Scheduling unrelated parallel batch processing machines with non-identical job sizes

Scheduling unrelated parallel batch processing machines to minimize makespan is studied in this paper. Jobs with non-identical sizes are scheduled on batch processing machines that can process several jobs as a batch as long as the machine capacity is not violated. Several heuristics based on best fit longest processing time (BFLPT) in two groups are proposed to solve the problem. A lower bound is also proved to evaluate the quality of the heuristics. Computational experiments were undertaken. These showed that J_SC-BFLPT, considering both load balance of machines and job processing times, was robust and outperformed other heuristics for most of the problem categories.     

Scheduling unrelated parallel batch processing machines with non-identical job sizes and unequal ready times

This research analyzes the problem of scheduling a set of n jobs with arbitrary job sizes and non-zero ready times on a set of m unrelated parallel batch processing machines so as to minimize the makespan. Unrelated parallel machine is a generalization of the identical parallel processing machines and is closer to real-world production systems. Each machine can accommodate and process several jobs simultaneously as a batch as long as the machine capacity is not exceeded. The batch processing time and the batch ready time are respectively equal to the largest processing time and the largest ready time among all the jobs in the batch. Motivated by the computational complexity and the practical relevance of the problem, we present several heuristics based on first-fit and best-fit earliest job ready time rules. We also present a mixed integer programming model for the problem and a lower bound to evaluate the quality of the heuristics. The small computational effort of deterministic heuristics, which is valuable in some practical applications, is also one of the reasons that motivates this study. The results show that the heuristic proposed in this paper has a superior performance compared to the heuristics based on ideas proposed in the literature.     

Minimizing total weighted tardiness on a single batch process machine with incompatible job families

The diffusion step in semiconductor wafer fabrication is very time consuming, compared to other steps in the process, and performance in this area has a significant impact on overall factory performance. Diffusion furnaces are able to process multiple lots of similar wafers at a time, and are therefore appropriately modeled as batch processing machines with incompatible job families. Due to the importance of on-time delivery in semiconductor manufacturing, we focus on minimizing the total weighted tardiness in this environment. The resulting problem is NP-Hard, and we decompose it into two sequential decision problems: assigning lots to batches followed by sequencing the batches. We develop several heuristics for these subproblems and test their performance.     

A discrete particle swarm optimization algorithm for the no-wait flowshop scheduling problem

In this paper, a discrete particle swarm optimization (DPSO) algorithm is presented to solve the no-wait flowshop scheduling problem with both makespan and total flowtime criteria. The main contribution of this study is due to the fact that particles are represented as discrete job permutations and a new position update method is developed based on the discrete domain. In addition, the DPSO algorithm is hybridized with the variable neighborhood descent (VND) algorithm to further improve the solution quality. Several speed-up methods are proposed for both the swap and insert neighborhood structures. The DPSO algorithm is applied to both 110 benchmark instances of Taillard [Benchmarks for basic scheduling problems. European Journal of Operational Research 1993;64:278–85] by treating them as the no-wait flowshop problem instances with the total flowtime criterion, and to 31 benchmark instances provided by Carlier [Ordonnancements a contraintes disjonctives. RAIRO Recherche operationelle 1978;12:333–51], Heller [Some numerical experiments for an M×J$\mathrm{M}\times \mathrm{J}$ flow shop and its decision-theoretical aspects. Operations Research 1960;8:178–84], and Revees [A genetic algorithm for flowshop sequencing. Computers and Operations Research 1995;22:5–13] for the makespan criterion. For the makespan criterion, the solution quality is evaluated according to the reference makespans generated by Rajendran [A no-wait flowshop scheduling heuristic to minimize makespan. Journal of the Operational Research Society 1994;45:472–8] whereas for the total flowtime criterion, it is evaluated with the optimal solutions, lower bounds and best known solutions provided by Fink and Voß [Solving the continuous flow-shop scheduling problem by metaheuristics. European Journal of Operational Research 2003;151:400–14]. The computational results show that the DPSO algorithm generated either competitive or better results than those reported in the literature. Ultimately, 74 out of 80 best known solutions provided by Fink and Voß [Solving the continuous flow-shop scheduling problem by metaheuristics. European Journal of Operational Research 2003;151:400–14] were improved by the VND version of the DPSO algorithm.     

Enhancing genetic algorithms for dependent job scheduling in grid computing environments

Genetic Algorithms (GAs) are stochastic search techniques based on principles of natural selection and recombination that attempt to find optimal solutions in polynomial time by manipulating a population of candidate solutions. GAs have been widely used for job scheduling optimisation in both homogeneous and heterogeneous computing environments. When compared with list scheduling heuristics, GAs can potentially provide better solutions but require much longer processing time and significant experimentation to determine GA parameters. This paper presents a GA for scheduling dependent jobs in grid computing environments. A?number of selection and pre-selection criteria for the GA are evaluated with an aim to improve GA performance in job scheduling optimization. A?Task Matching with Data scheme is proposed as a GA mutation operator. Furthermore, the effect of the choice of heuristics for seeding the GA is investigated.     

Scheduling of parallel machines with sequence-dependent batches and product incompatibilities in an automotive glass facility

This application is motivated by a complex real-world scheduling problem found in the bottleneck workstation of the production line of an automotive safety glass manufacturing facility. The scheduling problem consists of scheduling jobs (glass parts) on a number of parallel batch processing machines (furnaces), assigning each job to a batch, and sequencing the batches on each machine. The two main objectives are to maximize the utilization of the parallel machines and to minimize the delay in the completion date of each job in relation to a required due date (specific for each job). Aside from the main objectives, the output batches should also produce a balanced workload on the parallel machines, balanced job due dates within each batch, and minimal capacity loss in the batches. The scheduling problem also considers a batch capacity constraint, sequence-dependent processing times, incompatible product families, additional resources, and machine capability. We propose a two-phase heuristic approach that combines exact methods with search heuristics. The first phase comprises a four-stage mixed-integer linear program for building the batches; the second phase is based on a Greedy Randomized Adaptive Search Procedure for sequencing the batches assigned to each machine. We conducted experiments on instances with up to 100 jobs built with real data from the manufacturing facility. The results are encouraging both in terms of computing time—5 min in average—and quality of the solutions—less than 10 % relative gap from the optimal solution in the first phase and less than 5 % in the second phase. Additional experiments were conducted on randomly generated instances of small, medium, and large size.     

Two-machine flowshop scheduling problem with bounded processing times to minimize total completion time

We consider the two-machine flowshop scheduling problem where jobs have random processing times which are bounded within certain intervals. The objective is to minimize total completion time of all jobs. The decision of finding a solution for the problem has to be made based on the lower and upper bounds on job processing times since this is the only information available. The problem is NP-hard since the special case when the lower and upper bounds are equal, i.e., the deterministic case, is known to be NP-hard. Therefore, a reasonable approach is to come up with well performing heuristics. We propose eleven heuristics which utilize the lower and upper bounds on job processing times based on the Shortest Processing Time (SPT) rule. The proposed heuristics are compared through randomly generated data. The computational analysis has shown that the heuristics using the information on the bounds of job processing times on both machines perform much better than those using the information on one of the two machines. It has also shown that one of the proposed heuristics performs as the best for different distributions with an overall average percentage error of less than one.     

Unrelated parallel machine scheduling with setup times and a total weighted tardiness objective

This paper presents several search heuristics and their performance in batch scheduling of parallel, unrelated machines. Identical or similar jobs are typically processed in batches in order to decrease setup times and/or processing times. The problem accounts for allotting batched work parts into unrelated parallel machines, where each batch consists of a fixed number of jobs. Some batches may contain different jobs but all jobs within each batch should have an identical processing time and a common due date. Processing time of each job of a batch is determined according to the machine group as well as the batch group to which the job belongs. Major or minor setup times are required between two subsequent batches depending on batch sequence but are independent of machines. The objective of our study is to minimize the total weighted tardiness for the unrelated parallel machine scheduling. Four search heuristics are proposed to address the problem, namely (1) the earliest weighted due date, (2) the shortest weighted processing time, (3) the two-level batch scheduling heuristic, and (4) the simulated annealing method. These proposed local search heuristics are tested through computational experiments with data from dicing operations of a compound semiconductor manufacturing facility.     

Scheduling job classes on uniform machines

We study a scheduling problem with job classes on parallel uniform machines. All the jobs of a given class share a common due-date. General, non-decreasing and class-dependent earliness and tardiness cost functions are assumed. Two objectives are considered: (i) minmax, where the scheduler is required to minimize the maximum earliness/tardiness cost among all the jobs and (ii) minmax-minsum, where the scheduler minimizes the sum of the maximum earliness/tardiness cost in all job classes. The problem is easily shown to be NP-hard, and we focus here on the introduction of simple heuristics. We introduce LPT (Largest Processing Time first)-based heuristics for the allocation of jobs to machines within each class, followed by a solution of an appropriate non-linear program, which produces for this job allocation an optimal schedule of the classes. We also propose a lower bound, based on balancing the load on the machines. Our numerical tests indicate that the heuristics result in very small optimality gaps.     

Decomposition heuristics for minimizing earliness–tardiness on parallel burn-in ovens with a common due date

This paper considers a scheduling problem for parallel burn-in ovens in the semiconductor manufacturing industry. An oven is a batch processing machine with restricted capacity. The batch processing time is set by the longest processing time among those of all the jobs contained in the batch. All jobs are assumed to have the same due date. The objective is to minimize the sum of the absolute deviations of completion times from the due date (earliness–tardiness) of all jobs. We suggest three decomposition heuristics. The first heuristic applies the exact algorithm due to Emmons and Hall (for the nonbatching problem) in order to assign the jobs to separate early and tardy job sets for each of the parallel burn-in ovens. Then, we use job sequencing rules and dynamic programming in order to form batches for the early and tardy job sets and sequence them optimally. The second proposed heuristic is based on genetic algorithms. We use a genetic algorithm in order to assign jobs to each single burn-in oven. Then, after forming early and tardy job sets for each oven we apply again sequencing rules and dynamic programming techniques to the early and tardy jobs sets on each single machine in order to form batches. The third heuristic assigns jobs to the m early job sets and m tardy jobs sets in case of m burn-in ovens in parallel via a genetic algorithm and applies again dynamic programming and sequencing rules. We report on computational experiments based on generated test data and compare the results of the heuristics with known exact solution for small size test instances obtained from a branch and bound scheme.     

Mathematical modeling and heuristic approaches to flexible job shop scheduling problems

Scheduling for the flexible job shop is very important in both fields of production management and combinatorial optimization. However, it is quite difficult to achieve an optimal solution to this problem in medium and actual size problem with traditional optimization approaches owing to the high computational complexity. For solving the realistic case with more than two jobs, two types of approaches have been used: hierarchical approaches and integrated approaches. In hierarchical approaches assignment of operations to machines and the sequencing of operations on the resources or machines are treated separately, i.e., assignment and sequencing are considered independently, where in integrated approaches, assignment and sequencing are not differentiated. In this paper, a mathematical model and heuristic approaches for flexible job shop scheduling problems (FJSP) are considered. Mathematical model is used to achieve optimal solution for small size problems. Since FJSP is NP-hard problem, two heuristics approaches involve of integrated and hierarchical approaches are developed to solve the real size problems. Six different hybrid searching structures depending on used searching approach and heuristics are presented in this paper. Numerical experiments are used to evaluate the performance of the developed algorithms. It is concluded that, the hierarchical algorithms have better performance than integrated algorithms and the algorithm which use tabu search and simulated annealing heuristics for assignment and sequencing problems consecutively is more suitable than the other algorithms. Also the numerical experiments validate the quality of the proposed algorithms.     

A GRASP approach for makespan minimization on parallel batch processing machines

In this paper we consider the problem of scheduling a set of identical batch processing machines arranged in parallel. A Greedy Randomized Adaptive Search Procedure (GRASP) approach is proposed to minimize the makespan under the assumption of non-zero job ready times, arbitrary job sizes and arbitrary processing times. Each machine can process simultaneously several jobs as a batch as long as the machine capacity is not violated. The batch processing time is equal to the largest processing time among those jobs in the batch. Similarly, the batch ready time is equal to the largest ready time among those jobs in the batch. The performance of the proposed GRASP approach was evaluated by comparing its results to a lower bound and heuristics published in the literature. Experimental study suggests that the solution obtained from the GRASP approach is superior compared to other heuristics.     

Minimising makespan for two batch-processing machines with non-identical job sizes in job shop

In this article, the job shop scheduling problem with two batch-processing machines is considered. The machines have limited capacity and the jobs have non-identical job sizes. The jobs are processed in batches and the total size of each batch cannot exceed the machine capacity. The processing times of a job on the two machines are proportional. We show the problem of minimising makespan is NP-hard in the strong sense. Then we provide an approximation algorithm with worst-case ratio no more than 4, and the running time of the algorithm is O(n?log?n). Finally, the performance of the proposed algorithm is tested by different levels of instances. Computational results demonstrate the effectiveness of the algorithm for all the instances.     

Minimizing the makespan on a batch machine with non-identical job sizes: an exact procedure

A batch processing machine can simultaneously process several jobs forming a batch. This paper considers the problem of scheduling jobs with non-identical capacity requirements, on a single-batch processing machine of a given capacity, to minimize the makespan. The processing time of a batch is equal to the largest processing time of any job in the batch. We present some dominance properties for a general enumeration scheme and for the makespan criterion, and provide a branch and bound method. For large-scale problems, we use this enumeration scheme as a heuristic method.Scope and purposeUsually in classical scheduling problems, a machine can perform only one job at a time. Although, one can find machines that can process several jobs simultaneously as a batch. All jobs of a same batch have common starting and ending times. Batch processing machines are encountered in many different environments, such as burn-in operations in semiconductor industries or heat treatment operations in metalworking industries. In the first case, the capacity of the machine is defined by the number of jobs it can hold. In the second case, each job has a certain capacity requirement and the total size of a batch cannot exceed the capacity of the machine. Hence, the number of jobs contained in each batch may be different. In this paper, we consider this second case (which is more difficult) and we provide an exact method for the makespan criterion (minimizing the last ending time).     

Two-stage hybrid flowshop scheduling with simultaneous processing machines

Simultaneous processing machines, common in processing industries such as steel and food production, can process several jobs simultaneously in the first-in, first-out manner. However, they are often highly energy-consuming. In this paper, we study a new two-stage hybrid flowshop scheduling problem, with simultaneous processing machines at the first stage and a single no-idle machine with predetermined job sequence at the second stage. A mixed integer programming model is proposed with the objective of minimizing the total processing time to reduce energy consumption and improve production efficiency. We give a sufficient and necessary condition to construct feasible sequencing solutions and present an effective approach to calculate the time variables for a feasible sequencing solution. Based on these results, we design a list scheduling heuristic algorithm and its improvement. Both heuristics can find an optimal solution under certain conditions with complexity O(nlogn), where n is the number of jobs. Our experiments verify the efficiency of these heuristics compared with classical heuristics in the literature and investigate the impacts of problem size and processing times.     

Heuristic scheduling of jobs on parallel batch machines with incompatible job families and unequal ready times

This research is motivated by a scheduling problem found in the diffusion and oxidation areas of semiconductor wafer fabrication, where the machines can be modeled as parallel batch processors. We attempt to minimize total weighted tardiness on parallel batch machines with incompatible job families and unequal ready times of the jobs. Given that the problem is NP-hard, we propose two different decomposition approaches. The first approach forms fixed batches, then assigns these batches to the machines using a genetic algorithm (GA), and finally sequences the batches on individual machines. The second approach first assigns jobs to machines using a GA, then forms batches on each machine for the jobs assigned to it, and finally sequences these batches. Dispatching and scheduling rules are used for the batching phase and the sequencing phase of the two approaches. In addition, as part of the second decomposition approach, we develop variations of a time window heuristic based on a decision theory approach for forming and sequencing the batches on a single machine.     

Mean value analysis of re-entrant line with batch machines and multi-class jobs

We propose an approximate approach for estimating the performance measures of the re-entrant line with single-job machines and batch machines based on the mean value analysis (MVA) technique. Multi-class jobs are assumed to be processed in predetermined routings, in which some processes may utilize the same machines in the re-entrant fashion. The performance measures of interest are the steady-state averages of the cycle time of each job class, the queue length of each buffer, and the throughput of the system. The system may not be modeled by a product form queueing network due to the inclusion of the batch machines and the multi-class jobs with different processing times. Thus, we present a methodology for approximately analyzing such a re-entrant line using the iterative procedures based upon the MVA and some heuristic adjustments. Numerical experiments show that the relative errors of the proposed method are within 5% as compared against the simulation results.Scope and purposeWe consider a re-entrant shop with multi-class jobs, in which jobs may visit some machines more than once at different stages of processing, as observed in the wafer fabrication process of semiconductor manufacturing. The re-entrant line also consists of both the single-job machine and the batch machine. The former refers to the ordinary machine processing one job at a time, and the latter means the machine processing several jobs together as a batch at a time. In this paper, we propose an approximation method based on the mean value analysis for estimating the mean cycle time of each class of jobs, the mean queue length of each buffer, and the throughput of the system.     

Scheduling jobs on a single batch processing machine with incompatible job families and weighted number of tardy jobs objective

In this paper, we minimize the weighted and unweighted number of tardy jobs on a single batch processing machine with incompatible job families. We propose two different mixed integer linear programming (MILP) formulations based on positional variables. The second formulation does not contain a big-M coefficient. Two iterative schemes are discussed that are able to provide tighter linear programming bounds by reducing the number of positional variables. Furthermore, we also suggest a random key genetic algorithm (RKGA) to solve this scheduling problem. Results of computational experiments are shown. The second MILP formulation is more efficient with respect to lower bounds, while the first formulation provides better upper bounds. The iterative scheme is effective for the weighted case. The RKGA is able to find high-quality solutions in a reasonable amount of time.     

Scheduling hybrid flowshop with parallel batching machines and compatibilities

This paper considers a two-stage hybrid flowshop problem in which the first stage contains several identical discrete machines, and the second stage contains several identical batching machines. Each discrete machine can process no more than one task at time, and each batching machine can process several tasks simultaneously in a batch with the additional feature that the tasks of the same batch have to be compatible. A compatibility relation is defined between each pair of tasks, so that an undirected compatibility graph is obtained which turns out to be an interval graph. The batch processing time is equal to the maximal processing time of the tasks in this batch, and all tasks of the same batch start and finish together. The goal is to make batching and sequencing decisions in order to minimize the makespan. Since the problem is NP-hard, we develop several heuristics along with their worst cases analysis. We also consider the case in which tasks have the same processing time on the first stage, for which a polynomial time approximation scheme (PTAS) algorithm is presented.     

An improved NEH heuristic to minimize makespan in permutation flow shops

For over 20 years the NEH heuristic of Nawaz, Enscore, and Ham [A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem. Omega, The International Journal of Management Science 1983;11:91–5] has been commonly regarded as the best heuristic for solving the NP-hard problem of minimizing the makespan in permutation flow shops. The strength of NEH lies mainly in its priority order according to which jobs are selected to be scheduled during the insertion phase. Framinan et al. [Different initial sequences for the heuristic of Nawaz, Enscore and Ham to minimize makespan, idle time or flowtime in the static permutation flowshop problem. International Journal of Production Research 2003;41:121–48] presented the results of an extensive study to conclude that the NEH priority order is superior to 136 different orders examined. Based upon the concept of Johnson's algorithm, we propose a new priority order combined with a simple tie-breaking method that leads to a heuristic that outperforms NEH for all problem sizes.