Solving Fuzzy Attribute Quality Control Charts with proposed Ranking

The attribute quality control charts are one of the main useful tools to use in control of quality product in companies. In this paper utilizing the statistical procedures to find the attribute quality control charts for through fuzzified the real data which we got it from Baghdad Soft Drink Company in Iraq, by using triangular membership function to obtain the fuzzy numbers then employing the proposed ranking function to transform to traditional sample. Then, compare between crisp and fuzzy attribute quality control.

accurately and quickly, the results showed that fuzzy control chart are more accurate and economically faster in controlling the quality of production, leading to the detection of defective units during the production process, which helps to detect error quickly. Zadeh is the first from discover fuzzy set theory in (1965) [1]. Bradshaw (1983), for the first time used fuzzy sets as a basis for the explanation of the measurement of conformity of each product units with the specifications [2]. T. Raz and J. Wang (1990) have attempted to extend the use of control charts to allow for linguistic variables [3]. Ray Cheng et al (1995), proposed economic statistical np-control chart design [4]. F.Franceschine and D. Romano (1999), proposed a method for the online control of qualitative of the product/service using control charts for linguistic variables [5]. K.Latva-Kayra (2001), proposed EWMA and CUSUM with fuzzy control limits and their fuzzy combination is used [6]. M.Gulbayand C. Kahraman(2006),the direct fuzzy approach to fuzzy control charts without any distrotion, and fuzzy abnormal pattern rules based on the probabilities of fuzzy events is proposed [7]. Chih-Hsuan Wang-Way Kuo (2007), multiresolution relied on robust fuzzy clustering approach [8]. H.Moheb Alizadeh, A.R.Arshadi Khamseh and S.M.T Fatemi Ghomi (2010), developed multivariate variable control charts in fuzzy mode [9]. Osman Taylan, and Ibrahim A.Darrab (2012), describe the use of artificial intelligence (AI) methods such as fuzzy logic and neural networks in quality control and improvement [10]. Mohammad Hossein and IR (2014), provide a literature review of the control chart under a fuzzy environment with proposing several classifications and analyzes [11]. P.Fernández and other IR (2015), the use of fuzzy control charts becomes inevitable when statistical data considered are vague or affected by uncertainty [12]. M.Hadi and M.Mahmoudzadeh (2017), presented the fuzzy statistical process control development for attribute quality control chart by using Monte Carlo simulation method [13]. The aim of the study is applying the crisp control chart and fuzzy control chart for real data by utilizing of triangular membership function. Then employing the proposed ranking function to find the attribute quality control when (w=0.2 , λ=0.5) and (w=0.6 , λ=0.9). This paper is organized as follows. In section 2, showing the attribute control chart technique. In section 3, showing the fuzzy set theory. In section 4, introducing the new method of ranking function. In section 5, introduction the application of real data. In section 6, numerical results are shown. In section 7, conclusions are given.

2.Attributes Control Charts
In some cases, production units can be divided into two types defective and invalid production units and non-defective and valid production units, this means that the units produced are described by specific properties or characteristics. If assume the withdrawal of from random samples of equal sizes from a specific production process during regular interval and which distributes Binomial Distribution and assume that the defective ratios in production are (p) and the non-defective ratios are (1-p) then the ratios of defective values are between the two limits ( ̅ ∓ √ (1− ) ) where as ̅ = ∑ is the defective rate of proportions, the control chart is represented by three parallel lines: The middle limit of attribute control is: The upper limit of attribute control is: The lower limit of attribute control is: Where the proportion of defective is: If one or more defective proportions are outside the upper and lower control limits, then the production process is outside the control limit. Otherwise, the production process is under control.
3-Fuzzy set: [14] Let X be the universal set. A fuzzy set in X is a set of ordered pairs, A={(x, ( )); } , where : → [0,1] is called the membership set. α-Cats of a Fuzzy Set: [14] The crisp set that contains all the elements of X that have nonzero membership grades in a fuzzy set A is called the support of the set A, denoted by Supp(A). i.e.,Supp(A)={x :A(x) ≥ 0}. The membership function that we use it is as following In addition to three factories to manufacture carbon dioxide gas. The company is licensed to produce soft drinks from Pepsi Co. International and latter takes samples from markets and is examined to assess the quality of production and the company is adjacent to distribute its products in central and southern Iraq.

6.Numerical Results
The samples that we get it from Baghdad Soft Drinks Company are as follows:  Second compute the middle limit of attribute control in equation (1) CL= ̅ =  Then the fuzzified the data to make it vague numbers by using the membership function in equation (5) which are as follows: ( , , ) = ( − ̅ , , + ̅ ) (7) Therefore, applying the new ranking function in equation (7) to transform the fuzzy numbers to crisp numbers but the new ranking function depend upon w, λ∈[0,1]. Now, computing a new ranking function by utilizing the values of w, λ. The values of the w, λ are λ=0.5, w=0.2 . Then using equation (6) to find the ranking function  After that, compute the upper limit of attribute control in equation (2) UCL= ̅ + 3 * √ ̅ (1− ̅ ) =0.06414 Now, compute the lower limit of attribute control in equation (3) LCL= ̅ − 3 * √ ̅ (1− ̅ ) = −0.04433 Finally, drawing the charts of quality control with depend on the attributes control charts The values of the w, λ are λ=0.9, w=0.6 .
Then using equation (6) to find the ranking function

6.Conclusion
In beginning of employment the attribute quality control chart to calculate the proportion defective for all samples. Applying the triangular membership function to get the fuzzy numbers of defective for all samples. Then carrying out the proposed ranking function twice, the first through using (w=0.2 , λ=0.5), the second through using (w=0.6 , λ=0.9) to obtain the fuzzy number for all samples. After that, carrying out the fuzzy quality control to compute the proportion defective for all samples. Finally, comparing between crisp and fuzzy control charts for all samples of production are under control limits.