A new heuristic for scheduling the two-stage flowshop with additional resources

This paper deals with the problem of preemptive scheduling in a two-stage flowshop with parallel unrelated machines at the first stage and a single machine at the second stage. At the first stage, jobs use some additional resources which are available in limited quantities at any time. The resource requirements are of 0–1 type. The objective is the minimization of makespan. The problem is NP-hard. Heuristic algorithms are proposed which solve to optimality the resource constrained scheduling problem at the first stage of the flowshop, and at the same time, minimize the makespan in the flowshop by selecting appropriate jobs for simultaneous processing. Several rules of job selection are considered. The performance of the proposed heuristic algorithms is analyzed by comparing solutions with the lower bound on the optimal makespan. The extensive computational experiment shows that the proposed heuristic algorithms are able to produce near-optimal solutions in short computational time.     

Makespan distribution of permutation flowshop schedules

The makespan distribution of permutation flowshop schedules has been a topic of debate for almost fifty years. Many researchers have confirmed or doubted the famous claim that the makespan distribution of permutation flowshop schedules is asymptotically normal if the number of jobs is sufficiently large. This paper theoretically and empirically investigates the makespan distribution of permutation flowshop schedules and shows that the normality claim is not valid for the job-dominated and machine-dominated flowshops. Errors in the proof of normality of the makespan distribution of permutation flowshop schedules are pointed out. It is shown that the makespan distribution of a permutation flowshop scheduling problem depends on the number of jobs as well as the number of machines.     

Tabu search heuristic for two-machine flowshop with batch processing machines

Batch processing machines are frequently encountered in many industrial environments. A batch processing machine is one which can process several jobs simultaneously as a batch. The processing time of a batch is equal to the largest processing time of any job in the batch. This study deals with the problem of scheduling jobs in a flowshop with two batch processing machines such that the makespan is minimized. A heuristic based on Tabu search (TS) technique is proposed. The proposed heuristic is compared with a heuristic based on mixed integer linear programming (MILP). Because the complexity of the MILP-based heuristic is depended on the number of job batches, the comparison is under up-to-eight batches problem. In order to measure the proposed TS-based heuristic in larger batch problem, the relative error percentage with the lower bound (REP_LB) is used. The results show that the proposed heuristic is efficient and effective for the problems with relative large job sizes.     

The three-stage assembly flowshop scheduling problem

We consider the three-stage assembly flowshop scheduling problem with the objective of minimizing the makespan. The three-stage assembly problem generalizes both the serial three machine flowshop problem and the two-stage assembly flowshop scheduling problem and is therefore strongly NP-hard. We analyze the worst-case ratio bound for several heuristics for this problem. We also analyze the worst-case absolute bound for a heuristic based on compact vector summation techniques and we point out that, for a large number of jobs, this heuristic becomes asymptotically optimal.Scope and purposeThe three-stage assembly flowshop scheduling problem models situations which arise frequently in manufacturing when various fabrication operations are performed concurrently and then collected and transported into an assembly area for a final assembly operation. The main criterion for this problem is the minimization of the maximum job completion time (makespan). The objective of this paper is to derive algorithms for minimizing the makespan. In doing so, we also demonstrate the reduction of assembly flowshop problems to their embedded serial flowshop problems.     

Minimizing makespan on a three-machine flowshop batch scheduling problem with transportation using genetic algorithm

This paper studies the minimization of makespan in a three-machine flowshop scheduling problem in which a batch processing machine is located between two single processing machines on first and third stages. In this study also transportation capacity and transportation among machines times are explicitly considered.We establish a mixed integer programming model and propose a heuristic algorithm based on the basic idea of Johnson's algorithm. Since the problem under study is NP-hard, a genetic algorithm is also proposed to minimize makespan. The effectiveness of our solution procedures is evaluated through computational experiments. The results obtained from the computational study have shown that the genetic algorithm is a viable and effective approach that is capable to produce consistently good results.     

Setting a common due date in a constrained flowshop: A variable neighbourhood search approach

In this paper we study a due date setting problem in a flowshop layout. The problem consists of scheduling a set of jobs arriving to the system together with jobs already present (denoted as old jobs), in order to set a common due date for the new jobs. Since the old jobs have a common due date that must not be violated, our problem is a rescheduling problem with the objective of minimising the makespan of the new jobs (thus obtaining the tightest possible due date for the new jobs) and a constraint since the maximum tardiness of the old jobs must be equal to zero. This approach leads to an interesting scheduling problem in which two different objectives are considered, each one for a subset of the jobs that must be scheduled. To the best of our knowledge, this type of problems have been scarcely considered in the literature, and only for very specific purposes. Since our problem is clearly NP-hard, a new heuristic based on variable neighbourhood search (VNS) has been designed. The computational results show that our proposed heuristic outperforms two existing heuristic methods for similar problems in the literature.     

Minimizing makespan subject to minimum flowtime on two identical parallel machines

We consider the problem of scheduling jobs on two parallel identical machines where an optimal schedule is defined as one that gives the smallest makespan (the completion time of the last job) among the set of schedules with optimal total flowtime (the sum of the completion times of all jobs). We propose an algorithm to determine optimal schedules for the problem, and describe a modified multifit algorithm to find an approximate solution to the problem in polynomial computational time. Results of a computational study to compare the performance of the proposed algorithms with a known heuristic shows that the proposed heuristic and optimization algorithms are quite effective and efficient in solving the problem.Scope and purposeMultiple objective optimization problems are quite common in practice. However, while solving scheduling problems, optimization algorithms often consider only a single objective function. Consideration of multiple objectives makes even the simplest multi-machine scheduling problems NP-hard. Therefore, enumerative optimization techniques and heuristic solution procedures are required to solve multi-objective scheduling problems. This paper illustrates the development of an optimization algorithm and polynomially bounded heuristic solution procedures for the scheduling jobs on two identical parallel machines to hierarchically minimize the makespan subject to the optimality of the total flowtime.     

A generalized two-machine flowshop scheduling problem with processing time linearly dependent on job waiting-time

We consider a two-machine flowshop scheduling problem where the processing times are linearly dependent on the waiting times of the jobs. The objective is to minimize the makespan. A 0–1 mixed integer program and a heuristic algorithm are proposed. Some cases solved in polynomial time and computational experiments are also provided.     

The distributed permutation flowshop scheduling problem

This paper studies a new generalization of the regular permutation flowshop scheduling problem (PFSP) referred to as the distributed permutation flowshop scheduling problem or DPFSP. Under this generalization, we assume that there are a total of F identical factories or shops, each one with m machines disposed in series. A set of n available jobs have to be distributed among the F factories and then a processing sequence has to be derived for the jobs assigned to each factory. The optimization criterion is the minimization of the maximum completion time or makespan among the factories. This production setting is necessary in today's decentralized and globalized economy where several production centers might be available for a firm. We characterize the DPFSP and propose six different alternative mixed integer linear programming (MILP) models that are carefully and statistically analyzed for performance. We also propose two simple factory assignment rules together with 14 heuristics based on dispatching rules, effective constructive heuristics and variable neighborhood descent methods. A comprehensive computational and statistical analysis is conducted in order to analyze the performance of the proposed methods.     

Bi-criteria minimization for the permutation flowshop scheduling problem with machine-based learning effects

In traditional scheduling problems, the processing time for the given job is assumed to be a constant regardless of whether the job is scheduled earlier or later. However, the phenomenon named “learning effect” has extensively been studied recently, in which job processing times decline as workers gain more experience. This paper discusses a bi-criteria scheduling problem in an m-machine permutation flowshop environment with varied learning effects on different machines. The objective of this paper is to minimize the weighted sum of the total completion time and the makespan. A dominance criterion and a lower bound are proposed to accelerate the branch-and-bound algorithm for deriving the optimal solution. In addition, the near-optimal solutions are derived by adapting two well-known heuristic algorithms. The computational experiments reveal that the proposed branch-and-bound algorithm can effectively deal with problems with up to 16 jobs, and the proposed heuristic algorithms can yield accurate near-optimal solutions.     

Flowshop scheduling problems with transportation or deterioration between the batching and single machines

In this paper, we consider two new types of the two-machine flowshop scheduling problems where a batching machine is followed by a single machine. The first type is that normal jobs with transportation between machines are scheduled on the batching and single machines. The second type is that normal jobs are processed on the batching machine while deteriorating jobs are scheduled on the single machine. For the first type, we formulate the problem to minimize the makespan as a mixed integer programming model and prove that it is strongly NP-hard. Furthermore, a heuristic algorithm along with a worst case error bound is derived and the computational experiments are also carried out to verify the effectiveness of the proposed heuristic algorithm. For the second type, the two objectives are considered. For the problem with minimizing the makespan, we find an optimal polynomial algorithm. For the problem with minimizing the sum of completion time, we show that it is strongly NP-hard and propose an optimal polynomial algorithm for its special case.     

Heuristic algorithms for preemptive scheduling in a two-stage hybrid flowshop with additional renewable resources at each stage

This paper deals with the problem of preemptive scheduling in a two-stage flowshop with parallel unrelated machines and renewable resources at both the stages. The resource requirements are of a 0–1 type. The objective is the minimization of makespan. The problem is NP-hard. Four heuristic algorithms using linear programming are proposed for solving this problem. Performance of the algorithms is analyzed experimentally by comparing heuristic solutions with the lower bound on the optimal makespan. Statistical comparative analysis of the developed algorithms is carried out. The results of a computational experiment show that the proposed algorithms are able to produce good quality solutions in a small amount of computation time.     

Worst-case and numerical analysis of heuristic algorithms for flowshop scheduling problems with a time-dependent learning effect

In this paper, we investigate a time-dependent learning effect in a flowshop scheduling problem. We assume that the time-dependent learning effect of a job was a function of the total normal processing time of jobs scheduled before the job. The following objective functions are explored: the makespan, the total flowtime, the sum of weighted completion times, the sum of the kth power of completion times, and the maximum lateness. Some heuristic algorithms with worst-case analysis for the objective functions are given. Moreover, a polynomial algorithm is proposed for the special case with identical processing time on each machine and that with an increasing series of dominating machines, respectively. Finally, the computational results to evaluate the performance of the heuristics are provided.     

Scheduling flow lines with buffers by ant colony digraph

This work starts from modeling the scheduling of n jobs on m machines/stages as flowshop with buffers in manufacturing. A mixed-integer linear programing model is presented, showing that buffers of size n ? 2 allow permuting sequences of jobs between stages. This model is addressed in the literature as non-permutation flowshop scheduling (NPFS) and is described in this article by a disjunctive graph (digraph) with the purpose of designing specialized heuristic and metaheuristics algorithms for the NPFS problem. Ant colony optimization (ACO) with the biologically inspired mechanisms of learned desirability and pheromone rule is shown to produce natively eligible schedules, as opposed to most metaheuristics approaches, which improve permutation solutions found by other heuristics. The proposed ACO has been critically compared and assessed by computation experiments over existing native approaches. Most makespan upper bounds of the established benchmark problems from Taillard (1993) and Demirkol, Mehta, and Uzsoy (1998) with up to 500 jobs on 20 machines have been improved by the proposed ACO.     

A genetic algorithm and a simulated annealing algorithm combined with column generation technique for solving the problem of scheduling in the hybrid flowshop with additional resources

In this paper, a heuristic is proposed for solving the problem of scheduling in a two-stage flowshop with parallel unrelated machines and additional renewable resources at the first stage and a single machine at the second stage. Resource requirements are arbitrary integers. The availability of additional resources is limited at every moment. The objective is the minimization of makespan. The problem is NP-hard. The proposed heuristic combines column generation technique with a genetic algorithm (the heuristic algorithm HG) or a simulated annealing algorithm (the heuristic algorithm HS). The performance analysis is performed experimentally by comparing heuristic solutions to the lower bound on the optimal makespan. Results of the computational experiment show that both the heuristic algorithms yield good quality solutions using reasonable computation time and that HS outperforms HG for the most difficult problems.     

Heuristic algorithms for preemptive scheduling in a two-stage hybrid flowshop with additional renewable resources at each stage

This paper deals with the problem of preemptive scheduling in a two-stage flowshop with parallel unrelated machines and renewable resources at both the stages. The resource requirements are of a 0–1 type. The objective is the minimization of makespan. The problem is NP-hard. Four heuristic algorithms using linear programming are proposed for solving this problem. Performance of the algorithms is analyzed experimentally by comparing heuristic solutions with the lower bound on the optimal makespan. Statistical comparative analysis of the developed algorithms is carried out. The results of a computational experiment show that the proposed algorithms are able to produce good quality solutions in a small amount of computation time.     

Two-machine flowshop scheduling with bicriteria problem

This paper attempts to solve a two-machine flowshop bicriteria scheduling problem with release dates for the jobs, in which the objective function is to minimize a weighed sum of total flow time and makespan. To tackle this scheduling problem, an integer programming model with N²+3N variables and 5N constraints where N is the number of jobs, is formulated. Because of the lengthy computing time and high computing complexity of the integer programming model, a heuristic scheduling algorithm is presented. Experimental results show that the proposed heuristic algorithm can solve this problem rapidly and accurately. The average solution quality of the heuristic algorithm is above 99% and is much better than that of the SPT rule as a benchmark. A 15-job case requires only 0.018 s, on average, to obtain an ultimate or even optimal solution. The heuristic scheduling algorithm is a more practical approach to real world applications than the integer programming model.     

Permutation flowshop problems with bi-criterion makespan and total completion time objective and position-weighted learning effects

In this paper, we study the problem of minimizing the weighted sum of makespan and total completion time in a permutation flowshop where the processing times are supposed to vary according to learning effects. The processing time of a job is a function of the sum of the logarithms of the processing times of the jobs already processed and its position in the sequence. We present heuristic algorithms, which are modified from the optimal schedules for the corresponding single machine scheduling problem and analyze their worst-case error bound. We also adopt an existing algorithm as well as a branch-and-bound algorithm for the general m-machine permutation flowshop problem. For evaluation of the performance of the algorithms, computational experiments are performed on randomly generated test problems.     

Two-machine flowshop scheduling with flexible operations and controllable processing times

We consider a two-machine flowshop scheduling problem with identical jobs. Each of these jobs has three operations, where the first operation must be performed on the first machine, the second operation must be performed on the second machine, and the third operation (named as flexible operation) can be performed on either machine but cannot be preempted. Highly flexible CNC machines are capable of performing different operations. Furthermore, the processing times on these machines can be changed easily in albeit of higher manufacturing cost by adjusting the machining parameters like the speed and/or feed rate of the machine. The overall problem is to determine the assignment of the flexible operations to the machines and processing times for each operation to minimize the total manufacturing cost and makespan simultaneously. For such a bicriteria problem, there is no unique optimum but a set of nondominated solutions. Using ?-constraint$?\mathrm{-}\mathrm{constraint}$ approach, the problem could be transformed to be minimizing total manufacturing cost for a given upper limit on the makespan. The resulting single criterion problem can be reformulated as a mixed integer nonlinear problem with a set of linear constraints. We use this formulation to optimally solve small instances of the problem while a heuristic procedure is constructed to solve larger instances in a reasonable time.     

Two-machine flowshop scheduling with intermediate transportation under job physical space consideration

In this paper, we study a coordinated production–transportation scheduling problem in a two-machine flowshop environment where a single transporter may carry several jobs simultaneously as a batch between the machines. Each job has its own physical-space requirement while being loaded into the transporter. The goal is to minimize the makespan. For the jobs with the same size of physical space during transportation, we present a heuristic algorithm with an absolute worst-case ratio of 2 and a polynomial-time optimal algorithm for a special case with given job sequence. For the jobs having different size of physical storage space, a heuristic algorithm is constructed with an absolute worst-case ratio of 7/3 and asymptotic worst-case ratio of 20/9. Computational experiments demonstrate that the two heuristic algorithms developed are capable of generating near-optimal solutions quickly.