Abstract
In this paper, we consider a two-machine job-shop scheduling problem of minimizing total completion time subject to n jobs with two operations and equal processing times on each machine. This problem occurs e.g., as a single-track railway scheduling problem with three stations and constant travel times between any two adjacent stations. We present a polynomial dynamic programming algorithm of the complexity O ( n 5 ) and a heuristic procedure of the complexity O ( n 3 ) . This settles the complexity status of the problem under consideration which was open before and extends earlier work for the two-station single-track railway scheduling problem. We also present computational results of the comparison of both algorithms. For the 30,000 instances with up to 30 jobs considered, the average relative error of the heuristic is less than 1 % . In our tests, the practical running time of the dynamic programming algorithm was even bounded by O ( n 4 ) .
Highlights
We consider a two-machine job-shop scheduling problem
We present a new polynomially solvable case for the two-machine job-shop problem with minimizing total completion time based on dynamic programming [4]
We present some results of a numerical experiment, where we investigate the relative error of the heuristic algorithm H and the number of states considered in Algorithm dynamic programming algorithm (DP)
Summary
We consider a two-machine job-shop scheduling problem. Each job j ∈ N = {1, 2, . . . , n} consists of two operations, i.e., we have n j = 2 according to [1]. Mathematics 2019, 7, 301; doi:10.3390/math7030301 j∈ Nba www.mdpi.com/journal/mathematics We denote this problem by J2|n j = 2, p j1 = a, p j2 = b| ∑ Cj according to the traditional three-field notation α| β|γ for scheduling problems proposed by Graham et al [2], where α describes the machine environment, β gives the job characteristics and further constraints, and γ describes the objective function. We present a new polynomially solvable case for the two-machine job-shop problem with minimizing total completion time based on dynamic programming [4]. This extends an existing polynomial algorithm for the two-station single-track railway scheduling problem from [5] to the case of three stations.
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