Minimizing Energy Cost in Parallel Machines Considering Setup Time

In recent years, significant energy consumption and facing global warming have led to concern worldwide. Therefore, governments have turned to deterrent actions such as imposing daily tariffs in different intervals to tackle energy consumption. This article addresses unrelated parallel machine energy-efficient scheduling problems by considering sequence-independent setup times and energy consumption tariffs. The objective function is that jobs should be assigned to machines and processed in different intervals so that the cost of consumed energy becomes as less as possible. It should be noted that the assumed sequence-independent setup times are addressed in two different modes, setup times jointed to processing time and setup times disjointed from processing time. Design/methodology/approach: To optimize the total energy consumption cost in unrelated parallel machine scheduling problems with sequence-independent setup times jointed to processing time and disjointed from processing time, mixed-integer linear programming (MILP) models have been proposed from two different points of view. The first model has been formulated according to the predecessor jobs of a special job, while the second model has been conducted based on the immediate predecessor job. Also, a fix and relax heuristic (FRH) algorithm has been conducted to solve large- scale instances. All mathematical models and the heuristic algorithm have been coded in the Visual C# 2017 environment and implemented using the CPLEX 12.8 Concert Technology on a PC with 32GB RAM and Intel Corei7 4.0 GHz CPU (4 cores). Also, a sizeable number of instances have been solved to evaluate the efficiency of mathematical models and the heuristic algorithm and to ensure their accuracy.

According to numerical analysis, both mathematical models solved the instances of up to 20 jobs and 80 machines optimally for sequence-independent setup-times jointed to processing time, and sequence-independent disjointed from processing time problems. However, generally speaking, the mathematical model based on predecessor jobs was more efficient than another mathematical model, especially in terms of run time. Moreover, the proposed fix and relax-based heuristic algorithm solved instances of up to 20 machines and 190 jobs for the disjointed setup times problem, and up to 20 and 220 instances for the jointed setup times problem. It should be noted that all instances were generated analogously to the literature.

A vast number of exogenous factors contributed to the scheduling problems in the real world, which can disturb the scheduling process easily, frequent power outages, machine breakdown, and operator absence. Besides, considering all the real world's possibilities raises extreme complexity in problems. Therefore, similar to other studies, some assumptions were considered as follows: - machines are always available at all times; - idle is allowable for machines; - the energy consumption rate of various machines is different for each job; - each machine's energy consumption rate during processing and setups is different for each job, it is assumed as constant; - preemption is not allowed in the job's processing and setups; - all jobs are available at the beginning of the planning horizon; and - each machine can process or do the setup for only one job at a time.

Given that unrelated parallel machines are one of the most practical scheduling environments, this article can be effective in production sites and operation lines that contain such a kind of machine. Besides, unrelated parallel machines cover identical and related parallel machines. Consequently, this paper is the building blocks of cost-effective and environmentally friendly scheduling programs. Also, the application of unrelated parallel machines is not merely restricted to production problems. In other words, unrelated parallel machine scheduling problems can be used in other real-world cases, such as airplane scheduling and elevator scheduling. Originality/value - In this paper, unrelated parallel machine energy-efficient scheduling has been addressed considering sequence-independent setup times. Since it was a common belief that sequence-independent setup times could be included in processing times, sequence-independent setup times have been neglected so far. However, in this innovative study for the first time, an unrelated parallel machine energy efficient problem was investigated with sequence-independent setup times. Mathematical programming models and a heuristic algorithm were proposed for such a practical problem.