Makespan Minimization Research Articles

Modeling and scheduling hybrid open shops for makespan minimization

PurposeThis study aims to address the hybrid open shop problem (HOSP) with respect to the minimization of the overall finishing time or makespan. In the HOSP, we have to process n jobs in stages without preemption. Each job must be processed once in every stage, there is a set of mk identical machines in stage k and the production flow is immaterial.Design/methodology/approachComputational experiments carried out on a set of randomly generated instances showed that the minimal idleness heuristic (MIH) priority rule outperforms the longest processing time (LPT) rule proposed in the literature and the other proposed constructive methods on most instances.FindingsThe proposed mathematical model outperformed the existing model in the literature with respect to computing time, for small-sized instances, and solution quality within a time limit, for medium- and large-sized instances. The authors’ hybrid iterated local search (ILS) improved the solutions of the MIH rule, drastically outperforming the models on large-sized instances with respect to solution quality.Originality/valueThe authors formalize the HOSP, as well as argue its NP-hardness, and propose a mixed integer linear programming model to solve it. The authors propose several priority rules – constructive heuristics based on priority measures – for finding feasible solutions for the problem, consisting of adaptations of classical priority rules for scheduling problems. The authors also propose a hybrid ILS for improving the priority rules solutions.

Two-machine flow shop scheduling with a common due date to maximize total early work

This paper considers scheduling in a two-machine flow shop to maximize the total early work subject to a common due date. The early work of a job is a parameter defined as the total amount of the job that is processed before the specified due date. We mainly focus on the unweighted model in this paper, and propose a dynamic programming approach running in O(n2d2) time (compared with the previous result in O(n2d4) time for the weighted case discussed in the literature). Then we analyse the problem from an approximation point of view, in which we first show that Johnson’s algorithm, one of the most classical ones in flow shop scheduling, can only achieve the worst performance ratio for the considered problem (although it is an optimal one for makespan minimization). With the motivation of proposing better approximation algorithms, we further design a fully polynomial time approximation scheme (FPTAS). Finally, we point out that the approximation results also work for the weighted model - if a specific constraint is satisfied.

A new variable neighbourhood search with a constraint programming search strategy for the open shop scheduling problem with operation repetitions

This article presents a new variant for the open shop scheduling problem, the open shop scheduling problem with repetitions (OSSPR), where the jobs can be processed on any machine more than once (operation by operation). Thereby, all the jobs can be scheduled in an unconstrained way, substantially increasing the number of feasible solutions in comparison with the classical open shop. The OSSPR has many applications in automotive and maintenance actives. To solve the problem, a mixed-integer linear programming model is presented and a new constraint programming model is proposed. Since the problem under study is NP-hard, a new efficient variable neighbourhood search is proposed using variable search strategies through the proposed constraint programming model. The objective function is makespan minimization, and it uses the lower bound deviation as performance criterion. Computational results show very good performance of the proposed metaheuristic on the instances tested.

Reentrant flow shop with identical jobs and makespan criterion

The article focuses on the problem of a reentrant flow shop with identical jobs and makespan minimisation objective. After presenting an optimal integer programming formulation for the problem, we consider the case of unit processing times. We correct two results published a few decades ago regarding two greedy heuristic algorithms. In some cases, these algorithms lead to an optimal solution, but they are generally not optimal. We point out the advantages of each of the two heuristic algorithms and, by machine learning analysis, discuss features that differentiate between them.

Approximation Results for Makespan Minimization with Budgeted Uncertainty

We study approximation algorithms for the problem of minimizing the makespan on a set of machines with uncertainty on the processing times of jobs. In the model we consider, which goes back to Bertsimas et al. (Math. Program. 98(1-3), 49–71 2003), once the schedule is defined an adversary can pick a scenario where deviation is added to some of the jobs’ processing times. Given only the maximal cardinality of these jobs, and the magnitude of potential deviation for each job, the goal is to optimize the worst-case scenario. We consider both the cases of identical and unrelated machines. Our main result is an EPTAS for the case of identical machines. We also provide a 3-approximation algorithm and an inapproximability ratio of 2 − 𝜖 for the case of unrelated machines.

Optimal sequence for single server scheduling incorporating a rate-modifying activity under job-dependent linear deterioration

This study considers single processor scheduling problems incorporating a rate-modifying activity (RMA) as well as processing time deterioration rates. The RMA fully restores the processing speed of a processor. As discussed in recent literature, the optimal position of an RMA for a single processor scheduling problem with the objective of makespan minimization, 1|p[j]A=α[j]S[j],rm|Cmax, where the processing time of a job (p[j]A:[j]denotesthejthposition)is determined by its start time (S[j])and position-dependent rate (α[j]), can be found easily. This study considers six variants of the problem: (P1) 1|p[j]A=p¯+α[j]S[j],rm|Cmax, (P1+)1|p[j]A=p¯+α[j]S[j],drm|Cmax, (P2) 1|p[i,j]A=pj+α[i]Sj,rm|Cmax, (P2+)1|p[i,j]A=pj+α[i]Sj,drm|Cmax, (P3) 1|pjA=αjSj,rm|Cmax, and (P3+)1|pjA=αjSj,drm|Cmax, where p¯ is the common basic processing time, pjis the basic processing time for job j, [i,j] is the job j scheduled at the ith position, and drm is the deteriorating RMA. We show that (P1) and (P1+), where the common basic processing time is considered, can be converted into the problem 1|p[j]A=α[j]S[j],rm|Cmax, so that the optimal position of the RMA can be found easily. In the other variants, the optimal sequences of jobs and positions of the RMA must be determined. In (P2) and (P2+), the basic processing time for each job (pj)is considered, whereas, in (P3) and (P3+), the deterioration rate depends not on the position of the job (α[j]) but on the job itself (αj). We show that the optimal schedules for (P2), (P2+) and (P3), (P3+) can be found by solving the assignment problems and the knapsack problems, respectively.

A Comparative Analysis of Heuristic Metaheuristic and Exact Approach to Minimize Make Span of Permutation Flow Shop Scheduling

The study of permutation flow shop scheduling issues has proved fascinating. Make span reduction has been explored extensively throughout the years as one of the many factors. If there are more than two machines, finding a schedule that optimizes total completion times for Permutation Flow Shop issues is NP Hard. The well-known Nawaz–Enscore–Ham (NEH) heuristic is acknowledged as the top performing approach for the permutation flow shop scheduling issue till now, according to the make span minimization criteria. For large-scale problems, however, the NEH approach takes a long time to discover an estimated optimum solution. Many researchers have offered heuristics and meta-heuristics as the time it takes to compute grows exponentially in proportion to the size of the problem. For the permutation flow shop scheduling problem, we proposed a genetic algorithm with the criterion of minimizing make span as the criteria. The method's performance is compared to that of the well-known NEH algorithm. Computational experiments on benchmark data sets revealed that the proposed approach produces satisfactory solutions in short computational cycles. Furthermore, it just requires a few user-defined parameters, making it relevant to real-world flow shop scheduling challenges. The findings reveal that, when compared to an exact technique, the proposed genetic algorithm (GA) is capable of rapidly seeking optimal solutions for most small-sized instances. Furthermore, the proposed GA continues to function satisfactorily for medium and large-scale instances and generates solutions in a reasonable period.

A new efficient biased random key genetic algorithm for open shop scheduling with routing by capacitated single vehicle and makespan minimization

Over the last years, researchers have been paying particular attention to scheduling problems integrating production environments and distribution systems to adopt more realistic assumptions. This paper aims to present a new biased random key genetic algorithm with an iterated greedy local search procedure (BRKGA-IG) for open shop scheduling with routing by capacitated vehicles. We propose approximation and exact algorithms to obtain high-quality solutions in acceptable computational times. This paper presents a new integer linear programming model. The proposed integer model has not been addressed in the revised literature. The objective function adopted is makespan minimization, and we use the relative deviation as performance criteria. BRKGA-IG has a new decoding scheme for OSSP-VRP solutions, an intensive exploitation mechanism with an iterated greedy local search procedure, and a restart mechanism to reduce premature population convergence. With these new mechanisms, the extensive computational experience carried out shows that the proposed metaheuristic BRKGA-IG is promising in solving large-sized instances for the new proposed problem, outperforming all other tested methods.

Exact and heuristic procedures for the Heijunka-flow shop scheduling problem with minimum makespan and job replicas

In this paper, a new problem of job sequences in a workshop is presented, taking into account non-unit demands for the jobs and whose objective is to minimize the total completion time for all the jobs ({C}_{max}) satisfying a set of restrictions imposed on the problem to preserve the production mix. Two procedures are proposed to solve the new problem: Mixed Integer Linear Programming and a Metaheuristic based on Multistart and Local Search. The two proposed procedures are tested using instance set Nissan-9Eng.I, in both cases giving rise to highly satisfactory performance both in quality of solutions obtained and in the CPU times required. Through a case study of the Nissan engine manufacturing plant in Barcelona, our economic-productive analysis reveals that it is possible to save an average of € 1162.83 per day, manufacturing 270 engines, when we transform the current assembly line into a Heijunka-Flow Shop.

Accelerated Cuckoo Search With Extended Diversification and Intensification

Metaheuristics have been great to solve NP-hard class problems in the deterministic time, but due to so many parameter settings, they lack in generality (i.e., not easy to implement on all types of problems) and also lack in global search. But the cuckoo search (CS) algorithm has only one parameter as input and also has a good reachable probability to global solution due to Levy flight. But this algorithm lacks self-adaptive parameters and extended strategies. In this paper, a deep study and improvement of cuckoo search performance has been done by introducing self-adaptive step size, extended alien egg discovery replacement (on each dimension with the use of good neighbor study), and adaptive discovery probability, and it has been named accelerated cuckoo search (ACS). Then this ACS has been utilized as an example in the load balancing problem in cloud with minimum makespan time as an objective parameter to evaluate the performance of ACS over CS. Furthermore, to validate ACS superiority over CS in all problems, these have been successfully compared on a few benchmark functions.

A hybrid Jaya algorithm for solving flexible job shop scheduling problem considering multiple critical paths

As an extension of the classical job shop scheduling problem, flexible job shop scheduling problem (FJSP) is considered as a challenge in manufacturing systems for its complexity and flexibility. Meta-heuristic algorithms are shown effective in solving FJSP. However, the multiple critical paths issue, which has not been formally discussed in the existing literature, is discovered to be a primary obstacle for further optimization by meta-heuristics. In this paper, a hybrid Jaya algorithm integrated with Tabu search is proposed to solve FJSP for makespan minimization. Two Jaya operators are designed to improve solutions under a two-vector encoding scheme. During the local search phase, three approaches are proposed to deal with multiple critical paths and have been evaluated by experimental study and qualitative analyses. An incremental parameter setting strategy and a makespan estimation method are employed to speed up the searching process. The proposed algorithm is compared with several state-of-the-art algorithms on three well-known FJSP benchmark sets. Extensive experimental results suggest its superiority in both optimality and stability. Additionally, a real world scheduling problem, including six instances with different scales, is applied to further prove its ability in handling large-scale scheduling problems.

Reactive Procedure for Robust Project Scheduling under the Activity Disruptions

The baseline schedule may become infeasible when disruptions happen during the execution of project. This will lead to project delays and financial losses. In this paper, we developed a robust reactive optimization model. Our objective is to maximize the robustness of reactive schedule. Moreover, a robust reactive procedure is proposed for baseline schedule under a series of disruptions caused by activity duration. To test the proposed approach, sets of J60 and J90 instances from the Project Scheduling Library (PSPLIB) were used. Meanwhile, we tested the reactive project scheduling with the aim of minimizing makespan and compared the maximal robustness reactive schedule and the minimal makespan reactive schedule. We developed a reactive procedure involving genetic algorithm for reactive scheduling due to its non-deterministic polynomial hard (NP-hard). The results indicated that robust reactive scheduling has the advantage in maintaining the schedule’s ability of tolerating variabilities and reducing the impact of activity disruptions on the makespan. The research results can help project managers develop a robust schedule which can tolerate variabilities in a random environment.

Scheduling in the Random-Order Model

Makespan minimization on identical machines is a fundamental problem in online scheduling. The goal is to assign a sequence of jobs to m identical parallel machines so as to minimize the maximum completion time of any job. Already in the 1960s, Graham showed that Greedy is (2-1/m)-competitive. The best deterministic online algorithm currently known achieves a competitive ratio of 1.9201. No deterministic online strategy can obtain a competitiveness smaller than 1.88. In this paper, we study online makespan minimization in the popular random-order model, where the jobs of a given input arrive as a random permutation. It is known that Greedy does not attain a competitive factor asymptotically smaller than 2 in this setting. We present the first improved performance guarantees. Specifically, we develop a deterministic online algorithm that achieves a competitive ratio of 1.8478. The result relies on a new analysis approach. We identify a set of properties that a random permutation of the input jobs satisfies with high probability. Then we conduct a worst-case analysis of our algorithm, for the respective class of permutations. The analysis implies that the stated competitiveness holds not only in expectation but with high probability. Moreover, it provides mathematical evidence that job sequences leading to higher performance ratios are extremely rare, pathological inputs. We complement the results by lower bounds, for the random-order model. We show that no deterministic online algorithm can achieve a competitive ratio smaller than 4/3. Moreover, no deterministic online algorithm can attain a competitiveness smaller than 3/2 with high probability.

Scheduling Storage Process of Shuttle-Based Storage and Retrieval Systems Based on Reinforcement Learning

The Shuttle-Based Storage and Retrieval System (SBS/RS) has been widely studied because it is currently the most efficient automated warehousing system. Most of the related existing studies are focused on the prediction and improvement of the efficiency of such a system at the design stage. Hence, the control of existing SBS/RSs has been rarely investigated. In existing SBS/RSs, some empirical rules, such as storing loads column by column, are used to control or schedule the storage process. The question is whether or not the control of the storage process in an existing system can be improved further by using a different approach. The storage process is controlled to minimize the makespan of storing a series of loads into racks. Empirical storage rules are easy to control, but they do not reach the minimum makespan. In this study, the performance of a control system that uses reinforcement learning to schedule the storage process of an SBS/RS with fixed configurations is evaluated. Specifically, a reinforcement learning algorithm called the actor-critic algorithm is used. This algorithm is made up of two neural networks and is effective in making decisions and updating itself. It can also reduce the makespan relative to the existing empirical rules used to improve system performance. Experiment results show that in an SBS/RS comprising six columns and six tiers and featuring a storage capacity of 72 loads, the actor-critic algorithm can reduce the makespan by 6.67% relative to the column-by-column storage rule. The proposed algorithm also reduces the makespan by more than 30% when the number of loads being stored is in the range of 7-45, which is equal to 9.7%-62.5% of the systems' storage capacity.

Adaptation and parameters studies of CS algorithm for flow shop scheduling problem

Scheduling concerns the allocation of limited resources overtime to perform tasks to fulfill certain criterion and optimize one or several objective functions. One of the most popular models in scheduling theory is that of the flow-shop scheduling. During the last 40 years, the permutation flow-shop sequencing problem with the objective of makespan minimization has held the attraction of many researchers. This problem characterized as Fm/prmu/Cmax in the notation of Graham, involves the determination of the order of processing of n jobs on m machines. In addition, there was evidence that m-machine permutation flow-shop scheduling problem (PFSP) is strongly NP-hard for m ≥3. Due to this NP-hardness, many heuristic approaches have been proposed, this work falls within the framework of the scientific research, whose purpose is to study Cuckoo search algorithm. Also, the objective of this study is to adapt the cuckoo algorithm to a generalized permutation flow-shop problem for minimizing the total completion time, so the problem is denoted as follow: Fm | | Cmax. Simulation results are judged by the total completion time and algorithm run time for each instance processed.

Makespan minimization with OR-precedence constraints

We consider a variant of the NP-hard problem of assigning jobs to machines to minimize the completion time of the last job. Usually, precedence constraints are given by a partial order on the set of jobs, and each job requires all its predecessors to be completed before it can start. In this paper, we consider a different type of precedence relation that has not been discussed as extensively and is called OR-precedence. In order for a job to start, we require that at least one of its predecessors is completed—in contrast to all its predecessors. Additionally, we assume that each job has a release date before which it must not start. We prove that a simple List Scheduling algorithm due to Graham (Bell Syst Tech J 45(9):1563–1581, 1966) has an approximation guarantee of 2 and show that obtaining an approximation factor of 4/3 - varepsilon is NP-hard. Further, we present a polynomial-time algorithm that solves the problem to optimality if preemptions are allowed. The latter result is in contrast to classical precedence constraints where the preemptive variant is already NP-hard. Our algorithm generalizes previous results for unit processing time jobs subject to OR-precedence constraints, but without release dates. The running time of our algorithm is O(n^2) for arbitrary processing times and it can be reduced to O(n) for unit processing times, where n is the number of jobs. The performance guarantees presented here match the best-known ones for special cases where classical precedence constraints and OR-precedence constraints coincide.

Scheduling of Anaesthesia Operations in Operating Rooms.

This paper considers scheduling of surgical operations across multiple operating rooms subject to the limited availability of anaesthetists. The objective is to construct a feasible operations schedule that has the minimum makespan, i.e., the completion time of all operations. We abstract the problem into a theoretical server scheduling problem and formulate it in a mathematical form by proposing an integer programming model. Due to the intractability of its computing time, we circumvent the exact approaches and develop two approximation methods. Then, the steepest descent search is adopted for improving the solutions. Computational study suggests that the proposed methods can produce quality solutions in a few seconds.

Partially Informed Depth-First Search for the Job Shop Problem

We propose a partially informed depth-first search algorithm to cope with the Job Shop Scheduling Problem with makespan minimization. The algorithm is built from the well-known P. Brucker's branch and bound algorithm. We improved the heuristic estimation of Brucker's algorithm by means of constraint propagation rules and so devised a more informed heuristic which is proved to be monotonic. We conducted an experimental study across medium and large instances. The results show that the proposed algorithm reaches optimal solutions for medium instances taking less time than branch and bound and that for large instances it reaches much better lower and upper bounds when both algorithms are given the same amount of time.

The reversibility property in a job-insertion tiebreaker for the permutational flow shop scheduling problem

The best performing approximate methods proposed for the permutational flow shop scheduling problem with makespan minimization are the well known NEH constructive heuristic and the iterated greedy algorithm. Both methods are based on the successive insertion (or reinsertion) of jobs into a partial schedule, evaluating the makespan of the resulting schedule for all insertion positions, and selecting the insertion position that presents the shortest makespan. Frequently, there are many tied insertion positions that produce such shortest makespan. Thus, a tiebreaker must be used to discern a selection among the tied insertion positions. Many tiebreakers have been proposed in the literature for this case. These tiebreakers improve the results produced by approximate methods when embedded into them. In this paper we propose two new tiebreakers that use a weighted and an unweighted approximation of the idle time increment produced by inserting the job into each tied insertion position. They were designed considering the reversibility property of the PFSSP. Our computational experiments show that the proposed tiebreakers outperform tiebreakers from the literature when evaluated within the NEH heuristic and within the iterated greedy algorithm. The iterated greedy algorithms with the proposed tiebreakers embedded are the best approximate methods so far for the permutational flow shop scheduling problem.

Solving Non-Permutation Flow Shop Scheduling Problem with Time Couplings

In this paper, a non-permutation variant of the Flow Shop Scheduling Problem with Time Couplings and makespan minimization is considered. Time couplings are defined as machine minimum and maximum idle time allowed. The problem is inspired by the concreting process encountered in industry. The mathematical model of the problem and solution graph representation are presented. Several problem properties are formulated, including the time complexity of the goal function computation and block elimination property. Three solving methods, an exact Branch and Bound algorithm, the Tabu Search metaheuristic, and a baseline Genetic Algorithm metaheuristic, are proposed. Experiments using Taillard-based problem instances are performed. Results show that, for the Tabu Search method, the neighborhood based on the proposed block property outperforms other neighborhoods and the Genetic Algorithm under the same time limit. Moreover, the Tabu Search method provided high quality solutions, with the gap to the optimal solution for the smaller instances not exceeding 2.3%.