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Preventive maintenance for the flexible flowshop scheduling under
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uncertainty: A waste-to-energy system
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Hadi Gholizadeh a, Hamed Fazlollahtabar b, Amir M. Fathollahi-Fard c,, Maxim A. Dulebenets d
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aDépartement de Génie Mécanique, Université Laval, Québec, Canada
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bDepartment of Industrial Engineering, School of Engineering, Damghan University, Damghan, Iran;
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cDepartment of Electrical Engineering, École de Technologie Supérieure, University of Québec, 1100, Notre-Dame
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St. W., Montréal, Canada
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dDepartment of Civil & Environmental Engineering, Florida A&M University-Florida State University (FAMU-
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FSU) College of Engineering, 2525 Pottsdamer Street, Building A, Suite A124, Tallahassee, FL 32310-6046, USA
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Abstract
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Nowadays, an efficient and robust plan for maintenance activities can reduce the total cost significantly in
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the equipment-driven industry. Maintenance activities are directly associated with the impact on the plant
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output, production quality, production cost, safety, and the environmental performance. To address this
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challenge more broadly, this paper presents an optimization model for the problem of flexible flowshop
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scheduling in a series-parallel waste-to-energy (WTE) system. To this end, a preventive maintenance (PM)
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policy is proposed to find an optimal sequence for processing tasks and minimize the delays. To deal with
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the uncertainty of the flexible flowshop scheduling of waste-to-energy in practice, the work processing time
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is modeled to be uncertain in this study. Therefore, a robust optimization model is applied to address the
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proposed problem. Due to the computational complexity of this model, a novel scenario-based genetic
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algorithm is proposed to solve it. The applicability of this research is shown by a real-life case study for a
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WTE system in Iran. The proposed algorithm is compared against an exact optimization method and a
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canonical genetic algorithm. The findings confirm a competitive performance of the proposed method in
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terms of time savings that will ultimately save the cost of the proposed PM policy.
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Keywords: Flexible flowshop scheduling; Preventive maintenance; Robust optimization; Genetic
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algorithms; Waste-to-energy system.
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1. Introduction
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Scheduling domains are very active topics and well-studied with regards to the real-world applications in
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logistics and production systems (Alaswad, & Xiang, 2017; Bappy et al, 2019). As the scheduling
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academically contributes to a process of allocation for limited resources, the recent studies consider
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Corresponding author's email: amirmohammad.fathollahifard.1@ens.etsmtl.ca
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optimization models with one or more objectives, such as time, cost and reliability in a dynamic
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environment (Chen, 2019; Gholizadeh, et al., 2019; Zhang et al, 2021). In today’s business world, there is
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a great deal of attention to the timing and scheduling due to the fierce competition of manufacturers and
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service providers (Chang, 2018; Gholizadeh, et al., 2020b; Chowdhury et al, 2019). Without a doubt, an
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efficient scheduling model can help the industrial practitioners and organizers to address the financial
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concerns (Cui, 2020; Seidgar, et al., 2016; Fallahpour, et al., 2021a).
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The science of timing and scheduling can provide a link with a maintenance program to meet the recent
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challenges in industrial environments. As the high volume of machinery production in the single-use
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containers industry should be controlled, it is essential to implement a preventive maintenance program for
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the machinery (Cui, & Liang, 2018; Fallahpour, et al., 2021b; Wang et al, 2020). Based on this reason, the
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existing literature is very rich in providing a preventive maintenance (PM) program with optimization
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models to maintain the high-level reliability machines (Alaswad, & Xiang, 2017; Lu, & Zhou, 2017; Liu,
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et al., 2019). It goes without saying that although the preventive maintenance is not an independent part of
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the production process; the PM takes time to actually be used for a production system. The interplay
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between PM and production planning has led to a number of innovative works focusing on the integration
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of different production environments (Zandieh, et al., 2017). However, there is a research gap for a few
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previous studies considering the PM activities. The majority of studies in a planning horizon are looking at
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reducing the PM costs as their optimization goal.
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Step 1
Step 2
Step 3
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Figure 1. Three steps for waste-to-energy process
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It is essential to consider that the PM intervals required for each component influencing the
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reliability of the entire series-parallel system and achieve the adequate reliability levels. In a system with
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all sub-systems in series where each sub-system has other components in parallel is called a series-parallel
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system. In this regard, each component can play an important role to contribute to the reliability of the
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whole production system (Yahyatabar, & Najafi, 2017; Halvorsen-Weare, et al., 2017). With regards to
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these facts, this research article proposes a series-parallel production system for the waste-to-energy (WTE)
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system in a three-step process, including material processing, combustion and power generation. The
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proposed system is responsible for the generation of electrical energy for consumers in industrial and home
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sectors. In this research, contrary to the previous works, a three-step process is introduced, in which the
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received materials are processed, combustion is performed and finally the generated power is transferred to
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the users. The WTE process is shown in Figure 1. As most of the energy generation of companies is
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conducted in these three phases, an accurate planning of these steps reduces a significant amount of the
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company’s costs and reduces the lost profits.
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1.1. Main Motivation and challenges
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Wasting energy power caused by the adjustment of each combustion machine is another
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consequence of incorrect allocation of work to the machinery. Furthermore, each transfer at each step in
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addition to the cost of time lost and the cost of waste will also incur additional direct and indirect costs.
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These include the costs of depreciation of machinery, unemployment of production power from combustion
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time to energy transfer, the cost of combustion line replacement, etc. (Mojtahedi et al., 2021).
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Another challenge in the three-step waste-to-energy production process is the limited availability
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of machinery in the company (Pasha et al., 2021). High-volume production (both one-time and 24-hour) is
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common for the available machinery units in many industries. Therefore, a preventive maintenance program
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for machinery is required. In each planning period, a preventive maintenance plan is considered for each
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machine, depending on the machine’s operating hours and production rate. Although the production
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planning process has a well-rounded approach to meet the customer’s energy needs in the event of a
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production shortfall, it often overlooks the implementation of the maintenance plan during a given period
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of time and postpones the repairs to the next period as the production process continues the track. This risk,
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on the one hand, may put heavy costs on the production lines, and, moreover, make the machinery more
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depreciated.
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1.2. Significance of the research
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The waste to energy system including several components and processes, require an accurate
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maintenance system for being available. The availability of the system is significant since it generates
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energy to consumers and energy failure incurs loss. To have an efficient maintenance program it is
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necessary to follow an exact schedule to ensure the availability of the system. As explained earlier, waste
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to energy system can be resembled to a flowshop system while wastes movement, combustion lines and
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material processing chambers are like work stations that the material flows among them. To have an
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available WTE system, failures should be prevented applying a comprehensive preventive maintenance
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program.
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1.3. Research novelties
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With regards to the aforementioned needs to fill these gaps and to meet the existing challenges, this
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study implements an optimized schedule and sequence in the processing of tasks at each stage of energy
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production. The PM is also considered as a task to be processed on each machine during the course.
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Therefore, by creating a sequence in work processing and maintenance activities, this paper simultaneously
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achieves an optimal point for preventive maintenance programs of each machine and for reducing the delays
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in task completion.
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As mentioned earlier, a flexible flowshop system with the limited access to machinery due to
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planned repairs is one of the closest and most applicable systems contributing to workshop production
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industries (e.g., waste-to-energy system). Regarding the real-world setting of this production system, this
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research seeks to contribute to the tangible real-world science by approaching the fields of flexible flow
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generation theory to real-world issues and by applying its theoretical assumptions to optimize the planning
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process for scheduling activities. It also aims to minimize the delay in addressing the customer requests by
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properly allocating jobs to the machines as well as optimally allocating the PM program to the
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machines. The applicability of this research is shown by a real-life case study for the waste-to-energy
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company in the northern part of Iran. Last but not least, the studied problem is classified as NP-hard in a
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strong sense, and it is difficult to effectively address this problem using exact optimization methods in a
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reasonable computational time. Therefore, this study proposes a new metaheuristic algorithm as a solution
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approach, which is an extension of the classical genetic algorithm (GA) and is referred to as a modified
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scenario-based GA.
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In conclusion, the main contributions can be summarized as follows:
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• A robust optimization model is developed for PM scheduling under uncertainty.
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• A real-life case study for the waste-to-energy company in Iran is applied.
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• A modified scenario-based GA is developed to address the proposed problem.
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• Some efficient computational tests are done to evaluate the performance of our solution
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approach and to draw some practical insights using the results.
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The summary of other sections in this paper is as follows. Section 2 provides the literature review
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to highlight the recent advances in this field and identifies the literature gaps. Section 3 describes the
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problem studied herein and formulates a novel robust optimization model. Section 4 provides a
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comprehensive review of the proposed solution method (i.e., a scenario-based GA). The numerical
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experiments, results, sensitivity analysis, and managerial insights are given in Section 5. Section 6 presents
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the summary of this work, findings and future research opportunities.
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2. Literature Review
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Systems in industrial environments are designed to execute sequences of missions within a finite interval
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of time. However, the access to different resources, such as manpower, time and budget, is generally
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limited. It may not be possible to implement all the component maintenance activities for the system
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simultaneously (Tambe, & Kulkarni, 2016). Among all the options, only the most viable subset of
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maintenance actions is typically selected (Zhou, & Shi, 2019). Maatouk et al., (2019) aimed at minimizing
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the system performance cost in order to find the optimal maintenance reliability strategy in a multi-mode
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parallel system under the required availability constraints using different GA models. Furthermore, Diallo
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et al., (2018) investigated a new two-phase approach to optimally solve the selective maintenance problem
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for large and complex reliability systems. Zhou and Lu (2018) suggested a capacity failure rate based on
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the maintenance policy. They applied the capacity failure rate in order to clarify the influences of station
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failures on the system capacity. Moreover, the study defined the maintenance efficiency to examine the role
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of maintenance. Wang et al., (2019) focused on the PM costs and maintenance equipment. The main cause
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of breakdown maintenance cost was highlighted to be the result of accidental failures of the PM cycle.
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Therefore, the Weibull distribution with three parameters of the equipment was used to adjust the reliability
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model, and a differential evolutionary algorithm was developed for parameter optimization.
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Sequential production lines are one of the most complex production systems in the field of
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maintenance and repair due to the availability of many types of machinery and equipment. Therefore, in
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order to determine the optimum time for replacement of parts to reduce the output of defective parts, the
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PM planning needs to be developed (Eryilmaz, 2017). Various models have been designed to optimize the
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scheduling of replacement policies in different systems (Chen, 2019; Chang, 2018; Edwards, et al., 2017).
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The proposed replacement policies can be successfully applied in various settings. Lia et al. (2017)
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illustrated a single-machine-based integration model to deal with the requirements of production scheduling
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and PM policy. In addition, Edwards et al., (2017) established an infield route planner tool to obtain lower
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unit costs and reduce the risk of needy soil compaction.
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With respect to the maintenance cost, Zhang and Xie (2017) introduced an ameliorated progress
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factor model for imperfect maintenance, and its merit of fit contained practical grounds for the imperfect
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maintenance model. Next, Khatab (2018) defined a new maintenance policy under the circumstance of
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imperfect preventive maintenance. The optimal conditions were examined in the study, and a solution
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method was proposed. Zhou and Lu (2018) aimed to evaluate workstation reliability and presented a
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dynamic PM and OM policy for multi-station systems. Their main focus was on reducing the rate of station
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failure. The study proposed a cost-saving approach for PM and OM considered in the short and long terms,
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respectively. Lu and Zhou (2017) in their research developed a new PM approach for multi-step series-
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parallel systems aimed at improving product reliability and quality, considering the economic dependencies
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of PM planning.
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During recent years, metaheuristics and heuristics have been used more and more often to address
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the flexible flowshop problem (Escamilla, et al., 2016; Jiang, & Le, 2014). For example, many studies
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proposed different types of evolutionary algorithms based on the genetic algorithm (GA) for a flow-shop
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system with the objectives of energy consumption and makespan (Liu, et al., 2014; May, et al., 2015; Liao,
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et al., 2017; Liu et al., 2019). Moreover, Fu et al., (2019) recently proposed an energy-based permutation
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flowshop scheduling problem to minimize the makespan and the energy consumption simultaneously. They
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solved the developed optimization model by a storm optimization algorithm. Lu et al., (2021) added the
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social negative impact for working time of workers to the energy-efficient flowshop scheduling problem
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and used a multi-objective memetic algorithm to solve it.
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In recent years, many studies examine various approaches to solve production planning and PM
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scheduling problems. Miyata et al., (2019) associated the PM operations with dependent-sequence setup
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times and makespan minimization. Then, several maintenance levels (ML) were considered based on
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various parameters of a PM policy following the Weibull distribution. A set of constructive heuristics were
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proposed for the machine no-wait flowshop with the makespan minimization. Moreover, recent studies
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show a great deal of interest in production or maintenance aspects and propose various modeling
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alternatives for the formulations with deterministic parameters (Alimian, et al., 2019). In addition, a number
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of previous studies suggested that the buffer time may help to enhance the system’s robustness (Cui, &
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Liang, 2018, Jovanović, et al., 2017). However, the aforementioned studies do not consider the machine’s
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deterioration effect and the impact of PM.
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Alimian et al. (2019) aimed to propose an integrated production and maintenance planning in a
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multi-state system. The main contribution of the paper was the implementation of a robust optimization
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approach with the integrated problem and demand fluctuations. Halvorsen-Weare et al., (2017) introduced
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a metaheuristic approach, considering the uncertainty that was evaluated by means of simulation.
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Schrotenboer et al., (2020) defined a tactical maintenance planning model in offshore wind applications.
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Last but not least, Cui (2020) developed an integrated model for production scheduling and maintenance
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planning for a single machine by assuming the impact of failure uncertainty with the objective to minimize
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the weighted sum of quality robustness and solution robustness. Next, a three-stage algorithm was applied
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to solve the problem and compared with the exact solution achieved by CPLEX. Their findings confirmed
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that the proper capacity planning plays a crucial role to improve the efficiency of PM and to reduce the
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operating cost in manufacturing. It goes without saying that there are many relevant studies with a
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contribution to the flexible production scheduling under uncertainty (Naderi et al., 2009; 2011; Du et al.,
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2021); however, they did not consider the PM policies.
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Malekpour et al. (2021) introduced a multi-product hybrid flow shop system. Then, in order to
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employ the real data as a case study, this paper was conducted in the Alborz Tire Company (Iran) that
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contains multi-type machines. Needless to say, the fundamental objective in this investigation was to
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identify the product processing prioritization in workstations, according to the Nash bargaining model, to
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decrease makespan. Accordingly, researchers applied a simulation-optimization approach on the basis of
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discrete-event simulation and simulated annealing. The results of the case study show that makespan is
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reduced significantly for all players. This research contemplated competition among customers and
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bargaining strategies. Finally, the outcomes revealed that the makespan is decreased appropriately for all
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players.
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Ebrahimi et al. (2020) conducted a novel model technology of PM scheduling. The purpose of
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solving this investigation is to identify the period for which bag filters should be regarded offline for PM
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over a particular time and keep a specific level of reliability by decreasing maintenance costs. Next, to solve
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the model, a mathematical programming method (Benders' decomposition) and a meta-heuristic solution
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have been implemented. Eventually, to prove the importance and creativity of the suggested model and the
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performance of the algorithms, computational analysis was presented to a practical bag filters system in the
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cement company. Yu et al. (2020) introduced a model to cope with the scheduling problem in a hybrid flow
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shop with independent machines, machine eligibility, and sequence-dependent setup times (SDST) to
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decrease the total tardiness and total setup time. Then, evolutionary algorithms (EAs) were employed to
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resolve the problem. Firstly, four effective decoding algorithms relating several machine selection rules
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were extended, and accordingly, a multi-decoding framework (MDF) to reach multiple decoding algorithms
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was applied. Moreover, an NSGA-II model was developed to solve the problem in the “a posteriori”
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approach to reach the Pareto-optimal set. Additionally, this research represented the association of MDF
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and EAs in the decoding system and creating solutions adjusted to the user preference.
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Mao et al. (2020) developed the distributed permutation flowshop scheduling problem (DPFSP) by
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taking into account PM operation to impede machines from breaking down after the lengthy process. Plus,
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an iterated greedy (IG) algorithm was conducted to decrease total flow time. Additionally, a swapping
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operator was suggested for the beginning of the IG. Therefore, a local search was adopted to develop the
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solution produced in the construction phase, and a simulated annealing-like criterion was utilized to hinder
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local optimal situations. The integration of abovementioned algorithms led to the high performance in
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operations related to PM problems.
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Mao et al. (2021) expanded an investigation concerning DPFSP with PM operation (PM/DPFSP).
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Next, a multi-start iterated greedy (MSIG)) algorithm was formulated to reduce the makespan. Besides, an
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improved heuristic was proposed for the initialization and re-initialization by joining a dropout operation
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to NEH2 to create solutions with high performance and imperativeness. Finally, plenty of experiments were
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applied to examine the performance of the given MSIG. The computational outcomes showed that the
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suggested MSIG has great advantages in solving the PM/DPFSP. Zhang and Thang, (2021) conducted a
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MILP model with maintenance level limitations to reduce the total time. Concerning the methods, the latest
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PM decision strategy was introduced to show the execution time of PM activities. This current approach
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was applied to 15 constructive heuristics and 7 meta-heuristics to deal with these problems. The concluding
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analysis illustrated the importance of employing heuristics and meta-heuristics. An investigation by
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(Hosseini et al, 2020) was done to propose a greedy heuristic-based local search algorithm (GHLSA) for
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multi-component systems for a system maintenance schedule, intended to reduce system downtime costs.
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In this vein, the proposed algorithms are including the construction phase, an improvement phase, and a
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local research phase. Accordingly, the conducted algorithm led to a trade-off between exploration and
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exploitation of solutions. At the end, the outcomes for little (10 jobs) and big size (50 jobs) problems
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revealed that GHLSA works more efficiently both genetic algorithm and simulated annealing methods.
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Mao et al. (2020) formulated an improved discrete artificial bee colony (IDABC) algorithm as a solution
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for the suggested model, and makespan was considered as a fundamental criterion. Next, an improved NEH
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heuristic method was employed to initialize the population efficiently. Besides, a local search approach
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with insertion and swap operator was utilized to provide neighboring solutions in the bee stage and onlooker
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bee stage. Also, the parameters for the suggested IDABC were adjusted by a design of investigations and
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analysis of variance with contemplating large experiments to examine the efficiency of IDABC. In 2021,
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Fathollahi-Fard et al., (2021) developed a sustainable DPFSP considering job opportunities and lost
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working days which have been added to the energy-efficient DPFSP. They developed a learning-based
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social engineering optimizer to solve their problem and compared it with other recent and well-known
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metaheuristics.
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In conclusion, the research gaps based on the conducted literature review can be identified as
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follows:
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• Although many parallel production scheduling models have been developed, there is no
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comprehensive PM scheduling in series-parallel production systems;
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• Although there many similar uncertain models, there is no comprehensive model for the PM
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scheduling in series-parallel production systems;
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• Although most of PM scheduling models seek to minimize total costs, there are a few studies to
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minimize the delay in completing a set of jobs by properly allocating jobs to the machines.
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To fill aforementioned gaps, the main contributions of this study can be summarized as follows:
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✓ Providing a targeted framework for integrating the PM scheduling into a series-parallel production
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system which considers finding an optimal sequence for processing tasks to minimize delays in
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task completion;
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✓ Explicitly consider the contemplating uncertainty in the main deadline for delivery of any work
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and processing time of each job on each machine associated with a waste-to-energy production;
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✓ An extension to the GA is proposed which is quicker and more efficient than the canonical GA. A
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local search mechanism is added to improve the exploitive behavior of GA. Since the proposed
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model is a robust optimization, a modified scenario-based GA is developed to handle the
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uncertainty;
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✓ A set of computational experiments are performed to assess the performance of the developed
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solution approach and to draw some insights using the proposed mathematical model.
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3. Mathematical model
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3.1. Problem description
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This paper presents an optimization model for the problem of flexible flow system scheduling in a
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series-parallel production system for the waste-to-energy production system considering the PM and repair
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policy at all times. The objective of the proposed model aims to find an optimal sequence for work
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processing in order to minimize the total delay in processing all the arriving jobs. Since there is an ability
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to produce N different products in a planning period as well as a three-stage production, each machine is
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deployed in parallel with a flexible layout, and each station is deployed in series that eventually adopts a
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parallel series production system. Material processing, transfer jobs within the shop and gas cleaning can
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be viewed as jobs in the WTE system. On the other hand, grab crane, steam turbine, combustion furnaces
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and generators can be viewed as machines. The different products are allocated to a particular machine at
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each stage and then processed. The final product is produced and sent to the customer after completing the
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activities at all three stages. Although the machines at each workstation have the same manufacturing
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nature, there are differences in the processing of products on the machines, and no machine at each
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workstation is capable of handling all tasks. For each machine, a preventative maintenance plan is foreseen,
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which is unavoidable. But due to the working nature of the machines, it is possible to float the program
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running time, and the timing of starting the program and getting the machine out of reach can be changed.
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All the major processes that are completed within the WTE system are shown in Figure 2. The
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WTE process begins with collecting (1) the wasted products which are pre-sorted, separated and the
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recyclable items are removed to elsewhere (2). The first step is finished here, and after that the materials
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are fed into combustion incinerators (3) to be burnt (4) at about 850°C (Step 2). The heat in the combustion
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process enters the boiler (5) producing steam to power turbines and generating energy (electricity) to be
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transferred to the energy grids (6). This way Step 3 is also completed. Another use of the heat generated as
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a result of this process is for heating buildings of a given company (7). Useless gases are removed (8), and
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particles are filtered (9). The air cleaning system absorbs materials and performs a chemical treatment (10).
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Emissions are fully monitored for the environmental consideration (11). Ash is collected (12), purified by
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metal absorbent magnets (13), and moved to be used in construction (14).
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Figure 2. Graphical display of the problem
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The following assumptions have adopted for the problem studied herein:
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✓ The problem environment is set up in a three-stage period that consists of seven days.
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✓ The number of steps is predetermined.
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✓ The number of machines at each step is definite.
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✓ The processing time of each task on each machine for the WTE process is uncertain.
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✓ The deadline for delivery of any work is uncertain.
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✓ Machines at each step of the WTE process are different in parallel and unrelated to the processing
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time.
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✓ No machine can process more than one task at a time in any step of the WTE process.
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3.2. A definitive mathematical model
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The notations are presented in the appendix in supplementary materials F1. The definitive optimization
320
model is as follows:
321
1 1 1 1 1 1
0,
g
N
G M N T I
imkt gjimkt gjt
g j m k t i
Min Z Max ET X Td
= = = = = =
=−
(1)
imkt
ET
=
imkt imkt
NP T+
, , ,m i k t
(2)
( 1) ( 1)
, max , pm
imkt imkt im k t im k kt it imkt
NP Max CT ET ST t Y
−−
=+
, , ,m i k t
(3)
'
' ' '
''
( 1) ( 1)
11
11
gg
NN
GG
im k kt gjim k t g j imkt igg t
gj
gj
ST X X S
−−
==
==
=
, , ,m i k t
(4)
' ' ' '
( 1) ( 1)
1 1 , ' 1( ') , ' 1( ') 1 1
gi
NM
G N I T
imkt gjimkt gj i m k t i m k t
g j m m m m k k k k i t
CT X X ET
−−
= = = = = =
=
(5)
11
g
N
G
imkt gjit gjimkt
gj
T PT X
==
=
, , ,m i k t
(6)
1 1 1 1
i
MNT
gjimkt
m k t X
= = =
=
,,g i j
(7)
1 1 1 1
g
N
GT
gjimkt
g j t X
= = =
,,k i m
(8)
( 1)
1 1 1 1 1 1
gg
NN
G T G T
gjimkt gjim k t
g j t g j t
XX
+
= = = = = =
,im
kN
(9)
1
()
1 1 1 1
01
g
gg
iN
N N N
MT
gjim k p t
m k t j
pX
−
−
+
= = = =
=
=
,gi
(10)
gjimkt
X
,
0,1
imkt
Y
, , , , ,m i k t g j
(11)
imkt
ET
,
imkt
NP
,
( 1)im k kt
ST −
,
0
imkt
CT
, , ,m i k t
(21)
322
12
Equation (1) expresses the objective function of the problem, minimizing the total delay in
323
processing all the WTE combustion tasks. Equation (2) calculates the expected time of completing
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combustion or transfer task k on machine m in step i of the WTE process in period t. Equation (3) calculates
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the number of PM activities on combustion prior to processing task k on machine m in step i in period t.
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Equation (4) expresses the calculation of the startup time between processing tasks (k-1) and k on machine
327
m in step i in period t when moving the waste from the collection point to the combustion and transfer point.
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Equation (5) calculates the amount of time required in step (i-1) for completing task k on machine m in
329
period t. Constraints (6) estimate the processing time of task k on machine m in step i in period t for the
330
collection, combustion and transfer steps. Constraints (7) indicate that each operation of a task in the WTE
331
process is assigned exactly to one machine in each step during one of the periods. Constraints (8) ensure
332
that the processing two tasks cannot be completed by the same machine in the same step of the WTE process
333
during the same period. Constraints (9) indicate that the processing of a task assigned to a machine in WTE
334
cannot exceed the number of tasks processed on that machine. Constraints (10) ensure that the tasks from
335
a PM group are processed on a particular machine in each step. Finally, constraints (11) and (12) show the
336
range of decision and auxiliary variables.
337
338
3.3. Robust optimization
339
To handle uncertainties in parameters of real-life industrial systems, a variety of approaches were
340
introduced in the literature, such as application of lower/upper bounds, sample average approximation,
341
cardinality-constrained method, minimax regret, and others. Any of the aforementioned approaches are
342
suitable for specific circumstances depending of the conditions of the industrial system considered. For the
343
cases with a hierarchical structure where the state of each workstation may depend on the other
344
workstations, scenario planning would be an appropriate alternative. In this study, due to the uncertainties
345
in deadline for delivery of any work and processing time of each task on each machine per period, the robust
346
optimization is employed (Schrotenboer, et al., 2020; Gholizadeh et al., 2020; Zhang et al., 2020) as can be
347
represented as follows:
348
(13)
(14)
(15)
where
is the weight of risk;
is and infeasibility weight being set experimentally by the decision maker;
349
and are the probabilities of scenarios and .
350
13
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
Apply variable neighborhood search
to the subset
Initialize subset of the population to
perform variable neighborhood search
Start of msb-GA
Set initial parameters
Randomly initialize population G+N:
GP two-section chromosomes
Evaluate the fitness of each chromosome
Terminal condition
satisfied?
End of msb-GA
Output the best chromosome
Select two individuals from G+N,
and
, using the
tournament selection with possibility scenario
with possibility scenario ps
Update andby crossover
Update andby mutation
Evaluate the fitness of
and
Replace population with the best individuals from the offspring
Yes
Yes
Yes
Yes
No
14
Figure 3. The flowchart of msb-GA
374
375
4. Solution Method
376
One of the approaches to solve large-scale scheduling problems is the use of metaheuristic algorithms, such
377
as genetic algorithms (GAs), which have been applied by many of the previously conducted studies
378
(Fathollahi-Fard, et al., 2020a; Dulebenets, 2019; Kavoosi et al., 2019; Gholizadeh et al, 2020a ; Razavi et
379
al, 2020). Considering the reported efficiency of GAs for a wide range of scheduling decision problems,
380
this study will adopt a customized GA (named as “modified scenario-based GA” or msb-GA) as a solution
381
approach. GAs are recognized as stochastic search methods. The first step in solving a problem with
382
metaheuristic methods is to create an appropriate structure for potential solutions to the problem
383
(Fathollahi-Fard et al., 2020b; 2018; Gholizadeh & Fazlollahtabar, 2020). In the proposed GA, a local
384
search is used based on variable neighborhood search as opposed to the classic GA algorithm, aiming to
385
improve the algorithmic performance.
386
The neighborhood is changed systematically through a local search process to explore the solution
387
space dynamically. A variable neighborhood search is applied to a subset of the population within a domain
388
after performing crossover, mutation, and fitness evaluation to search for better solutions under different
389
scenarios. Figure 3 provides an overview of this algorithmic framework and explicitly shows where the
390
proposed variable neighborhood search procedure is performed.
391
392
393
Figure 4. Example of the solution structure for the msb-GA algorithm
394
395
The solution structure used in this research is made up of two sections (see Figure 4). It shows a
396
solution for a problem with 7 tasks, 3 groups, 3 steps and 3 scenarios. Groups 1 and 3 have 2 tasks,
397
while group 2 has 3 tasks. According to the second section of the solution structure, pre-work PM operations
398
15
(2, 3) occur in the second stage in the likely scenario (i.e., scenario 2). Furthermore, the following
399
observations can be made based on the provided solution structure example:
400
1) The first section of the solution structure indicates that the task groups will be assigned in the
401
following order “2”, “3”, and “1” (i.e., group “2” tasks will be completed first, while group “1” tasks
402
will be completed last).
403
2) The second section of the solution structure indicates that the tasks in group “2” will be assigned in
404
the following order “1”, “3”, and “2” (i.e., task “1” will be completed first, while task “2” will be
405
completed last).
406
407
Here, the PM groups and tasks are processed based on the processing plan in the chromosome. To
408
assign a task, if the task in PM group g was processed by a machine, then the task is assigned to the same
409
machine or, otherwise, it is processed on a machine to complete the task earlier (Moosavi et al., 2021). At
410
each step, the PM groups and tasks are allocated based on the completion time of the last step. An example
411
of the chromosome decoding is presented in Figure 5.
412
413
414
Figure 5. Decoding of a chromosome
415
416
Finally, the objective function value (OFV) is computed, and the fitness function is obtained as
417
follows:
418
1
() ()
fX OFV X
=
. (16)
419
420
It should be noted that the parent selection operator that was adopted in this study is based on a
421
tournament between any two randomly selected individuals, and the one with a higher fitness value is
422
chosen to undergo crossover and mutation (such a tournament selection scheme is generally referred to as
423
“binary tournament”) (Whitley, 1994). In addition, the crossover and mutation operators are discussed and
424
16
introduced in supplementary materials F2. Finally, a pseudo-code of the proposed solution algorithm is
425
given in Figure 6.
426
427
Generate a set of random chromosomes.
Define the encoding scheme for these solutions.
Calculate the objective function.
X*=the best solution based on the objective function.
while (t< maximum number of iteration)
Select a pair of chromosomes as parents.
Perform crossover and mutation to generate new chromosomes.
Merge the all chromosomes and select the new population.
Update the X* if there is better solution.
Update the Pareto optimal solutions.
t=t+1;
end while
return X*
Figure 6. The pseudo-code of the proposed solution
428
5. A real-life case study and computations
429
The present research focuses on a waste-to-energy (WTE) power system generation as a case study to verify
430
our proposed mathematical model and imply one of its applications in a real-life environment. In particular,
431
the proposed methodology was applied to a company that is located in the northern part of Iran. As stated
432
before, the WTE is a three-stage process that involves material feeding, combustion processes and power
433
transfer. A combustion process in WTE is very important, and the impact of different types of waste on
434
combustion is substantial. Thus, the optimization of energy obtained from each type of waste could help
435
the policy makers in waste collection and generated energy estimation. Different types of waste are
436
collected, and the combustion process for each type is conducted. The aim is to evaluate the delays in task
437
processing for this integrated system while scheduling the energy obtained from combustion of each type
438
of waste. Three process lines are considered being differentiated by the equipment and machines. The first
439
step includes a process line handling 7 types of waste. In the second combustion line, 9 types of waste are
440
considered, while the third line is able to handle 5 types of waste. Generally speaking, the WTE with a
441
waste combustion process can be represented as a series-parallel system that is depicted in Figure 7.
442
Furthermore, in any WTE line (especially, the combustion line), various failures may occur, such
443
as failures in equipment and machines, materials handling, human resource mistakes, and chemical process
444
errors. These failures impact the overall system reliability. In line with the company’s development
445
strategies and with regard to the economic benefits and legal pressure, the company decided to implement
446
a preventive maintenance model for scheduling of its energy production planning. Considering the
447
deadlines for delivery of different tasks, variability of processing time of each task on each combustion
448
17
machine, and the existing uncertainty of these parameters, three scenarios were modeled in this study,
449
including the following:
450
• Scenario (1) with a low probability () of 25% for failure events;
451
• Scenario (2) with an average probability () of 55% for failure events;
452
• Scenario (3) with a high probability () of 70% for failure events.
453
454
Figure 7. A configuration of a series-parallel WTE system
455
456
The data was collected from the energy production system for the WTE company in Iran. The
457
proposed problem involves uncertainty features in the production framework of WTE and may require more
458
comprehensive parameters to adequately capture the key operational characteristics, which can be further
459
explored as a part of future research.
460
461
5.1. Generation of the simulated instances
462
To show the validation of the model, we have used both exact method and our msb-GA for solving the
463
proposed model. By performing a series of Taguchi pre-experiments to select the algorithmic parameters
464
(Gholizadeh et al., 2019; Fazlollahtabar, & Gholizadeh, 2020; Govindan & Gholizadeh, 2021), the msb-
465
18
GA parameters were set as follows: crossover probability , mutation probability ,
466
population size and maximum number of generations . In order to evaluate the
467
performance of the proposed model and algorithm, we have designed 10 test problems in small size, 10 test
468
problems in medium size and 30 test problems in large size.
469
470
5.2. Numerical example
471
To confirm that the proposed model is feasible, an illustrative example of 10 test problems with a medium
472
size has been selected and solved with GAMS. The data used in this illustrative example, which includes 7
473
tasks from 3 PM groups in 5 steps, are shown in Tables 1 and 2. According to Figure 8 and Table 3, the
474
optimal results for the PM intervals of each machine and the processed tasks are shown. As can be seen,
475
tasks (1, 1) and (1, 2) are processed in the first PM distance of the machine (2, 1), and three tasks (2, 2), (2,
476
3) and (2, 1) are processed with a delay, and the optimal target value in this case is 406.15 units of time.
477
Based on the provided illustrative example, it can be concluded that the proposed optimization model is
478
able to generate feasible schedules that capture practical considerations.
479
480
Table 1. Input data for the considered tasks
481
J(1,1)
J(1,2)
J(2,1)
J(2,2)
J(3,1)
J(3,2)
gjt
Td
1000
1000
1000
1000
1000
1000
gjit
PT
i
= 1
23
24
48
108
40
55
i
= 2
200
215
253
149
226
215
i
= 3
194
0
59
86
83
92
i
= 4
155
120
215
219
93
76
i
= 5
112
64
210
118
91
130
482
Table 2. Input data for the considered machines
483
Step 2
Step 2
Step 3
Step 4
Step 5
'
igg t
S
26
25
20
25
18
pmit
t
48
36
43
35
48
i
M
1
2
1
2
1
484
Table 3. The results of validation
485
Machine
PM group 1
PM group 2
PM group 3
No.
imkt
ET
tasks
imkt
ET
tasks
imkt
ET
tasks
M(1, 1)
147
J(1,1)→J(1,2)→J(3,1)→J(3,2)
222
J(2,2)→J(2,3)→J(2,1)
M(2, 1)
428
J(1,1)→J(1,2)
298
J(2,2)→J(2,3)
266
J(2,1)
M(2, 2)
223
J(3,1)
227
J(3,2)
M(3, 1)
194
J(1,1)
269
J(3,1)→J(3,2)→J(2,2)
298
J(2,3)→J(2,1)
M(4, 1)
286
J(1,1)→J(1,2)
M(4, 2)
387
J(3,1)→J(3,2) →J(2,2)
241
J(2,3)
230
J(2,1)
M(5, 1)
177
J(1,1)→J(1,2)
338
J(3,1) →J(3,2) →J(2,2)
300
J(2,3)→J(2,1)
19
486
Figure 8. The results of model validation
487
488
Table 4. Generated data for the model parameters
489
Parameter
Data generation
'
igg t
S
Uniform(10,40)
gjit
PT
Uniform(10,300)
gjt
Td
Uniform(800,1500)
pmit
t
Uniform(40,200)
490
5.3. Results and discussion
491
As stated earlier, the model parameters were generated according to the actual company data following a
492
set of uniform random distributions, which are shown in Table 4. This section of the manuscript compares
493
the exact optimization method (i.e., GUROBI executed within the GAMS environment) and the proposed
494
msb-GA algorithm for different test problems to determine the accuracy of msb-GA. Table 5 summarizes
495
the results of a comparative analysis between GAMS and msb-GA for the robust optimization model and
496
provides the following information for each small-size and medium-size problem instance: (1) the problem
497
size; (2) the optimal objective function value provided by GAMS – OS; (3) the computational time required
498
by GAMS; (4) the best objective function value provided by msb-GA (over 10 replications); (5) the average
499
20
objective function value provided by msb-GA (over 10 replications); and (6) the average computational
500
time required by msb-GA (over 10 replications). Along with the aforementioned indicators, two additional
501
factors were estimated to evaluate the proposed msb-GA algorithm for each problem instance, including
502
Percentage Relative Error (PRE) and Related Percentage Deviation (RPD) (see Table 5), used the following
503
equations:
504
505
(18)
where and are the optimum value from the GAMS software and the best objective
506
value obtained by the proposed msb-GA, respectively.
507
508
(19)
509
Note that only small-size and medium size problems instances were considered due to declining
510
performance of GAMS (i.e., GUROBI could not converge within a reasonable amount of computational
511
time for large-size problems instances). It can be concluded from Table 5 that the proposed msb-GA
512
provided good-quality solutions for each one of the generated 20 problem instances. In particular, the
513
maximum PRE value of msb-GA did not exceed 4.5% over the considered small-size and medium size
514
problems instances (see Figure 9). Furthermore, msb-GA demonstrated stability in solution quality, as the
515
maximum RPD value of msb-GA did not exceed 5.3% over the considered small-size and medium size
516
problems instances (see Figure 9).
517
The average computational time values required by GAMS and msb-GA to solve the robust
518
optimization model are reported in Table 5 and also illustrated in Figure 10 for the ease of comparison for
519
each small-size and medium-size problem instance. It can be observed that GAMS was very sensitive to
520
increasing problem size and required up to 12,038.40 seconds (or 3.34 hours) to solve some of the medium-
521
size problem instances. The msb-GA computational time increased with the problem size as well. However,
522
the maximum msb-GA computational time did not exceed 52.7 seconds over the generated small-size and
523
medium-size problem instances. Such a computational time can be viewed as acceptable from the practical
524
standpoint. In conclusion, based on the conducted analysis, msb-GA was found to be more competitive than
525
the exact optimization method (i.e., GUROBI executed within the GAMS environment) for the flexible
526
flow system scheduling problem in the WES production system when considering both solution quality and
527
computational time criteria.
528
529
21
Table 5. Results of comparison between GAMS and msb-GA solutions
Problem
group
RPD
PRE
msb-GA
GAMS
Problem size
Time
(Seconds)
Avg.
Best
Time
(Seconds)
OS
0
0
18.3
273.6
273.6
20.3
273.6
2
3
1
2
3
2
Small size
0
0
18.8
239.45
239.45
21.1
239.45
2
3
2
3
2
2
0
0
19.5
230.44
230.44
22.6
230.44
3
3
1
2
3
3
1.8
0
19.2
214.01
210.23
36.14
210.23
3
3
2
3
4
3
2.2
0
21.4
327.70
320.65
76.5
320.65
4
3
1
2
2
4
3.4
0
22.3
280.08
270.87
103.22
270.87
4
3
2
3
4
4
2
0
22.8
412.25
404.17
195.76
404.17
5
3
1
2
3
5
3.5
0
21.9
314.80
304.15
348.58
304.15
5
3
2
3
2
5
1.3
1.5
24.6
469.83
463.80
736.17
456.95
6
3
1
2
4
6
4.7
2.6
26.7
382.86
365.68
1053.85
356.41
6
3
2
3
2
6
3.4
1.07
28.5
419.96
406.15
1766.56
401.85
7
3
3
5
7
5
Medium size
4.1
3
30.1
401.07
385.27
2623.45
374.05
7
4
3
4
5
7
2.6
2.4
31.6
384.32
374.58
3415.91
365.8
8
3
4
5
6
8
1.5
1
33.4
553.32
545.15
4896.16
539.75
8
4
4
5
7
8
2.8
3.8
35
498.59
485.01
5623.51
467.25
9
3
5
6
8
9
3.2
4.1
38.6
565.20
547.67
6854.24
526.1
9
5
5
6
4
9
5.3
4.5
42.1
638.39
606.26
8069.65
580.15
10
4
3
5
5
10
4.9
2.3
44.5
672.37
640.96
9345.33
626.55
10
5
4
5
6
10
3.6
3.6
48.3
699.06
674.77
10560.51
651.32
11
6
5
7
7
11
4
3.2
52.7
778.84
748.88
12038.40
725.66
11
5
6
7
8
11
22
Figure 9. Comparison of the PRE and RPD values of msb-GA
Figure 10. Comparison of the msb-GA computational time (in seconds) with GAMS
23
Table 6. Comparison between GA and msb-GA
GA elapsed time
(Seconds)
GA
objective function
msb-GA elapsed
time (Seconds)
msb-GA objective
function
Problem
No.
139.87
871.23
135.4
868.62
1
145.86
1076.93
141.2
1073.71
2
154.02
1357.72
149.1
1353.66
3
157.33
1690.42
152.3
1685.36
4
162.08
2132.63
156.9
2126.25
5
164.04
2674.72
158.8
2666.72
6
170.86
2872.50
165.4
2863.91
7
174.16
3281.19
168.6
3271.38
8
177.99
3853.36
172.3
3841.83
9
182.74
4395.36
176.9
4382.21
10
186.46
4854.05
180.5
4839.53
11
190.18
5375.75
184.1
5359.67
12
195.44
5671.94
189.2
5654.98
13
203.50
6176.14
197
6157.67
14
211.56
6915.34
204.8
6894.66
15
224.57
7534.01
217.4
7511.48
16
237.18
8284.83
229.6
8260.05
17
249.06
8782.39
241.1
8756.12
18
256.42
9229.46
248.23
9201.85
19
261.49
9567.35
253.14
9538.73
20
267.81
9841.81
259.25
9812.37
21
273.25
10221.71
264.52
10191.14
22
297.72
10670.65
288.21
10638.73
23
326.79
11617.42
316.35
11582.67
24
348.31
12022.11
337.18
11986.15
25
382.35
12395.61
370.14
12358.53
26
409.75
12901.25
396.66
12862.66
27
451.15
13311.19
436.74
13271.38
28
501.29
13726.20
485.28
13685.14
29
537.49
14142.78
520.32
14100.48
30
24
5.4. Comparison of the proposed method against the canonical GA
To evaluate the efficiency and performance of the proposed msb-GA in terms of solution quality and
computational time for all the considered problem instances (including large-size problem instances), an
additional comparative analysis was conducted. In particular, the developed msb-GA algorithm was
compared against the canonical GA. Note that the developed msb-GA was specifically evaluated against
the canonical GA, as the latter algorithm has been widely used in the previous studies over the years [45].
However, as a part of the future research, the developed msb-GA can be compared to the alternative
metaheuristics as well. The results of the performed analysis are reported in Table 6, summarizing the
following information: (1) problem instance number; (2) the average objective function value provided by
msb-GA and GA (over 10 replications); and (3) the average computational time required by msb-GA and
GA (over 10 replications). It can be observed that the developed msb-GA clearly outperformed the
canonical GA for each one of the instances. The average objective function value of msb-GA comprised
7,226.58 time units over all the considered problem instances. On the other hand, the average objective
function value of GA was found to be 7,248.26 time units over all the considered problem instances.
Figure 11. Comparison of the msb-GA computation time (in seconds) with GA
Along with solution quality, it is important to assess the computational time required by msb-GA
to solve real-life problem instances. Throughout the numerical experiments, the differences in the
computational time values required by msb-GA and GA were estimated for each one of the generated
problem instances (see Figure 11). It can be observed that the developed msb-GA outperformed the
canonical GA for the majority of the considered problem instances. The average computational time of
25
msb-GA comprised 246.55 seconds over all the considered problem instances. On the other hand, the
average computational time of GA was found to be 254.69 seconds over all the considered problem
instances. More importantly, the computational time savings offered by msb-GA increased with increasing
problem size. In conclusion, the developed msb-GA algorithm can be viewed as a competitive solution
approach in terms of both solution quality and computational time for various problem instances (including
large-size problem instances that cannot be effectively tackled by exact optimization methods). Therefore,
it can be used by the relevant decision makers to solve complex flexible flow system scheduling problems
in series-parallel production systems as the one that is investigated in this study.
5.5. Sensitivity analyses
5.5.1. Robustness analysis
To investigate the robustness of the model, we first compare the results of the robust model against a
deterministic outcome. Then, we conduct a sensitivity analysis on the penalty weights. Moreover, the
effects of scenario changes are assessed as well.
i) Comparing robust and deterministic models
To compare the results obtained by the robust and deterministic models, ten experiments were designed
and solved in a deterministic mode. For comparison purpose, the mean and standard deviation values were
computed for the deterministic and robust objective functions. The comparison factor was the total delay
in processing all the WTE combustion tasks, and thus the lower total delay standard deviation could provide
better robustness. Four penalty weights were considered with regards to the feasibility of the model. The
differences between the results obtained using the deterministic model (the general form of the model
without uncertainty) and the results obtained using the robust model (including uncertainty in the model)
are presented in Table 7.
Table 7. Comparing the results of problem in two models (deterministic, robust)
Realization
1
2
3
4
5
6
7
8
9
10
Mean
Standard
Deviation
Deterministic
258.44
177
236.75
186.25
294.18
234
402.12
298.9
476
350.5
291.41
95.10
Robust (penalty weights=50)
288
195.25
241.14
199.38
337.5
285.12
445.42
320.15
481
375.16
316.81
96.30
Robust (penalty weights=100)
270.72
183.53
226.67
187.41
317.25
268.01
399.89
300.94
452.14
352.65
295.92
87.89
Robust (penalty weights=150)
292.37
198.21
244.80
202.41
342.63
289.45
431.88
325.01
488.31
380.86
319.59
94.92
Robust (penalty weights=200)
308.16
208.91
258.02
213.37
361.12
305.07
455.19
342.56
514.67
401.42
336.84
100.04
26
Table 7 shows the results from the conducted experiments for deterministic and robust models.
The robust model with penalty weights of 100 and 150 outperformed the deterministic model and returned
lower total delay standard deviation values, suggesting the efficiency of the proposed robust model.
However, when penalty weights were set to 50 and 200, the deterministic model had better performance.
Thus, the proposed robust model would be able to overcome the weaknesses of the deterministic model.
ii) Effects of scenario changes
To indicate the performance, more machines would be available to handle the available tasks that would
reduce the associated delays in task processing. Solving the WTE optimization problem to obtain a robust
decision repeatedly is inferred as a stochastic optimization problem that performs best in the reference
sample. Therefore, the number of considered scenarios was initially set to s = 100. However, an additional
analysis was conducted as a part of this study to determine the effects of changes in the number of scenarios
on the objective function values returned by msb-GA. In particular, the number of scenarios was steadily
decreased from 100 scenarios to 5 scenarios, and the results obtained by msb-GA are reported in Figure
12.
Figure 12. Effects of scenario changes for objective functions
It can be observed that increasing number of scenarios increased the total objective function value
over all the considered scenarios but substantially reduced the objective function deviation over these
scenarios. The interaction of decision variables and constraints may cause a conflict among the scenarios,
27
and thus consideration of a large number of scenarios (or even the entire feasible set of scenarios depending
on the computational time required) would be appropriate for the proposed robust mathematical model.
5.5.2. Time analysis
In this section, we analyze the sensitivity of the proposed mathematical model to changing values of the
PM time and processing time. All the parameters of the proposed model were fixed throughout the
conducted analysis, and only changes from -20% to +20% were made for the unit PM time and processing
time. Based on the results obtained in Figure 13, it can be observed that increasing PM time leads to an
increase in the processing time implying that the maintenance time directly impacts the WTE processes in
various steps. Also, implementing a preventive maintenance scheduling configuration for series-parallel
WTE systems under uncertainty fulfills a better chance of long-term profitability in production planning.
On the other hand, due to the inconsistency of scenarios and delays in performing tasks in WTE systems,
the slope of processing time changes was found to be the same in all cases, but the slope of PM time changed
for some of the considered cases (i.e., a sharper slope was observed for +10% and +20% cases as compared
to -10% and -20% cases). These results provide the insights regarding the impact of maintenance time on
WTE systems and can be potentially used to reduce short-term maintenance costs.
Figure 13. Sensitivity analysis for the processing time and PM time
5.6. Managerial Insights
Based on the findings from this research, the following recommendations are suggested for practical
considerations. It is obvious that production managers are very careful in the implementation of PM
28
policies, as they have a very large financial cost due to the organization of such PM policies and various
uncertain factors under different scenarios. The application of a preventive maintenance scheduling
configuration for series-parallel WTE systems under uncertainty fulfills a better chance of effective
production planning, leading to a long-term profitability. Increasing task processing times and maintenance
times are expected to incur additional costs in a short term that would further lead to a reduced profit. The
findings from this study indicate that by optimizing the current production capacity of the equipment
employed in the WTE process, the facility managers can optimally introduce the PM time for each machine
to effectively compete in the energy market at the lowest possible cost.
6. Conclusion and future works
Nowadays, in discrete production industries, such as waste-to-energy systems, scheduling is of a particular
importance due to the processing of different tasks performed on multiple machines in several successive
steps. The problem of flexible flowshop scheduling is one of the most common challenges in production
systems. Therefore, in this paper, an optimization model for flexible flowshop scheduling problem for a
waste-to-energy (WTE) system was developed. An effort was made to consider the preventive maintenance
and repair policy at all times in order to find an optimal sequence for processing tasks and minimize the
total delay in processing tasks for the proposed WTE energy system. The problem was formulated as a
mixed integer mathematical programming model. Furthermore, a robust formulation was presented to
account for potential uncertainties in the parameter values. Due to the problem complexity, a modified
scenario-based genetic algorithm (msb-GA) was developed.
A real-life waste-to-energy power system located in the northern part of Iran was used as a case
study. The developed msb-GA was compared against the exact optimization approach (i.e., GUROBI
executed using the GAMS environment). It was found that the msb-GA algorithm returned the solutions
that were close to the optimal ones but required much less computational time. Moreover, the proposed
msb-GA was compared against the canonical GA. The average objective function value of msb-GA
comprised 7,226.58 time units, while the average objective function value of GA was found to be 7,248.26
time units. The msb-GA outperformed the GA in terms of the average computational time as well (4.11
minutes versus 4.25 minutes). Hence, the developed msb-GA algorithm can be viewed as a competitive
solution approach in terms of both solution quality and computational time. Therefore, it can be used by
the relevant decision makers to solve complex flexible flow system scheduling problems in series-parallel
production systems as the one that was investigated in this study.
This study can be extended in several directions that include, but are not limited to, the following:
• Application of rigorous problem-solving techniques (e.g., Bender’s decomposition, advanced
relaxation methods);
29
• Analysis of multi-objective extensions when decision makers have to address multiple objectives
that are conflicting in nature;
• Evaluation of the proposed robust approach against the alternative methods for modeling
uncertainties (e.g., application of lower/upper bounds, sample average approximation, cardinality-
constrained method, minimax regret);
• Comparison of the developed msb-GA algorithm against the alternative metaheuristics (e.g., red
deer algorithm, social engineering optimizer, ant colony optimization, etc.);
• Consideration of different industry applications.
Conflict of interest statement
Not applicable.
Ethical Approval:
The authors declare that there is no conflict of interest.
Consent to Participate:
The authors declare that they agree with the participate of the journal.
Consent to Publish:
The authors declare that they agree with the publication of this paper in this journal.
Authors Contributions:
Hadi Gholizadeh: Conceptualization; Formal analysis; Investigation; Methodology; Software;
Validation; Original draft; Visualization;
Hamed Fazlollahtabar: Original draft; Conceptualization, Methodology, Project Admiration;
Amir M. Fathollahi-Fard: Investigation; Supervision; Review & Editing;
Maxim A. Dulebenets: Review & Editing;
Availability of data and materials:
The authors declare that the data are not available and can be presented upon the requested of the
readers.
30
Appendix
The appendix is available in supplementary materials.
References
Alaswad, S., & Xiang, Y. (2017). A review on condition-based maintenance optimization models for
stochastically deteriorating system. Reliability Engineering & System Safety, 157, 54-63.
Alimian, M., Saidi-Mehrabad, M., & Jabbarzadeh, A. (2019). A robust integrated production and preventive
maintenance planning model for multi-state systems with uncertain demand and common cause
failures. Journal of Manufacturing Systems, 50, 263-277.
Bappy, M. M., Ali, S. M., Kabir, G., & Paul, S. K. (2019). Supply chain sustainability assessment with
Dempster-Shafer evidence theory: Implications in cleaner production. Journal of Cleaner Production,
237, 117771.
Chen, Y. L. (2019). Optimal scheduling replacement policies for a system with multiple random works.
Communications in Statistics-Theory and Methods, 48(3), 676-688.
Chang, C. C. (2018). Optimal age replacement scheduling for a random work system with random lead
time. International Journal of Production Research, 56(16), 5511-5521.
Chowdhury, N. A., Ali, S. M., Mahtab, Z., Rahman, T., Kabir, G., & Paul, S. K. (2019). A structural model
for investigating the driving and dependence power of supply chain risks in the readymade garment
industry. Journal of Retailing and Consumer Services, 51, 102-113.
Cui, W. (2020). Approximate Approach to Deal with the Uncertainty in Integrated Production Scheduling
and Maintenance Planning. Journal of Shanghai Jiaotong University (Science), 25(1), 106-117.
Cui, N. F., & Liang, Y. Y. (2018). Robust project scheduling based on the integrated optimization between
resource flow network and time buffers. Systems Engineering—Theory & Practice, 38(1), 102-112.
Diallo, C., Venkatadri, U., Khatab, A., & Liu, Z. (2018). Optimal selective maintenance decisions for large
serial k-out-of-n: G systems under imperfect maintenance. Reliability Engineering & System Safety,
175, 234-245.
Dulebenets, M. A. (2019). A Delayed Start Parallel Evolutionary Algorithm for just-in-time truck
scheduling at a cross-docking facility. International Journal of Production Economics, 212, 236-258.
https://doi.org/10.1016/j.ijpe.2019.02.017
Du, Y., Li, J. Q., Luo, C., & Meng, L. L. (2021). A hybrid estimation of distribution algorithm for
distributed flexible job shop scheduling with crane transportations. Swarm and Evolutionary
Computation, 62, 100861.
31
Ebrahimi, M., Ghomi, S. M. T. F., & Karimi, B. (2020). Application of the preventive maintenance
scheduling to increase the equipment reliability: Case study-bag filters in cement factory. Journal of
Industrial & Management Optimization, 16(1), 189.
Eryilmaz, S. (2017). Computing optimal replacement time and mean residual life in reliability shock
models. Computers & industrial engineering, 103, 40-45.
Edwards, G. T. C., Hinge, J., Skou-Nielsen, N., Villa-Henriksen, A., Sørensen, C. A. G., & Green, O.
(2017). Route planning evaluation of a prototype optimized infield route planner for neutral material
flow agricultural operations. Biosystems Engineering, 153, 149e157.
Escamilla, J., Salido, M. A., Giret, A., & Barber, F. (2016). A metaheuristic technique for energy-efficiency
in job-shop scheduling. The Knowledge Engineering Review, 31(5), 475-485.
Fallahpour, A., Nayeri, S., Sheikhalishahi, M., Wong, K. Y., Tian, G., & Fathollahi-Fard, A. M. (2021a).
A hyper-hybrid fuzzy decision-making framework for the sustainable-resilient supplier selection
problem: a case study of Malaysian Palm oil industry. Environmental Science and Pollution
Research, 1-21.
Fallahpour, A., Wong, K. Y., Rajoo, S., & Tian, G. (2021b). An evolutionary-based predictive soft
computing model for the prediction of electricity consumption using multi expression programming.
Journal of Cleaner Production, 283, 125287.
Fathollahi-Fard, A. M., Hajiaghaei-Keshteli, M., Tian, G., & Li, Z. (2020a). An adaptive Lagrangian
relaxation-based algorithm for a coordinated water supply and wastewater collection network design
problem. Information Sciences. 512, 1335-1359. https://doi.org/10.1016/j.ins.2019.10.062
Fathollahi-Fard, A. M., Hajiaghaei-Keshteli, M. & Tavakkoli-Moghaddam, R., (2020b). Red deer
algorithm (RDA): a new nature-inspired meta-heuristic, Soft Computing, 10.1007/s00500-020-
04812-z.
Fathollahi-Fard, A. M., Hajiaghaei-Keshteli, M., & Tavakkoli-Moghaddam, R. (2018). The social
engineering optimizer (SEO). Engineering Applications of Artificial Intelligence, 72, 267-293.
Fathollahi-Fard, A. M., Woodward, L., & Akhrif, O. (2021). Sustainable distributed permutation flow-shop
scheduling model based on a triple bottom line concept. Journal of Industrial Information
Integration, 100233.
Fazlollahtabar, H., & Gholizadeh, H. (2020). Fuzzy possibility regression integrated with fuzzy adaptive
neural network for predicting and optimizing electrical discharge machining parameters. Computers
& Industrial Engineering, 140, 106225.
Fu, Y., Tian, G., Fathollahi-Fard, A. M., Ahmadi, A., & Zhang, C., (2019). Stochastic multi-objective
modelling and optimization of an energy-conscious distributed permutation flow shop scheduling
32
problem with the total tardiness constraint. Journal of Cleaner Production. 226, 515-525.
https://doi.org/10.1016/j.jclepro.2019.04.046
Gholizadeh, H., Fazlollahtabar, H., & Gholizadeh, R. (2019). A modified branch and bound algorithm for
a vague flow-shop scheduling problem. Iranian Journal of Fuzzy Systems, 16(4), 55-64.
Gholizadeh, H., Tajdin, A., & Javadian, N. (2020a). A closed-loop supply chain robust optimization for
disposable appliances. Neural computing and applications, 32(8), 3967-3985.
Gholizadeh, H., Fazlollahtabar, H., & Khalilzadeh, M. (2020b). A robust fuzzy stochastic programming for
sustainable procurement and logistics under hybrid uncertainty using big data. Journal of Cleaner
Production, 258, 120640.
Gholizadeh, H., & Fazlollahtabar, H. (2020). Robust optimization and modified genetic algorithm for a
closed loop green supply chain under uncertainty: Case study in melting industry. Computers &
Industrial Engineering, 147, 106653.
Govindan, K., & Gholizadeh, H. (2021). Robust network design for sustainable-resilient reverse logistics
network using big data: A case study of end-of-life vehicles. Transportation Research Part E:
Logistics and Transportation Review, 149, 102279.
Halvorsen-Weare, E. E., Norstad, I., Stålhane, M., & Nonås, L. M. (2017). A metaheuristic solution method
for optimizing vessel fleet size and mix for maintenance operations at offshore wind farms under
uncertainty. Energy Procedia, 137, 531-538.
Hosseini, S., Kalam, S., Barker, K., & Ramirez-Marquez, J. E. (2020). Scheduling multi-component
maintenance with a greedy heuristic local search algorithm. Soft Computing, 24(1), 351-366.
Kavoosi, M., Dulebenets, M. A., Abioye, O. F., Pasha, J., Wang, H., & Chi, H. (2019). An augmented self-
adaptive parameter control in evolutionary computation: A case study for the berth scheduling
problem. Advanced Engineering Informatics, 42, 100972. https://doi.org/10.1016/j.aei.2019.100972
Khatab, A. (2018). Maintenance optimization in failure-prone systems under imperfect preventive
maintenance. Journal of Intelligent Manufacturing, 4, 1–11.
Jiang, Z., & Le, Z. (2014). Study on multi-objective flexible job-shop scheduling problem considering
energy consumption. Journal of Industrial Engineering and Management, 7(3), 589-604.
Jovanović, P., Kecman, P., Bojović, N., & Mandić, D. (2017). Optimal allocation of buffer times to increase
train schedule robustness. European Journal of Operational Research, 256(1), 44-54.
Lu, B., & Zhou, X. (2017). Opportunistic preventive maintenance scheduling for serial-parallel multistage
manufacturing systems with multiple streams of deterioration. Reliability Engineering & System
Safety, 34, 176-189.
33
Liu, C. H., & Huang, D. H. (2014). Reduction of power consumption and carbon footprints by applying
multi-objective optimisation via genetic algorithms. International Journal of Production Research,
52(2), 337-352.168, 116-127.
Liu, Q., Dong, M., Chen, F. F., Lv, W., & Ye, C. (2019). Single-machine-based joint optimization of
predictive maintenance planning and production scheduling. Robotics and Computer-Integrated
Manufacturing, 55, 173-182.
Liao, W., Zhang, X., & Jiang, M. (2017). Multi-objective group scheduling optimization integrated with
preventive maintenance. Engineering Optimization, 49(11), 1890-1904.
Lu, C., Gao, L., Gong, W., Hu, C., Yan, X., & Li, X. (2021). Sustainable scheduling of distributed
permutation flow-shop with non-identical factory using a knowledge-based multi-objective memetic
optimization algorithm. Swarm and Evolutionary Computation, 60, 100803.
May, G., Stahl, B., Taisch, M., & Prabhu, V. (2015). Multi-objective genetic algorithm for energy-efficient
job shop scheduling. International Journal of Production Research, 53(23), 7071-7089.
Maatouk, I., Jarkass, I., Châtelet, E., & Chebbo, N. (2019). Preventive Maintenance Optimization and
Comparison of Genetic Algorithm Models in a Series–Parallel Multi-State System. Journal of
Intelligent Systems, 28(2), 219-230.
Malekpour, H., Hafezalkotob, A., & Khalili-Damghani, K. (2021). Product processing prioritization in
hybrid flow shop systems supported on Nash bargaining model and simulation-optimization. Expert
Systems with Applications, 180, 115066.
Mao, J., Hu, X., Pan, Q. K., Miao, Z., He, C., & Tasgetiren, M. F. (2020, July). An improved discrete
artificial bee colony algorithm for the distributed permutation flowshop scheduling problem with
preventive maintenance. In 2020 39th Chinese Control Conference (CCC) (pp. 1679-1684). IEEE.
Mao, J., Hu, X., Pan, Q. K., Miao, Z., He, C., & Tasgetiren, M. F. (2020, July). An iterated greedy algorithm
for the distributed permutation flowshop scheduling problem with preventive maintenance to
minimize total flowtime. In 2020 39th Chinese Control Conference (CCC) (pp. 1507-1512). IEEE.
Mao, J. Y., Pan, Q. K., Miao, Z. H., & Gao, L. (2021). An effective multi-start iterated greedy algorithm to
minimize makespan for the distributed permutation flowshop scheduling problem with preventive
maintenance. Expert Systems with Applications, 169, 114495.
Miyata, H. H., Nagano, M. S., & Gupta, J. N. (2019). Integrating preventive maintenance activities to the
no-wait flow shop scheduling problem with dependent-sequence setup times and makespan
minimization. Computers & Industrial Engineering, 135, 79-104.
Moosavi, J., Naeni, L. M., Fathollahi-Fard, A. M., & Fiore, U. (2021). Blockchain in supply chain
management: a review, bibliometric, and network analysis. Environmental Science and Pollution
Research, 1-15.
34
Mojtahedi, M., Fathollahi-Fard, A. M., Tavakkoli-Moghaddam, R., & Newton, S. (2021). Sustainable
vehicle routing problem for coordinated solid waste management. Journal of Industrial Information
Integration, 23, 100220.
Naderi, B., Ghomi, S. F., Aminnayeri, M., & Zandieh, M. (2011). Modeling and scheduling open shops
with sequence-dependent setup times to minimize total completion time. The International Journal of
Advanced Manufacturing Technology, 53(5-8), 751-760.
Naderi, B., Zandieh, M., Balagh, A. K. G., & Roshanaei, V. (2009). An improved simulated annealing for
hybrid flowshops with sequence-dependent setup and transportation times to minimize total
completion time and total tardiness. Expert systems with Applications, 36(6), 9625-9633.
Pasha, J., Dulebenets, M. A., Fathollahi-Fard, A. M., Tian, G., Lau, Y. Y., Singh, P., & Liang, B. (2021).
An integrated optimization method for tactical-level planning in liner shipping with heterogeneous
ship fleet and environmental considerations. Advanced Engineering Informatics, 48, 101299.
Razavi, N., Gholizadeh, H., Nayeria, S., & Ashrafi, T. A. (2020). A robust optimization model of the field
hospitals in the sustainable blood supply chain in crisis logistics. Journal of the Operational Research
Society, 1-26.
Schrotenboer, A. H., Ursavas, E., & Vis, I. F. (2020). Mixed Integer Programming models for planning
maintenance at offshore wind farms under uncertainty. Transportation Research Part C: Emerging
Technologies, 112, 180-202.
Seidgar, H., Zandieh M., & Mahdavi, I., (2016) Bi-objective optimization for integrating production and
preventive maintenance scheduling in two-stage assembly flow shop problem, Journal of Industrial
and Production Engineering, 33:6, 404-425, DOI:10.1080/21681015.2016.1173599
Tambe, P. P., & Kulkarni, M. S. (2016). Selective maintenance optimization under schedule and quality
constraints. International Journal of Quality & Reliability Management, 24, 635-657.
Wang, X., Li, L., & Xie, M. (2019). Optimal preventive maintenance strategy for leased equipment under
successive usage-based contracts. International Journal of Production Research, 57(18), 5705-5724.
Wang, W., Tian, G., Chen, M., Tao, F., Zhang, C., Abdulraham, A. A., ... & Jiang, Z. (2020). Dual-objective
program and improved artificial bee colony for the optimization of energy-conscious milling
parameters subject to multiple constraints. Journal of Cleaner Production, 245, 118714.
Whitley, D. (1994). A genetic algorithm tutorial. Statistics and computing, 4(2), 65-85.
Xiao, L., & Xia, T. (2018). Opportunistic Replacement Optimization for Multi-Component System Based
on Programming Theory. Journal of Shanghai Jiaotong University (Science), 23(1), 77-84.
Yahyatabar, A., & Najafi, A. A. (2017). A quadratic reproduction based Invasive Weed Optimization
algorithm to minimize periodic preventive maintenance cost for series-parallel systems. Computers
& Industrial Engineering, 110, 436-461.
35
Yu, C., Andreotti, P., & Semeraro, Q. (2020). Multi-objective scheduling in hybrid flow shop: Evolutionary
algorithms using multi-decoding framework. Computers & Industrial Engineering, 147, 106570.
Zandieh, M., Khatami, A. R., & Rahmati, S. H. A. (2017). Flexible job shop scheduling under condition-
based maintenance: Improved version of imperialist competitive algorithm. Applied Soft Computing,
58, 449-464.
Zhang, M., & Xie, M. (2017). An ameliorated improvement factor model for imperfect maintenance and
its goodness of fit. Technometrics, 59(2), 237–246.
Zhou, X., & Lu, B. (2018). Preventive maintenance scheduling for serial multi-station manufacturing
systems with interaction between station reliability and product quality. Computers & Industrial
Engineering, 122, 283-291.
Zhou, X., & Shi, K. (2019). Capacity failure rate based opportunistic maintenance modeling for series-
parallel multi-station manufacturing systems. Reliability Engineering & System Safety, 181, 46-53.
Zhang, C., Tian, G., Fathollahi-Fard, A. M., & Li, Z., (2020). Interval-valued Intuitionistic Uncertain
Linguistic Cloud Petri Net and its Application in Risk Assessment for Subway Fire Accident, IEEE
Transactions on Automation Science and Engineering, 10.1109/TASE.2020.3014907.
Zhang, C., Fathollahi-Fard, A. M., Li, J., Tian, G., & Zhang, T. (2021). Disassembly Sequence Planning
for Intelligent Manufacturing Using Social Engineering Optimizer. Symmetry, 13(4), 663.
Zhang, Z., & Tang, Q. (2021). Integrating preventive maintenance to two-stage assembly flow shop
scheduling: MILP model, constructive heuristics and meta-heuristics. Flexible Services and
Manufacturing Journal, 1-48.
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