A constraint programming approach to tool allocation and production scheduling in flexible manufacturing systems

This paper presents a constraint programming (CP) methodology to deal with the scheduling of flexible manufacturing systems (FMSs). The proposed approach, which consists of both a model and a search strategy, handles several features found in industrial environments, such as limitations on number of tools in the system, lifetime of tools, as well as tool magazine capacity of machines. In addition, it tackles the problem in a integrated way by considering tool planning and allocation, machine assignment, part routing, and task timing decisions altogether in the approach. The formulation, which is able to take into account a variety of objective functions, has been successfully applied to the solution of test problems of various sizes and degrees of difficulty.     

基于粗糙规划的不确定加工时间的并行机调度

针对并行机调度中的不确定工件加工时间,提出用粗糙变量表示不确定量,并由此建立该问题的粗糙期望值规划模型.提出一种应用于调度问题的进化规划算法,改进了针对并行机问题的编码方式和变异方法.采用粗糙模拟的方法计算个体的适应值,即粗糙期望估计值,并加以不同规模的算例进行仿真实验.仿真结果表明,改进进化规划算法得到的解优于遗传算法得到的解.     

A new heuristic for scheduling the two-stage flowshop with additional resources

This paper deals with the problem of preemptive scheduling in a two-stage flowshop with parallel unrelated machines at the first stage and a single machine at the second stage. At the first stage, jobs use some additional resources which are available in limited quantities at any time. The resource requirements are of 0–1 type. The objective is the minimization of makespan. The problem is NP-hard. Heuristic algorithms are proposed which solve to optimality the resource constrained scheduling problem at the first stage of the flowshop, and at the same time, minimize the makespan in the flowshop by selecting appropriate jobs for simultaneous processing. Several rules of job selection are considered. The performance of the proposed heuristic algorithms is analyzed by comparing solutions with the lower bound on the optimal makespan. The extensive computational experiment shows that the proposed heuristic algorithms are able to produce near-optimal solutions in short computational time.     

Using quadratic programming to solve high multiplicity scheduling problems on parallel machines

We introduce and analyze several models of schedulingn different types (groups) of jobs onm parallel machines, where in each group all jobs are identical. Our main goal is to exhibit the usefulness of quadratic programming approaches to solve these classes of high multiplicity scheduling problems, with the total weighted completion time as the minimization criterion. We develop polynomial algorithms for some models, and strongly polynomial algorithms for certain special cases. In particular, the model in which the weights are job independent, as well as the generally weighted model in which processing requirements are job independent, can be formulated as an integer convex separable quadratic cost flow problem, and therefore solved in polynomial time. When we specialize further, strongly polynomial bounds are achievable. Specifically, for the weighted model with job-independent processing requirements if we restrict the weights to be machine independent (while still assuming different machine speeds), anO(mn+n logn) algorithm is developed. If it is also assumed that all the machines have the same speed, the complexity of the algorithm can be improved toO(m logm+n logn). These results can be extended to related unweighted models with variable processing requirements in which all the machines are available at time zero. The research of Frieda Granot was partially supported by Natural Sciences and Engineering Research Council of Canada Grant 5-83998. The research of Jadranka Skorin-Kapov was partially supported by National Science Foundation Grant DDM-8909206.     

Heuristic algorithms for preemptive scheduling in a two-stage hybrid flowshop with additional renewable resources at each stage

This paper deals with the problem of preemptive scheduling in a two-stage flowshop with parallel unrelated machines and renewable resources at both the stages. The resource requirements are of a 0–1 type. The objective is the minimization of makespan. The problem is NP-hard. Four heuristic algorithms using linear programming are proposed for solving this problem. Performance of the algorithms is analyzed experimentally by comparing heuristic solutions with the lower bound on the optimal makespan. Statistical comparative analysis of the developed algorithms is carried out. The results of a computational experiment show that the proposed algorithms are able to produce good quality solutions in a small amount of computation time.     

Time-constrained project scheduling with adjacent resources

We develop a decomposition method for the Time-Constrained Project Scheduling Problem (TCPSP) with adjacent resources. For adjacent resources the resource units are ordered and the units assigned to a job have to be adjacent. On top of that, adjacent resources are not required by single jobs, but by job groups. As soon as a job of such a group starts, the adjacent resource units are occupied, and they are not released before all jobs of that group are completed. The developed decomposition method separates the adjacent resource assignment from the rest of the scheduling problem. Test results demonstrate the applicability of the decomposition method. The presented decomposition forms a first promising approach for the TCPSP with adjacent resources and may form a good basis to develop more elaborated methods.     

Dominance rules for the parallel machine total weighted tardiness scheduling problem with release dates

We address the parallel machine total weighted tardiness scheduling problem with release dates. We describe dominance rules and filtering methods for this problem. Most of them are adaptations of dominance rules based on solution methods for the single-machine problem. We show how it is possible to deduce whether or not certain jobs can be processed by a particular machine in a particular context and we describe techniques that use this information to improve the dominance rules. On the basis of these techniques we describe an enumeration procedure and we provide experimental results to determine the effectiveness of the dominance rules.     

Non-permutation flowshop scheduling with dual resources

Typically, in order to process jobs in a flowshop both machines and labor are required. However, in traditional scheduling problems, labor is assumed to be plentiful and only machine is considered to be a constraint. This assumption could be due to the lower cost of labor compared to machines or the complexity of dual-resource constrained problems. In this paper a mathematical model is developed to minimize the work-in-process inventory while maximizing the service level in a flowshop with dual resources. The model focuses on optimizing a non-permutation flowshop. There are different skill levels considered for labor and the setup times on machines are sequence-dependent. Jobs are allowed to skip one or more stages in the flowshop. Job release and machine availability times are considered to be dynamic. The problem is solved in two layers. The outer layer is a search algorithm to find the schedule of jobs on the machine (traditional flowshop scheduling problem) and the inner layer is a three-step heuristic to find a schedule of jobs on labor in accordance to the machine schedule. Three different search algorithms are developed to solve the proposed NP-hard problem. First algorithm can solve a permutation flowshop while the other two are developed to solve a non-permutation flowshop. The comparison between the optimal solution and the search algorithms in small examples shows a good performance of the algorithms with an average deviation of only 2.00%. An experimental design analyzes the effectiveness and efficiency of the algorithms statistically. The results show that non-permutation algorithms perform better than the permutation algorithm, although the former are less efficient. The effectiveness and efficiency in all three algorithms have an inverse relation. To the best of our knowledge, this research is the first of its kind to provide a comprehensive mathematical model for dual resource flowshop scheduling problem.     

A branch-and-price algorithm for the general case of scheduling parallel machines with a single server

We consider the strongly NP-hard problem of scheduling two-operation non-preemptable jobs on two identical parallel machines. A single server, that can handle at most one job at a time, is available to carry out the first (or setup) operation. The second operation, to be carried out on the same machine but without the server, must be executed immediately after the setup. The objective is to minimize the makespan. We apply a column generation method to a population of partial schedules, in turn generated by some well known heuristics, to achieve effective and efficient solutions. We compare the performance of this method with those proposed earlier and also suggest future work.     

Parallel machine scheduling problems in green manufacturing industry

Manufacturing companies are now more conscious about the environment. As such, there are more concerns in reducing the consumption of energy and the production of pollutants. Reduced consumption of energy will save cost, while reduction of pollutants will decrease the cost of cleaning up the environment. This paper considers scheduling problems that arise in green manufacturing companies. Suppose the manufacturing company has a set of parallel machines. Each machine has a cost per unit time that differs from machine to machine. The cost here is the sum of the energy cost and the clean up cost. A set of jobs is to be processed by these machines. Our goal is to find a schedule that minimizes the makespan (schedule length) or the total completion time, subject to the constraint that the total cost is not more than a given threshold value. We propose efficient heuristics and show, by computational experiments, that they perform very well in practice.     

Block models for scheduling jobs on two parallel machines with a single server

We consider the problem of scheduling a set of non-preemptable jobs on two identical parallel machines such that the makespan is minimized. Before processing, each job must be loaded on a machine, which takes a given setup time. All these setups have to be done by a single server which can handle at most one job at a time. For this problem, we propose a mixed integer linear programming formulation based on the idea of decomposing a schedule into a set of blocks. We compare the results obtained by the model suggested with known heuristics from the literature.     

Parallel machine scheduling problem with preemptive jobs and transportation delay

In this research the parallel machine scheduling problem with preemption by considering a constant transportation delay for migrated jobs and minimization of makespan as the criterion i.e., P_m|pmtn(delay)|C_max is investigated. It is assumed that when a preempted job resumes on another machine, it is required a delay between the processing time of the job on these two machines. This delay is called transportation delay. We criticize an existing mathematical model for the research problem in the literature [Boudhar M, Haned A. Preemptive scheduling in the presence of transportation times. Computers & Operations Research 2009; 36(8):2387–93]. Then, we prove that there exists an optimal schedule with at most (m−1) preemptions with transportation among machines for the problem. We also propose a linear programming formulation and an exact algorithm for the research problem in case of equal transportation delay. The experiments show that the proposed exact algorithm performs better than the proposed mathematical model.     

Scheduling jobs with equal processing times and time windows on identical parallel machines

We present a linear programming approach to the problem of scheduling equal processing time jobs with release dates and deadlines on identical parallel machines. The known algorithm with complexity O(n ³log log n) of B. Simons schedules all the jobs while minimizing both the maximum completion time and the mean flow time. Our approach permits also to minimize the weighted sum of completion times and total tardiness in polynomial time for the problems without deadlines. The complexity status of these problems was open. Contract/grant sponsor: Alexander von Humboldt Foundation.     

Algorithms for propagating resource constraints in AI planning and scheduling: Existing approaches and new results

This paper summarizes the main existing approaches to propagate resource constraints in Constraint-Based scheduling and identifies some of their limitations for using them in an integrated planning and scheduling framework. We then describe two new algorithms to propagate resource constraints on discrete resources and reservoirs. Unlike most of the classical work in scheduling, our algorithms focus on the precedence relations between activities rather than on their absolute position in time. They are efficient even when the set of activities is not completely defined and when the time window of activities is large. These features explain why our algorithms are particularly suited for integrated planning and scheduling approaches. All our algorithms are illustrated with examples. Encouraging preliminary results are reported on pure scheduling problems as well as some possible extensions of our framework.     

A branch and bound algorithm for an identical parallel machine scheduling problem with a job splitting property

We focus on the problem of scheduling n independent jobs on m identical parallel machines with the objective of minimizing total tardiness of the jobs considering a job splitting property. In this problem, it is assumed that a job can be split into sub-jobs and these sub-jobs can be processed independently on parallel machines. We develop several dominance properties and lower bounds for the problem, and suggest a branch and bound algorithm using them. Computational experiments are performed on randomly generated test problems and results show that the suggested algorithm solves problems of moderate sizes in a reasonable amount of computation time.     

Unrelated parallel machine scheduling with dedicated machines and common deadline

This paper addresses a scheduling problem in the manufacturing of Polyvinyl Chloride pipes. There are two main attributes of PVC pipes: diameter and color. Each attribute has a corresponding attribute setup time and usually has several different levels. Each extruder produces different PVC pipe products based on the diameters as large, middle and small. The alternatives exist between these extruders, where the large and the middle type extruders can be used to produce the PVC pipes with the other diameters; the small type extruders can be used to produce the PVC pipes with middle diameters but cannot produce those with large diameters. The processing times are longer in all of the alternatives among different types of extruders. The objective is to minimize the total completion time for the unrelated parallel machine problem.Three dedicated machine heuristics are proposed herein for the problem and have been evaluated by comparing with the current scheduling method used in the case plant. The computational results show that the proposed constructive heuristics outperform the current scheduling method with significant improvements and can be used to solve large-size problems in reasonable computational times.     

A hybrid approach to the single line scheduling problem with multiple products and sequence-dependent time

The present paper presents a hybrid approach for solving manufacturing scheduling problems, based on the integration between Constraint Logic Programming (CLP) and Genetic Algorithm (GA) approaches. The proposed methodology is applied to a single line with multiple products and sequence-dependent time. This system model derives from a real case of a company producing sheets for catalytic converters. A sensitivity analysis of the hybrid methodology is carried out to compare the performance of the CLP, GA and integrated CLP–GA approaches.     

Two-machine flowshop scheduling with flexible operations and controllable processing times

We consider a two-machine flowshop scheduling problem with identical jobs. Each of these jobs has three operations, where the first operation must be performed on the first machine, the second operation must be performed on the second machine, and the third operation (named as flexible operation) can be performed on either machine but cannot be preempted. Highly flexible CNC machines are capable of performing different operations. Furthermore, the processing times on these machines can be changed easily in albeit of higher manufacturing cost by adjusting the machining parameters like the speed and/or feed rate of the machine. The overall problem is to determine the assignment of the flexible operations to the machines and processing times for each operation to minimize the total manufacturing cost and makespan simultaneously. For such a bicriteria problem, there is no unique optimum but a set of nondominated solutions. Using ?-constraint$?\mathrm{-}\mathrm{constraint}$ approach, the problem could be transformed to be minimizing total manufacturing cost for a given upper limit on the makespan. The resulting single criterion problem can be reformulated as a mixed integer nonlinear problem with a set of linear constraints. We use this formulation to optimally solve small instances of the problem while a heuristic procedure is constructed to solve larger instances in a reasonable time.