کمینه‌سازی هزینۀ انرژی در ماشین‌های موازی با درنظرگرفتن زمان آماده‌سازی

در سال‌های اخیر، افزایش چشم‌گیر مصرف انرژی و همچنین مواجه‌شدن با پدیدۀ گرمایش زمین، باعث نگرانی‌هایی در سطح جهان شده است؛ ازاین‌رو، دولت‌ها با سیاست‌هایی مانند تعیین تعرفۀ مصرف انرژی در بازه‌های زمانی مختلف روز، سعی در کنترل مصرف انرژی دارند. محیط‌های تولیدی نیز به‌عنوان مصرف‌کنندگان بزرگ انرژی، از این قضیه مستثنا نیستند. ماشین‌های موازی، یکی از پرکاربردترین محیط‌های ماشینی در تولید است؛ اما تاکنون پژوهش‌هایی در پیشینۀ موضوع دیده نشده است که زمان‌بندی همراه با زمان آماده‌سازی را در این محیط، با هدف کمینه‌سازی هزینۀ انرژی در حالت وجود تعرفۀ مصرف، در بازه‌های زمانی مختلف بررسی کرده باشد؛ بنابراین در مقالۀ حاضر، مسئلۀ زمان‌بندی کارای انرژی ماشین‌های موازی غیر مرتبط، با زمان‌های آماده‌سازی مستقل از توالی، در دو حالت منفصل از پردازش و متصل به پردازش، با تعرفۀ مصرف انرژی در بازه‌های زمانی مختلف بررسی می‌شود. برای هرکدام از این دو حالت، دو مدل ریاضی ارائه شده که در هر دو حالت، نمونه‌هایی تا ابعاد 20 ماشین و 80 کار، به‌صورت بهینه حل شده است. برای حل مسائل در ابعاد بزرگ‌تر، از الگوریتم ابتکاری مبتنی بر تثبیت و آزادسازی استفاده شده است. این الگوریتم برای هرکدام از مسائل، با آماده‌سازی منفصل از پردازش و متصل به پردازش به‌ترتیب، نمونه‌های تا 20 ماشین و 190 کار و نمونه‌هایی تا 20 ماشین و 220 کار را حل کرده است.

Minimizing Energy Cost in Parallel Machines Considering Setup Time

Purpose: 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 energyefficient scheduling problems by considering sequenceindependent 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 sequenceindependent 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 sequenceindependent setup times jointed to processing time and disjointed from processing time, mixedinteger 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 largescale 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. Findings: According to numerical analysis, both mathematical models solved the instances of up to 20 jobs and 80 machines optimally for sequenceindependent setuptimes jointed to processing time, and sequenceindependent 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 relaxbased 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. Research limitations/implications: 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 × 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. Practical implications: 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 costeffective 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 realworld cases, such as airplane scheduling and elevator scheduling.Originality/value In this paper, unrelated parallel machine energyefficient scheduling has been addressed considering sequenceindependent setup times. Since it was a common belief that sequenceindependent setup times could be included in processing times, sequenceindependent 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 sequenceindependent setup times. Mathematical programming models and a heuristic algorithm were proposed for such a practical problem.