Abstract
Quality improvement is one of the most important requirements to strengthen a competitive position in our markets today. So improving the quality, will lead to decrease variations, shrinkages and so production costs hence the customers acquire the appropriate products and services to use. Control charts have an effective usage field to keep the process under control. Control charts are illustrated as graphical analysis method which defines the products whether to stay in the acceptable limits or not and as a graphical analysis technique that specifies a signal in the state of product to be out of that limits. In this paper by detecting basic concept and essentials beyond the control charts usage and the improvement; so it combined with fuzzy approach to detect the optimal limits. Hence the application of the proposed fuzzy control chart is in Al–Dura Refinery to monitor variable quality characteristics. The proposed fuzzy control chart is under vague, imprecise, uncertain, and incomplete data and based on α-level fuzzy midrange for α – cut approach. As a result of the application, it’s rational to say that constructing fuzzy control charts have a more flexible, a more convenient mathematical characterization concept and have more reasonable results than the traditional quality control chart techniques.
Keywords: Statistical process control, Fuzzy logic, Fuzzy transformation techniques, Fuzzy control charts.
Received: 15 March 2017 / Revised: 21 April 2017 / Accepted: 2 May 2017 / Published: 8 May 2017
Contribution/ Originality
The paper's primary contribution is finding a way to improve the quality of petroleum products in Al- Dura refinery / Baghdad/ Iraq. This can be achieved by applying fuzzy control charts to monitor the petroleum products specification which is rarely mentioned in relative literature.
1. INTRODUCTION
Quality of manufacturing products and services is the primary key factor for the success and competitiveness of organizations and there are literally different definitions of quality. Based on the definition of Montgomery [1 ] quality is characterized as “inversely proportional to variability”. This definition of quality is rooted in the belief that an increase in the variability of key characteristics of a product or service results in a reduction in its quality. Hence, quality control techniques, especially Statistical Process Control, have absorbed significant amount of attention as an effective tool in reducing variability of processes and improving quality [2 ]. Quality control (QC) is a set of planned activities which include procedures and tests to achieve a specific specification of final products [3 ]. Quality Control focuses on the conversion of inputs into outputs . The purpose of quality control is to assure that processes are performing in an acceptable manner. This is accomplished by monitoring process output using statistical techniques [4 ]. Statistical process control is a dynamic monitoring method where product quality is actively measured and simultaneously charted while manufactured goods are being mass produced [5 ]. A major objective of statistical process control is to quickly detect the occurrence of assignable causes of process shifts so that investigation of the process and corrective action may be undertaken before many nonconforming units are manufactured [1 ]. The aim of this paper is to propose a fuzzy control chart to monitor the process of petroleum products because there are limited numbers of scientific papers about that. The proposed fuzzy control chart is validated by a case study applied in Al-Dura Refinery. This paper is organized as follows. In the second part it is mentioned a basic concepts about fuzzy numbers and fuzzy transformation techniques. In the third part, mentioned a related research of fuzzy control charts .in the fourth part mentioned the framework of the proposed methodology. In the fifth part, all mathematical models applied in this paper are listed. The next parts presents a case study based on real data collected from Al-Dura refinery to apply the proposed methodology. And then results of the proposed control charts with different approaches are compared with each other. In the last part, conclusions and findings have been interpreted.
2. FUZZY NUMBERS
Trapezoidal and triangular fuzzy numbers are the most famous shapes of fuzzy numbers.
Where a, b, c, and d are real numbers and u (x) is: membership function of (x).
Fig-1. A Triangular Fuzzy Number (TRIFN).
- Fuzzy Transformation Techniques:
Fuzzy transformation techniques are used to transform the fuzzy numbers into crisp values. The four fuzzy measures of central tendency, fuzzy mode, α-level fuzzy midrange, fuzzy median and fuzzy average, which well-known in descriptive statistics, are given below:
- The fuzzy mode, f mode: The fuzzy mode of a fuzzy set F is the value of the base variable where the membership function equals 1. This is stated as:
It should be mentioned that there is no specific basis supporting any one precisely or the selection between them. Generally, the first two methods are easier to calculate than the last two when the membership function is nonlinear. Additionally, the fuzzy mode may lead to biased results when the membership function is extremely asymmetrical. The fuzzy midrange is more flexible because one can choose different levels of membership (α) of interest. If the area under the membership function is considered to be an appropriate measure of fuzziness, the fuzzy median is suitable [7 ].
3. LITERATURE SURVEY
In the literature there exist many papers about fuzzy control charts. Numerical examples using the data of real case studies are also given to highlight the practical usage of the proposed approaches as illustrated below:
Moraditadi [8 ] found out that fuzzy individual x and moving range (IX-MR) control chart in the uncertainty case for process parameters and data based on Fuzzy mode for triangular and trapezoidal fuzzy number coded by MATLAB software. It was found that the fuzzy control chart was more sensitive than the traditional control charts and more consistent with the actual situation. As well, fuzzy mode method was more sensitive for trapezoidal than triangular fuzzy numbers Moraditadi [8 ]. Chen and Yu [9 ] applied fuzzy zones instead of crisp ones to describe the monitored zones; and a fuzzy rule to construct the corresponding fuzzy zone control chart. It was found that the proposed fuzzy zone control chart could achieve better performance against the classical control charts the monitoring of process variation Chen and Yu [9 ]. Tadi and Darestani [10 ] Investigated the fuzzy IX-MRcontrol chart when Data transformed into trapezoidal fuzzy number using fuzzy mode and Fuzzy Rules approaches. The results were grouped into fourcategories (in control, out of control, rather in control and rather out of control). It was found that fuzzy control chart is more sensitivethan classical control chart Tadi and Darestani [10 ]. Sogandi, et al. [11 ] developed a new fuzzycontrol chart for monitoring attribute quality characteristics based on α-level fuzzy midrange approach. It was found the performance and comparative results of the proposed fuzzy control chart was measured in terms of average run length (ARL) by Mont Carlo simulation Sogandi, et al. [11 ]. Sorooshian [12 ]tried an approach based on fuzzy set theory monitoring attribute quality characteristics which considers uncertainty and vagueness and for this purpose. It was found that the proposed approach has a better performance and detects abnormal shifts in the process, especially in small shifts and small sample size, faster than current related approaches [12 ].
4. THE PROPOSED METHODOLOGY
In this paper the proposed methodology as shown in Fig (4) has been focus on applying traditional control charts and fuzzy control charts. The main steps of the proposed methodology list as follows:
- Problem Definition and Data Collection.
- Selection of the control chart type.
- Establish the control limits.
- Analysis of the control chart
- Decision making
Fig-2.The Framework of Proposed Methodology.
5. MATHEMATICAL MODEL
In this section basic concepts of traditional control charts, fuzzy numbers and proposed fuzzy control charts.
A –Traditional and S control chart
For traditional and S control chart the control limits are calculated as follows:
B. - Fuzzy - S control chart
The study lists all the formulas of establishing the control limits of the control charts for Fuzzy – S control charts, α - Cut Fuzzy
control charts and α - Level Fuzzy Midrange for α - Cut Fuzzy
Control Chart Control Limits as illustrated bellow:
Fuzzy Chart Control Limits
Fuzzy Chart Control Limits
Control Limits for α – cut control charts
The control limits for α - cut fuzzy control charts based on standard deviation are obtained as follows:
Control Limits for α – cut fuzzy control charts
The control limits and center line for α- Cut Fuzzy control chart can be obtained as follows:
C- Determining the Process Condition
The process condition of the sample which identify the process whether in control or not as illustrated below:
a- Level Fuzzy Midrange for a- Cut FuzzyControl Chart based on Standard Deviation
Then, the condition of process control for each sample can be defined as:
α - Level Fuzzy Midrange for α - Cut Fuzzy Control Chart
Then, the condition of process control for each sample can be defined as:
6. CASE STUDY I
The study selects the petroleum products to monitor the quality of it because of the importance of the oil industry. The study applied the traditional – S control chart and Fuzzy
control charts to monitor the Kerosene product the sulphur content specification as shown below:
A. The Constructing of Traditional – S Control Chart
The study implements traditional control charts for Kerosene product the flash point specification by using MINITAB software 16.as illustrated in Fig(3).
The study implements traditional control charts for Kerosene product the flash point specification by using MINITAB software 16.as illustrated in Fig(3).
B. The Construction of Fuzzy Control Charts:
Hence by applying the mathematical models and formulas mentioned above the Arithmetic Mean and Standard deviation for Fuzzy Data where calculated and by using Excel software13. The condition of process control for each sample can be identified by the condition rule mentioned above to detect the process control of the sample as shown in table (2).
Table-1. The Fuzzy Triangular Data
No |
xa1 |
xb1 |
xc1 |
xa2 |
xb2 |
xc2 |
xa3 |
xb3 |
xc3 |
xa4 |
xb4 |
xc4 |
1 |
0.29 |
0.32 |
0.33 |
0.31 |
0.34 |
0.35 |
0.30 |
0.33 |
0.34 |
0.34 |
0.35 |
0.36 |
2 |
0.20 |
0.23 |
0.29 |
0.26 |
0.3 |
0.36 |
0.23 |
0.32 |
0.35 |
0.23 |
0.32 |
0.38 |
3 |
0.28 |
0.32 |
0.35 |
0.30 |
0.31 |
0.33 |
0.26 |
0.29 |
0.31 |
0.29 |
0.32 |
0.34 |
4 |
0.33 |
0.33 |
0.35 |
0.33 |
0.35 |
0.37 |
0.32 |
0.34 |
0.35 |
0.32 |
0.34 |
0.36 |
5 |
0.31 |
0.32 |
0.34 |
0.32 |
0.33 |
0.34 |
0.33 |
0.33 |
0.34 |
0.31 |
0.32 |
0.33 |
6 |
0.23 |
0.32 |
0.34 |
0.15 |
0.23 |
0.25 |
0.27 |
0.29 |
0.31 |
0.25 |
0.31 |
0.37 |
7 |
0.23 |
0.25 |
0.26 |
0.25 |
0.27 |
0.29 |
0.25 |
0.26 |
0.27 |
0.25 |
0.27 |
0.28 |
8 |
0.23 |
0.26 |
0.28 |
0.17 |
0.2 |
0.22 |
0.25 |
0.29 |
0.31 |
0.21 |
0.27 |
0.29 |
9 |
0.27 |
0.28 |
0.29 |
0.25 |
0.27 |
0.28 |
0.24 |
0.26 |
0.27 |
0.25 |
0.27 |
0.28 |
10 |
0.13 |
0.18 |
0.47 |
-0.04 |
0.23 |
0.35 |
0.23 |
0.4 |
0.69 |
0.15 |
0.38 |
0.55 |
11 |
0.28 |
0.28 |
0.29 |
0.25 |
0.27 |
0.28 |
0.24 |
0.26 |
0.27 |
0.25 |
0.27 |
0.28 |
12 |
0.30 |
0.32 |
0.43 |
0.15 |
0.23 |
0.27 |
0.18 |
0.29 |
0.40 |
0.22 |
0.31 |
0.42 |
13 |
0.21 |
0.22 |
0.25 |
0.16 |
0.2 |
0.22 |
0.16 |
0.21 |
0.26 |
0.20 |
0.25 |
0.30 |
14 |
0.20 |
0.23 |
0.23 |
0.21 |
0.22 |
0.25 |
0.22 |
0.25 |
0.28 |
0.20 |
0.23 |
0.25 |
15 |
0.23 |
0.27 |
0.31 |
0.18 |
0.22 |
0.23 |
0.21 |
0.24 |
0.25 |
0.18 |
0.22 |
0.23 |
16 |
0.21 |
0.23 |
0.25 |
0.20 |
0.22 |
0.24 |
0.22 |
0.25 |
0.26 |
0.22 |
0.23 |
0.24 |
17 |
0.26 |
0.27 |
0.28 |
0.19 |
0.22 |
0.25 |
0.23 |
0.24 |
0.26 |
0.18 |
0.22 |
0.24 |
18 |
0.17 |
0.22 |
0.27 |
0.16 |
0.2 |
0.25 |
0.16 |
0.21 |
0.22 |
0.24 |
0.25 |
0.30 |
19 |
0.13 |
0.18 |
0.20 |
0.20 |
0.22 |
0.27 |
0.11 |
0.16 |
0.21 |
0.14 |
0.19 |
0.24 |
20 |
0.27 |
0.27 |
0.29 |
0.26 |
0.28 |
0.30 |
0.26 |
0.28 |
0.28 |
0.27 |
0.29 |
0.31 |
21 |
0.18 |
0.22 |
0.23 |
0.23 |
0.24 |
0.28 |
0.23 |
0.27 |
0.29 |
0.27 |
0.28 |
0.32 |
22 |
0.17 |
0.24 |
0.31 |
0.22 |
0.28 |
0.35 |
0.24 |
0.31 |
0.32 |
0.29 |
0.3 |
0.37 |
23 |
0.27 |
0.32 |
0.38 |
0.26 |
0.28 |
0.33 |
0.22 |
0.27 |
0.29 |
0.25 |
0.3 |
0.36 |
24 |
0.17 |
0.22 |
0.27 |
0.16 |
0.2 |
0.25 |
0.17 |
0.21 |
0.26 |
0.22 |
0.25 |
0.30 |
Table-2. The Process Control Condition for- Level Fuzzy Midrange For
- Cut (Fuzzy
Control Chart Based on Standard deviation and Fuzzy
Control Chart)
No |
Process Control |
Process Control |
||
1 |
0.33 |
out of control |
0.01 |
in control |
2 |
0.29 |
in control |
0.04 |
in control |
3 |
0.31 |
in control |
0.02 |
in control |
4 |
0.34 |
out of control |
0.01 |
in control |
5 |
0.33 |
out of control |
0.01 |
in control |
6 |
0.28 |
in control |
0.05 |
in control |
7 |
0.26 |
in control |
0.01 |
in control |
8 |
0.25 |
in control |
0.04 |
in control |
9 |
0.27 |
in control |
0.01 |
in control |
10 |
0.31 |
in control |
0.02 |
in control |
11 |
0.27 |
in control |
0.01 |
in control |
12 |
0.29 |
in control |
0.05 |
in control |
13 |
0.22 |
in control |
0.03 |
in control |
14 |
0.23 |
in control |
0.01 |
in control |
15 |
0.23 |
in control |
0.03 |
in control |
16 |
0.23 |
in control |
0.01 |
in control |
17 |
0.24 |
in control |
0.02 |
in control |
18 |
0.22 |
in control |
0.03 |
in control |
19 |
0.19 |
out of control |
0.03 |
in control |
20 |
0.28 |
in control |
0.01 |
in control |
21 |
0.25 |
in control |
0.03 |
in control |
22 |
0.28 |
in control |
0.03 |
in control |
23 |
0.29 |
in control |
0.03 |
in control |
24 |
0.22 |
in control |
0.02 |
in control |
7. CASE STUDYII
The study applied the traditional – S control chart and Fuzzy
control charts to monitor the Benzene product the octane number specification as shown below:
A. The Constructing of Traditional – S Control Chart.
The study implements traditional control charts for Benzene product the octane number specification by using MINITAB software 16.as illustrated in Fig(5).
The study implements traditional control charts for Benzene product the octane number specification by using MINITAB software 16.as illustrated in Fig(5).
B. The Construction of Fuzzy Control Charts:
To construct fuzzy control chart the data must be transformed to fuzzy triangular number (a, b, c) each observation must be transformed to fuzzy triangular number as shown in table (3). Hence by applying the mathematical models and formulas mentioned above the Arithmetic Mean and Standard deviation for Fuzzy Data where calculated and by using Excel software13. The condition of process control for each sample can be identified by the condition rule mentioned above to detect the process control of the sample as shown in table (4).
Table-3. The Fuzzy Triangular Data
No |
xa1 |
xb1 |
xc1 |
xa2 |
xb2 |
xc2 |
xa3 |
xb3 |
xc3 |
xa4 |
xb4 |
xc4 |
1 |
81.09 |
83 |
83.74 |
80.54 |
82.3 |
82.72 |
80.09 |
82 |
82.74 |
83.58 |
84 |
84.42 |
2 |
83.04 |
83.5 |
84.48 |
81.72 |
83 |
83.98 |
81.11 |
82.5 |
82.96 |
80.61 |
82 |
83.39 |
3 |
80.69 |
82 |
83.31 |
81.76 |
82 |
82.76 |
81.93 |
83 |
83.76 |
80.93 |
82 |
83.07 |
4 |
82.76 |
83 |
84.07 |
81.01 |
82 |
83.31 |
80.93 |
82 |
82.76 |
80.93 |
82 |
82.99 |
5 |
80.93 |
82 |
83.31 |
80.75 |
82 |
82.63 |
81.84 |
82 |
83.07 |
79.93 |
81 |
81.76 |
6 |
78.80 |
82 |
82.71 |
79.05 |
82 |
82.71 |
82.29 |
83 |
83.71 |
77.23 |
79.5 |
81.77 |
7 |
78.25 |
80 |
81.24 |
78.39 |
80 |
81.75 |
79.76 |
81 |
82.24 |
77.25 |
79 |
79.68 |
8 |
77.76 |
79 |
79.82 |
78.76 |
80 |
80.82 |
81.34 |
83 |
83.82 |
77.37 |
80 |
80.82 |
9 |
79.76 |
81 |
82.24 |
78.39 |
80 |
80.68 |
77.25 |
79 |
79.68 |
78.25 |
80 |
81.24 |
10 |
79.36 |
79.5 |
80.26 |
78.28 |
79 |
79.32 |
78.56 |
79 |
79.76 |
78.88 |
79.5 |
80.26 |
11 |
80.61 |
81 |
82.24 |
78.39 |
80 |
80.90 |
77.25 |
79 |
79.90 |
78.25 |
80 |
80.68 |
12 |
79.36 |
79.5 |
80.26 |
78.43 |
79 |
79.32 |
78.24 |
79 |
79.76 |
78.88 |
79.5 |
80.26 |
13 |
80.53 |
81 |
81.83 |
77.02 |
79 |
79.96 |
76.85 |
79 |
80.52 |
76.85 |
79 |
80.52 |
14 |
79.69 |
81 |
81.16 |
80.76 |
81 |
82.07 |
78.69 |
80 |
81.07 |
79.69 |
81 |
81.76 |
15 |
80.09 |
82 |
83.91 |
79.09 |
81 |
81.59 |
79.40 |
81 |
81.59 |
77.09 |
79 |
79.59 |
16 |
80.24 |
81 |
81.76 |
80.01 |
81 |
81.76 |
78.93 |
80 |
80.24 |
80.76 |
81 |
81.24 |
17 |
81.41 |
82 |
82.59 |
79.40 |
81 |
82.38 |
80.41 |
81 |
81.90 |
77.09 |
79 |
80.20 |
18 |
78.47 |
79.5 |
80.53 |
78.05 |
79 |
80.03 |
77.97 |
79 |
79.16 |
79.77 |
80 |
81.03 |
19 |
77.93 |
79 |
79.42 |
79.58 |
80 |
81.07 |
77.93 |
79 |
80.07 |
77.93 |
79 |
80.07 |
20 |
79.34 |
79.5 |
80.53 |
78.05 |
79 |
80.03 |
77.97 |
79 |
79.23 |
78.97 |
80 |
81.26 |
21 |
77.44 |
79 |
79.49 |
79.01 |
79.5 |
81.06 |
79.44 |
81 |
81.74 |
80.51 |
81 |
82.56 |
22 |
77.93 |
79 |
80.07 |
78.01 |
79 |
80.07 |
77.93 |
79 |
79.24 |
79.76 |
80 |
81.07 |
23 |
77.93 |
79 |
80.31 |
78.45 |
79 |
80.07 |
77.93 |
79 |
79.42 |
78.93 |
80 |
81.31 |
24 |
77.93 |
79 |
80.07 |
78.01 |
79 |
79.42 |
78.01 |
79 |
80.07 |
79.45 |
80 |
81.07 |
Table-4. The Process Control Condition for- Level Fuzzy Midrange For a- Cut (Fuzzy
Control Chart Based on Standard deviation and Fuzzy
Control Chart)
No |
Process Control |
Process Control |
||
1 |
82.6 |
out of control |
1.04 |
in control |
2 |
82.71 |
out of control |
0.75 |
in control |
3 |
82.26 |
out of control |
0.51 |
in control |
4 |
82.3 |
out of control |
0.62 |
in control |
5 |
81.76 |
in control |
0.62 |
in control |
6 |
81.33 |
in control |
1.48 |
in control |
7 |
79.91 |
in control |
0.94 |
in control |
8 |
80.28 |
in control |
1.75 |
in control |
9 |
79.84 |
in control |
0.93 |
in control |
10 |
79.29 |
in control |
0.37 |
in control |
11 |
79.89 |
in control |
1.01 |
in control |
12 |
79.28 |
in control |
0.38 |
in control |
13 |
79.38 |
in control |
1.15 |
in control |
14 |
80.68 |
in control |
0.58 |
in control |
15 |
80.52 |
in control |
1.39 |
in control |
16 |
80.68 |
in control |
0.62 |
in control |
17 |
80.71 |
in control |
1.36 |
in control |
18 |
79.38 |
in control |
0.65 |
in control |
19 |
79.25 |
in control |
0.63 |
in control |
20 |
79.4 |
in control |
0.62 |
in control |
21 |
80.14 |
in control |
1.16 |
in control |
22 |
79.26 |
in control |
0.66 |
in control |
23 |
79.27 |
in control |
0.57 |
in control |
24 |
79.25 |
in control |
0.6 |
in control |
8. RESULTS AND DISCUSSION
Table-5. The Results of Applying Traditional and Fuzzy Control Charts
9. CONCLUSION
- The decreasing of the gap in the construction and interpretation of fuzzy control charts is achieved. α-cut of fuzzy sets is successfully applied so as to reflect the tightness of an inspection that can be subjectively set by the quality controller accordingly to the importance of an inspection. The proposed approach is efficient in detecting process shifts.
- The fuzzy approach is convenient to detect the signals in the variable control charts, because its gives some flexibility to the control limits. Since the plotted values are close to the upper, lower and center line may cause false alarms with traditional control charts, fuzzy control limits can provide more flexibility for controlling a process.
- In regard with the occurrence of a mean shift in a real process, hence the proposed fuzzy control chart can help researchers in the production management area to more accurately describe the implementation of fuzzy theory in a control chart. In practice, the quality control department of a factory can construct a specific fuzzy control chart according to its specific environment by following the steps illustrated in this study and further more improve the performance of the control charts with the usage of computers, thus reducing the amount of unqualified products.
- Constructing fuzzy control charts have more flexible than traditional control chart approach and give more meaning results than traditional quality chart technique, hence the proposed fuzzy control chart have better performance as compared with the traditional control chart. Finally for future work , the study can be extended by using multivariate fuzzy control charts to monitor more quality characteristic and it can be extended by monitoring the process capability.
| Funding: This study received no specific financial support. |
| Competing Interests: The authors declare that they have no competing interests. |
| Contributors/Acknowledgement: All authors contributed equally to the conception and design of the study. |
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