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دسترسی آزاد مقاله
1 - Electromagnetism-like algorithm for fuzzy flow shop batch processing machines scheduling to minimize total weighted earliness and tardiness
S. Molla-Alizadeh-‎Zavardehi‎ R. Tavakkoli-‎Moghaddam F. Hosseinzadeh ‎Lotfi‎‎In this paper, we study a flow shop batch processing machines scheduling problem. The fuzzy due dates are considered to make the problem more close to the reality. The objective function is taken as the weighted sum of fuzzy earliness and fuzzy tardiness. In order چکیده کامل‎In this paper, we study a flow shop batch processing machines scheduling problem. The fuzzy due dates are considered to make the problem more close to the reality. The objective function is taken as the weighted sum of fuzzy earliness and fuzzy tardiness. In order to tackle the given problem, we propose a hybrid electromagnetism-like (EM) algorithm, in which the EM is hybridized with a diversification mechanism and effective local search to enhance the efficiency of the algorithm. The proposed algorithms are evaluated by comparison against two existing well-known EMs in the literature. Additionally, we propose some heuristics based on the earliest due date (EDD) to solve the given problem. The proposed hybrid EM algorithm is tested on sets of various randomly generated instances. For this purpose, we investigate the impacts of the rise in problem sizes on the performance of the developed algorithm. Through the analysis of the experimental results, the highly effective performance of the proposed algorithm is shown against the two existing well-known EMs from the literature and proposed ‎EDDs.‎ پرونده مقاله -
دسترسی آزاد مقاله
2 - بکارگیری رویه جستجوی تصادفی تطابقی حریصانه برای زمانبندی مسئله جریان کارگاهی بدون صف های میانی با استفاده از تبدیل به مسئله فروشنده دوره گرد
Javad Behnamian Ronak Mohammadi Omid Rezaeiهدف از این مقاله یافتن توالی بهینه به منظور کمینه کردن فاصله زمانی ساخت برای مسئله زمانبندی جریان کارگاهی بدون صفهای میانی میباشد. مسائل زمانبندی بدون انتظار در آن دسته از محیطهای تولیدی رخ میدهد که در آن یک کار میبایست از آغاز تا پایان بر روی یک ماشین یا چند ماشین بدون چکیده کاملهدف از این مقاله یافتن توالی بهینه به منظور کمینه کردن فاصله زمانی ساخت برای مسئله زمانبندی جریان کارگاهی بدون صفهای میانی میباشد. مسائل زمانبندی بدون انتظار در آن دسته از محیطهای تولیدی رخ میدهد که در آن یک کار میبایست از آغاز تا پایان بر روی یک ماشین یا چند ماشین بدون وقفه پردازش شود. از آنجایی که ساختار این مسئله شباهت بسیاری با مسئله فروشنده دورهگرد دارد، در تحقیق حاضر از یک رویکرد جدید جهت بدست آوردن دیرکردها کمک گرفته شده به گونه ای که با هدف یافتن توالی بهینه عملیاتی که کمترین فاصله زمانی ساخت را داراست از ماتریس دیرکردهای بدست آمده از مسئله فروشنده دورهگرد استفاده شده است. همچنین از الگوریتم جستجوی تصادفی تطابقی حریصانه برای حل مسئله تعیین توالی جریان کارگاهی بدون صفهای میانی استفاده و کارایی آن پس از تعیین پارامتر از طریق روش فاکتوریل، با الگوریتم کلونی مورچگان مقایسه شده است. پرونده مقاله -
دسترسی آزاد مقاله
3 - زمانبندی دو هدفه جریان کارگاهی مختلط با تقریب پارتو در یک منطقه مشخص
Seyed Mostafa Mousaviاین تحقیق، مساله زمانبندی تولید در محیط جریان کارگاهی مختلط با زمان های آماده سازی وابسته به توالی و با هدف مینیمم کردن ماکزیمم زمان تکمیل کارها و جمع زمان های تاخیر را مورد بررسی قرار می دهد. در گذشته مسائل دو هدفه با یافتن تقریب پارتو از کل فضای مساله (بدون هیچ محدودی چکیده کاملاین تحقیق، مساله زمانبندی تولید در محیط جریان کارگاهی مختلط با زمان های آماده سازی وابسته به توالی و با هدف مینیمم کردن ماکزیمم زمان تکمیل کارها و جمع زمان های تاخیر را مورد بررسی قرار می دهد. در گذشته مسائل دو هدفه با یافتن تقریب پارتو از کل فضای مساله (بدون هیچ محدودیتی) حل شده است. محدودیت در این تحقیق یافتن تقریب پارتو در یک منطقه مشخص شده است. به منظور حل مساله، الگوریتم ژنتیک چند هدفه مبتنی بر رتبه بندی پارتو مورد استفاده قرار گرفته است. در ساختار الگوریتم، دو استراتژی انتخاب جواب برای آرشیو جهت تولید پارتو در یک منطقه مشخص پیشنهاد شده است. پس از تولید مسائل نمونه، الگوریتم ژنتیک با سه استراتژی (دو استراتژی پیشنهادی و استراتژی عمومی در ادبیات) اجرا شده است. استراتژی مناسب براساس جواب-های موثر در آرشیو تعیین شده است. نتایج نشان دهنده این واقعیت است که استراتژی های پیشنهاد شده عملکرد بهتری نسبت به استراتژی در ادبیات نشان داده اند. پرونده مقاله -
دسترسی آزاد مقاله
4 - مقایسه کارایی روش های "سیستم کلونی مورچگان" و "برنامه ریزی خطی" در مدل سازی مسأله زمان- بندی تولید جریانی
Said Esfandyari Ali Morovati Sharif Abadi Seyed Habibolah Mirghafouri Hamid Reza Kadkhodazadehهر چند که برنامه ریزی خطی در دنیای واقع کاربردهای زیادی دارد، اما در برخورد با مسائل پیچیده و سخت عدم کارایی خود را نشان داده است. با پیشرفت علم و رویارویی با مشکلات مختلف، تمایل به حل مسائل در حجم زیاد در زمان کوتاه بیشتر شده است. روش های ابتکاری و فوق ابتکاری جدیدترین چکیده کاملهر چند که برنامه ریزی خطی در دنیای واقع کاربردهای زیادی دارد، اما در برخورد با مسائل پیچیده و سخت عدم کارایی خود را نشان داده است. با پیشرفت علم و رویارویی با مشکلات مختلف، تمایل به حل مسائل در حجم زیاد در زمان کوتاه بیشتر شده است. روش های ابتکاری و فوق ابتکاری جدیدترین دستاورد برنامه ریزی غیرخطی در حل این گونه مسائل هستند. یکی از حوزه هایی که نیاز به برنامه ریزی در حجم بالا دارد زمان بندی تولید در مسائل سخت می باشد. این مقاله به مدل سازی و مقایسه دو روش برنامه ریزی خطی و الگوریتم سیستم مورچگان در زمان بندی تولید جریانی منعطف با توجه به متغیرهای تعداد ماشین و سفارش پرداخته است؛ مبنای مقایسه در این پژوهش شاخص های زمان پردازش، تعداد محدودیت، بهینگی و حجم حافظه اشغال شده مربوط به اعداد تصادفی می باشد. در این مقاله از روش پژوهششبه آزمایشی استفاده شده است، ابزار آزمایش به ترتیب نرم افزارهای سی شارپ و لینگو برای الگوریتم مورچگان و برنامه ریزی خطی است. نتایج به دست آمده نشان می دهد که مدل برنامه ریزی خطی درتعداد ماشین و سفارش پایین کارایی بالاتری دارد، اما با افزایش ماشین و سفارش با توجه به شاخص های در نظر گرفته شده، الگوریتم سیستم مورچگان کارایی بالاتر خود را نشان می دهد. پرونده مقاله -
دسترسی آزاد مقاله
5 - Scheduling of a flexible flow shop with multiprocessor task by a hybrid approach based on genetic and imperialist competitive algorithms
Javad Rezaeian Hany Seidgar Morteza KianiThis paper presents a new mathematical model for a hybrid flow shop scheduling problem with multiprocessor tasks in which sequence dependent set up times and preemption are considered. The objective is to minimize the weighted sum of makespan and maximum tardiness. Thre چکیده کاملThis paper presents a new mathematical model for a hybrid flow shop scheduling problem with multiprocessor tasks in which sequence dependent set up times and preemption are considered. The objective is to minimize the weighted sum of makespan and maximum tardiness. Three meta-heuristic methods based on genetic algorithm (GA), imperialist competitive algorithm (ICA) and a hybrid approach of GA and ICA are proposed to solve the generated problems. The performances of algorithms are evaluated by computational time and Relative Percentage Deviation (RPD) factors. The results indicate that ICA solves the problems faster than other algorithms and the hybrid algorithm produced best solution based on RPD. پرونده مقاله -
دسترسی آزاد مقاله
6 - Modeling and scheduling no-idle hybrid flow shop problems
Mehdi Yazdani Bahman NaderiAlthough several papers have studied no-idle scheduling problems, they all focus on flow shops, assuming one processor at each working stage. But, companies commonly extend to hybrid flow shops by duplicating machines in parallel in stages. This paper considers the prob چکیده کاملAlthough several papers have studied no-idle scheduling problems, they all focus on flow shops, assuming one processor at each working stage. But, companies commonly extend to hybrid flow shops by duplicating machines in parallel in stages. This paper considers the problem of scheduling no-idle hybrid flow shops. A mixed integer linear programming model is first developed to mathematically formulate the problem. Using commercial software, the model can solve small instances to optimality. Then, two metaheuristics based on variable neighborhood search and genetic algorithms are developed to solve larger instances. Using numerical experiments, the performance of the model and algorithms are evaluated.Although several papers have studied no-idle scheduling problems, they all focus on flow shops, assuming one processor at each working stage. But, companies commonly extend to hybrid flow shops by duplicating machines in parallel in stages. This paper considers the problem of scheduling no-idle hybrid flow shops. A mixed integer linear programming model is first developed to mathematically formulate the problem. Using commercial software, the model can solve small instances to optimality. Then, two metaheuristics based on variable neighborhood search and genetic algorithms are developed to solve larger instances. Using numerical experiments, the performance of the model and algorithms are evaluated. پرونده مقاله -
دسترسی آزاد مقاله
7 - New Heuristic Algorithm for Flow Shop Scheduling with 3 Machines and 2 Robots Considering the Breakdown Interval of Machines and Robots Simultaneously
mahdi eghbali Mohammad Saidi Mehrabad hassan halehScheduling is an important subject of production and operations management area. In flow-shop scheduling, the objective is to obtain a sequence of jobs which when processed in a fixed order of machines, will optimize some well defined criteria. The concept of transporta چکیده کاملScheduling is an important subject of production and operations management area. In flow-shop scheduling, the objective is to obtain a sequence of jobs which when processed in a fixed order of machines, will optimize some well defined criteria. The concept of transportation time is very important in scheduling. Transportation can be done by robots. In situations that robots are used to transport materials (material handler), breakdown of the machines and robots have a significant role in the production concern. This paper deals with minimization of the total elapsed time for flow shop scheduling problem (number of machine=3) in which the effect of machine and robots breakdown interval are considered simultaneously. Furthermore, by providing an example, the proposed algorithm is described. A summary and future works conclude the paper پرونده مقاله -
دسترسی آزاد مقاله
8 - A Mathematical Model and a Solution Method for Hybrid Flow Shop Scheduling
Esmaeil Najafi Bahman Naderi Hassan Sadeghi Mehdi YazdaniThis paper studies the hybrid flow shop scheduling where the optimization criterion is the minimization of total tardiness. First, theproblem is formulated as a mixed integer linear programming model. Then, to solve large problem sizes, an artificial immune algorithmhyb چکیده کاملThis paper studies the hybrid flow shop scheduling where the optimization criterion is the minimization of total tardiness. First, theproblem is formulated as a mixed integer linear programming model. Then, to solve large problem sizes, an artificial immune algorithmhybridized with a simple local search in form of simulated annealing is proposed. Two experiments are carried out to evaluate the modeland the algorithm. In the first one, the general performance of the model and the proposed algorithm is experimented. In the next one, thepresented algorithm is compared against some other algorithms. The results support high performance of the proposed algorithm. پرونده مقاله -
دسترسی آزاد مقاله
9 - The Preemptive Just-in-time Scheduling Problem in a Flow Shop Scheduling System
Javad Rezaeian Sadegh Hosseini-Kia Iraj MahdaviFlow shop scheduling problem has a wide application in the manufacturing and has attracted much attention in academic fields. From other point, on time delivery of products and services is a major necessity of companies’ todays; early and tardy delivery times will چکیده کاملFlow shop scheduling problem has a wide application in the manufacturing and has attracted much attention in academic fields. From other point, on time delivery of products and services is a major necessity of companies’ todays; early and tardy delivery times will result additional cost such as holding or penalty costs. In this paper, just-in-time (JIT) flow shop scheduling problem with preemption and machine idle time assumptions is considered in which objective function is minimizing the sum of weighted earliness and tardiness. A new non-linear mathematical model is formulated for this problem and due to high complexity of the problem meta-heuristic approaches have been applied to solve the problem for finding optimal solution. The parameters of algorithms are set by Taguchi method. Each parameter is tested in three levels. By implementation of many problems with different sizes these levels are determined .The proposed model is solved by three meta-heuristic algorithms: genetic algorithm (GA), imperialist competitive algorithm (ICA) and hybrid of GA and ICA. To evaluate the performance of the proposed algorithms many test problems have been designed. The Computational results indicate the superiority of the performance of hybrid approach than GA and ICA in finding thebest solution in reasonable computational time. پرونده مقاله -
دسترسی آزاد مقاله
10 - Economic Lot Sizing and Scheduling in Distributed Permutation Flow Shops
Mohammad Alaghebandha Bahman Naderi Mohammad MohammadiThis paper addresses a new mixed integer nonlinear and linear mathematical programming economic lot sizing and scheduling problem in distributed permutation flow shop problem with number of identical factories and machines. Different products must be distributed between چکیده کاملThis paper addresses a new mixed integer nonlinear and linear mathematical programming economic lot sizing and scheduling problem in distributed permutation flow shop problem with number of identical factories and machines. Different products must be distributed between the factories and then assignment of products to factories and sequencing of the products assigned to each factory has to be derived. The objective is to minimize the sum of setup costs, work-in-process inventory costs and finished products inventory costs per unit of time. Since the proposed model is NP-hard, an efficient Water Cycle Algorithm is proposed to solve the model. To justify proposed WCA, Monarch Butterfly Optimization (MBO), Genetic Algorithm (GA) and combination of GA and simplex are utilized. In order to determine the best value of algorithms parameters that result in a better solution, a fine-tuning procedure according to Response Surface Methodology is executed. پرونده مقاله -
دسترسی آزاد مقاله
11 - Three Hybrid Metaheuristic Algorithms for Stochastic Flexible Flow Shop Scheduling Problem with Preventive Maintenance and Budget Constraint
Sadigh Raissi Ramtin Rooeinfar Vahid Reza GhezavatiStochastic flexible flow shop scheduling problem (SFFSSP) is one the main focus of researchers due to the complexity arises from inherent uncertainties and also the difficulty of solving such NP-hard problems. Conventionally, in such problems each machine’s job pr چکیده کاملStochastic flexible flow shop scheduling problem (SFFSSP) is one the main focus of researchers due to the complexity arises from inherent uncertainties and also the difficulty of solving such NP-hard problems. Conventionally, in such problems each machine’s job process time may encounter uncertainty due to their relevant random behaviour. In order to examine such problems more realistically, fixed interval preventive maintenance (PM) and budget constraint are considered.PM activity is a crucial task to reduce the production efficiency. In the current research we focused on a scheduling problem which a job is processed at the upstream stage and all the downstream machines get busy or alternatively PM cost is significant, consequently the job waits inside the buffers and increases the associated holding cost. This paper proposes a new more realistic mathematical model which considers both the PM and holding cost of jobs inside the buffers in the stochastic flexible flow shop scheduling problem. The holding cost is controlled in the model via the budget constraint. In order to solve the proposedmodel, three hybrid metaheuristic algorithms are introduced. They include a couple of well-known metaheuristic algorithms which have efficient quality solutions in the literature. The two algorithms of them constructed byincorporationof the particle swarm optimization algorithm (PSO) and parallel simulated annealing (PSA) methods under different random generation policies. The third one enriched based on genetic algorithm (GA) with PSA. To evaluate the performance of the proposed algorithms, different numerical examples are presented. Computational experiments revealed that the proposed algorithms embedboth desirable accuracy and CPU time. Among them, the PSO-PSAП outperforms than other algorithms in terms of makespan and CPU time especially for large size problems. پرونده مقاله -
دسترسی آزاد مقاله
12 - Minimizing the operational costs in a flexible flow shop scheduling problem with unrelated parallel machines
Ali Hassani Seyed Mohamad Hasan Hosseini Foroogh BehrooziThis paper investigates a flexible flow shop scheduling problem with the aim of minimizing the operational costs as a new objective function. In this production system, there are some unrelated parallel machines with different performances and different technology level چکیده کاملThis paper investigates a flexible flow shop scheduling problem with the aim of minimizing the operational costs as a new objective function. In this production system, there are some unrelated parallel machines with different performances and different technology levels in the first stage and each other stage consists of a single machine. Setup times are assumed as sequence-dependent and are need when a machine starts to process a new job. Some of the parallel machines in the first stage are multifunctional and can do several processes on jobs. So, the jobs that are assigned to these machines do not need to be processed in some next stages. This problem is described with an example, and its parameters and decision variables are defined. Then a mathematical model based on mixed-integer linear programming (MIP) is developed to solve the problem in small-sized scales. As this problem is discussed in an Np-hard environment, the Genetic Algorithm (GA) is applied to solve the considered problem on practical-sized scales. Due to the result, the operational costs conflict with makespan as a common objective function in scheduling problems. Therefore, the supplementary analysis has been presented considering a restriction on the makespan. پرونده مقاله -
دسترسی آزاد مقاله
13 - Scheduling on flexible flow shop with cost-related objective function considering outsourcing options
Mojtaba Enayati Ebrahim Asadi-Gangraj Mohammad Mahdi PaydarThis study considers outsourcing decisions in a flexible flow shop scheduling problem, in which each job can be processed by either an in-house production line or outsourced. The selected objective function aims to minimize the weighted sum of tardiness costs, in-house چکیده کاملThis study considers outsourcing decisions in a flexible flow shop scheduling problem, in which each job can be processed by either an in-house production line or outsourced. The selected objective function aims to minimize the weighted sum of tardiness costs, in-house production costs, and outsourcing costs with respect to the jobs due date. The purpose of the problem is to select the jobs that must be processed in-house, schedule processing of the jobs in-house, and finally select and assign other jobs to the subcontractors. We develop a mixed-integer linear programming (MILP) model for the research problem. Regarding the complexity of the research problem, the MILP model cannot be used for large-scale problems. Therefore, four metaheuristic algorithms, including SA, GA, PSO, hybrid PSO-SA, are proposed to solve the problem. Furthermore, some random test problems with different sizes are generated to evaluate the effectiveness of the proposed MILP model and solution approaches. The obtained results demonstrate that the GA can obtain better solutions in comparison to the other algorithms. پرونده مقاله -
دسترسی آزاد مقاله
14 - Solving Group Scheduling Problem in No-wait Flow Shop with Sequence Dependent Setup Times
Abolfazl Adressi Reza Bashirzadeh Vahid Azizi Saeed Tasouji HassanpourDifferent manufacturing enterprises use regularly scheduling algorithms in order to help meeting demands over time and reducing operational costs. Nowadays, for a better useofresources and manufacturingin accordance withcustomer needs and given the level ofcompetitionbe چکیده کاملDifferent manufacturing enterprises use regularly scheduling algorithms in order to help meeting demands over time and reducing operational costs. Nowadays, for a better useofresources and manufacturingin accordance withcustomer needs and given the level ofcompetitionbetweencompanies, employing asuitablescheduling programhasa double importance. Conventional productionmethods are constantly substituted with new ones for improving the efficiency and effectiveness of the entire production system. In this paper, two Meta-heuristic algorithms, Genetic and simulated annealing, have been used in order to solve the group scheduling problem of jobs in a single stage No-wait flow shop environment in which setup times are sequence dependent,. The purpose of solving the proposed problem is to minimize the maximum time needed to complete the jobs (Makespan). The results show that Genetic algorithm is efficient in problems with small and large dimensions, with respect to time parameter of problem solving. پرونده مقاله -
دسترسی آزاد مقاله
15 - Modelling and robust scheduling of two-stage assembly flow shop under uncertainty in assembling times
Maryam seyedhamzeh hossein amoozad khalili Seyed Mohammad Hassan Hosseini mortaza honarmand azimi Kamaladdin RahmaniThis paper focuses on robust scheduling for an assembly flow shop where assembling times are uncertain. The considered manufacturing environment is a two-stage production system that consists of a processing stage followed by an assembly stage. There are two parallel ma چکیده کاملThis paper focuses on robust scheduling for an assembly flow shop where assembling times are uncertain. The considered manufacturing environment is a two-stage production system that consists of a processing stage followed by an assembly stage. There are two parallel machines in the first stage to process the components followed by an assembly stage wherein, the components are assembled to products. The majority of scheduling research considers a deterministic environment with pre-known and fixed data. However, in real-world condition, several kinds of uncertainties should be considered. In this article, the scheduling problem is tackle under uncertainty and the assembling time of each product at the second stage is the source of uncertainty. The problem is described and formulated as a mixed-integer linear programing model under deterministic condition. After that, the uncertainty issue is discussed and the robust scheduling procedure is introduced to minimize the maximum completion time of all products based on the min–max regret approach. To solve the robust scheduling an exact method is proposed to solve the problem on the small scales. Moreover, two approximate methods are modified and used to solve this NP-hard problem on the small and practical scales. The performances of the proposed methods are evaluated by several numerical examples taken from valid references. Computational results indicate that the proposed robust scheduling methods provide effective hedges against processing time uncertainty while maintaining excellent expected makespan performance. پرونده مقاله -
دسترسی آزاد مقاله
16 - Improved teaching–learning-based and JAYA optimization algorithms for solving flexible flow shop scheduling problems
Raviteja Buddala Siba Sankar MahapatraFlexible flow shop (or a hybrid flow shop) scheduling problem is an extension of classical flow shop scheduling problem. In a simple flow shop configuration, a job having ‘g’ operations is performed on ‘g’ operation centres (stages) with each sta چکیده کاملFlexible flow shop (or a hybrid flow shop) scheduling problem is an extension of classical flow shop scheduling problem. In a simple flow shop configuration, a job having ‘g’ operations is performed on ‘g’ operation centres (stages) with each stage having only one machine. If any stage contains more than one machine for providing alternate processing facility, then the problem becomes a flexible flow shop problem (FFSP). FFSP which contains all the complexities involved in a simple flow shop and parallel machine scheduling problems is a well-known NP-hard (Non-deterministic polynomial time) problem. Owing to high computational complexity involved in solving these problems, it is not always possible to obtain an optimal solution in a reasonable computation time. To obtain near-optimal solutions in a reasonable computation time, a large variety of meta-heuristics have been proposed in the past. However, tuning algorithm-specific parameters for solving FFSP is rather tricky and time consuming. To address this limitation, teaching–learning-based optimization (TLBO) and JAYA algorithm are chosen for the study because these are not only recent meta-heuristics but they do not require tuning of algorithm-specific parameters. Although these algorithms seem to be elegant, they lose solution diversity after few iterations and get trapped at the local optima. To alleviate such drawback, a new local search procedure is proposed in this paper to improve the solution quality. Further, mutation strategy (inspired from genetic algorithm) is incorporated in the basic algorithm to maintain solution diversity in the population. Computational experiments have been conducted on standard benchmark problems to calculate makespan and computational time. It is found that the rate of convergence of TLBO is superior to JAYA. From the results, it is found that TLBO and JAYA outperform many algorithms reported in the literature and can be treated as efficient methods for solving the FFSP. پرونده مقاله -
دسترسی آزاد مقاله
17 - MILP models and valid inequalities for the two-machine permutation flowshop scheduling problem with minimal time lags
Imen Hamdi Saïd ToumiIn this paper, we consider the problem of scheduling on two-machine permutation flowshop with minimal time lags between consecutive operations of each job. The aim is to find a feasible schedule that minimizes the total tardiness. This problem is known to be NP-hard in چکیده کاملIn this paper, we consider the problem of scheduling on two-machine permutation flowshop with minimal time lags between consecutive operations of each job. The aim is to find a feasible schedule that minimizes the total tardiness. This problem is known to be NP-hard in the strong sense. We propose two mixed-integer linear programming (MILP) models and two types of valid inequalities which aim to tighten the models’ representations. One of them is based on dominance rules from the literature. Then, we provide the results of extensive computational experiments used to measure the performance of the proposed MILP models. They are shown to be able to solve optimally instances until the size 40-job and even several larger problem classes, with up to 60 jobs. Furthermore, we can distinguish the effect of the minimal time lags and the inclusion of the valid inequalities in the basic MILP model on the results. پرونده مقاله
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