Lot sizing in case of defective items with investments to increase the speed of quality control Zsuzsanna Hauck, József Vörös n University of Pécs, Faculty of Business and Economics Pécs, Rákóczi 80, Hungary article info Article history: Received 8 January 2014 Accepted 22 April 2014 Keywords: Lot sizing Quality checking Backlogs Optimization abstract In many cases the quality of each item in a lot is checked. Speeding up the quality checking process increases the responsiveness of the system and saves cost. The percentage of defective items is a random variable and two models are proposed. In one of the models the system remains always at the same state, while in the other one after each order cycle, the state of the system may change, thus the percentage of defective items may be different in consecutive periods. In both cases the speed of the quality checking is a variable, and procedures are provided to find the optimal lot sizes and screening speed for general and specific investment cost functions. The characteristics of the two model settings will largely be different when the percentage of defective items is high. Among the important managerial insights gained is that a high unit backlogging cost, especially spurs the system to invest more intensively into improving the quality checking process. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction We consider an inventory system when products arrive in lots and the quality of each item in a lot is checked in order to decide whether it is acceptable or not. Defective items accumulating during the screening process are transported back to the supplier in a lot, reworked, or sold, and the good ones are used to satisfy demand. Demand is constant over time. The issue is interesting when we intend to determine the optimal ordering quantity, or when we plan the optimal production quantity. In producing medical instruments for example, when software is installed, the quality of hardware is always checked, i.e. the quality of each item is checked. The portion of defective items is considered to be a random variable. This basic concept was defined by Salameh and Jaber [15], who have inspired significant number of new papers, and this work directly belongs to this stream as well. Not long ago, Khan et al. [8] wrote a comprehensive summary on EOQ models including the problem of defective items, and this fact gives the flexibility of not summarizing the main results again but focusing on the relevant issues only. Salameh and Jaber [15] explicitly determined the optimum ordering quantity by taking the mini- mum of the expected value of the inventory and setup costs over unit time. Later, Maddah and Jaber [13] suggested a new process in which we have to minimize the ratio of the expected value of inventory and setup costs occurring in a cycle and the expected length of a cycle. Vo ro s [18] pointed out that different model settings can be aligned to each of these procedures. The original Salameh and Jaber [15] procedure gives the optimal lot size for the model when the system randomly gets into a state at the beginning, but the consecutive cycles inherit this state. On the other hand, when cycles are independent and may get into different states in each cycle, the Maddah and Jaber [13] procedure gives the optimum lot size. Both approaches use an important assumption, namely that p r1  z, where p is a random variable defined in [0, 1], denoting the fraction of the defective items in a lot, while z is a positive number, and 0 oz o1. Both models assume that the fraction of defective items is low enough to avoid shortages. In our case, when we assume that the speed of the quality checking process is a decision variable, this assumption will be easily violated and shortages will occur frequently. Papchristos and Konstantaras [14] and later Khan et al. [8] in their summary expressed that the condition to avoid shortages mentioned above is not sufficient to prevent non planned shortages. Papchristos and Konstantaras [14] also pointed out that even when p is replaced in the constraint by its expected value, the condition is still not sufficient to prevent non planned shortages. They expressed the view that there is no simple sufficient condition to prevent non planned shortages. Let us note that shortages may easily occur due to machine break downs as well, and the production-inventory-maintenance literature handles the problem in many ways. Jonrinaldi and Zhang [7] for example, similarly to Berthaut et al. [1], protect the supply system by developing safety stocks (and consequently shortages Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/omega Omega http://dx.doi.org/10.1016/j.omega.2014.04.004 0305-0483/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. E-mail addresses: hauckzs@ktk.pte.hu (Zs. Hauck), voros@ktk.pte.hu (J. Vörös). Please cite this article as: Hauck Zs, Vörös J. Lot sizing in case of defective items with investments to increase the speed of quality control. Omega (2014), http://dx.doi.org/10.1016/j.omega.2014.04.004i Omega ∎ (∎∎∎∎) ∎∎∎–∎∎∎