Mathematical models for a batch scheduling problem to minimize earliness and tardiness

Journal of Industrial Engineering and Management JIEM, 2018 – 11(3): 390-405 – Online ISSN: 2013-0953 – Print ISSN: 2013-8423 https://doi.org/10.3926/jiem.2541 Mathematical Models for a Batch Scheduling Problem to Minimize Earliness and Tardiness Basar Ogun1 1 2 , Çigdem Alabas-Uslu2 Eczacibasi Yapi Gerecleri Furniture Plant (Turkey) Marmara University, Department of Industrial Engineering (Turkey) , Received: November 2017 Accepted: March 2018 Abstract: Purpose: Today’s manufacturing facilities are challenged by highly customized products and just in time manufacturing and delivery of these products. In this study, a batch scheduling problem has been addressed to enable on-time completion of customer orders in a lean manufacturing environment. The problem is optimizing the partitioning of product components into batches and scheduling of the resulting batches where each customer order is received as a set of products made of various components. Design/methodology/approach: Three different mathematical models for minimization of total earliness and tardiness of customer orders are developed to provide on-time completion of customer orders and also, to avoid excess final product inventory. The first model is a non-linear integer programming model whereas the second is a linearized version of the first. Finally, to solve larger sized instances of the problem, an alternative linear integer model is presented. Findings: Computational study using a suit set of test instances showed that the alternative linear integer model is able to solve all test instances in varying sizes within quite shorter computer times compared to the other two models. It has also been showed that the alternative model is able to solve moderate sized real-world problems. Originality/value: The problem under study differentiates from existing batch scheduling problems in the literature owing to the inclusion of new circumstances that are present in real-world applications. Those are: customer orders consisting of multi-products made of multi-parts, processing of all parts of the same product from different orders in the same batch, and delivering the orders only when all related products are completed. This research also contributes to the literature of batch scheduling problem by presenting new optimization models. Keywords: batch scheduling, earliness and tardiness, mixed integer programming formulation, on-time delivery 1. Introduction A batch-processing machine (BPM) can simultaneously process several jobs in a batch, that is, different jobs in the same batch are processed at the same time. BPM scheduling problem is defined as a combination of sequencing and partitioning problems with an objective function to be optimized (Albers & Brucker, 1993). The aspect of partitioning is to find a partition of all jobs into batches whereas the sequencing aspect is to find a sequence of the batches formed by the partitioning. -390- Journal of Industrial Engineering and Management – https://doi.org/10.3926/jiem.2541 BPM scheduling problem addressed in this study mainly stems from scheduling of a cutting machine in furniture manufacturing. The short-term production planning problem considers the following input data: customer orders which consist of a set of products, bill of materials which involves a set of components for each product, due dates of the customer orders, capacity of BPM (cutting machine) in terms of number of components and finally process time of BPM. Decision problem is to cluster the components of products into batches that have limited capacity and also to find a production sequence for these batches. The problem intends to minimize the sum of the weighted sum of early and late completion times of customer orders by determining optimum composition of each batch in terms of both type and number of components as well as optimum production sequence of these batches. To emphasize the importance of on-time completion of customer orders, we have called the problem BPM-On-time scheduling. One of the differentiating aspect of BPM-On-time scheduling problem arises from its policy of order of deliveries. According to this policy, a customer order cannot be delivered unless all products ordered by the same customer are completed. The other aspect is that products are made of several components and all components of a product belonging to the same customer order must be processed simultaneously because of technical requirements. In other words, products of a customer order can be partitioned into different batches but components of a product in the same customer order cannot be partitioned. This constraint is different from the constraint of incompatible job families considered in the literature since it does not allow processing of jobs from different job families simultaneously. Existing literature, which is closely related to the completion of customer orders just in time, focuses on either processing of jobs in batches to minimize due date related objectives or delivering of completed jobs in batches to minimize inventory and delivery costs. The latter is called batch delivery and was first introduced by Cheng and Kahlbacher (1993). Batch delivery problem mainly considers finding optimal partitions of jobs into batches for delivery. In BPM-On-time scheduling problem however, a customer order is a given set of products and it must be delivered once all the products ordered are completed. Therefore, BPM-On-time scheduling problem should not be treated in batch scheduling literature as batch delivery. In this study, we aim to contribute to the literature of batch scheduling by emphasizing the differentiating constraints of BPM-On-time scheduling problem. We also aim to contribute by suggesting optimization models to BPM-On-time scheduling problem since there is room for further research. In this study, firstly, a nonlinear integer programming formulation to the problem, called Model-NL, is provided. Then, Model-NL is converted into an integer linear programming (ILP) model, named Model-PL, using the piecewise linearizing method. Finally, we suggest an alternative ILP model, Model-A, to deal with difficulties of both Model-NL and Model-PL. The proposed models, Model-A, Model-NL and Model-PL are compared using a suit set of test instances. The experimental studies confirm that Model-A strongly outperforms the other models in terms of both run time and problem size which can be solvable optimally. Section 2 addresses the related works in the literature. In section 3, BPM-On-time scheduling problem is described and the three mathematical models are explained in detail. Section 4 contains experimental study in which generation of the test instances and results obtained by the models are given. A case study from furniture industry is presented in Section 5 to show the performance of suggested Model-A in real life. Finally, conclusions and future directions are discussed in the last section. 2 (...truncated)