Permutation Schedules Research Articles

Empirical Analysis of Characteristics of a Two-Machine Robotic Flowshop with Intermediate Operations.

In this paper we deal with a two-machine robotic unit of flowshop type, in which each of n jobs is processed on the first machine and later on the second machine. Transportation of the jobs in this system is performed by robots. There is an intermediate station between the two machines for intermediate operations such as washing, chip disposal, cooling, drying and/or quenching. It is already known that the scheduling problem of minimizing the maximum completion time (i.e., the makespan) can be solved in O(n2) time if only permutation schedules are allowed. Minimizing the maximum completion time leads to increasing efficiency of the system, and permutation schedules may be more practical than non-permutation ones in industry. In this paper, we analyze the characteristics of the system in terms of the optimal permutation schedule, and report the numerical results. In particular, we examine the effect of transportation times of the robots to the efficiency, and the number of work-in-processes in the intermediate station.

Approximate Scheduling Algorithms with Performance Guarantee for a Two-Machine Robotic Unit with an Intermediate Station

In this paper we consider the scheduling problem of minimizing the maximum completion time (i.e., the makespan) for a two-machine robotic unit of flowshop type, in which each of n jobs is processed on the first machine and later on the second machine. There is an intermediate station between the two machines for intermediate operations such as washing, chip disposal, cooling, drying and/or quenching. If only permutation schedules are allowed (i.e., sequences of jobs on the two machines have to be the same), the problem can be solved in polynomial time, although non-permutation problem (i.e., the original problem) is NP-hard in the strong sense. It is already known that, if the optimal permutation schedule is used as an approximate solution for the non-permutation problem, it gives a maximum completion time within twice the optimal. In this paper, we present a different approximate algorithm which is based on a relaxation to the single-machine problem with delivery times, and show that its non-permutation schedule also gives a maximum completion time within twice the optimal. Moreover, we examine its approximation performance by means of numerical experiments, comparing with the optimal permutation scheduling.

Analysis and comparison of multimodal cancer treatments

We analyse the sequence in which the three most commonly prescribed cancer treatments--surgery (S), chemotherapy (C) and radiotherapy (R)--should be administered. A system of ordinary differential equations is formulated that captures the various local and systemic effects of the three modes of treatment, as well as the first-order effects of the inter-relationship between the primary tumour and the distant metastatic tumours, including primary tumour shedding and the primary tumour's effect on the rate of angiogenesis in the metastatic tumours. Under a set of stated assumptions on the parameter values, we find the exact cancer cure probability (subject to toxicity constraints) for the six permutation schedules (i.e. SCR, CSR, CRS, SRC, RSC, RCS) and for two novel schedules, SRCR and RSCR, that apply radiotherapy in disjoint, optimally timed portions. We show analytically that SRCR and RSCR are the two best-performing (i.e. highest cure probability) schedules among the eight considered. Further, SRCR is shown to be optimal among all possible schedules, provided a modest condition is satisfied on the delay of initial angiogenesis experienced by the patient's dormant tumours.

Analysis and comparison of multimodal cancer treatments

We analyse the sequence in which the three most commonly prescribed cancer treatments--surgery (S), chemotherapy (C) and radiotherapy (R)--should be administered. A system of ordinary differential equations is formulated that captures the various local and systemic effects of the three modes of treatment, as well as the first-order effects of the inter-relationship between the primary tumour and the distant metastatic tumours, including primary tumour shedding and the primary tumour's effect on the rate of angiogenesis in the metastatic tumours. Under a set of stated assumptions on the parameter values, we find the exact cancer cure probability (subject to toxicity constraints) for the six permutation schedules (i.e. SCR, CSR, CRS, SRC, RSC, RCS) and for two novel schedules, SRCR and RSCR, that apply radiotherapy in disjoint, optimally timed portions. We show analytically that SRCR and RSCR are the two best-performing (i.e. highest cure probability) schedules among the eight considered. Further, SRCR is shown to be optimal among all possible schedules, provided a modest condition is satisfied on the delay of initial angiogenesis experienced by the patient's dormant tumours.

A performance analysis of dispatching rules and a heuristic in static flowshops with missing operations of jobs

An experimental investigation of the performance of dispatching rules and a heuristic for scheduling in static flowshops with missing operations is undertaken in this study. The measure of performance is the minimization of total flow time of jobs. Permutation schedules are generated by using the heuristic for scheduling. General schedules, which can be permutation or non-permutation schedules, are obtained by using dispatching rules. Four dispatching rules, including a new dispatching rule, are considered. Two types of flowshops are studied: one with no missing operations of jobs and another with missing operations of jobs. In the latter type of flowshops, jobs with varying number of missing operations are considered. An extensive investigation of the performance of the dispatching rules and the heuristic is carried out. It is observed that the heuristic minimizes total flow time of jobs more than dispatching rules up to a certain level of missing operations of jobs in flowshops, after which dispatching rules perform better. The performance of the heuristic and the dispatching rules in terms of minimizing the makespan as a secondary measure is also reported.

Scheduling a flowline manufacturing cell with sequence dependent family setup times

This paper considers the problem of scheduling part families and jobs within each part family in a flowline manufacturing cell where the setup times for each family are sequence dependent and it is desired to minimize the makespan while processing parts (jobs) in each family together. Lower bounds on the optimal makespan value and efficient heuristic algorithms for finding permutation schedules are proposed and empirically evaluated as to their effectiveness in finding optimal permutation schedules.

On the Asymptotic Optimality of the SPT Rule for the Flow Shop Average Completion Time Problem

Consider a flow shop with M machines in series, through which a set of jobs are to be processed. All jobs have the same routing, and they have to be processed in the same order on each of the machines. The objective is to determine such an order of the jobs, often referred to as a permutation schedule, so as to minimize the total completion time of all jobs on the final machine. We show that when the processing times are statistically exchangeable across machines and independent across jobs, the Shortest ProcessingTime first (SPT) scheduling rule, based on the total service requirement of each job on all M machines, is asymptotically optimal as the total number of jobs goes to infinity. This extends a recent result of Kaminsky and Simchi-Levi (1996), in which a crucial assumption is that the processing times on all M machines for all jobs must be i.i.d.. Our work provides an alternative proof using martingales, which can also be carried out directly to show the asymptotic optimality of the weighted SPT rule for the Flow Shop Weighted Completion Time Problem.

The two-machine stochastic flowshop problem with arbitrary processing time distributions

We treat the two-machine flowshop problem with the objective of minimizing the expected makespan when the jobs possess stochastic durations of arbitrary distributions. We make three contributions in this paper: (1) we propose an exact approach with exponential worst-case time complexity. We also propose approximations which are computationally modest in their requirements. Experimental results indicate that our procedure is within less than 1 % of the optimum; and (2) we provide a more elementary proof of the bounds on the project completion time based on the concepts of ‘control networks’; and (3) we extend the ‘reverse search’ procedure of Avis and Fukuda [1[ to the context of permutation schedules.

Off-line permutation embedding and scheduling in multiplexed optical networks with regular topologies

There are two basic approaches for establishing a connection in a reconfigurable switched optical network, whose links are multipiexed with virtual channels (e.g., wavelengths or time slots). One is called path multiplexing (PM), in which the same virtual channel has to be used on each link along a path, and the other is link multiplexing (LM), in which different virtual channels may be used. We focus on the problem of off-line permutation embedding and scheduling as a part of the comparative study of the multiplexing approaches. Specifically, we determine the minimum number of virtual channels per link needed for a given network to be rearrangeably nonblocking in PM and LM, respectively. We also examine the schedule length of a permutation in PM and LM when the network is blocking as a result of having an insufficient number of virtual channels per link. We found that PM and LM are equally effective in linear arrays, and LM is slightly more effective than PM in rings, meshes (grids), tori, and hypercubes.

Minimizing total weighted completion time in a proportionate flow shop

We study the special case of the m machine flow shop problem in which the processing time of each operation of job j is equal to pj; this variant of the flow shop problem is known as the proportionate flow shop problem. We show that for any number of machines and for any regular performance criterion we can restrict our search for an optimal schedule to permutation schedules. Moreover, we show that the problem of minimizing total weighted completion time is solvable in O(n2) time. © 1998 John Wiley & Sons, Ltd.

A new constructive heuristic for the flowshop scheduling problem

This paper presents a simple constructive heuristic (HFC) for the flowshop makespan problem which is capable of producing non-permutation schedules when it deems it appropriate. HFC determines the order of any two jobs in the final schedule based on their order in all two-machine problems embedded in the problem. Computational experiments indicate that HFC performs as well as NEH which is the currently best available constructive heuristic on problems where a permutation schedule is expected to be optimal. However, HFC outperforms NEH on problems where a non-permutation schedule may be optimal.

A flowshop scheduling problem with two operations per job

This paper considers a special case of the flowshop scheduling problem where each job requires only two operations on specified machines and shows that this problem is NP-hard in the strong sense even if the first operation of all jobs is processed on the same machine and the number of machines performing the second operation equals two. For the case when the first operation of all jobs is performed on the same machine, it is sufficient to consider only permutation schedules for minimizing any regular measure of performance. Five polynomially bounded heuristic algorithms are described for minimizing makespan for this case and their performance in finding a minimum makespan schedule is theoretically and empirically evaluated.

A variant of the permutation flow shop model with variable processing times

We consider a modification of the classical flow shop problem with infinite buffer capacities. The processing times on various machines are variable: the more one delays the beginning of an operation, the longer is its processing time. There are many industrial applications for this model, for instance in the planning of machine maintenance or service and in steel production where the material will cool during the waiting periods and has to be reheated for the subsequent process. The problem turns out to be already NP-hard in the strong sense for two processors, and in general the optimal schedule is not a permutation schedule. In the literature, a restricted problem is considered where one wants to determine the optimal release times for the jobs, once their order is given. We shall solve this m-machine flow shop problem by a greedy placement algorithm. For the two-machine case, where also the optimal order of the jobs has to be determined, several approximate solution methods are analyzed.

Variants of the Two Machine Flow Shop Problem connected with factorization of matrix functions

In this paper we consider a number of variants of the Two Machine Flow Shop Problem. In these variants the makespan is given and the problem is to find a schedule that meets this makespan, thereby minimizing the infeasibilities of the jobs in a prescribed sense: In the max-variant the maximum infeasibility of the jobs is to be minimized, whereas in the sum-variant the objective is to minimize the sum of the infeasibilities of the jobs. For both variants observations about the structure of the optimal schedules are presented. In particular, it is proved that every instance of these problems has an optimal permutation schedule. It is also shown that the max-variant can be solved by Johnson's Rule. For the sum-variant this is not the case: For solving this problem to optimality something quite different is necessary. Both variants are connected with factorization problems for certain rational matrix functions. The factorizations involved are optimal in some sense and generalize the notion of complete factorization. In this way a connection is established between job scheduling theory on one hand, and mathematical systems theory on the other.

Completion times in multipurpose batch plants with set-up, transfer and clean-up times

Once, the number of batches, batch sizes, and production paths have been determined, and the tasks performed in the units have been sequenced, the determination of completion times is the last step in the scheduling of operations. In this paper, we deal with this problem considering that non permutation schedules may occur in multipurpose batch plants, once the task sequence in the units and the set-up, load, operation, unload and clean-up times are given. Finite wait (FW) interstage policy is assumed, meaning that intermediates may wait temporarily in discontinuous units if the waiting time does not exceed the maximum stability time for the intermediate, and assuming that no intermediate storage is available. A recursive procedure is presented to determine the start times and the duration of the possible waiting times in the discontinuous units. Then completion times are calculated adding the subtasks duration to the start times.

Environmental considerations in batch production scheduling

Increasingly strict environment regulations stimulate the adoption of clean technologies and materials recycling in the process industry. The emphasis is now on improving process efficiency, minimising effluents and optimising waste treatment and disposal practices to ensure compliance with discharge and disposal legislation. Some of the waste is consequence of process inefficiency, as occurs with part of the product changeover waste. This paper refers to the product changeover waste which is processing-sequence dependent. A procedure for reducing the amount of this waste is proposed. It is based on the generation of non permutation schedules, where the order in which the products are processed may be different in all the recipe stages, and passing of products between stages is allowed.

The Two-Stage Assembly Scheduling Problem: Complexity and Approximation

This paper introduces a new two-stage assembly scheduling problem. There are m machines at the first stage, each of which produces a component of a job. When all m components are available, a single assembly machine at the second stage completes the job. The objective is to schedule jobs on the machines so that the makespan is minimized. We show that the search for an optimal solution may be restricted to permutation schedules. The problem is proved to be NP-hard in the strong sense even when m = 2. A schedule associated with an arbitrary permutation of jobs is shown to provide a worst-case ratio bound of two, and a heuristic with a worst-case ratio bound of 2 − 1/m is presented. The compact vector summation technique is applied for finding approximation solutions with worst-case absolute performance guarantees.

Flowshop scheduling with dominant machines

This paper examines two special cases of the n-job, m-machine permutation scheduling flowshop problem. The first case assumes an increasing series of dominating machines; while the second case assumes a decreasing series of dominating machines. Efficient solution procedures for finding the optimal permutation schedules for various performance measures, including maximum flowtime, mean flowtime, mean completion time of machines, the number of tardy jobs, maximum lateness, and maximum tardiness, are developed.

Minimum tardiness scheduling in flow shops: Construction and evaluation of alternative solution approaches

Abstract Surveys of industrial scheduling practice show that meeting customer due dates is a critical concern for many manufacturing systems. While there is considerable research on the effectiveness of scheduling rules in job shops, very little work is reported on flow shops. Although scheduling rules developed for job shops can be applied in other systems as well, we show that the inherent structure of a flow shop can be utilized to construct effective solution procedures. In particular, for the total tardiness problem, we develop conditions for local optimality in a 2-machine flow shop, and use these results to generate an efficient improvement heuristic procedure. We also construct solution methods that utilize the notion of shifting bottlenecks. Our computational experience reveals the superiority and robustness of the proposed approaches under a variety of problem scenarios. The results of this study also indicate the need to look beyond permutation schedules for solving the flow shop tardiness problem. Furthermore, they suggest that the criterion for determining machine criticality needs to take the scheduling objective into consideration. While it appears intuitive to impute criticality on the basis of machine workloads and, therefore, emphasize the bottleneck machine for generating the shop schedule, this approach may frequently result in inferior system performance.

An efficient heuristic based on machine workload for the flowshop scheduling problem with setup and removal

We are concerned in this paper with solving ann jobs,M machines flowshop scheduling problem where the objective function is the minimization of the makespan. We take into account setup, processing and removal times separately. After drawing up a synthesis of existing work which addresses this type of problems, we propose a new heuristic algorithm which is based on the machine workload to find an efficient permutation schedule. Computational experiences show that our algorithm yields excellent results, particularly when bottleneck machines are present.