Purpose In recent years, significant energy consumption and facing global warming have led to concern worldwide. Therefore, governments have turned to deterrent actions such as imposing daily tariffs in different intervals to tackle energy consumption. This article addresses unrelated parallel machine energy-efficient scheduling problems by considering sequence-independent setup times and energy consumption tariffs. The objective function is that jobs should be assigned to machines and processed in different intervals so that the cost of consumed energy becomes as less as possible. It should be noted that the assumed sequence-independent setup times are addressed in two different modes, setup times jointed to processing time and setup times disjointed from processing time. Design/methodology/approach: To optimize the total energy consumption cost in unrelated parallel machine scheduling problems with sequence-independent setup times jointed to processing time and disjointed from processing time, mixed-integer linear programming (MILP) models have been proposed from two different points of view. The first model has been formulated according to the predecessor jobs of a special job, while the second model has been conducted based on the immediate predecessor job. Also, a fix and relax heuristic (FRH) algorithm has been conducted to solve large-scale instances. All mathematical models and the heuristic algorithm have been coded in the Visual C# 2017 environment and implemented using the CPLEX 12. 8 Concert Technology on a PC with 32GB RAM and Intel Corei7 4. 0 GHz CPU (4 cores). Also, a sizeable number of instances have been solved to evaluate the efficiency of mathematical models and the heuristic algorithm and to ensure their accuracy. Findings According to numerical analysis, both mathematical models solved the instances of up to 20 jobs and 80 machines optimally for sequence-independent setup-times jointed to processing time, and sequence-independent disjointed from processing time problems. However, generally speaking, the mathematical model based on predecessor jobs was more efficient than another mathematical model, especially in terms of run time. Moreover, the proposed fix and relax-based heuristic algorithm solved instances of up to 20 machines and 190 jobs for the disjointed setup times problem, and up to 20 and 220 instances for the jointed setup times problem. It should be noted that all instances were generated analogously to the literature. Research limitations/implications A vast number of exogenous factors contributed to the scheduling problems in the real world, which can disturb the scheduling process easily, frequent power outages, machine breakdown, and operator absence. Besides, considering all the real world's possibilities raises extreme complexity in problems. Therefore, similar to other studies, some assumptions were considered as follows:-machines are always available at all times,-idle is allowable for machines,-the energy consumption rate of various machines is different for each job,-each machine's energy consumption rate during processing and setups is different for each job, it is assumed as constant,-preemption is not allowed in the job's processing and setups,-all jobs are available at the beginning of the planning horizon,and-each machine can process or do the setup for only one job at a time. Practical implications Given that unrelated parallel machines are one of the most practical scheduling environments, this article can be effective in production sites and operation lines that contain such a kind of machine. Besides, unrelated parallel machines cover identical and related parallel machines. Consequently, this paper is the building blocks of cost-effective and environmentally friendly scheduling programs. Also, the application of unrelated parallel machines is not merely restricted to production problems. In other words, unrelated parallel machine scheduling problems can be used in other real-world cases, such as airplane scheduling and elevator scheduling. Originality/value-In this paper, unrelated parallel machine energy-efficient scheduling has been addressed considering sequence-independent setup times. Since it was a common belief that sequence-independent setup times could be included in processing times, sequence-independent setup times have been neglected so far. However, in this innovative study for the first time, an unrelated parallel machine energy efficient problem was investigated with sequence-independent setup times. Mathematical programming models and a heuristic algorithm were proposed for such a practical problem.

Economic lot size scheduling problem (ELSP) for a multi-product single machine system is a classical problem. This paper considers ELSP with budgetary constraint as an important aspect of such systems. In the real world situations the available funds for investment in inventory is limited. By adopting the common cycle time approach to ELSP, we obtain the optimal common cycle which minimizes the total inventory ordering and holding costs for the case of nonzero setup times. One aspect of the scheduling is to decide what should be the sequence of production runs and how the idle times shall be distributed in the common cycle time. For such a sequencing problem, we consider two cases: a) the common cycle time is given, and b) the common cycle time is a decision variable. In the literature, scheduling rules are introduced for both cases, which assume that the total idle time is located at the end of each cycle. This paper relaxes this assumption and provides: i) a rule to optimize the production sequence and the length of idle times before (or after) producing each item, for both cases (a) and (b), and ii) the optimal common cycle for case (b). The presented rule is interestingly general, simple and easy-to-apply.

In this paper we adopt the common cycle approach to economic lot scheduling problem and minimize the maximum aggregate inventory. We allow the occurrence of the idle times between any two consecutive products and consider limited capital for investment in inventory. We assume the setup times are negligible. To achieve the optimal investment in inventory we first find the idle times that minimize the maximum aggregate inventory for a given sequence of production runs and for any arbitrary cycle time T. Then, we show that these values of idle times, for the given sequence, are also optimal idle times for any other sequence. The result is an easy-to-apply rule that greatly simplifies the task of scheduling to achieve minimum required investment in inventory.

Purpose: Lot streaming, which has much attention in recent years, is an effective technique to increase production efficiency in a production system by splitting a job into several smaller parts in a multi-stage production system. But important assumptions that exist in the real-world scheduling environment are always ignored. Hence, in this paper, these assumptions are discussed and the results are reviewed. In this paper, the aim is solving a multi objective mathematical model for solving hybrid flow shop scheduling problem with lot-streaming, setup time and transportation time.Methodology: At first, a multi objective mathematical programming model is presented for solving the problem. Then, by wighting method, the multi objective model convert to single objective model and GAMS software is used to solve the small size problems to show the performance of the mathematical mothel. Inspired by previous studies, two multi objective metaheuristic algorithms based on the genetic algorithm is used to solve the large-scale problems. To illustrate the performance of the proposed metaheuristic algorithms, the obtained results of the algorithms compared with GAMS outputs in single mode.Findings: To validate the proposed model, a sample is solved using GAMS software and compared with the genetic algorithm. The obtained results show the performance of the mathematical model. Then, two proposed algorithms are used to solve the large-scale problems. For this purpose, 30 instance problems are randomly generated and six indicators are used to compare the algorithms. After performing the experiments and comparing the algorithms with each other, the results show NRGA algorithm performs bether than NSGA-II.Originality/Value: In this paper, for solving a multi objective hybrid flow shop scheduling problem with lot-streamingm mathematical model with the aim of minimizing the makespan and total tardiness, the sequence-dependent setup time and the transportation time constraints between consecutive stages are considered. Since the problem is NP-hard, NSGA-II and NRGA algorithms were used to solve the proposed problem.

Production scheduling is one of the very important problems that industry and production are confronted with it. Production scheduling is often planned in the industrial environments while productivity in production can improve significantly the expansion of simultaneous optimization of the scheduling plan. Production scheduling and production are two areas that have attracted much attention in the industry literature and production and research in the operation systems. In this study, the problem of single-machine scheduling with linear earliness and tardiness costs considering the work failure, energy consumption restriction, and the allowed idleness have been investigated and a new nonlinear mathematical model has been presented for the single-machine scheduling problem. Considering complexity in solution, this problem has been regarded as NP-hard problem. However, using methods that produce optimized results, it is just suitable for small size problems. Based on this, a genetic algorithm has been presented for solving this problem in average and large sizes. Numerical samples show that the presented algorithm is effective and efficient.

In this paper, a flexible job shop scheduling problem (FJSP) with assembly operations and sequence dependent setup time is studied. In this problem, each product is produced from assembling a set of several different parts. At first, the parts are processed in a flexible job shop system. setup time is needed when a machine starts processing the parts or it changes items. Then in the second stage, the parts are assembled and products are produced. The assembly operation cannot be started for a product until the set of parts are completed in machining operations. In this paper, we presented a mathematical model for a flexible job shop scheduling problem with assembly operations and sequence dependent setup time. The objective is to minimize the completion time of all products (makespan). Since the problem is NP-hard, one particle swarm optimization (PSO) algorithm and two hybrid metaheuristic algorithms based on particle swarm optimization are proposed. The proposed hybrid algorithms are called, respectively, hybrid particle swarm optimization with a variable neighborhood search algorithm (HPSOVNS) and hybrid particle swarm optimization with a simulated annealing algorithm (HPSOSA). In these hybrid algorithms, we used particle swarm optimization (PSO) algorithm for global exploration at search space and variable neighborhood search (VNS)/ simulated annealing (SA) algorithm for local search at around solutions obtained in the each iteration. In order to evaluate and validate the performance of the proposed algorithms, we are designed numerical experiments and results are compared with hybrid genetic algorithm and tabu search (HGATS) presented by Li and Gao. For this purpose, the proposed mathematical model is coded in GAMS software and the proposed metaheuristic algorithms are coded in MATLAB software. For obtaining better and more sustainable results of the metaheuristic algorithms, Minitab software was used to design the experiments and assign the best level to the size of problems. For the problems in the small size, the optimal solution is obtained by GAMS software. Then a randomized complete block design considered to compare the ability of algorithms at finding the best solution for medium and large problems. Computational results revealed that for medium and large problems the HPSOVNS algorithm outperforms the HPSOSA, PSO and HGATS algorithms.

One of the earliest results in flow shop scheduling theory is an algorithm given by Johnson’s [1] for scheduling jobs on two or three machines to minimize the total elapsed time whenever the processing times of jobs are random. The present paper is an attempt to develop a heuristic algorithm for two stages specially structured flow shop scheduling in which the processing times of the jobs are not completely random, but bear a well defined relationship to one another to minimize the utilization time of machines and hence their rental cost under a specified rental policy. Further the processing times and independent set up times, each are associated with probabilities. A computer programme followed by a numerical illustration is given to validate the proposed algorithm.