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.

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

This study proposes an exact algorithm for the general single-machine scheduling problem without machine idle time to minimize the total job completion cost. Our algorithm is based on the Successive Sublimation Dynamic Programming (SSDP) method. Its major drawback is heavy memory usage to store dynamic programming states, although unnecessary states are eliminated in the course of the algorithm. To reduce both memory usage and computational efforts, several improvements to the previous algorithm based on the SSDP method are proposed. Numerical experiments show that our algorithm can optimally solve 300 jobs instances of the total weighted tardiness problem and the total weighted earliness-tardiness problem, and that it outperforms the previous algorithms specialized for these problems.

An exact algorithm for single-machine scheduling without machine idle time

Exact algorithms for a generalization of the order acceptance and scheduling problem in a single-machine environment

http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/160225/2/j.cor.2012.07.004.pdf

An exact algorithm for the single-machine total weighted tardiness problem with sequence-dependent setup times

We address the NP-hard single-machine problem of scheduling n independent jobs so as to minimize the sum of α times total completion time and β times total earliness with β > α, which can be rewritten as an earliness–tardiness problem. Postponing jobs by leaving the machine idle may then be advantageous. The allowance of machine idle time between the execution of jobs singles out our problem from most concurrent research on problems with earliness penalties. Solving the problem to optimality poses a computational challenge, since the possibility of leaving the machine idle has a major effect on designing a branch-and-bound algorithm in general, and on computing lower bounds in particular. We present a branch-and-bound algorithm which is based upon many dominance rules and various lower bound approaches, including relaxation of the machine capacity, data manipulation, and Lagrangian relaxation. The algorithm is shown to solve small instances with up to 20 jobs.

Single-machine earliness-tardiness scheduling with idle time

This paper proposes an efficient exact algorithm for the general single-machine scheduling problem where machine idle time is permitted. The algorithm is an extension of the authors' previous algorithm for the problem without machine idle time, which is based on the SSDP (Successive Sublimation Dynamic Programming) method. We first extend our previous algorithm to the problem with machine idle time and next propose several improvements. Then, the proposed algorithm is applied to four types of single-machine scheduling problems: the total weighted earliness-tardiness problem with equal (zero) release dates, that with distinct release dates, the total weighted completion time problem with distinct release dates, and the total weighted tardiness problem with distinct release dates. Computational experiments demonstrate that our algorithm outperforms existing exact algorithms and can solve instances of the first three problems with up to 200 jobs and those of the last problem with up to 80 jobs.

/pdf/a-dynamic-programming-based-exact-algorithm-for-general-qr9js23z2e.pdf

A dynamic-programming-based exact algorithm for general single-machine scheduling with machine idle time

In this paper we propose an exact algorithm for the general single-machine scheduling problem where machine idle time is permitted. The algorithm is an extension of the authorspsila previous algorithm for the problem without machine idle time, which is based on the SSDP (successive sublimation dynamic programming) method. In this algorithm a lower bound is computed by applying dynamic programming to a Lagrangian relaxation of the original problem and then it is successively improved by imposing additional constraints on the relaxation until the gap between lower and upper bounds diminishes. Unnecessary dynamic programming states are eliminated in the course of the algorithm to reduce both computational efforts and memory usage. Experimental results show that the proposed algorithm can solve 200 jobs instances of the single-machine total weighted earliness-tardiness problem.

Shuji Fujikuma

Papers

An exact algorithm for single-machine scheduling without machine idle time

A dynamic-programming-based exact algorithm for general single-machine scheduling with machine idle time

An efficient exact algorithm for general single-machine scheduling with machine idle time

6A2 A BRANCH-AND-BOUND ALGORITHM BASED ON LARGRANGIAN RELAXATION FOR SINGLE-MACHINE SCHEDULING(Technical session 6A: Machine and shop scheduling 1)