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Article Excerpt
1. Introduction
Quality control is fundamental to the production process. A firm that is not wise enough to implement a quality control program will not survive in today's global competitive environment. If a defective unit reaches a customer, the firm's ability to compete is compromised. This can be realized, for example, in a customer's decision to go elsewhere. Moreover, a competitor might use this information to spread reports of inadequate quality among customers.
Practitioners and researchers use several approaches and methods in the area of quality control. Among these are ones that are based on performing inspections and making decisions according to their results. In these methodologies there are two inspection paradigms that are differentiated by the way in which inspections are made.
(1.) On-line inspection: units are inspected during the manufacturing process.
(2.) Off-line inspection: units are inspected after the manufacturing process is completed.
Off-line inspection differs from sampling methods inasmuch as the order in which the products are produced is maintained. In sampling methods, when the production process is completed a random sample is chosen for inspection. After these units are inspected, a decision is made whether to accept or reject the whole batch (some sampling methods allow, under certain circumstances, for additional samples to be taken). In off-line inspection methods, a particular (non-random) item from the batch is inspected, and based on the inspection result, a decision is made concerning what to do next.
On-line inspection regimes are considered more economical and effective than off-line inspection ones. However, there are situations in which on-line inspections are infeasible (or impractical); thus off-line inspections are performed.
As will be reviewed below, to the best of our knowledge, it would seem that all sound work in the off-line inspection literature assumes that the inspection process is error-free. Clearly this is not the case: the inspection result may not indicate the true condition of the inspected unit. The effect of inspection errors when inspection are performed online was tested and found to be significant, as described in Kwei and Schneider (1987) and Maghsoodloo (1987). In contrast, our research explores the effect of inspection errors in an off-line inspection regime.
The particular process, which we investigate, sequentially produces a batch of units. The process itself is subject to random failures. The process starts in the IN state, during which conforming units are produced. During the processing of a unit, a failure can occur with known probability. If a failure does occur, the process shifts to the OUT state where non-conforming units are produced. After shifting to the OUT state, the process remains there until the end of the batch. The unit that was produced during the failure is called the transition unit.
This particular process appears in the food industry as written in Raz et al. (2000). Actually, it is also typical of almost every process that deals with intermixtures: small containers are filled from a big tank which contains the intermixture (chemicals, food, drinks, etc.). The intermixture ingredients should be at certain proportions, otherwise the container is not qualified for sale. The small containers are produced in a batch in which a big tank is emptied into them. During the process one (or more) of the intermixture ingredients, for some reason, may no longer meet required proportions and the remaining intermixture is not fit for sale.
In order to ensure the quality of the outgoing products, the production order of the units is preserved and quality inspections are performed. The inspections themselves are subject to inspection errors, where two kinds of errors are possible.
1. Type I error - a conforming unit is erroneously identified as a non-conforming one.
2. Type II error - a non-conforming unit is erroneously identified as a conforming one.
It is assumed that each item can only be inspected once.
If inspection errors did not exist, we could determine the exact position of the transition unit in the batch. Moreover, we could do so optimally with the policy developed in He et al. (1996). However, because inspection errors in fact do occur, the transition unit cannot be identified with certainty, but may be identified with a certain confidence. Our goal is to minimize the expected number of inspections needed to identify the transition unit with a given confidence level (1). We determine the optimal inspection policy. Furthermore, given that it may take an inordinate amount of time to compute the optimal policy, we also investigate heuristic policies.
This paper is organized as follows: we begin by reviewing the literature in Section 2. Then we give a detailed description of the process, objective function, and model in Section 3. Afterward, in Section 4, using dynamic programming we specify how to find the optimal policy. In Section 5 we develop four heuristics. We describe the experimental design, its results and implications in Section 6. Finally, we present conclusions and topics for future research in Section 7.
2. Literature review
Hassin (1984) was one of the first to refer to off-line inspection in his research, though he used the terminology of dichotomous search. He found a search strategy that minimizes the expected number of inspections until identifying the transition unit for a process with a constant failure rate where that last unit is known to be non-conforming. He et al. (1996) extended Hassin's work to the case where the quality of the last unit is unknown. It should be noted that both Hassin (1984) and He et al. (1996) also developed heuristics.
Herer and Raz (2000) investigated parallel inspections; they developed both an optimal policy as well as heuristic policies based on the theory of information developed by Shannon and Weaver (1949). Raz et al. (2000) took offline inspection research in a new direction by investigating economic optimization rather than identifying the transition unit. They found an inspection policy that minimizes the sum of inspection and penalty costs. Finkelshtein et al. (2005) developed an optimal policy for off-line inspection in a regenerative process. Their model, which can be viewed as an extension of Raz et al. (2000), is also based on economic optimization. In another extension of Raz et al. (2000), Bendavid and Herer (2006) considered a process in which non-conforming units can be produced in the IN state and conforming units in the OUT state.
Sheu et al. (2003) attempted to extend the research of Raz et al. (2000) to consider inspection errors. Their analysis is flawed. In addition, the interested reader is encouraged to compare the paper by Sheu et al. (2003) with that of Raz et al. (2000). Wang (2007) notes the flaw in Sheu et al. (2003) and attempts to present a corrected optimal policy. Their analysis assumes that after a unit is inspected and found to be conforming (non-conforming) then all units preceding (following) the inspected unit are accepted (rejected). Tzimerman (2006) found that a better policy is possible if one considers all the inspection results before accepting or rejecting units. However, the last two approaches differ in their objectives. While Wang (2007) investigates the economic approach of inspections, Tzimerman (2006) deals with finding the transition unit with a given confidence level.
3. Model formulation
3.1. Notation
The following notation will be used to describe the model (we follow the notation of Raz et al. (2000), whenever possible):
N = Number of units in the batch.
[P.sub.j] = Probability that the process will remain in control during the production of unit j given the process was in control during the production of unit j - 1.
1 - [p.sub.j] = Probability of a failure during the production of unit j.
[FNU.sub.j] = Probability that unit j is the transition unit (i.e., the first non-conforming unit), that is [FNU.sub.j] = ([[product].sub.m=1.sup.j-1]pm)(1-[p.sub.j][for all]j > 1 and [FNU.sub.1] = [p.sub.1]. Note that 1 - [[SIGMA].sub.j=1.sup.N] [FNU.sub.j] is the probability that...
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