PDF《INDEX》

Appl.

Comput. Math.

7 (2008), no.1, pp.143-161 A CONSISTENT CONFIDENCE INTERVAL FOR FUZZY CAPABILITY INDEX A. PARCHAMI ?, M. MASHINCHI ?, PARTOVI NIA ?, § Abstract. Fuzzy process capability indices are used to determine whether a production process is capable of producing items within speci?cation tolerance, where instead of precise quality we have two membership functions for speci?cation limits. In practice these indices are estimated using sample data and it is of interest to obtain con?dence limits for fuzzy capability index given a sample estimate. After introducing 100(1 ? α)% fuzzy con?dence interval by Parchami et al. in

2005 and 2006, for fuzzy capability index ? Cp, our observation leads us to propound an open problem about the consistency property for interval estimation of ? Cp. In this paper we rede?ne some concepts about fuzzy con?dence interval and then we show the consistency property of the fuzzy con?dence interval proposed by Parchami et al. in

2006 holds for almost every α in [0, 1]. Keywords: Fuzzy process capability index, Fuzzy con?dence interval, Triangular fuzzy num- ber, Weak law of large numbers. 1. PRELIMINARIES When we use the precise speci?cation limits, several statistics such as Cp, Cpm, Cpk and Cpk are used to estimate the capability of a manufacturing process, which in most cases it is assumed we have a large sample from a normal population [6]. If we introduce vagueness into speci?cation limits, we face quite new and interesting problems and the classical capability indices could not be applied. For such cases Yongting [18] introduced a process capability index Cp as a real number and it was used by Sadeghpour-Gildeh [16]. Lee investigated a process capability index, Cpk, as a fuzzy set [7]. Parchami et al. introduced fuzzy process capability indices as fuzzy numbers and discussed relations that governing between them when speci?cation limits are fuzzy rather than crisp [11, 12, 14]. The organization of this paper is as follows. In Section 2, we review traditional and fuzzy capability indices and then we review ranking functions in Section 3. In Section 4, we reintroduce a fuzzy con?dence interval for fuzzy capability index ? Cp and then we prove an open problem presented by Parchami at al. in [11, 14]. A conclusion presented in the ?nal section. Let R be the set of real numbers. Assume F(R) be the set of all real valued continuous functions from R to [0, 1], i.e. F(R) = {A|A : R → [0, 1], A is a continuous function}. Also suppose that FT (R) = {Ta,b,c|a, b, c ∈ R, a ≤ b ≤ c}, where Ta,b,c(x) = ? ? ? (x ? a)/(b ? a) if a ≤ x <

b, (c ? x)/(c ? b) if b ≤ x <

c,

0 elsewhere. ? ? ? (1) ?Department of Statistics, Faculty of Mathematics and Computer Sciences, Shahid Bahonar University of Kerman, Kerman, Iran,fax: 0098-341-3221080 e-mail: mashinchi@mail.uk.ac.ir. ?Ecole Polytechnique Federale de Lausanne (EPFL) CH-1015 Lausanne, Switzerland §Manuscript received

15 February, 2008.

143 144 APPL. COMPUT. MATH., VOL. 7, NO.1,

2007 Any A ∈ F(R) is called a fuzzy set on R and any Ta,b,c ∈ FT (R) is called a triangular fuzzy number, which we sometimes write as T(a, b, c). We assume T(a, a, a) be I{a}, the indicator function of a. The following de?nition could be given by using the extension principle [10]. De?nition 1. Let T(a, b, c) ∈ FT (R), k ∈ R and k ≥ 0. The operation ? on FT (R) is de?ned as follows k ? T(a, b, c) = T(a, b, c) ? k = T(ka, kb, kc). (2) 2. PROCESS CAPABILITY INDEX 2.1. Traditional Capability Index. Process capability compares the output of a process with the speci?cation limits by using capability indices. Frequently, this comparison is made by forming the ratio of the width between the process speci?cation limits with the width of the natural tolerance limits, as measured by