Makespan Objective Function Research Articles

Robust job-shop scheduling under deterministic and stochastic unavailability constraints due to preventive and corrective maintenance

This paper presents a modeling and solution approach for a robust job-shop scheduling problem (JSP) under deterministic and stochastic machine unavailability caused, respectively, by planned preventive maintenance (PM) and unplanned corrective maintenance (CM) following random breakdowns. The goal is to optimize the sequence of jobs and run-based planned preventive maintenance tasks in a robust manner while considering the degradation of machines over time. The time to failure for each machine is assumed to follow a Weibull distribution. The robust objective function to be minimized is a weighted sum of the expected values of the makespan and the gross positive deviation between the actual and planned start times of the jobs, as proxies for quality robustness and solution robustness, respectively. Two metaheuristic algorithms that aim at providing a good balance between performance quality and solution robustness are developed. In both algorithms, the “true” makespan objective function (with both PM and CM) is approximated using three surrogate measures. Genetic algorithm (GA) is first used to optimize the surrogate functions, then the fittest solutions from the three oracles are simulated with random breakdown scenarios and the best among them is introduced as an elite member in the next GA iteration. The first algorithm terminates as soon as the solution obtained is worse than the best know solution or when the maximum number of iterations is reached, whereas the second algorithm applies a rule inspired by Simulated Annealing for termination. Numerical experimentation on benchmark instances from the literature showed excellent performance of the proposed algorithms in terms of average marginal improvement and runtime. These results show that the proposed framework can generate high-quality robust job shop schedules under deterministic and stochastic machine unavailability constraints. Moreover, the sensitivity analysis results recommended some key insights and directions for future research.

Robust Job Shop Scheduling with Condition-Based Maintenance and Random Breakdowns

This paper presents a simulation-optimization approach for solving the Job Shop Scheduling Problem (JSSP) under both planned and unplanned machine unavailability. Two maintenance policies are considered: Condition-Based Maintenance (CBM) and Corrective Maintenance (CM). The real makespan objective function is first approximated using several surrogate functions that are independently optimized using Genetic Algorithm (GA), before their best solutions are evaluated through simulation with stochastic degradation of machines, random breakdowns, and uncertain CBM and CM duration. To ensure schedule robustness, a weighted average of the expected makespan and its 90th percentile is used as the evaluation criterion, and the best schedule is added to an elite list to initiate the next iteration. A stopping rule inspired by Simulated Annealing (SA) is employed to prevent premature conversion. Numerical experimentation on random instances showed that the proposed approach can reach high-quality solutions effectively.

HEART: Unrelated parallel machines problem with precedence constraints for task scheduling in cloud computing using heuristic and meta‐heuristic algorithms

SummaryCloud computing is becoming a profitable technology because of it offers cost‐effective IT solutions globally. A well‐designed task scheduling algorithm ensures the optimal utilization of clouds resources and reducing execution time dynamically. This research article deals with the task scheduling of inter‐dependent subtasks on unrelated parallel computing machines in a cloud computing environment. This article considers two variants of the problem‐based on two different objective function values. The first variant considers the minimization of the total completion time objective function while the second variant considers the minimization of the makespan objective function. Heuristic and meta‐heuristic (HEART) based algorithms are proposed to solve the task scheduling problems. These algorithms utilize the property of list scheduling algorithm of unrelated parallel machine scheduling problem. A mixed integer linear programming (MILP) formulation has been provided for the two variants of the problem. The optimal solution is obtained by solving MILP formulation using A Mathematical Programming Language (AMPL) software. Extensive numerical experiments have been performed to evaluate the performance of proposed algorithms. The solutions obtained by the proposed algorithms are found to out‐perform the existing algorithms. The proposed algorithms can be used by cloud computing service providers (CCSPs) for enhancing their resources utilization to reduce their operating cost.

Decomposition algorithms for the integrated process planning and scheduling problem

There are several algorithms to solve the integrated process planning and scheduling (IPPS) problem (i.e., flexible job shop scheduling with process plan flexibility) in the literature. All the existing algorithms for IPPS are heuristic-based search methods and no research has investigated the use of exact solution methods for this problem. We develop several decomposition approaches based on the logic-based Benders decomposition (LBBD) algorithm. Our LBBD algorithm allows us to partition the decision variables in the IPPS problem into two models, master-problem and sub-problem. The master-problem determines process plan and operation-machine assignment, while the sub-problem optimizes sequencing and scheduling decisions. To achieve faster convergence, we develop two relaxations for the optimal makespan objective function and incorporate them into the master-problem. We analyze the performance and further enhance the algorithm with two ideas, a Benders optimality cut based on the critical path and a faster heuristic way to solve the sub-problem. 16 standard benchmark instances available in the literature are solved to evaluate and compare the performances of our algorithms with those of the state-of-the-art methods in the literature. The proposed algorithm either results in the optimal solution or improves the best-known solutions in all the existing instances, demonstrating its superiority to the existing state-of-the-art methods in literature.

Solving the single crane scheduling problem at rail transshipment yards

We consider single-crane scheduling at rail transshipment yards, in which gantry cranes move containers between trains, trucks and a storage area. The single-crane scheduling problem arises at single-crane transshipment terminals and as a subproblem of the multiple-crane scheduling problem. We consider a makespan objective function, which is equivalent to minimizing the train dwell time in the yard, and introduce time windows for container moves, for example, as a customer service promise. Our proposed decomposition algorithm with integrated dynamic branch-and-cut or dynamic programming solves practically relevant instances within short time limits.

A Tabu Search Algorithm for the Stage Shop Problem

This paper presents stage shop problem which is a special case of the general shop. The stage shop is a more realistic generalization of the mixed shop problem. In the stage shop problem, each job has several stages of operations. In order to solve the stage shop problem with makespan objective function, a tabu search algorithm is developed. In addition, an existing lower bound of the job shop is adapted to the new problem and the computational results have been compared to it. The proposed TS algorithm has reached the optimal solutions for about half of the problem instances

A GA/TS algorithm for the stage shop scheduling problem

This paper presents a special case of the general shop called stage shop problem. The stage shop is a more realistic generalization of the mixed shop problem. In the stage shop problem, each job has several stages of operations. In order to solve the stage shop problem with makespan objective function, an existing neighborhood of job shop is used. In this neighborhood, few enhanced conditions are proposed to prevent cycle generation. In addition, a new neighborhood for operations that belong to the same job is presented. These neighborhoods are applied to the stage shop problem in a tabu search framework. A genetic algorithm is used to obtain good initial solutions. An existing lower bound of the job shop is adapted to our problem and the computational results have been compared to it. Our algorithm has reached the optimal solutions for more than half of the problem instances.

Minimizing Makespan in No-Wait Job Shops

In this paper, we study polynomial time approximation schemes (PTASes) for the no-wait job shop scheduling problem with the makespan objective function. It is known that the problem is MaxSNP-hard in the case when each job is allowed to have three operations or more. We show that if each job has at most two operations, the problem admits a PTAS if the number of machines is a constant (i.e., not part of the input). If the number of machines is not a constant, we show that the problem is hard to approximate within a factor better than 5/4.

Different behaviour of a double branch-and-bound algorithm on [formula omitted] and [formula omitted] problems

In this paper we face the permutation flow-shop scheduling problem with a makespan objective function in two variants, with and without storage space between machines. We use an improved branch and bound algorithm, suitable for parallel computation, to solve these problems, and auxiliary heuristics to attain an initial good solution. The auxiliary heuristics proposed are built by two steps: in the first step a permutation is obtained; in the second step a local search procedure is applied. The improvement obtained by the local search procedure on NEH heuristic as first step is shown. Since the flow-shop scheduling problem with storage space is a relaxation of the problem without storage space, some elements and procedures developed for that problem can be used in both problems. In particular, some bounding procedures, for instance Nabeshima or Lageweg bounding schema, can be adapted. Moreover, the reversibility property holds on both problems. Consequently a double branch and bound algorithm can be applied simultaneously to the direct and the inverse instances. The same sets of data are submitted to heuristics and to the double branch-and-bound algorithm, LOMPEN, assuming first they are instances of flow-shop scheduling problem with storage space and later they are instances of flow-shop scheduling problem without storage space. The algorithms are coded in a similar way; therefore the behaviour and performance can be compared.

Complexity of flow shop scheduling problems with transportation constraints

In most manufacturing and distribution systems, semi-finished jobs are transferred from one processing facility to another by transporters such as Automated Guided Vehicles, robots and conveyors, and finished jobs are delivered to warehouses or customers by vehicles such as trucks. This paper investigates two-machine flow shop scheduling problems taking transportation into account. The finished jobs are transferred from the processing facility and delivered to customers by truck. Both transportation capacity and transportation times are explicitly taken into account in these models. We study the class of flow shop problems by analysing their complexity. For the makespan objective function, we prove that this problem is strongly NP-hard when the capacity of a truck is limited to two or three parts with an unlimited buffer at the output of the each machine. This problem with additional constraints, such as blocking, is also proven to be strongly NP-hard.

Concurrent flowshop scheduling to minimize makespan

This paper considers the concurrent flowshop scheduling problem with the makespan objective function. In this problem each job consists of several components each of which is processed on a dedicated flowshop. Equivalently, each component requires a series of consecutive operations in a prespecified order as in the standard serial flowshop. We assume that each component has the same number of indexed operations and that these operations must be processed in the same order for each component. We first show that this problem is strongly NP-hard and then, using compact vector summation techniques, we construct schedules using a quadratic time heuristic with an absolute performance guarantee independent of the number of jobs. This makes our schedules asymptotically optimal as the number of jobs becomes very large.

Preemptive job-shop scheduling problems with a fixed number of jobs

It is shown that the two machine preemptive job-shop problem with mean flow-time or makespan objective function and three jobs is NP-hard. This contrasts the fact that the nonpreemptive versions of these problems are polynomially solvable if the number of jobs is arbitrary but fixed. It is also shown that the preemptive problems can be solved pseudopolynomially if both the number of machines and the number of jobs is fixed.

Scheduling on a two-machine flowshop subject to random breakdowns with a makespan objective function

Two-machine flowshop scheduling problems have been discussed in the literature extensively under the assumption that machines are continuously available. We address the problem of minimizing makespan in a two-machine flowshop when the machines are subject to random breakdowns. We first show that it is sufficient to consider the same sequence of the jobs on each machine. After providing an elimination criterion for minimizing makespan with probability 1, we show that under appropriate conditions Johnson's algorithm stochastically minimizes makespan.

Solvable Cases of the No-Wait Flow-Shop Scheduling Problem

The no-wait flow-shop scheduling problem (NWFSSP) with a makespan objective function is considered. As is well known, this problem is 𝒩𝒫-hard for three or more machines. Therefore, it is interesting to consider special cases, i.e. special structured processing time matrices, that allow polynomial time solution algorithms. Furthermore, it is well known that the NWFSSP with a makespan objective function can be formulated as a travelling salesman problem (TSP). It is observed that special structured processing time matrices for the NWFSSP lead to special structured distance matrices for which the TSP is polynomially solvable. Using this observation, it is shown that some NWFSSPs with fixed processing times on all except two machines are well solvable while the others are conjectured to be 𝒩𝒫-hard. Also, it is shown that NWFSSPs with a mean completion time objective function restricted to semi-ordered processing time matrices are easily solvable.