An optimal operating policy for the production system with rework

Abstract

Rework, the transformation of defective products generated during production period into serviceable ones, is an important issue in reverse logistics. This paper studies a production inventory system with rework where a stationary demand is satisfied either by production setup with new raw materials or by rework setup with defective items coming from production process. Through the formulation of mathematical models, we find simultaneously the optimal number of production and rework setups in a cycle, their sequence, and the economic production quantity of each setup. Numerical examples are solved to illustrate the validity of the solution procedure proposed.

Introduction

Almost all manufacturing companies are facing an ever-increasing pressure to produce high quality products at low costs. Nevertheless, some defective units are unexpectedly produced, which are to be reworked, scrapped or sold at reduced prices. Usually rework is done to transform them into serviceable ones rather than scrap due to economic reason as seen in semiconductor, glass, steel, pharmaceutical, chemical and food industry. For instance, in steel plants, some unqualified steel products always exist and result in a profit loss, which can be carried back into the furnace to obtain new ones by re-melting and rework through the same process. Another example of rework is in chemistry or pharmacy industry. Flapper, Fransoo, Broekmeulen, and Inderfurth (2002) gave a review for planning and control of rework in process industries.

The earliest approach in the area of joint determination of production and remanufacturing lot sizes was made by Schrady (1967). He analyzed the problem in the traditional Economic Order Quantity (EOQ) setting: deterministic and continuous demand and return, infinite production and recovery rates, 1 production lot with n recovery lots (in short, (1, n) policy). Mabini, Pintelon, and Gelders (1992) extended it to a multi-item case. Then (m, 1) (m production lots with 1 recovery lot) and (1, n) policy were examined by Nahmias and Rivera, 1979, Koh et al., 2002 and Teunter, 2001, Teunter, 2004. Also, Richter, 1996a, Richter, 1996b, Richter, 1997, Richter and Dobos, 1999 and Dobos and Richter, 2000, Dobos and Richter, 2003, Dobos and Richter, 2004, presented their findings on (m, n) policy (m production lots with n recovery lots) with infinite or finite production and recovery rates. Recently, Choi, Hwang, and Koh (2007) studied a recovery system in which a stationary demand is satisfied by recovered products as well as newly purchased products. They treated the sequence of orders for newly purchasing products and setups for recovery process within a cycle as a decision variable. They assumed a constant and continuous external return stream of recoverable items from consumers or terminal markets. Fig. 1 shows the general framework of a production system with rework under this study. It is assumed that due to an unreliable underlying manufacturing process defective items are generated during the production process and they are classified as reworkable inventories through inspection process. Later, they are reworked to “as-new” condition in an EPQ setting. A constant demand is satisfied by either new-manufactured items from production process or reworked ones through rework process. Therefore, all the models mentioned previously are unable to cope with the situation in Fig. 1 where the reverse flow of defective units occurs only during production periods.

In the production system with rework, let cycle be defined as the time span between two successive time points where both the serviceable and reworkable inventories become zero. Let (m, n) policy be defined as the production policy of having m production setups and n rework setups in a cycle. Kim (1981) studied (m, 1) policy for a production system with rework. With (1, 1) policy, Lindner and Buscher (2002) studied a similar system that has a limitation on capacities. Inderfurth et al., 2003, Inderfurth et al., 2005 dealt with deteriorating items. Compared to (m, 1) policy, (m, n) policy results in additional decision variables, that increase the complexities of the system. Fig. 2 shows inventory levels of serviceable and reworkable products with (3, 2) policy in a cycle under two different sequences of production and rework setups. As shown in Fig. 2b, changing the sequence of the production orders and rework setups in a cycle can reduce the inventory holding cost for reworkable products. Therefore, (3, 2) policy alone may not be optimal.

In other words, even though they have the same number of production and rework setups in a cycle, differences in the sequences result in different inventory fluctuations. Therefore, in addition to the values of (m, n), its sequence is an important decision variable in the optimal operating policy of the production system with rework. In this paper, we study the production system with rework with (m, n) policy through the development of mathematical models. We find simultaneously the optimal number of production and rework setups in a cycle, their sequence, and the economic production quantity (EPQ) of each setup. The remainder of this paper is organized as follows. In Section 2, the cost model is developed. Then Section 3 proposes a solution procedure to find an optimal production and rework setup policy. Section 4 reports the results of computational experiments to validate the model and the solution procedure. Finally, conclusions appear in Section 5.