Stochastics and Statistics Optimal means for continuous processes in series Shokri Z. Selim, Walid K. Al-Zu’bi ⇑ Department of Systems Engineering, King Fahd University of Petroleum and Minerals, 5067, Dhahran 31261, Saudi Arabia article info Article history: Received 9 January 2010 Accepted 9 October 2010 Available online 21 October 2010 Keywords: Quality control Process target levels Properties of single machine targeting problems abstract We discuss the problem of determining the means of a set of processes in series. Each process generates a random quality characteristic that in turn has lower and upper specification limits. Depending on the value of the quality characteristic, an item can be reworked, scrapped or forwarded to the next process. An item is reworked at the same stage. The processes are continuously running, hence we develop the ‘‘long term” probabilities of meeting specifications, and of violating each limit. These are used to con- struct the profit function to be maximized. We present a recursive form of the profit function that yields a very efficient method for determining the means. The method relies on solving single stage problems. Next, we turn our attention to the single stage problem and show that if the quality characteristics are normally distributed, then a local optimum is also global. Finally, we present a very fast solution method for this problem. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Process targeting is an important problem in production eco- nomics and quality control. In process targeting, it is assumed that process parameters or machine settings are variables, thus, the objective of the problem is to find the optimum values of process parameters or machine settings that will achieve certain econom- ical objectives. A number of models have been proposed in the lit- erature for determining an optimum process target (mean and/or variance) – as discussed in the next section. Products in modern day production systems are often processed through multi-stage production systems, however, the great majority of process target models in the literature were derived assuming a single-stage production process, except for Al-Sultan (1994), Al-Sultan and Pulak (2000), and Bowling et al. (2004). Most process target models have been developed using short- term probability of the state of an item in the process, i.e. probabil- ities of meeting or violating the specifications. Short-term probabilities represent a snap-shot of the process and apply to pro- ducing a single item. However, in a continuous process using long term probabilities is recommended to describe system dynamics. Bowling et al. (2004) introduced a Markovian approach to study production systems where the output of a stage could be accepted, reworked, or scraped. The approach is supposed to generate the absorption probabilities into scrap, rework, and accept states for each production stage. We show in this paper that these probabil- ities are partially generated through that approach. In addition, we develop a model for N-stage serial production system using an alternative method. This paper is organized as follows. Section 2 presents literature review. In Section 3, we examine the two-stage production system considered by Bowling et al. (2004). Then we present an alternative approach to model N-stage production system. We show that the profit function can be cast in a recursive form that results in a very efficient solution method in which we solve successive single stage problems. We prove that for a single stage problem with normal quality characteristic, any local solution is also global. Then we present a very fast algorithm for solving this problem. Numerical examples are given for illustrative purposes in Section 4. The con- clusion follows in the last section. 2. Literature review The initial process targeting problem addressed is the ‘‘can fill- ing problem”. The first real attempt to tackle this problem was in Springer (1951) who considered the problem of finding the optimal process mean for a canning process when both upper and lower control limits are specified. Bettes (1962) studied the same prob- lem as in Springer (1951) except that only the lower limit was specified. Hunter and Kartha (1977) present a model to determine the optimal process target under the assumption that the products meeting the requirement are sold in a regular market at a fixed price, while the underachieved products are sold at a reduced price in a secondary market. Nelson (1978) determined approximate solutions to the model proposed by Hunter and Kartha (1977). Nelson (1979) considered the same problem by Springer (1951). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.10.013 ⇑ Corresponding author. Tel.: +966 56 0555294. E-mail addresses: selim@kfupm.edu.sa (S.Z. Selim), zwalidk@kfupm.edu.sa (W.K. Al-Zu’bi). European Journal of Operational Research 210 (2011) 618–623 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor