits the solution of many instances of the P|| w j T j problem with up to 100 jobs, and having 2 or 4 machines. This is the first time that medium-sized instances of the P|| w j T j have been solved to optimality.

The time dependent traveling salesman problem: polyhedra and algorithm

“…We use the same dynamic programming procedure described in [26] to fix x t i,j variables by reduced costs. The procedure is very effective.…”

“…• The formulation in [23], called here PQ, can be generalized to provide very effective formulations to be used in branch-cut-and-price algorithms for several Vehicle Routing Problem (VRP) variants [24] (including "nasty" cases, like the heterogenous fleet VRP [25]) and also complex single and multi-machine scheduling problems [26]. The TDTSP facet-defining inequalities studied in this paper can be readily generalized and used on those problems.…”

Column generation for extended formulations

“…Valerio de Carvalho [23] proposed to solve the linear relaxation of (21-26) by column generation: iteratively solve a partial formulation stemming from a restricted set of variables F i uv , collect the dual solution π associated to (22), solve pricing problem (16), transform its solution, x * , into a path flow that can be decomposed into a flow on the arcs, a solution to (17)(18)(19)(20) which we denote by f (x * ), and add in (21-26) the missing arc flow variables {F i vu : f i vu (x * ) > 0}, along with the missing flow balance constraints active for f (x * ).…”

“…Observe that (17)(18)(19)(20) is only an approximation of the extended network flow formulation associated to the dynamic programming recursion to solve a 0-1 knapsack problem. A dynamic programming recursion for the bounded knapsack problem yields state space {(j, b) : j = 0, .…”

Dantzig-Wolfe Reformulation for the Network Pricing Problem with Connected Toll Arcs

“…A dual variable smoothing approach stabilization procedure to speedup column generation has been introduced by Wentges (1997) and further developed by Pessoa et al (2010). The basic idea is that at each iteration of the column generation procedure, we use a convex combination of these current values with the best known estimates.…”