Construction of Control Charts Using Fuzzy Multinomial Quality

Vahid Amirzadeh

Construction of Control Charts Using Fuzzy Multinomial Quality

Vahid Amirzadeh

2008, Journal of Mathematics and Statistics

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Control charts are the simplest type of on-line statistical process control techniques. One of the basic control charts is p-chart. In classical p-charts, each item classifies as either "nonconforming" or "conforming" to the specification with respect to the quality characteristic. In practice, one may classify each item in more than two categories such as "bad", "medium", "good", and "excellent". Based on this, we introduce a fuzzy multinomial chart (FM-chart) for monitoring a multinomial process. Control limits of FM-chart are obtained by using the multinomial distribution and the degrees of membership which one assigned to the distinct categories. The comparison of the FM-chart and the p-chart based on a food production process illustrates that the FM-chart leads to better results.

INTRODUCTION

Control charts are widely used for monitoring and examining a production process. The power of control charts lies in their ability to detect process shifts and to identify abnormal conditions in the process. This makes possible the diagnosis of many production problems and often reduces losses and brings substantial improvements in product quality. In 1924, Walter Shewhart designed the first control chart and proposed a general model for control charts as follows: Let w be a sample statistic that measures some quality characteristic of interest. Moreover, suppose that the mean of w is w µ and the standard deviation of w is where k is the "distance" of the control limits from the center line, expressed in standard deviation units.

Control charts are constructed and operated with data collected from the process. The data collected should represent the various levels of the quality characteristic associated with the product. The characteristics might be measurable on numerical scales, such as length, weight, voltage, etc., in which case control charts for variables are used. These included the X -chart for controlling the process average and the R -chart (or S -chart) for controlling the process variability. If the quality-related characteristics cannot be represented in numerical form, such as characteristics for appearance, softness, colour, etc., then control charts for attributes are used [4] . Product units are classified either as "conforming" or "nonconforming", depending upon whether or not they meet specification. The p -chart is used to monitor the fraction nonconforming units. In p -chart, control limits calculate by using the normal approximation.

Linguistic scales are commonly used in industry to express properties or characteristic of products. Typically, the conformity to specifications on a quality standard is evaluated onto a two-state scales, e.g. acceptable or unacceptable, good or bad, and so on. However the binary classification might not be appropriate in many situations, where product quality can assume more intermediate states. The assignment of weights, to reflect the degree of severity of product nonconformity has been adopted in many circumstances. Different numbers of weights are assigned to each class and the total number of weights is monitored with some control charts for defectives [9] . This approach requires the ability to classify each state uniquely into one of several mutually exclusive classes with well-defined boundaries between them. Quite often, there is some vagueness in the judgment applied by human quality inspectors, especially when dealing with characteristics that are evaluated subjectively. The vagueness present in linguistic variables may be treated formally with the help of fuzzy set theory.

Zadeh [10] In 1965, introduced the notion of fuzzy sets. After that, there have been efforts to apply it in statistics [7,8] . When products are classified into mutually exclusive linguistic categories, fuzzy control charts are used. Different procedures are proposed to construct these charts. Raz and Wang [5,9] developed fuzzy control charts for linguistic data which are mainly based on membership and probabilistic approaches. Kanagawa et al. [3] proposed an assessment of intermediate quality levels instead of the traditional binary judgment. Gulby et al. [2] proposed α -level fuzzy control charts for attributes in order to reflect the vagueness of data and tightness of inspection. This work attempts to construct a new fuzzy multinomial control chart (namely FM -chart) for linguistic variables. To this end, the control limits of the FMchart are introduced. The FM -chart is able to deal with a linguistic variable which is classified in more than two categories.

Therefore the FM -chart provides more information than p -chart. This fact is illustrated by an example from a production process. [11] .

MATERIALS AND METHODS

For example, on a production line, a visual control of the corking and closing process might have the following assessment possibilities [1] :

1. "reject" if the cork does not work; 2. "poor quality" if the cork works but has some defects; 3. "medium quality" if the cork works and has no defects, but it has some aesthetic flaws; 4. "good quality" if the cork works and has no defects, but has only a few aesthetic flaws; 5. "excellent quality" if the cork works and has neither defects nor aesthetic flaws of any kind.

The monitoring of production, using a sampling control technique, is aimed at recognizing and, possibly, correcting unfavorable trends and out of control conditions. In order to do this, the five classifications listed above could have different degrees of membership. For example, one may assign to the five quality levels 1-5, the degrees of membership: 1, 0.75, 0.5, 0.25, and 0, respectively. In other words, if linguistic variable L be "the quality of the corking and closing process", then L ={(reject,1), (poor quality, 0.75), (medium quality, 0.5), (good quality, 0.25), (excellent quality, 0)}. Although the numerical conversion of verbal information simplifies subsequent analysis, it also gives rise to two problems. The first is concerned with the validity of encoding a discrete verbal scale into a numerical form. This approach introduces properties that were not present in the original linguistic scale (for example, is it legitimate to assume that the difference between the "reject" state and the "poor quality" state is the same as that between "medium" and "good quality" states?). The second is related to the absence of consistent criteria to select the values of the degree of membership. It is obvious that changing the values of the degree of membership may determine a change in obtained results [ 1,6] . In order to minimize these problems, it is recommended that the number of categories with their degrees of membership be arrived at after discussion with experts on the process concerned.

RESULTS AND DISCUSSION

Fuzzy multinomial control chart: In the following, we propose a new approach for construction of a control chart. The statistical principles underlying the fuzzy multinomial control chart ( FM -chart) are based on the multinomial distribution.

be a linguistic variable. In addition suppose that the production process is operating in a stable manner, and i p is the probability that an item is i l,, . ,...

, 1 k i =

Moreover, successive items produced are independent.

Assume that a random sample of n items of the product

, be the number of items of the product that are

We introduce the control limits for FM -chart as:

where k (usually k =3) is the "distance" of the control limits from the center line. The following theorem shows how to compute ( ) , be the number of items of the product

has a binomial distribution with the mean i np and variance ( )

, respectively, and

An illustrative example:

In food process industry, packaging of a frozen food is important quality characteristic that has to be monitored [4]. The product (3) item's packaging, may be classified by an expert team as either "excellent", "good", "medium" or "bad" with the degrees of membership 0, 0.25, 0.5, and 1, respectively. For control of the quality packging process, 30 samples of size 50 are selected. The data with i L and i p are given in Table 1. Suppose that the process is in control in the period corresponding to first ten samples. The sample proportions for the base period estimate as follows:

Table 1

The data of samples of size 50

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