A new heuristic for scheduling the two-stage flowshop with additional resources

Abstract

This paper deals with the problem of preemptive scheduling in a two-stage flowshop with parallel unrelated machines at the first stage and a single machine at the second stage. At the first stage, jobs use some additional resources which are available in limited quantities at any time. The resource requirements are of 0–1 type. The objective is the minimization of makespan. The problem is NP-hard. Heuristic algorithms are proposed which solve to optimality the resource constrained scheduling problem at the first stage of the flowshop, and at the same time, minimize the makespan in the flowshop by selecting appropriate jobs for simultaneous processing. Several rules of job selection are considered. The performance of the proposed heuristic algorithms is analyzed by comparing solutions with the lower bound on the optimal makespan. The extensive computational experiment shows that the proposed heuristic algorithms are able to produce near-optimal solutions in short computational time.

Introduction

This paper deals with the problem of preemptive scheduling in a two-stage flowshop with parallel unrelated machines at the first stage and a single machine at the second stage. At the first stage, jobs use some additional renewable resources which are available in limited quantities at any time. The resource requirements are of 0–1 type. The objective is the minimization of makespan.

During the last years, the flowshops with multiple processors (FSMP), also called hybrid flowshops, have received considerable attention from researchers. The multiprocessor flowshop is a system which consists of a set of two or more processing centers (processing stages) with at least one center having two or more parallel machines. A job in such a system consists of a sequence of operations processed at successive stages, and all jobs pass through processing stages in the same order. At a stage with parallel machines a job can be processed on any machine and a machine can work on at most one job at a time. Most literature on multiprocessor flowshop addresses the minimum makespan problems under the assumption that preemptions of jobs are not allowed and parallel machines are identical, and include among others, Gupta, 1988, Chen, 1995, Haouari and M’Hallah, 1997, Nowicki and Smutnicki, 1998, Brah and Loo, 1999, Linn and Zhang, 1999. Only a handful of papers concern makespan minimization problem in the flowshop with parallel machines that are not identical (Adler et al., 1993, Ruiz and Maroto, 2006, Suresh, 1997) and in all these papers jobs are considered to be non-preemptive. To the best of our knowledge, the additional renewable resources have not been considered so far in the literature in the context of multiprocessor flowshop scheduling.

In the one stage environment, the problem of scheduling parallel machines under the resource constraints has been widely investigated in the literature. A number of papers in this area concern preemptive scheduling of unrelated machines. For the case of 0–1 resource requirements, a two-stage approach was proposed by Słowiński, 1980, Słowiński, 1981 in which the solution of an LP problem provides input data for the procedure finding an optimal schedule. Some extensions of this approach were proposed in De Werra, 1984, De Werra, 1988. The case of arbitrary resource requirements can be optimally solved by means of a column generation algorithm (Słowiński, 1980) which alternately solves an LP problem with columns (a column determines an assignment of machines to jobs during some period of time) generated so far, and creates a new column, satisfying resource constraints, which improves the current solution. A column generation technique combined with a genetic algorithm was applied in (Figielska, 1998, Figielska, 1999) to solving resource-constrained problems with setup times and costs. In (Hindi & Toczyłowski, 1995), a searching procedure combined with tabu search technique was used for solving a resource constrained scheduling problem with setup times. In (Figielska, 2006) a genetic algorithm was developed for a problem in which non-preemptive jobs as well as setups use some amounts of additional renewable resources. Some works address parallel machine scheduling problems with resource-dependent processing times (Józefowska et al., 1998, Józefowska et al., 2002, Shabtay and Kaspi, 2006). The survey of scheduling under resource constraints can be found in (Błażewicz, Cellary, Słowiński, & Węglarz, 1987).

In the multiprocessor flowshop environment the resource constrained scheduling problem was not considered so far in the literature. However, in real-life systems in majority of cases a cell with parallel machines is a part of a larger system such as, for example, multiprocessor flowshop, so the problem of resource constrained scheduling arises also in such systems and a special approach to solving this problem is needed.

In this paper, we extend the research on resource constrained parallel machine scheduling and on multiprocessor flowshop scheduling by including the additional renewable resources in the problem of scheduling in the flowshop with parallel machines.

We consider the problem of scheduling in a two-stage flowshop with parallel machines at the first stage and a single machine at the second stage. At the first stage, jobs use additional renewable resources, which are available in limited quantities at any time. This problem can be described as follows. There are n preemptive jobs to be processed at two-stages in the same technological order, first at stage 1 then at stage 2. At stage 1 there are m parallel unrelated machines, stage 2 has one machine. A job upon finishing its processing at stage 1 is ready to be processed at stage 2; it may be processed at stage 2 when the machine is available there, or it may reside in a buffer space of unlimited capacity following stage 1 until the machine at stage 2 becomes available. At stage 1, a job can be processed on any of the parallel machines, and its processing times may be different on different machines. The processing times of job j (j = 1, …, n) are equal to p_ij (if it is executed on machine i (i = 1, …, m)) and s_j time units, respectively, at stage 1 and at stage 2. The processing of a job on a machine of stage 1 may be interrupted at any moment and resumed later on the same or another machine. Jobs for their processing at stage 1, besides machines, require additional resources. There are l types of additional resources. A resource of type r (r = 1, …, l) is available in an amount limited to W_r units at a time. The total usage of resource r at any moment by jobs simultaneously executed on parallel machines cannot exceed the availability of this resource. Each job, during its processing at stage 1, uses 0 or 1 unit of the resource of each type. All required resources are granted to a job before its processing begins or resumes and they are returned by the job after finishing its processing at a stage of the flowshop or in the case of its preemption. The objective is to find a feasible schedule which minimizes makespan, C_max, which is equal to the maximum job completion time at stage 2.

The considered problem is NP-hard in the strong sense since the problem of preemptive scheduling in the two-stage flowshop with two identical parallel machines at one stage and one machine at another is NP-hard in the strong sense (Hoogeveen, Lenstra, & Veltman, 1996).

The problem of scheduling in multiprocessor flowshop arises in real-life systems that are encountered in a variety of industries, e.g., in ceramic industry (Ruiz & Maroto, 2006), metallurgical industry (Narasimhan and Mangiameli, 1987, Narasimhan and Panwalkar, 1984), chemical industry (Fortemps et al., 1996, Riane, 1998), paper industry (Sherali, Sarin, & Kodialam, 1990), wood industry (Riane, 1998) food, cosmetics and textile industries (Sherali et al., 1990). The multiprocessor scheduling problem can be also met in computer systems and telecommunication networks (Hunsucker & Shah, 1994). In the multiprocessor flowshop there are more than one parallel machines and jobs can be processed through any one of the machines at each processing stages. Because jobs that are simultaneously processed on parallel machines at a stage of the flowshop may use the same resource, the problem of resource constrained scheduling arises when the amount of the available resource is limited. This takes place when for example the number of workers attending the machines, or the number of tools that are used by simultaneously executed jobs, is limited. The problem of preemptive production arises for example in the textile industry (Serafini, 1996) where processing of any job (the article to be woven) on one of the parallel machines (the looms) may be interrupted (preempted) and resumed on the other machine.

For solving the considered problem we propose heuristic algorithms based on linear programming. These algorithms first solve to optimality the resource constrained preemptive scheduling problem for the first stage of the flowshop. While solving the problem at the first stage, jobs for simultaneous processing on parallel machines are chosen so as to minimize the makespan in the flowshop. Given the schedule at the first stage, the schedule at the second stage is constructed using ready times of jobs, which are equal to the corresponding completion times at the first stage, and processing times of jobs at the second stage. The selection of jobs is performed according to 5 selection rules. Thus, 5 heuristic algorithms are designed for the considered problem. Performance of the algorithms is analyzed by comparing the heuristic solutions with the lower bound on the optimal makespan.