Scheduling with flexible resources in parallel workcenters to minimize maximum completion time
Introduction
This paper addresses the assignment of jobs and secondary resources to parallel workcenters with the objective of minimizing the schedule's makespan. The problem is called the Unassigned Parallel Machine Flexible Resource Scheduling Problem (UPMFRS) and was originally addressed by Daniels et al. [1]. In this environment, the time to process a job is dependent on the number of secondary resources assigned to the workcenter (machine). Also, at the start of the schedule jobs can be assigned to any of the workcenters. The workcenter represents fixed type resources while the secondary resource is flexible and must be assigned to a workcenter for it to function properly. There are two versions of the UPMFRS problem, the dynamic case, where resources can be assigned to any of the machines every time a job is completed, and the static case, where resource allocation decisions are fixed for the duration of the schedule. A related problem proposed earlier by Daniels et al. [2] was called the Parallel Machine Flexible Resource Scheduling Problem (PMFRS) where jobs have specific machine assignments (therefore not controlled by the scheduling process) and the only decision faced was the allocation of the secondary (flexible) resource to the machines.
This research contributes to the scheduling field by extending the knowledge in the area of planning with dual resources in a multi-machine setting. This is a highly relevant problem in manufacturing environments as it includes the two most common resource types: stationary type resources, such as workcenters and assembly equipments, and the non-stationary flexible resource, in most cases people. Scheduling research has often ignored the second type of resource, even when it has been demonstrated by researchers (as shown below) that it results in better solutions, and more importantly, that is a more accurate representation of a large number of real-world production systems.
The allocation of flexible resources in parallel and multi-machine scheduling problems has received attention in the last few years. Olafsson and Shi [3] provided a new formulation to the PMFRS problem, as well as a new heuristic methodology called Nested Partitions. A problem similar to the PMFRS was addressed by Kellerer and Strusevich [4] for parallel machines dedicated to a job set (e.g., a product family) where some jobs required additional renewable resources, although this had no effect on processing speed. In this case, the dual constraint effect is only a requirement to production for that specific job. Jensen [5] considered parallel workcenters and investigated the impact of labor flexibility on flowtime, mean tardiness, and root mean squared tardiness, concluding that controlling labor assignment has significant impact on these measures.
The objective of this paper is to further understand the effect of a secondary flexible resource (called FRs) in a parallel workcenter setting for the makespan criterion by evaluating the performance of various scheduling approaches. The evaluation considers an experimental setting similar to that used in [1] and a second experimental setting where there is a higher level of resource flexibility. This was accomplished by using a larger pool of flexible resources and a change in the representation of the relationship between processing times and the assignment of these resources. The research also investigates the effect of shutting down some of the machines in order to evaluate having additional units of the flexible resource on the available machines, and therefore process jobs at their faster speed modes. The rest of the paper is organized as follows. The next section presents the problem formulation, which is followed by a discussion of the solution approaches, an example of the approaches, the experimental setting, computational results and conclusions.
Section snippets
Problem notation and lower bound
The problem addressed in this paper is described as follows: there are n single operation jobs, to be processed on m identical machines, . Jobs are available at time 0 and cannot be preempted or divided. The processing time of job j in mode k, is and to achieve , there must be units of resource allocated to job j for its duration. Each job can be processed in modes: and the mode number is the same as the number of resources processing a job. The
Solution methodologies
This section presents the heuristic procedures used to assign jobs and flexible resources to the machines. Two of these procedures were presented in Daniels et al. [1], while the others are being proposed as alternative methods.
Numerical example
This section illustrates the heuristics presented in Section 3 with a numerical example. For examples on the DH and TS rules refer to [1]. The following parameters are assumed: , and job processing times as in Table 1, where the associated speeds are and .
The application of the SF procedure assigned r to the three machines as and jobs are loaded into the machines using the LPT index of resulting in the assignment
Experimental setting
The existing and proposed heuristics were coded in Microsoft's Visual Basic for Applications on an Excel platform and implemented on a Pentium 4 personal computer. To evaluate the performance of the algorithms two sets of experiments were conducted, one using as a basis the experimental parameters from [1] for their “large problems” experiment called E1, and a second with parameters that emulate a larger production shop with a higher number of flexible resources, called E2. The aim was to
Results
The lower bound (LB) makespan for each problem instance was found by full enumeration of the resource assignment alternatives. In the search for the lower bound, the shut down of machines was considered to the rounded up value of as shutting down more machines would leave some of the FRs unassigned, which resulted in an inferior result. The evaluation of the heuristics was based on two measures: first the error, determined by , and second, the percentage of time a
Conclusions and future research
This paper addressed the static case of the Unassigned Parallel Machine Flexible Resource Scheduling problem, providing a method to estimate the lower bound, presenting new heuristics, and comparing them to previous methods. The proposed heuristics combined traditional methods used for the parallel machine makespan problem with two new strategies to assign the flexible resources. The computational results demonstrated that the proposed methods worked very well, outperforming the previously
Acknowledgements
The authors want to thank Dr. Johnny Ho from the University of Texas at El Paso and two anonymous reviewers for their insightful comments toward the improvement of the paper.
References (11)
- et al.
Heuristics for parallel machine flexible resource scheduling problems with unspecified job assignment
Computers and Operations Research
(1999) The impact of resource flexibility and staffing decisions on cellular and departmental performance
European Journal of Operational Research
(2000)- et al.
Makespan minimization for two uniform parallel machines
International Journal of Production Economics
(2003) - et al.
Makespan minimization on uniform parallel machines with release times
European Journal of Operational Research
(2004) - et al.
Multiprocessor scheduling: combining LPT and MULTIFIT
Discrete Applied Mathematics
(1988)
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