Objective Of Minimizing Makespan Research Articles

A NOTE ON HARDNESS OF MULTIPROCESSOR SCHEDULING WITH SCHEDULING SOLUTION SPACE TREE

We study the computational complexity of the non-preemptive scheduling problem of a listof independent jobs on a set of identical parallel processors with a makespan minimizationobjective. We make a maiden attempt to explore the combinatorial structure showing theexhaustive solution space of the problem by defining the Scheduling Solution Space Tree(SSST) data structure. The properties of the SSST are formally defined and characterizedthrough our analytical results. We develop a unique technique to show the problemNP using the SSST and the Weighted Scheduling Solution Space Tree (WSSST) datastructures. We design the first non-deterministic polynomial-time algorithm named MagicScheduling (MS) for the problem based on the reduction framework. We also define anew variant of multiprocessor scheduling by including the user as an additional inputparameter. We formally establish the complexity class of the variant by the reductionprinciple. Finally, we conclude the article by exploring several interesting open problemsfor future research investigation.

Twin-crane scheduling during seaside workload peaks with a dedicated handshake area

To enable the efficient division of labor in container yards, many large ports apply twin cranes, two identical automated stacking cranes each dedicated to one of the transfer zones on the seaside and landside. The use of a handshake area, a bay of containers that separates the dedicated areas of the two cranes, is a simple means to avoid crane interference. Inbound containers arriving in the transfer zone of one crane and dedicated to a stacking position of the other crane’s area are placed intermediately in the handshake area by the first crane and then taken over by the second crane, and vice versa for outbound containers. Existing research only evaluates simple heuristics and rule-based approaches for the coordination of twin cranes interconnected by a handshake area. For this setting, accounting for precedence constraints due to stacking containers in the handshake area, we derive exact solution procedures under a makespan minimization objective. In this way, a comprehensive computational evaluation is enabled, which benchmarks heuristics with optimal solutions and additionally compares alternative crane settings (i.e., without workload sharing and cooperation with flexible handover). We further provide insights into where to position the handshake area. Our results reveal that when planning is too simple (i.e., with a rule-based approach), optimality gaps become large, but with sophisticated optimization, the price of a simplified crane coordination via a handshake area is low.

An ant colony optimization approach for the proportionate multiprocessor open shop

Multiprocessor open shop makes a generalization to classical open shop by allowing parallel machines for the same task. Scheduling of this shop environment to minimize the makespan is a strongly NP-Hard problem. Despite its wide application areas in industry, the research in the field is still limited. In this paper, the proportionate case is considered where a task requires a fixed processing time independent of the job identity. A novel highly efficient solution representation is developed for the problem. An ant colony optimization model based on this representation is proposed with makespan minimization objective. It carries out a random exploration of the solution space and allows to search for good solution characteristics in a less time-consuming way. The algorithm performs full exploitation of search knowledge, and it successfully incorporates problem knowledge. To increase solution quality, a local exploration approach analogous to a local search, is further employed on the solution constructed. The proposed algorithm is tested over 100 benchmark instances from the literature. It outperforms the current state-of-the-art algorithm both in terms of solution quality and computational time.

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.

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.

Algorithms for hierarchical and semi-partitioned parallel scheduling

We propose a model for scheduling jobs in a parallel machine setting that takes into account the cost of migrations by assuming that the processing time of a job may depend on the specific set of machines among which the job is migrated. For the makespan minimization objective, the model generalizes classical scheduling problems such as unrelated parallel machine scheduling, as well as novel ones such as semi-partitioned and clustered scheduling. In the case of a hierarchical family of machines, we derive a compact integer linear programming (ILP) formulation for the job assignment subproblem, and show how to turn arbitrary ILP solutions into valid schedules. We also derive a polynomial-time 2-approximation algorithm for the problem. Extensions that incorporate memory capacity constraints are also discussed.

A mathematical model for integrated operating room and surgical member scheduling considering lunch break

Operating rooms (ORs) are among the most influential departments of a hospital that a major portion of its expenditures and revenues originate from it. Due to the limited resources and the presence of different stockholders, effective management, and the optimal planning of this department are challenging. This paper develops a mathematical model to address the integrated problem of the operating room and surgical member scheduling with the objective of makespan minimization. In the proposed model, several aspects, including the availability and necessity of surgical members and equipment, and lunch break consideration, are incorporated. Finally, a case study related to an OR department in a general hospital is provided to assess the applicability and performance of the proposed model.

An Improved Branch-and-Bound Algorithm for the One-Machine Scheduling Problem with Delayed Precedence Constraints

In this paper, we discuss the one-machine scheduling problem with release and delivery times with the minimum makespan objective. Both heuristics and branch-and-bound algorithms have been formulated for the problem. One such branch-and-bound algorithm solves the problem and a variation that requires a delay between the completion of one job and the start of another (delayed precedence constraints). This paper analyzes key components of this branch-and-bound algorithm and proposes an improved heuristic to be used in conjunction with a different search strategy. Computational experiments demonstrate that the modifications lead to substantial improvements in running time and number of iterations on the one-machine problem instances both with and without delayed precedence constraints.

A genetic programming hyper-heuristic for the distributed assembly permutation flow-shop scheduling problem with sequence dependent setup times

Abstract In this paper, a genetic programming hyper heuristic (GP-HH) algorithm is proposed to solve the distributed assembly permutation flow-shop scheduling problem with sequence dependent setup times (DAPFSP-SDST) and the objective of makespan minimization. The main idea is to use genetic programming (GP) as the high level strategy to generate heuristic sequences from a pre-designed low-level heuristics (LLHs) set. In each generation, the heuristic sequences are evolved by GP and then successively operated on the solution space for better solutions. Additionally, simulated annealing is embedded into each LLH to improve the local search ability. An effective encoding and decoding pair is also presented for the algorithm to obtain feasible schedules. Finally, computational simulation and comparison are both carried out on a benchmark set and the results demonstrate the effectiveness of the proposed GP-HH. The best-known solutions are updated for 333 out of the 540 benchmark instances.

Bottleneck-Based Heuristic for Permutation Flowshop Scheduling

This paper addresses the permutation flowshop scheduling problem with objective of makespan minimization. It proposes a heuristic that apply bottleneck-based identification technique and Nawaz, Enscore and Ham (NEH) insertion technique. This experiment was carried out using measurement of four-machines and six-jobs simulated in Microsoft Excel Simple Programming. The computational result show that bottleneck-based (BNB) achieved 60 % and more of optimum solution and heuristic validation result prove that the proposed heuristic resulting to be much better than existing heuristic by achieving more best solution of 6 sets out of 10 sets replication.

Examining additive manufacturing in supply chain context through an optimization model

This study explores the problem of integrated jobs and vehicles scheduling in a two-stage supply chain, where parts are processed on additive manufacturing (AM) machines and then distributed to customers. In this study, we develop an optimization model for the problem with the objective of makespan minimization. Besides, several structural properties and lower bound formulation are provided to explain the main characteristics of the problem. In this regard, this study also contributes to the existing academic literature by investigating the two-stage supply chain scheduling problem with the additive manufacturing technology. Because the addressed problem belongs to non-deterministic polynomial-time hardness (NP-hard) problem class, a best-fit heuristic-based approach and several selection rules are developed to solve the problem. Each selection rule is designed by considering a distinct structural property of the problem and, thus, each of which is considered to be a different algorithm. A comprehensive experimental study is conducted through random instances, which are generated for small- and large-sized problems by considering real-production data. According to the computational results, the best-fit capacity utilization based selection (BFCUBS) algorithm is superior to others and yields a substantial improvement in the makespan. The reason behind this fact is that the high utilization of capacity enables a large number of jobs to be completed in a short time. Besides, as the number of jobs decreases and the capacity utilization rates increases all algorithms provide better results.

Analysis of flow shop scheduling anomalies

Anomalies in flow shop scheduling are rare; only two anomalies have been reported. We present five new anomalies for the permutation flow shop models with the minimum makespan objective and seven more anomalies for the minimum total flow time objective. These anomalies (including the existing ones) are divided into three types corresponding to an increased processing time of a single operation, the addition of a job and the addition of a machine. We derive two properties which, when satisfied, eliminate the possibility of certain anomalies. We conclude that restrictions such as no-delay schedules, no job waiting or no machine idle time (after it starts processing) often result in anomalies. We also show that anomalies can also occur in non-permutation flow shops (four new anomalies presented).

An effective genetic algorithm with a critical-path-guided Giffler and Thompson crossover operator for job shop scheduling problem

This work presents an effective genetic algorithm with a critical-path-guided Giffler and Thompson crossover operator for job shop scheduling problem with the objective of makespan minimization (GA-CPG-GT). Even though passing important traits from parents to offspring is known to be an important feature of any uniform crossover operator, most of the proposed operators adopt random exchange of genetic materials between parents; this is probably due to the fact that it is tricky to identify the genetic materials that hold the important traits. For that reason, in this work, a new selective exchange of genetic materials is proposed. In the proposed approach, at first, the genetic materials that hold the important traits are identified according to some domain specific information provided by the critical paths of the parents, and then the exchange is made on favor of them. The properties of critical path are usually utilized by the local search methods such as tabu search and simulating annealing; however, in this work, they are utilized in the global search method GA during the crossover operator. The implications of the proposed approach are investigated using the Giffler and Thompson crossover operator, which is a uniform crossover combined with the G&T algorithm. The proposed approach is tested on 55 benchmarks, with the proposed selective exchange, and without it using the random one, and also compared with other 5 similar works reported in the literature. The computational results validate the enhancements accomplished by the proposed selective exchange, and show the superiority of the proposed algorithm over the compared works in terms of solution quality, and validate its effectiveness.

A Social Spider Optimization Algorithm for Hybrid Flow Shop Scheduling with Multiprocessor Task

Hybrid flow shop scheduling with multiprocessor task (HFSPMT) is a strongly NP-hard problem that has a wide range of applications in several real-life industrial environments. Due to its essential complexity and practical relevance, HFSPMT has been of an increasing interest for researchers and industrialists in the last two decades. Social spider optimization (SSO) is a recent swarm intelligence algorithm that has been rarely applied on combinatorial optimization problems. In order to make a small step toward having an idea about its effectiveness in solving combinatorial optimization, in this work, a new SSO algorithm is proposed to solve HFSPMT problem with the objective of makespan minimization. The proposed algorithm is experimented on benchmark instances and compared with other existing swarm intelligence algorithms in the literature. The obtained results and the comparisons show that the performance of the proposed algorithm is highly competitive in terms of solution quality.

An approximation algorithm for vehicle routing with compatibility constraints

We study a multiple-vehicle routing problem with a minimum makespan objective and compatibility constraints. We provide an approximation algorithm and a nearly-matching hardness of approximation result. We also provide computational results on benchmark instances with diverse sizes showing that the proposed algorithm (i) has a good empirical approximation factor, (ii) runs in a short amount of time and (iii) produces solutions comparable to the best feasible solutions found by a direct integer program formulation.

Permutation flowshop scheduling with time lag constraints and makespan criterion

Abstract A permutation flowshop with time lag constraints requires that the time lag between consecutive operations of a job must be in the given interval. In this study, the scheduling of such flowshop with makespan minimization objective is investigated. We prove that this problem has the reversibility property, and present a two-stage constructive heuristic with time complexity O( n 2 m ), where n and m are the numbers of jobs and machines, respectively. The first stage generates a rank of jobs based on an equivalent formulation of the makespan, and the second stage constructs a schedule through an insertion mechanism. We apply the reversibility property to reduce the time complexity of insertion procedures in both two stages, and find a better solution by solving both the original problem and its reversed problem. To verify the effectiveness and efficiency of this heuristic, we compare its performance to the mixed integer linear programming (MILP) formulation of this problem for small-scale problems, and conduct a comprehensive computational study on the Taillard’s benchmark.

New index priority rules for no-wait flow shops

We derive an index priority rule for an m-machine no-wait flow shop with the minimum makespan objective and specially-structured job processing times. Our rule generalizes the previously known rules for two special cases of the problem. We also derive an index priority rule for a two-machine no-wait flow shop with the minimum weighted total job completion time objective when all jobs have the same processing time on the first machine. We then show that additional index priority rules can be derived for the latter problem with other scheduling objectives. Finally, we discuss extensions to flow shops with blocking and no-wait job shops and open shops.

Decomposition Methods for the Parallel Machine Scheduling Problem with Setups

We study the unrelated parallel machine scheduling problem with sequence and machine-dependent setup times and the objective of makespan minimization. Two exact decomposition-based methods are proposed based on logic-based Benders decomposition and branch and check. These approaches are hybrid models that make use of a mixed-integer programming (MIP) master problem and a specialized solver for travelling salesman subproblems. The master problem is used to assign jobs to machines, whereas the subproblems find optimal schedules on each machine given the master problem assignments. Computational results show that the decomposition models are able to find optimal solutions up to four orders of magnitude faster than the existing state of the art as well as solve problems six times larger than an existing MIP model. We further investigate the solution quality versus runtime trade-off for large problem instances for which the optimal solutions cannot be found and proved in a reasonable time. We demonstrate that the branch-and-check hybrid algorithm is able to produce better schedules in less time than the state-of-the-art metaheuristic, while also providing an optimality gap.

Job selection in two-stage shops with ordered machines

We consider job selection problems in two-stage flow shops, open shops and job shops. The objective is to select the best job subset with a given cardinality to minimize either the total job completion time or the maximum job completion time (makespan). An O(n2) algorithm is available for the two-stage flow shop with ordered machines and the minimum makespan objective; we utilize this algorithm to solve the same problem for the total job completion time objective. We then propose an O(n1) algorithm for the two-machine open shop job selection problem with ordered machines and the minimum makespan objective. We also consider a two-machine job shop in which the first operation of each job is no longer than the second one. We show that the job selection problem in this job shop with the minimum makespan objective is ordinary NP-hard and that the problem becomes solvable in O(nlogn) time with the additional assumption of ordered jobs.

Heuristics and metaheuristics for the distributed assembly permutation flowshop scheduling problem with sequence dependent setup times

We consider a Distributed Assembly Permutation Flowshop Scheduling Problem with sequence dependent setup times and the objective of makespan minimization. The problem consists of two stages, production and assembly. The first stage comprises f identical factories, where each factory is a flowshop that produces jobs which are later assembled into final products through an identical assembly program in a second assembly stage made by a single machine. Both stages have sequence dependent setup times. This is a realistic and complex problem and therefore, we propose two simple heuristics and two metaheuristics to solve it. A complete calibration and analysis through a Design Of Experiments (DOE) approach is carried out. In the process, important knowledge of the studied problem is obtained as well as some simplifications for the powerful Iterated Greedy methodology which results in a simpler approach with less parameters. Finally, the performance of the proposed methods is compared through extensive computational and statistical experiments.