Diagnostic errors and repetitive sequential classifications in on-line process control by attributes

Abstract

The procedure of on-line process control by attributes, known as Taguchi’s on-line process control, consists of inspecting the mth item (a single item) at every m produced items and deciding, at each inspection, whether the fraction of conforming items was reduced or not. If the inspected item is non-conforming, the production is stopped for adjustment. As the inspection system can be subject to diagnosis errors, one develops a probabilistic model that classifies repeatedly the examined item until a conforming or b non-conforming classification is observed. The first event that occurs (a conforming classifications or b non-conforming classifications) determines the final classification of the examined item. Proprieties of an ergodic Markov chain were used to get the expression of average cost of the system of control, which can be optimized by three parameters: the sampling interval of the inspections (m); the number of repeated conforming classifications (a); and the number of repeated non-conforming classifications (b). The optimum design is compared with two alternative approaches: the first one consists of a simple preventive policy. The production system is adjusted at every n produced items (no inspection is performed). The second classifies the examined item repeatedly r (fixed) times and considers it conforming if most classification results are conforming. Results indicate that the current proposal performs better than the procedure that fixes the number of repeated classifications and classifies the examined item as conforming if most classifications were conforming. On the other hand, the preventive policy can be averagely the most economical alternative rather than those ones that require inspection depending on the degree of errors and costs. A numerical example illustrates the proposed procedure.

Introduction

Taguchi et al., 1989, Taguchi et al., 2004 presented an economical design to monitor on-line process control for attributes. The inspection system is automatic and allows sampling and verifying only a single item at each time. In general the proposed system can be implemented in high-speed electronics manufacturing facilities, where the testing equipment is connected to a central computer and data is taken automatically as produced items are being tested. The procedure of on-line control to monitor the process was studied by many authors such as Nayebpour and Woodall, 1993, Gong and Tang, 1997, Borges et al., 2001, Wang and Yue, 2001, Dasgupta, 2003, Trindade et al., 2007. An excellent survey of process maintenance policies can be found in Wang (2002). The purpose is to introduce an inspection policy that minimizes average costs by means of shifts in process parameters. In a random time the 100% conforming fraction process (State I: p₁ = 1) shifts to another process with conforming fraction lower than 100% (State II: p₂ < p₁). The control consists of inspecting the mth item at every m produced ones. If the inspected item is considered conforming, the production goes on; otherwise, it is stopped for adjustment. Taguchi’s approach (Taguchi et al., 1989, Taguchi et al., 2004) does not assume explicitly a probability function for the shift of the parameter from p₁ = 1 to p₂ < p₁ and many simplifications and approximations were used to calculate the average cost and the optimum value of sampling interval m, (hereafter denoted as m°).

Nayebpour and Woodall (1993) developed an alternative procedure in which the shifts from State I to State II were described by a geometric distribution. They concluded that their approach is more adequate than Taguchi’s (Taguchi et al., 1989), mainly if p₂ > 0. Their main objection to Taguchi’s approach is the use of a uniform distribution to describe the shifts from State I to State II. This assumption allows obtaining a more simplified cost function; however, it does not describe well the practical situations.

Wang, 2007, Borges et al., 2001 pointed out that the inspection procedures discussed in Taguchi et al., 1989, Nayebpour and Woodall, 1993 may present diagnosis errors and compromise the determination of the optimum inspection interval (m°). These authors presented an approach incorporating classification errors and their economical impact if they were not taken into account. However they did not consider the realization of repeated (and independent) classifications on the inspected item as a possible way to reduce the average cost. Greenberg and Stokes, 1995, Ding et al., 1998, Ding and Gong, 2008 demonstrated that repeated classifications can produce savings when errors are present in a classification system.

Trindade et al. (2007) assumed the possibility of making r(r ⩾ 1 and integer) classifications on the inspected item. If at least (w = ⌊r/2⌋ + 1) in r classifications are conforming, then the inspected item is declared as conforming. The design consists of determining the optimum values: interval sampling (m) and the number of repeated classifications (r). However this policy (proposed by Trindade et al. (2007)) might be making unnecessary classifications to reach a final judgment (as conforming or non-conforming) of the inspected item. For example: r = 5 and the first three classifications were conforming. These results were enough to judge the inspected item as conforming, without the need to make the additional two classifications. This procedure is more economical because it is not necessary to perform all r (fixed) repeated classifications at every inspected item.

This paper expands the procedure discussed in Quinino and Ho (2004), namely, the inspected item is classified repeatedly (and independently) until a (integer and a ⩾ 1) conforming classifications or b (integer and b ⩾ 1) non-conforming classifications are observed (admitting the possibility of misclassification errors). The examined item is judged as conforming if a conforming classifications are observed firstly, or it is judged as non-conforming if b non-conforming classifications are observed firstly. So at most (a + b − 1) classifications are needed to make the judgment of the examined item. The main purpose is to find values of a, b and m that minimize the average cost of the control system. It is also desirable to compare these results with those obtained from a simple preventive maintenance approach (that is, when adjustments are previously determined and no inspections are made).

This paper is organized as follows: Section 2 describes the probabilistic model of the repeated classification process. In Section 3, the economical model is developed to determine the optimum designs. In Section 4, the preventive adjustment approach is evaluated and compared with the one proposed in this article. A numerical example to illustrate the proposed model is discussed in Section 5 and the conclusions and final remarks are presented in Section 6.