33 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

 

  

 

 

Solving Fuzzy Attribute Quality Control Charts with proposed Ranking 

Function 

 
 

 

Department of Mathematics, College of Science for woman, University of Baghdad, Baghdad, Iraq. 

 

 

 

 

 Abstract  

    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. 

 

Keywords: Quality Control, Fuzzy Set Theory, Attributes Control Charts, Ranking         

function.  

 

1.Introduction 
    Statistical procedures are one of the main useful tools in supervision the production 

process by using control charts and sample inspection plans and these two procedures depend 

on the random variations which happen in terminology then determine the production process 

is subject to specifications of quality control or not, since the units produced differ in quality, 

if these imbalances and deviations are minor, the production is acceptable but if these 

differences and deviations exceed certain limits. The production is not acceptable. 

Statistical methods were used in the field of quality control for the first time in 1924 by 

researcher (Shewart). The company makes the objective in manufacturing the items which 

vacant from defective and congruent to specifications, then utilizing the control charts to 

esteem the statistical techniques to control the quality production. 

Control Chart is one of the most common statistical methods used in terms of monitoring the 

changes that occur during the stages of product process, it is determined by whether the 

process is statistically accurate by the observation recorded from the samples drawn. 

The data was taken from one of the important production plants is a laboratory for the 

production of drinking water bottles to detect the specifications and efficiency of production 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Doi: 10.30526/34.2.2611 

Article history: Received,25,February,2020, Accepted  23,June,2020, Published in April 2021 

 

Esraa Dhafer Thamer 

Israadafer60@gmail.com 

Iden Hasan Hussein 

Idenalkanani@yahoo.com 

 

mailto:Israadafer60@gmail.com
mailto:Idenalkanani@yahoo.com


  

34 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

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 

𝑛𝑖 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 (�̅� ∓ √
𝑝(1−𝑝)

𝑛
 ) where as �̅� =

∑𝑝𝑖

𝑛
  is the defective rate of  proportions, 

the control chart is represented by three parallel lines: 

 The middle limit of attribute control is: 

CL=�̅� =
∑𝑝𝑖

𝑛
                               (1) 

The upper limit of attribute control is: 



  

35 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

UCL= (�̅� + √
𝑝(1−𝑝)

𝑛
 )                  (2) 

The lower limit of attribute control is: 

LCL= (�̅� − √
𝑝(1−𝑝)

𝑛
 )                    (3) 

Where the proportion of defective is:  

P=
defective

production
                               (4) 

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,𝜇𝐴(𝑥));𝑥 𝜖 𝑋} , where 𝜇𝐴 :𝑋 → [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 non-

zero membership grades in a fuzzy set A is called the support of the set A, denoted by 

Supp(A). i.e.,Supp(A)={x 𝜖 𝑋 :A(x) ≥ 0}. 

The membership function that we use it is as following 

𝜇𝐴(𝑥) =

{
 

 
𝜆(𝑥−𝑎)

(𝑏−𝑎
             𝑎 ≤ 𝑥 ≤ 𝑏

𝜆                     𝑥 = 𝑏
𝜆(𝑐−𝑥)

(𝑐−𝑏)
           𝑏 ≤ 𝑥 ≤ 𝑐 

              (5) 

4-Ranking function: [8] Let Ã, �̃� 𝜖 E, define is ranking of A, B by R(.) on E, i.e  

R(Ã) > R(�̃�)  ↔ Ã > �̃� 

R(Ã) < R(�̃�)  ↔ Ã < �̃� 

R(Ã)= R(�̃�)  ↔ Ã ≈ �̃� 

Considure that the triangular fuzzy numbers represented by Ã= (a, b, c), where b is the all 

sample, a is the lift width and c is the right width. 

Now, presented the arbitrary fuzzy numbers Ã(α) by an ordered pair of function 

[Ã𝐿(𝛼),Ã𝑈(𝛼)] , where Ã𝐿(𝛼) is a bounded left continuous nondecreasing function over [0,1] 

and  Ã𝑈(𝛼) is a bounded left continuous nonincreasing function over [0,1] , Ã𝐿(𝛼) ≤ Ã𝑈(𝛼). 

Where Ã𝐿(𝛼) = 𝑖𝑛𝑓{𝑥|Ã(𝑥) ≥ 𝛼} and Ã𝑈(𝛼) = 𝑠𝑢𝑝{𝑥|Ã(𝑥) ≥ 𝛼}. 

Now, the utilizing the triangular membership function to find the new ranking function which 

is as follows: 

𝛼 =
𝜆(𝑥−𝑎)

(𝑏−𝑎)
   α=

𝜆(𝑐−𝑥)

(𝑐−𝑏)
 

𝑥 =
𝛼

𝜆
(𝑏 − 𝑎) + 𝑎  𝑥 = 𝑐 −

𝛼

𝜆
((𝑐 − 𝑏) 

Ã𝐿(𝛼) = 𝑎 +
𝛼

𝜆
(𝑏 − 𝑎)   Ã𝑢(𝛼) = 𝑐 −

𝛼

𝜆
(𝑐 − 𝑏)  

R(Ã)= 
∫ [�̌�𝐿(𝛼)+ �̌�𝑢(𝛼)] 𝑑𝛼
𝜆
𝑤

∫ 𝛼 𝑑𝛼
𝜆
𝑤

 =
∫ [𝑎+

𝛼

𝜆
(𝑏−𝑎)+𝑐−

𝛼

𝜆
(𝑐−𝑏)] 𝑑𝛼

𝜆
𝑤

∫ 𝛼 𝑑𝛼
𝜆
𝑤

  

        =
∫ [(𝑎+𝑐)+

𝛼

𝜆
(2𝑏−𝑎−𝑐)] 𝑑𝛼

𝜆
𝑤

∫ 𝛼 𝑑𝛼
𝜆
𝑤

 =
[(𝑎+𝑏)𝛼+

𝛼2

2𝜆
(2𝑏−𝑎−𝑐)]|

𝜆
𝑤

𝛼2

2
|
𝜆
𝑤

 



  

36 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

        =
(𝑎+𝑐)(𝜆−𝑤)+

(𝜆2−𝑤2)

2𝜆
(2𝑏−𝑎−𝑐)

𝜆2−𝑤2

2

 = 
(𝑎+𝑐)(𝜆−𝑤)+

(𝜆−𝑤)(𝜆+𝑤)

2𝜆
(2𝑏−𝑎−𝑐)

(𝜆−𝑤)(𝜆+𝑤)

2

 

        =
2𝜆(𝑎+𝑏)+(𝜆+𝑤)(2𝑏−𝑎−𝑐)

2𝜆
 *

2

(𝜆+𝑤)
 =
2𝜆(𝑎+𝑏)+(𝜆+𝑤)(2𝑏−𝑎−𝑐)

𝜆(𝜆+𝑤)
 

        =
𝜆(2𝑏+𝑎+𝑐)+𝑤(2𝑏−𝑎−𝑐)

𝜆(𝜆+𝑤)
                             (6) 

 5.Application  

      Baghdad Soft Drinks Company is one of the most important companies operating in the 

province of Baghdad /Zaafaraniya and the establishment of this company dates back to the 

1960s as one of the establishments of the Ministry of Industry and Minerals where issued a 

founding license from the Directorate General of Industrial Development no. 474 in 

1961/12/21. However, it became a mixed company in 1989 in accordance with companies 

Law no. 36 0f 1983 and procedures of establishing the company were completed by issuing 

the certificate of incorporation under the decision of the Registrar of companies in the 

Ministry of Commerce no. (m.s/3315) in 1989/3/23. 

The company has 5 factories, each of which contains production lines: 

a) Degla factory: consists of four production lines. 

b) Euphrates factory: consists of four production lines. 

c) Shatt al-Arab factory: consists of two production lines. 

d) Rafidain factory: consist of one production line. 

e) Aquafina factory: consists of two production lines. 

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: 

 Table 1. contain defective and production from company   

Now, applying attributes control charts in all samples of Table (1)  

First, find (p) in equation (4). 

𝑃1 =
2613

1035510
= 0.002523 and so that 

 

n defective production n defective production n defective production 

1 2613 1035510 11 3953 1205130 21 224 179970 

2 3337 1161630 12 910 809400 22 920 590268 

3 3087 1118460 13 1932 1064520 23 778 1007520 

4 2650 1104840 14 452 366990 24 512 406200 

5 1974 896250 15 947 654330 25 263 528000 

6 2566 1150950 16 3544 1115640 26 896 1204500 

7 3417 1138590 17 1632 1120320 27 731 749592 

8 1405 772566 18 845 769740 28 444 808500 

9 1837 982566 19 1562 1054260 29 2054 1118700 

10 2539 1121220 20 1602 1060890 30 312 314130 



  

37 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

Table 2. for the value (P) 

n p n p n p 

1 0.002523394 11 0.003280144 21 0.001244652 

2 0.002872688 12 0.00112429 22 0.001558614 

3 0.002760045 13 0.001814902 23 0.000772193 

4 0.002398537 14 0.001231641 24 0.001260463 

5 0.00220251 15 0.001447282 25 0.000498106 

6 0.002229463 16 0.003176652 26 0.000743877 

7 0.00300108 17 0.001456727 27 0.000975197 

8 0.001818615 18 0.001097773 28 0.000549165 

9 0.001869595 19 0.001481608 29 0.00183606 

10 0.002264498 20 0.001510053 30 0.000993219 

 

 

Second compute the middle limit of attribute control in equation (1) 

CL= �̅� =
∑ 𝑃𝑖
30
1

30
 = 

0.051993

30
 = 0.001733 

∗After that, compute the upper limit of attribute control in equation (2) 

UCL=�̅� + 3√
�̅�(1−�̅�)

𝑛
=0.024515 

Now, compute the lower limit of attribute control in equation (3) 

LCL=�̅� − 3 ∗ √
�̅�(1−�̅�)

𝑛
= −0.02105 

Finally, drawing the charts of quality control with depend on the attributes control charts 

 

Figure 1. Contain the charts to the original values 

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

P CL UCL LCL



  

38 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

Then the fuzzified the data to make it vague numbers by using the membership function in 

equation (5) which are as follows: 

(𝑎,𝑏,𝑐) = (𝑎𝑙𝑙 𝑠𝑎𝑚𝑝𝑙𝑒𝑠 − �̅�,𝑎𝑙𝑙 𝑠𝑎𝑚𝑝𝑙𝑒𝑠 ,𝑎𝑙𝑙 𝑠𝑎𝑚𝑝𝑙𝑒𝑠 + �̅�)                  (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 

 

Table 3. Fuzzy ranking function of defective samples 

n defective n defective n defective 

1 14931.42857 11 22588.57143 21 1280 

2 19068.57143 12 5200 22 5257.142857 

3 17640 13 11040 23 4445.714286 

4 15142.85714 14 2582.857143 24 2925.714286 

5 11280 15 5411.428571 25 1502.857143 

6 14662.85714 16 20251.42857 26 5120 

7 19525.71429 17 9325.714286 27 4177.142857 

8 8028.571429 18 4828.571429 28 2537.142857 

9 10497.14286 19 8925.714286 29 11737.14286 

10 14508.57143 20 9154.285714 30 1782.857143 

Now, applying attributes control charts in all samples when λ=0.5, w=0.2 

First, find (p) in equation (4). 

𝑃1 =
14931.42857

1035510
= 0.014419396 and so that 

Table 4. For the value (P) when λ=0.5, w=0.2 

n p n p n p 

1 0.014419396 11 0.01874368 21 0.007112296 

2 0.016415357 12 0.006424512 22 0.008906366 

3 0.015771686 13 0.010370871 23 0.004412532 

4 0.013705928 14 0.00703795 24 0.007202645 

5 0.012585774 15 0.008270183 25 0.00284632 

6 0.012739786 16 0.018152297 26 0.004250726 

7 0.01714903 17 0.008324152 27 0.005572555 

8 0.010392085 18 0.00627299 28 0.003138086 

9 0.010683397 19 0.008466331 29 0.01049177 

10 0.012939986 20 0.008628874 30 0.005675539 

Second compute the middle limit of attribute control in equation (1) 



  

39 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

CL= �̅� =
∑ 𝑃𝑖
30
1

30
 = 0.009903 

After that, compute the upper limit of attribute control in equation (2) 

UCL=�̅� + 3 ∗ √
�̅�(1−�̅�)

𝑛
=0.06414 

Now, compute the lower limit of attribute control in equation (3) 

LCL=�̅� − 3 ∗ √
�̅�(1−�̅�)

𝑛
= −0.04433 

Finally, drawing the charts of quality control with depend on the attributes control charts 

 

Figure 2. Contain the charts when λ=0.5, w=0.3 

Now, computing a new ranking function by utilizing the values of w, λ. 

The values of the w, λ are λ=0.9, w=0.6 . 

Then using equation (6) to find the ranking function  

Table 5. Fuzzy ranking function of defective samples 

n defective n Defective n defective 

1 6968 11 10541.33333 21 597.3333333 

2 8898.666667 12 2426.666667 22 2453.333333 

3 8232 13 5152 23 2074.666667 

4 7066.666667 14 1205.333333 24 1365.333333 

5 5264 15 2525.333333 25 701.3333333 

6 6842.666667 16 9450.666667 26 2389.333333 

7 9112 17 4352 27 1949.333333 

8 3746.666667 18 2253.333333 28 1184 

9 4898.666667 19 4165.333333 29 5477.333333 

10 6770.666667 20 4272 30 832 

Now, applying attributes control charts in all samples when λ=0.9, w=0.6 

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

P CL UCL LCL



  

40 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

First, find (p) in equation (4). 

𝑃1 =
6968

1035510
= 0.006729051 and so that 

Table 6. For the value (P) when λ=0.5, w=0.2 

n p n P n p 

1 0.006729051 11 0.008747051 21 0.003319072 

2 0.0076605 12 0.002998106 22 0.004156304 

3 0.00736012 13 0.00483974 23 0.002059182 

4 0.0063961 14 0.003284377 24 0.003361234 

5 0.005873361 15 0.003859419 25 0.001328283 

6 0.005945234 16 0.008471072 26 0.001983672 

7 0.008002881 17 0.003884604 27 0.002600526 

8 0.00484964 18 0.002927395 28 0.00146444 

9 0.004985585 19 0.003950955 29 0.004896159 

10 0.00603866 20 0.004026808 30 0.002648585 

Second compute the middle limit of attribute control in equation (1) 

CL= �̅� =
∑ 𝑃𝑖
30
1

30
 =0.004621604  

After that, compute the upper limit of attribute control in equation (2) 

UCL=�̅� + 3 ∗ √
�̅�(1−�̅�)

𝑛
= 0.041770943 

Now, compute the lower limit of attribute control in equation (3) 

LCL=�̅� − 3 ∗ √
�̅�(1−�̅�)

𝑛
= −0.032527735 

Finally, drawing the charts of quality control with depend on the attributes control charts 

 

Figure 3. Contain the charts when λ=0.9, w=0.6 

 

 

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

P CL UCL LCL



  

41 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 

 

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.  

  

References  

1.Zadeh, L.A., Fuzzy sets, Information and control. 1965, 8, 3, 338-353. 

2.Bradshaw, Jr, C.W., A fuzzy set theoretic interpretation of economic control limits. 

European Journal of Operational Research. 1983, 13, 403-408. 

3.WANG, J.-H. and T. RAZ, On the construction of control charts using linguistic variables. 

The International Journal of Production Research, 1990. 28,3,477-487. 

4.Wang, R.C. ; Chen,C.H. Economic statistical np‐control chart designs based on fuzzy 

optimization. International Journal of Quality & Reliability Management, 1995. 

5.Franceschini, F. ; Romano, D. Control chart for linguistic variables: a method based on the 

use of linguistic quantifiers. International Journal of Production Research, 1999,37,16, 3791-

3801. 

6.Latva-Käyrä, K., Fuzzy logic and SPC,in Industrial Applications of Soft Computing., 

Springer. 2001, 197-210. 

7.Gülbay, M. ; Kahraman, C. Development of fuzzy process control charts and fuzzy 

unnatural pattern analyses. Computational statistics & data analysis, 2006,51,1,434-451. 

8.Hajjari, T. ; Barkhordary, M. Ranking fuzzy numbers by sign length.in 7th Iranian 

Conference on Fuzzy Systems, Iran. 2007. 

9.MOHEB, A.H., ARSHADI, K.A. ; FATEMI, G.S. Fuzzy development of multivariate 

variable control charts using the fuzzy likelihood ratio test. 2010. 

10.Taylan, O. ; Darrab, I.A.Fuzzy control charts for process quality improvement and product 

assessment in tip shear carpet industry. Journal of Manufacturing Technology Management, 

2012. 

11.Sabegh, M.H.Z., et al., A literature review on the fuzzy control chart; classifications & 

analysis. International Journal of Supply and Operations Management, 2014,1,2 ,167. 

  

12.Pastuizaca Fernández, M.N., A. Carrión García, and O. Ruiz Barzola, Multivariate 

multinomial T 2 control chart using fuzzy approach. International Journal of Production 

Research, 2015, 53,7, 2225-2238. 

13.Madadi, M. ; Mahmoudzadeh, M. A fuzzy development for attribute control chart with 

Monte Carlo simulation method. Management Science Letters, 2017, 7,11,555-564. 

14.George, M., Fuzzy Mathmatics_ Application in Economic, Campus Books International. 

2008, 555-564.