Open Shop Scheduling Research Articles

Two- and three-machine open shop scheduling using LAPT-like rules

In the paper, open shop scheduling problems with the objective to minimize the makespan and fixed or proportionally deteriorating operation processing times are considered. We solve the problems using three rules which are modifications of the Longest Alternate Processing Time (LAPT) rule that earlier was proved to be optimal for two-machine open shop scheduling problem with the same objective and fixed operation processing times. We prove that the first of the modified rules is optimal for two-machine open shop scheduling problem with proportionally deteriorating operation processing times. We also show, based on results of conducted numerical experiments, that the next two modified rules generate near-optimal schedules for three-machine open shop problems, regardless whether operation processing times are fixed or proportionally deteriorating.

A complexity analysis and algorithms for two-machine shop scheduling problems under linear constraints

We study several two-machine shop scheduling problems, namely flow shop, job shop and open shop scheduling problems under linear constraints. In these problems, the processing times of two stages of jobs are also decision variables and satisfy a system of linear constraints. The goal of each problem is to determine the processing time of each job, and to schedule the jobs to the shop machine such that the makespan, i.e., the completion time of all jobs, is minimized. These problems can find application in various areas, such as industrial production, advertising and automotive maintenance. We study the computational complexity and propose polynomial-time optimal or approximation algorithms for them. In particular, we show that although a two-machine flow shop scheduling problem and a two-machine job shop scheduling problem without recirculation can be solved in polynomial time, the problems where processing times satisfy linear constraints are generally NP-hard in the strong sense. Then, we design algorithms for various settings of these problems. We design polynomial-time algorithms for them when there are a fixed number of constraints. For the general case, we first propose a simple 2-approximation algorithm, and then design a polynomial-time approximation schemes. In contrast to the previous two problems, we show that the two-machine open shop scheduling problem under linear constraints can be solved in polynomial time.

Open shop scheduling games

This paper takes a game theoretical approach to open shop scheduling problems to minimize the sum of completion times. We assume that there is an initial schedule to process the jobs (consisting of a number of operations) on the machines and that each job is owned by a different player. Thus, we can associate a cooperative TU-game to any open shop scheduling problem, assigning to each coalition the maximal cost savings it can obtain through admissible rearrangements of jobs’ operations. A number of different approaches to admissible schedules for a coalition are introduced and, in the main result of the paper, a core allocation rule is provided for games arising from unit (execution times and weights) open shop scheduling problems for the most of these approaches. To sharpen the bounds of the set of open shop scheduling problems that result in games that are balanced, we provide two counterexamples: one for general open shop problems and another for further relaxations of the definition of admissible rearrangements for a coalition.

Three-machine open shop with a bottleneck machine revisited

The paper considers the three-machine open shop scheduling problem to minimize the makespan. In the model, each job consists of two operations, one of which is to be processed on the bottleneck machine, which is the same for all jobs. A new linear-time algorithm to find an optimal non-preemptive schedule is developed. The suggested algorithm considerably simplifies the only previously known method as it straightforwardly exploits the structure of the problem and its key components to yield an optimal solution.

Optimization for cooperative task planning of heterogeneous multi-robot systems in an order picking warehouse

This article investigates the cooperative task planning problem of heterogeneous multi-robot systems in an order picking warehouse, with challenging characteristics such as heterogeneous agents, interrelated utilities and complex intertask dependencies. These characteristics bring great challenges for modelling and optimization of the problem. This study establishes a novel mapping mechanism to describe the problem as open shop scheduling with sequence-dependent set-up and transportation times, and develops an efficient mixed-integer linear programming model to obtain the optimal task planning scheme for small-size problems. To increase the size of solvable problems and improve the performance of the solution, an effective hybrid artificial bee colony algorithm is proposed, which introduces some concepts from the tabu search algorithm and the artificial immune algorithm. The effectiveness of the proposed method is confirmed by simulation experiments.

Complex scheduling network: an objective performance testing platform for evaluating vital nodes identification algorithms

With the widespread application of complex network theory, identifying vital nodes is an important part of complex network analysis, which has been a key issue in analyzing the characteristics of network structure and functions. Although many centrality measures have been proposed to identify vital nodes, there is still a lack of an objective performance testing platform for evaluating vital nodes identification algorithms. This study introduces the complex scheduling network as an objective performance testing platform where the identification of vital nodes directly determines the decision-making in each decision scenario through the development of heuristic algorithms upon centrality measures, and therefore, the optimization goals of scheduling problems can be regarded as an objective index for evaluating vital node identification algorithms. Finally, the undirected network obtained by the open shop scheduling problem is taken as an example to analyze and evaluate the performance of various vital node identification algorithms.

An improved bat optimization algorithm to solve the tasks scheduling problem in open shop

The open shop scheduling problem involves a set of activities that should be run on a limited set of machines. The purpose of scheduling open shops problem is to provide a timetable for implementation of the entire operation so that the total execution time is reduced. The tasks scheduling problem in open shops is important in many applications due to the arbitrariness of the processing sequence of each job and lack of a prioritization of its operations. This is an NP-hard problem and obtaining an optimal solution to this problem requires a high time complexity. Therefore, heuristic techniques are used to solve these problems. In this paper, we investigate the tasks scheduling problem in open shops using the Bat Algorithm (BA) based on ColReuse and substitution meta-heuristic functions. The heuristic functions are designed to increase the rate of convergence to the optimal solution. To evaluate the performance of the proposed algorithm, standard open shop benchmarks were used. The results obtained in each benchmark are compared with those of the previous methods. Finally, after analyzing the results, it was found that the proposed BA had a better performance and was able to generate the best solution in all cases.

Four Decades of Research on the Open-Shop Scheduling Problem to Minimize the Make-span

One of the basic scheduling problems, the open-shop scheduling problem has a broad range of applications across different sectors. The problem concerns scheduling a set of jobs, each of which has a set of operations, on a set of different machines. Each machine can process at most one operation at a time and the job processing order on the machines is immaterial, i.e., it has no implication for the scheduling outcome. The aim is to determine a schedule, i.e., the completion times of the operations processed on the machines, such that a performance criterion is optimized. While research on the problem dates back to the 1970s, there have been reviving interests in the computational complexity of variants of the problem and solution methodologies in the past few years. Aiming to provide a complete road map for future research on the open-shop scheduling problem, we present an up-to-date and comprehensive review of studies on the problem that focuses on minimizing the make-span, and discuss potential research opportunities.

A Stochastic Model for Prioritized Outpatient Scheduling in a Radiology Center

This paper discussed the scheduling problem of outpatients in a radiology center with an emphasis on priority. To more compatibility to real-world conditions, we assume that the elapsed times in different stages to be uncertain that follow from the specific distribution function. The objective is to minimize outpatients’ total spent time in a radiology center. The problem is formulated as a flexible open shop scheduling problem and a stochastic programming model. By considering the specific distribution function for uncertain variables, deterministic mixed integer linear programming (MILP) is developed such that the proposed problem can be solved by a linear programming solver in small size. Besides an effective heuristic method is proposed for the moderate size problem. To indicate the applicability of the proposed model, it has been applied to a real radiology center. The results from the proposed optimization models indicate an increase in outpatients’ satisfaction, as well as the improving of the efficiency and productivity of the radiology center.

Bi-Objective Dynamic Multiprocessor Open Shop Scheduling: An Exact Algorithm

An important element in the integration of the fourth industrial revolution is the development of efficient algorithms to deal with dynamic scheduling problems. In dynamic scheduling, jobs can be admitted during the execution of a given schedule, which necessitates appropriately planned rescheduling decisions for maintaining a high level of performance. In this paper, a dynamic case of the multiprocessor open shop scheduling problem is addressed. This problem appears in different contexts, particularly those involving diagnostic operations in maintenance and health care industries. Two objectives are considered simultaneously—the minimization of the makespan and the minimization of the mean weighted flow time. The former objective aims to sustain efficient utilization of the available resources, while the latter objective helps in maintaining a high customer satisfaction level. An exact algorithm is presented for generating optimal Pareto front solutions. Despite the fact that the studied problem is NP-hard for both objectives, the presented algorithm can be used to solve small instances. This is demonstrated through computational experiments on a testbed of 30 randomly generated instances. The presented algorithm can also be used to generate approximate Pareto front solutions in case computational time needed to find proven optimal solutions for generated sub-problems is found to be excessive. Furthermore, computational results are used to investigate the characteristics of the optimal Pareto front of the studied problem. Accordingly, some insights for future metaheuristic developments are drawn.

A self-tuning variable neighborhood search algorithm and an effective decoding scheme for open shop scheduling problems with travel/setup times

In this paper, we study Open Shop Scheduling Problems (OSSPs) that involve (1) travel times between machines and/or (2) sequence-dependent setup times. First, we propose a new decoding scheme on the well-known permutation list representation and study its properties. Second, we describe an effective Variable Neighborhood Search (VNS) algorithm which incorporates the proposed decoding scheme and that uses a self-tuning routine to set its most important parameter. Last, we tested the performance of the algorithm on several sets of instances: the first two sets consisted of classical instances of OSSPs extended with randomly generated both travel times and anticipatory sequence-dependent setup times. The third set of problems were instances of OSSPs with travel times previously presented in the literature. The last set of problems consisted of classical OSSP of the literature and was used mainly to corroborate our results. The solutions of the proposed VNS were compared with the solutions of constraint programming (CP) algorithms, previous solutions and with the optimal solutions where available. The results revealed three important things: First, the decoding strategy was the factor that had the greatest influence on the performance of the VNS algorithm. Second, the proposed self-tuning VNS algorithm was robust and very easy to adapt to a variety of OSSPs. Third, the algorithm exhibited consistent and very competitive performance in terms of computer time and solution quality in all sets of instances.

Scatter search with path relinking for multiprocessor open shop scheduling

Maintenance and health care diagnostic systems are generally composed of different workstations pertaining to technologically different processes. A workstation is composed of one or more parallel machines. In such systems, the multiprocessor open shop scheduling problem is commonly encountered. It is concerned with assigning processing intervals for each job on machines that need to be selected in each requested workstation. Meanwhile, jobs do not require a specific order for visiting workstations. This paper considers a static, deterministic version of the problem in which jobs do not have to visit all workstations, the workstations do not necessarily have identical machines, and the processing times depend on both the job and the machine. The objective is to minimize the maximum completion time (makespan) which is commensurate with maximizing the utilization of the available machines. To the best of our knowledge, this problem structure has not been considered in the literature before despite its existence in real-life applications. Since it is NP-hard problem, efficient heuristics are needed to generate near optimal solutions in practically acceptable computational times. In this paper, two neighborhood search functions and two solution combination functions are developed and used within a scatter search with path relinking metaheuristic, along with a new distance definition between solutions. Computational experiments are conducted first to select the best levels of the metaheuristic parameters. Then, computational experiments are conducted on specially designed instances that take into consideration different settings of the studied problem. This is followed by computational experiments on a set of benchmark instances of the proportionate multiprocessor open shop scheduling problem which is a special case of the studied problem for which other metaheuristics have been developed in the literature. Results show that the developed metaheuristic is capable of generating optimal or near-optimal solutions for different configurations of the studied problem. In addition, it generates competitive results for the proportionate case compared to the available metaheuristics with 18 new upper bounds; among them seven are optimal.

Two-stage open-shop scheduling with a two-machine flow shop as a stage: approximation algorithms and empirical experiments

We study a scheduling environment that finds many real-world manufacturing applications, in which there is a close connection between a hybrid multiprocessor open shop and multiple parallel identical flow shops. In this environment, there is an extended two-stage open shop, where in one stage we have a set of parallel identical machines, while in the other we have a two-machine flow shop. Our objective is to minimize the makespan, that is, the latest completion time of all jobs. We pursue approximation algorithms with provable performance, and we achieve a 2-approximation when the number of parallel identical machines is constant or is part of the input; we also design a 5/3-approximation for the special case where there is only one machine in the multiprocessor stage, which remains weakly NP-hard. Our empirical experiments show that both approximation algorithms perform much better in simulated instances; their average ratios over the proposed lower bound are around 1.5 and 1.2, respectively.

Improved Scheduling for the Three-Machine Proportionate Open Shop and Mixed Shop Minimum Makespan Problems

Open shop scheduling problems have many practical applications, in which the jobs can be in any order with the only restriction that their durations do not overlap with each other. In some cases, the order of a subset of jobs dictates the flow shop model while the remaining jobs can be processed in any order according to the open shop model, and the corresponding problem is called mixed shop scheduling. For every job, if its processing times on all machines are the same, then the shop is called proportionate shop. In this article we investigate the three-machine proportionate open shop and mixed shop minimum makespan problems proposed by Koulamas and Kyparisis. Our result improves the previous work in three ways. First, for both the open shop and the mixed shop scheduling problems, we derive additional sufficient conditions under which the corresponding problems are solvable in polynomial time. Second, for the non-solvable cases of the open shop scheduling problem we present an improved 13/12-approximation algorithm. Third, for the non-solvable cases of the mixed shop scheduling problem we present an improved approximation algorithm with the worst-case ratio bound of (7/6 + ϵ). All the algorithms proposed in this article run in polynomial time.

An Efficient Relax-and-Solve Heuristic for Open-Shop Scheduling Problem to Minimize Total Weighted Earliness-Tardiness

An open-shop scheduling problem to minimize the total weighted earliness-tardiness of job-on-machine operations is studied. Each operation has its own due date, earliness weight and tardiness weight. An efficient relax-and-solve heuristic is developed. A feature that makes the proposed heuristic distinctive from the existing heuristics of this kind is that it operates by iterative relaxation and optimization of a series of sub-problems. Performance of the proposed heuristic is verified on a set of 72 benchmark instances with up to 200 operations, adapted from the literature on a similar job-shop scheduling problem. The heuristic delivers best solutions for nearly 71\% of the instances (i.e., 51 out of 72 instances), including the best solution for all instances with 150 and 200 operations. Its running time is around three minutes, while CLPEX was run for 30 minutes.

The mean flow time in open shop scheduling

PurposeThe purpose of this paper is to minimize the mean flow time in open shop scheduling problem (OSSP). The scheduling problems consist of n jobs and m machines, in which each job has O operations. The processing time for 50 OSSP was generated using a linear congruential random number.Design/methodology/approachDifferent evolutionary algorithms are used to minimize the mean flow time of OSSP. This research study used simulated annealing (SA), Discrete Firefly Algorithm and a Hybrid Firefly Algorithm with SA. These methods are referred as A1, A2 and A3, respectively.FindingsA comparison of the results obtained from the three methods shows that the Hybrid Firefly Algorithm with SA (A3) gives the best mean flow time for 76 percent instances. Also, it has been observed that as the number of jobs increases, the chances of getting better results also increased. Among the first 25 problems (i.e. job ranging from 3 to 7), A3 gave the best results for 13 instances, i.e., for 52 percent of the first 25 instances. While for the last 25 problems (i.e. Job ranging from 8 to 12), A3 gave the best results for all 25 instances, i.e. for 100 percent of the problems.Originality/valueFrom the literature it has been observed that no researchers have attempted to solve OOSPs using Firefly Algorithm (FA). In this research work an attempt has been made to apply the FA and its hybridization to solve OSSP. Also the research work carried out in this paper can also be applied for a real-time Industrial problem.

Bounded colouring motivated by the limited resource partially concurrent open shop problem

Partially Concurrent Open Shop Scheduling (PCOSS) is a relaxation of the well-known Open Shop Scheduling (OSS), where some of the operations that refer to the same job may be processed concurrently. Here we extend the study of the PCOSS model by considering an additional resource, which is limited to several units. We deal with the case of preemption PCOSS, where a few polynomial algorithms are known for its OSS counterpart. The scheduling problem is equivalent to the problem of conflict graph colouring. The restriction on the number of resource units bounds the size of colour classes. We thus study the problem of bounded vertex colouring. A new bound for the makespan is presented, and it is shown that for a 2-bounded colouring the bound is attainable. Some insight about a special case of PCOSS, composed of identical jobs, is given. The model correlates to a real-life timetabling project of assigning technicians to vehicles in a garage, with an additional resource, such as vehicle lift.

Open-shop scheduling for unit jobs under precedence constraints

We study open-shop scheduling for unit jobs under precedence constraints, where if one job precedes another job then it has to be finished before the other job can start to be processed. For the three-machine open-shop to minimize the makespan, we first present a simple 5/3-approximation algorithm based on a partition of the job set into agreeable layers using the natural layered representation of the precedence graph, which is directed acyclic. We then show a greedy algorithm to reduce the number of singleton-job layers, resulting in an improved partition, which leads to a 4/3-approximation algorithm. Both approximation algorithms apply to the general m-machine open-shops too.

A genetic algorithm for scheduling open shops with sequence-dependent setup times

In the open shop scheduling problem with sequence-dependent setup times, there is no established order for the processing of the jobs, which leads to a large number of possible solutions for the scheduling problem. Furthermore, there is a setup time between two consecutive operations, which depends on the job previously processed. In this work we propose a hybrid genetic algorithm for the OSSP with sequence-dependent setup times and total completion time minimization as objective function. Our proposal uses two novel constructive heuristics which are combined for the generation of the initial population. We carry out an extensive computational experience using problem instances taken from the related literature to evaluate the performance of the proposed algorithms as compared to existing heuristics for the problem. The quality of the solutions and the CPU time required are used as performance criteria. Among them, the genetic algorithm with direct decoding presents a smaller value for the average relative percentage deviation in comparison with the electromagnetic algorithm proposed by Naderi et al. (2011). The computational results prove the excellent performance of the proposed metaheuristic for the tested instances, resulting in the most efficient algorithm so-far for the problem under consideration.

Dynamic programming approach for solving the open shop problem

This paper deals with the open shop scheduling problem (OSP) with makespan minimization. An exact dynamic programming algorithm is proposed for solving the OSP to optimality. This approach is applied to the OSP for the first time. Computational results show that the proposed algorithm is able to solve moderate benchmark instances.