Batching and Scheduling with Tardiness Penalties

Journal of Scheduling, 1998

We address the problem of scheduling n jobs on a batching machine to minimize regular scheduling criteria that are non-decreasing in the job completion times. A batching machine is a machine that can handle up to b jobs simultaneously. The jobs that are processed together form a batch, and all jobs in a batch start and complete at the same time. The processing time of a batch is equal to the largest processing time of any job in the batch. We analyse two variants: the unbounded model, where b¿n; and the bounded model, where b¡n. For the unbounded model, we give a characterization of a class of optimal schedules, which leads to a generic dynamic programming algorithm that solves the problem of minimizing an arbitrary regular cost function in pseudopolynomial time. The characterization leads to more e cient dynamic programming algorithms for speciÿc cost functions: a polynomial algorithm for minimizing the maximum cost, an O(n 3) time algorithm for minimizing the number of tardy jobs, an O(n 2) time algorithm for minimizing the maximum lateness, and an O(n log n) time algorithm for minimizing the total weighted completion time. Furthermore, we prove that minimizing the weighted number of tardy jobs and the total weighted tardiness are NP-hard problems. For the bounded model, we derive an O(n b(b−1)) time dynamic programming algorithm for minimizing total completion time when b ¿ 1; for the case with m di erent processing times, we give a dynamic programming algorithm that requires O(b 2 m 2 2 m) time. Moreover, we prove that due date based scheduling criteria give rise to NP-hard problems. Finally, we show that an arbitrary regular cost function can be minimized in polynomial time for a ÿxed number of batches.

Journal of Engineering and Technological Sciences

This paper addresses an integrated production and delivery batch scheduling problem for a make-to-order environment over daily time period, where the holding costs of in-process and completed parts at a supplier location and of completed parts at a manufacturer location are distinguished. All orders of parts with different due dates from the manufacturer arrive at the same time. The parts are produced in production batches and subsequently the completed parts are delivered in delivery batches using a capacitated vehicle in order to be received at the respective due dates. This study was aimed at finding an integrated schedule of production and delivery batches so as to meet the due date at minimum total cost consisting of the corresponding holding cost and delivery cost. The holding cost is a derivation of the so-called actual flow time (AFT), while the delivery cost is assumed to be proportional to the number of deliveries. The problems can be formulated as an integer non-linear programming model, and the global optimal solution can be obtained using optimization software. A heuristic algorithm is proposed to cope with the computational time problem using software. The numerical experiences show that the proposed algorithm yields near global optimal solutions.

2010

This study considers the scheduling problem observed in the burn-in operation of semiconductor final testing, where jobs are associated with release times, due dates, processing times, sizes, and non-agreeable release times and due dates. The burn-in oven is modeled as a batch-processing machine which can process a batch of several jobs as long as the total sizes of the jobs do not exceed the machine capacity and the processing time of a batch is equal to the longest time among all the jobs in the batch. Due to the importance of ontime delivery in semiconductor manufacturing, the objective measure of this problem is to minimize total weighted tardiness. We have formulated the scheduling problem into an integer linear programming model and empirically show its computational intractability. Due to the computational intractability, we propose a few simple greedy heuristic algorithms and meta-heuristic algorithm, simulated annealing (SA). A series of computational experiments are conducted to evaluate the performance of the proposed heuristic algorithms in comparison with exact solution on various small-size problem instances and in comparison with estimated optimal solution on various real-life large size problem instances. The computational results show that the SA algorithm, with initial solution obtained using our own proposed greedy heuristic algorithm, consistently finds a robust solution in a reasonable amount of computation time.

International Journal of Industrial Engineering Computations, 2011

This study investigates minimizing the number of weighted tardy jobs on a single machine when jobs are delivered to either customers or next station in various size batches. In real world, this issue may happen within a supply chain in which delivering goods to customers entails costs. Under such circumstances, keeping completed jobs to deliver in batches may result in reducing delivery costs; nevertheless, it may add to the tardy jobs, which in turn leads to higher costs. In literature review, minimizing the number of weighted tardy jobs is known as NP-Hard problem, so the present issue aiming at minimizing the costs of delivering, in addition to the aforementioned objective function, remains an NP-Hard problem. In this study, the issue is assessed where the customers are numerous, and a mathematical model is presented. We also present a meta-heuristic method based on simulated annealing (SA) and the performance of the SA is examined versus exact solutions.

This paper is about minimizing the tardiness of a single machine batch processing. It is a scheduling problem, which consist of minimizing the tardiness of jobs assigned in batches to be processed by machines that have size and time limitations. This problem is common in manufacturing operations such as heat treatment and wafer burn-in operations. It is well studied in the literature and many problems can be created from the basic batch processing machine problem. This research aims to solve the problem by formulating and comparing the heuristic simulated annealing and CPLEX in term of accuracy and processing time.

Operations Research Letters, 1997

We address the single-machine batch scheduling problem which arises when there are job families and setup requirements exist between these families; our objective is to minimize the maximum lateness. As our main result, we give an improved dynamic program for the solution of the problem. © 1997 Elsevier Science B.V.

international journal of industrial engineering computations, 2012

This paper presents a mathematical model for the problem of minimizing the maximum lateness on a single machine when the deteriorated jobs are delivered to each customer in various size batches. In reality, this issue may happen within a supply chain in which delivering goods to customers entails cost. Under such situation, keeping completed jobs to deliver in batches may result in reducing delivery costs. In literature review of batch scheduling, minimizing the maximum lateness is known as NP-Hard problem; therefore the present issue aiming at minimizing the costs of delivering, in addition to the aforementioned objective function, remains an NP-Hard problem. In order to solve the proposed model, a Simulation annealing meta-heuristic is used, where the parameters are calibrated by Taguchi approach and the results are compared to the global optimal values generated by Lingo 10 software. Furthermore, in order to check the efficiency of proposed method to solve larger scales of problem, a lower bound is generated. The results are also analyzed based on the effective factors of the problem. Computational study validates the efficiency and the accuracy of the presented model.

International Journal of Production Economics, 2000

This paper deals with lot sizing and scheduling for a single-stage, single-machine production system where setup costs and times are sequence dependent. A large-bucket mixed integer programming (MIP) model is formulated which considers only e$cient sequences. A tailor-made enumeration method of the branch-and-bound type solves problem instances optimally and e$ciently. The size of solvable cases ranges from 3 items and 15 periods to 10 items and 3 periods. Furthermore, it will become clear that rescheduling can neatly be done.

2001

In multiproduct batch plants, the processing tasks required to complete the production of different items share manufacturing resources such as raw materials, intermediates, manpower, equipment and utilities (steam, electricity, cooling water, etc). Such resources are usually available by limited amounts that cannot be exceeded at any time of the scheduling period. This type of restriction is computationally costly when a continuous-time representation is applied to model the short-term scheduling of multiproduct batch plants. To meet such constraints, it becomes important to monitor the resource requirement profile over the entire planning horizon to exclude from the problem feasible space those schedules exceeding at least one of the resource capacities. Most of current continuous-time based methodologies ignore the resource capacity constraints. Manufacturing resources are usually classified into two major groups: renewable and nonrenewable resources. A renewable resource like units or manpower becomes again available for use after ending the processing task to which is currently assigned. Schedules involving the execution of simultaneous tasks featuring a total resource requirement larger than the available capacity is to be discarded by a proper problem representation. To this end, 0-1 decision variables and additional constraints have been defined to forbid running simultaneous processing tasks if, by doing that, some shortage in a resource capacity arises. A typical case in industry is the number of production lines running in parallel being constrained by the labor capacity. Among non-renewable resources, finite initial inventories and especially the reception of open orders of raw materials and intermediates during the period to be scheduled are challenging real-world capacity constraints to be considered by the proposed mathematical formulation. In this paper, it has been developed a new MILP mathematical formulation for the short-term scheduling of multiproduct batch plants subject to resource capacity constraints usually encountered in the manufacturing industry. The proposed model has been solved by using the modeling system GAMS and the solver OSL (IBM, 1991). A significant number of examples involving up to 15 jobs and limited availability of raw materials, utilities and manpower have been successfully solved. Results show an important reduction in the number of variables with regards to current continuous-time approaches and a good computational efficiency.

IIE Transactions, 2001

The problem of scheduling n jobs on a single machine in batches to minimize some regular cost functions is studied. Jobs within each batch are processed sequentially so that the processing time of a batch is equal to the sum of the processing times of the jobs contained in it. Jobs in the same batch are completed at the same time when the last job of the batch has ®nished its processing. A constant setup time precedes the processing of each batch. The number of jobs in each batch is bounded by some value b. If b`n, then the problem is called bounded. Otherwise, it is unbounded. For both the bounded and unbounded problems, dynamic programming algorithms are presented for minimizing the maximum lateness, the number of late jobs, the total tardiness, the total weighted completion time, and the total weighted tardiness when all due dates are equal, which are polynomial if there is a ®xed number of distinct due dates or processing times. More ecient algorithms are derived for some special cases of both the bounded and unbounded problems in which all due dates and/or processing times are equal. Several special cases of the bounded problem are shown to be NP-hard. Thus, a comprehensive classi®cation of the computational complexities of the special cases is provided. 0740-817X Ó 2001``IIE''