Using penalty function and Tabu search to solve cell formation problems with fixed cell cost

Abstract

In this paper, an integrated approach for manufacturing cell formation with fixed charge cost is presented. The solution of the problem includes not only cell formation decisions but also cell set-up decisions. A mixed integer non-linear programming model is formulated to solve the problem. The NP-hardness of the problem makes direct solution computationally prohibitive for real-life applications. A heursitic algorithm was developed to solve the problem efficiently based on the features of the model and model duality analysis. Tabu search was used to find the optimal or sub-optimal solutions of the problem. Numerical examples are presented.

Scope and purpose

Today's manufacturing industry is facing strong competition in providing high quality and low cost products to ever demanding consumers. Cellular manufacturing is one of the widely used approaches to improve manufacturing productivity to achieve this purpose. In cellular manufacturing, machine tools are grouped into different manufacturing cells for improved efficiency and flexibility. This research proposes a mathematical programming model to form the cells so that material handling cost, machine tool cost and eventually the overall production cost can be minimized. We also developed an efficient solution method to solve the complicated mathematical model.

Introduction

Cellular manufacturing (CM) is a widely studied approach for organizing machines and people into groups to produce a variety of parts in part families. CM is also an effective approach to implement flexible manufacturing systems (FMS) and is normally associated with automated batch production [1]. Successful implementation of CM will result in reduced set-up times, reduced material flow and in-process inventory, better system management, improved production efficiency and product quality [2], [3]. In the last 30 years, various CM problems have been studied by many researchers as summarized in [4], [5]. More recently, CM research has been expanded to developing integrated models and methods as discussed in [6], [7]. For example, Atmani et al. [8] presented a model for simultaneous cell formation and operation allocation. Lockwood et al. [9] studied CM scheduling problems. Production planning in CM systems was discussed in [10], [11]. A variety of effective methods were developed to investigate and solve CM problems. These include traditional mathematical programming [12] and simulation studies [13], [14]. Non-traditional methods such as neural networks, Tabu search, genetic search and searches using simulated annealing have also been used to search for near optimal solutions for various CM system problems. Recent work can be found in [15], [16], [17], [18]. Some of the methods were compared in [19] for their effectiveness and efficiency in solving CM problems. In the research presented in this paper, we also followed the integrated approach to study a cellular manufacturing problem with fixed charge cost. In most existing cell formation models, the number of manufacturing cells has been given and the cells have been identified. The solutions of the problems were then to allocate proper machines to the cells and to decide the part families to be processed. Objectives of such models are generally to minimize certain criteria such as material handling cost or dis-similarities among part families. The integer programming model developed in this paper considers a more general situation where the number of cells is to be determined by the solution of the model. The model is similar to that of a fixed charge problem [20] with decision variables for cell formation. The objective of the model is to minimize the total cost including inter-cell material handling cost, fixed cell set-up cost and fixed machine and operating costs in the system. Detailed description of the problem and the development of the integer programming model are given in the next section. Due to the NP-hardness of the problem we employed an embedded optimization technique to transform the original model to a parametric LP problem following the approach developed in [21]. The parametric model was then transformed into a unified computable penalty problem. This solution procedure and model transformations are presented in Section 3. We used a Tabu search [22] procedure to find optimal or near-optimal solutions of the transformed problem. As shown in recent literature, Tabu search is an effective search process to quickly arrive in optimal or near-optimal solutions of large-scale combinatorial optimization problems. Details of applying the Tabu search process to solve the transformed cell formation model are presented and discussed in Section 4. We tested the developed methods using several numerical examples. Data and computational results of two examples are presented in Section 5. In Section 6, we present the summary, conclusions and our plans for future research in this area.