Production, Manufacturing and LogisticsThe value of setup cost reduction and process improvement for the economic production quantity model with defects
Introduction
An enormous body of research has stemmed from the original economic production quantity (EPQ) model. The basic model determines the optimal lot size when demand and production rates have known, deterministic values. The manufacturing process is assumed to be perfect; production occurs without defects. One branch of the literature has accounted for imperfections in the manufacturing process. Much of this work has focused on the benefits of smaller lot sizes in imperfect systems, and the relationship between quality improvement (improving yield) and reducing setup costs. Wright and Mehrez [17] provide a taxonomy and survey of the work relating inventory and quality issues.
Perhaps the first paper to relate quality and lot size was written by Porteus [11]. Porteus describes a system that begins each production run in control (i.e. producing only good units). As each unit is produced, there is a probability p that the system goes out of control, at which time all subsequent units (until the end of the production run) are defective. The time until the process goes out of control therefore follows a geometric distribution. Porteus used this model to study the optimal setup investment in relation to reducing the probability p of the process going out of control. Other researchers have used this model to investigate the economic benefits of reducing setup costs (see [6], [5]). More recently, Affisco et al. [1] investigated investment in quality improvement and setup cost reduction in a joint supplier–customer system with defects produced at a known constant rate. Rosenblatt and Lee [14] present a model where the process goes out of control after some exponentially distributed time. Once out of control, the process produces units at a constant defect rate α.
We extend prior work in this area by providing a general method for modeling production yields. Several papers (see [2], [1]) model systems in which defects are produced at a known constant rate. Others, such as Porteus, model the production of defects as a random process. Yano and Lee [18] provide a representative survey of research involving lot sizing with random yields, together with a useful description of the modeling issues involved. They list several methods for modeling random yield that are represented in the literature, which include:
- 1.
modeling defects as a Bernoulli process, where each unit is defective with probability p (see [4], [19]);
- 2.
describing a distribution for the overall fraction of defective units (see [4], [15], [9]);
- 3.
specifying a distribution for the time in which the process remains in control (i.e. producing good units), after which the process is out of control (a fraction of the units are defective).
In the first two methods, the fraction of defective units is stochastically constant with respect to the length of the production run. In the third case it is stochastically increasing with the duration of the run (see [4], [11], [13]).
Yano and Lee [18] also point out that expected cost is the dominant performance measure among models described by the literature. In this context, our approach generalizes the three methods described above, as well as the case in which defects are produced at a known rate which may vary over time. That is, we allow any random yield function as long as all defective units are repaired instantly at some per unit cost. If all units are not repaired instantly, then one must allow for the possibility of lost sales or backorders, or impose conditions on the random yield function such that the effective production rate always exceeds the demand rate.
It should be noted that while our approach generalizes several prior methods, some recent research has addressed other aspects of the EPQ problem such as stochastic demand [16], coordination between buyer and seller [3], [7] and choosing an inspection schedule [8], [12].
We focus on systems in which the process deteriorates over time, although we point out where our results may be generalized (in the opposite direction) for a process that improves over time. We develop properties of the optimal production quantity and optimal cost and consider the options of investing in reducing setup cost and improving process quality. Expressions for the marginal value of setup cost reduction and process improvement are also developed.
This paper is organized as follows. In Section 2 we present our model formulation and establish results regarding the optimal solution. In Section 3 we investigate the percentage cost penalty of using the standard economic production quantity even when the process is imperfect. In Sections 4 Optimization of setup cost reduction, 5 Optimization of process quality improvement we explore optimizing investment in setup cost reduction and process improvement. Concluding remarks are offered in Section 6.
Section snippets
Model formulation and notation
We consider the production system of a single item on a single machine which produces some (time-varying) fraction of defective items. Specifically, we assume that the fraction of defective goods produced is described by u(x) at any time x, with 0 ⩽ u(x) ⩽ 1. As noted above, we focus on the situation in which the process deteriorates over time, u′(x) ⩾ 0. All defective units are repaired instantaneously at a per unit cost s. If repair takes some constant time then the analysis either remains the same
Cost penalty of using the EPQ
Despite the strong underlying assumptions, the EPQ can often provide a reasonable solution for more complicated settings (see [16]). We now examine the percentage cost penalty of using the EPQ when the production process is imperfect. Defineand
The percentage cost penalty if using the EPQ rather than the optimal quantity is given by
Analyzing Δ for general u(x) appears to be difficult. We first present general expressions for the change in
Optimization of setup cost reduction
In this section we consider investing in reducing the setup cost. First, we develop an expression for the marginal value of setup cost reduction for general u(x). We analyze how the cost parameters K, h and s affect this marginal value. We then consider the linear deterioration case and a specific functional form for investment in setup cost reduction.
From Eq. (14) and the definition a = KD/R, the marginal change in optimal cost cost as a function of K is
Eq. (58) together
Optimization of process quality improvement
In this section we consider investing in improving process quality. We examine two kinds of process improvement: (1) a constant downward shift ϵ in the defect rate u(x) such that the new process produces defects at a rate , and (2) a scale factor p such that the new defect rate is .
We first develop expressions for the marginal values of each kind of process improvement for general u(x), analyzing how the cost parameters K, h and s affect these marginal values. We then
Conclusions
In this paper we consider the classical economic production quantity model with defects produced according to some time-varying function u(x). Our results hold for any random yield function as long as units are repaired instantly. With a deteriorating process, the optimal run length is shorter than the run length implied by the EPQ. Furthermore, the faster the process deteriorates the shorter the optimal run length. For the special case of linear deterioration, the cost penalty for using the
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2019, International Journal of Production EconomicsCitation Excerpt :The first stream assumes that the proportion of defective items follows a probability distribution that is independent of the lot size and identically distributed across batches; see (Silver, 1976; Shih, 1980; Salameh and Jaber , 2000; Goyal and Cardenas-Barron, 2002; Papachristos and Konstantaras, 2006; Khan et al., 2011). The second stream of research considers the random generation of defective items when a production system is in an out-of-control state, either following the production of a unit (Porteus, 1986) or after producing for some time (Rosenblatt and Lee, 1986; Freimer et al., 2006). The third stream assumes that the number defective within a batch follows a Binomial distribution (Levitan, 1960; Grosfeld-Nir and Ronen, 1993; Nasr et al., 2017).
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2016, Applied Mathematical ModellingCitation Excerpt :Shih and Wang [34] considered a model in which system may similarly be in or out of control, and they minimized the total cost per product per cycle time. In the model of Freimer et al. [35] the system produces time varying proportion of defective parts which can be repaired at some cost. They propose investments to improve process quality and reduce setup cost.
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2014, Applied Mathematical ModellingCitation Excerpt :Thus, they presented a model that simultaneously optimizes the reorder point, lot size, lead time and process quality level. Freimer et al. [8] have studied the impact of imperfect items on EPQ decisions. They considered two types of investment in process improvement that are: (i) improving process quality and (ii) reducing setup costs.
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2013, European Journal of Operational ResearchCitation Excerpt :Vörös (1999) considered a model in which it is assumed that the production process is interrupted once quality issues are detected. Freimer et al. (2006) assumed instead that the process deteriorates over time, with defective units being repaired instantaneously at a known unit cost. In particular their study established how the marginal value of setup cost reduction (or process improvement) is related to the optimal run length.