A Cutting Plane Algorithm for the Single Machine Scheduling Problem with Release Times

We propose a mixed integer programming formulation for the single machine scheduling problem with release times and the objective of minimizing the weighted sum of the start times The basic formulation involves start time and sequence determining variables and lower bounds on the start times Its linear programming relaxation solves problems in which all release times are equal For the general problem good lower bounds are obtained by adding additional valid inequalities that are violated by the solution to the linear programming relaxation We report computational results and suggest some modi cations based on including additional variables that are likely to give even better results Supported by NATO Collaborative Research Grant No Supported by NSF Research Grant No ISI Supported by the Netherlands Organization for Scienti c Research through NATO Science Fellowship Grant No N A Cutting Plane Algorithm for the Single Machine Scheduling Problem with Release Times G L Nemhauser Georgia Institute of Technology Atlanta M W P Savelsbergh Eindhoven University of Technology Abstract We propose a mixed integer programming formulation for the single machine scheduling problem with release times and the objective of minimizing the weighted sum of the start times The basic formulation involves start time and sequence determining variables and lower bounds on the start times Its linear programming relaxation solves problems in which all release times are equal For the general problem good lower bounds are obtained by adding additional valid inequalities that are violated by the solution to the linear programming relaxation We report computational results and suggest some modi cations based on including additional variables that are likely to give even better resultsWe propose a mixed integer programming formulation for the single machine scheduling problem with release times and the objective of minimizing the weighted sum of the start times The basic formulation involves start time and sequence determining variables and lower bounds on the start times Its linear programming relaxation solves problems in which all release times are equal For the general problem good lower bounds are obtained by adding additional valid inequalities that are violated by the solution to the linear programming relaxation We report computational results and suggest some modi cations based on including additional variables that are likely to give even better results Introduction Recently developed polyhedral methods have yielded substantial progress in solving many important NP hard combinatorial optimization problems Some well known examples are the traveling salesman problem Gr otschel and Padberg a Gr otschel and Pad berg b the acyclic subgraph problem J unger and large scale integer programming problems Crowder Johnson and Padberg See Ho man and Pad berg and Nemhauser and Wolsey for general descriptions of this approach However for mixed integer problems in particular machine scheduling polyhedral methods have not been nearly so successful Investigation and development of polyhedral methods for machine scheduling problems is important because traditional combinatorial algorithms do not perform well on certain problem types in this class for instance job shop scheduling The major di culty is obtaining tight lower bounds which are needed to prove optimality or even optimality within a speci ed tolerance Relatively few papers and reports have been written in this area Balas pio neered the study of scheduling polyhedra with his work on the facial structure of the job shop scheduling problem Queyranne completely characterized the polyhedron as sociated with the nonpreemptive single machine scheduling problem Dyer and Wolsey examined several formulations for the single machine scheduling problem with release times Queyranne and Wang generalized Queyranne s results to the non preemptive single machine scheduling problem with precedence constraints Sousa and Wolsey investigated time indexed formulations for several variants of the nonpre emptive single machine scheduling problem Finally Wolsey compared di erent formulations for the single machine scheduling problem with precedence constraints In this paper we propose a formulation that involves start time and sequence deter mining variables for the nonpreemptive single machine scheduling problem with release times and we develop a cutting plane algorithm based on this formulation and several classes of valid inequalities The paper is organized as follows In the next section we formally introduce the single machine scheduling problem with release times and pro pose a mixed integer programming formulation In the subsequent sections we discuss a linear relaxation various classes of valid inequalities separation heuristics and the cutting plane algorithm we have implemented In the nal sections we present compu tational results and possible enhancements that are based on using additional variables and column generation The single machine scheduling problem with release times A set J of n jobs has to be processed without interruption on a single machine that can handle at most one job at a time Each job j J becomes available at its release time rj and requires a processing time pj The problem is to nd a feasible schedule that minimizes the weighted sum of the completion times In the sequel we assume that both rj and pj are nonnegative integers and the jobs are numbered in order of nondecreasing release time i e r r rn For any ordering of the jobs there exists one feasible schedule that dominates all others In this schedule called an active schedule each job is processed as early as possible given the processing order If t j denotes the start time of job j the active schedule for is t r t j max r j t j p j for j n The above observation shows that we can restate the nonpreemptive single machine problem with release times as nd a permutation and associated active schedule for which the objective function is minimum Therefore to obtain a valid formulation it su ces to nd a linear inequality description of the set of permutations and of the active schedule associated with a given permutation Let ij be equal to if job i precedes j and otherwise Then B n n is a permutation if and only if it satis es