Mathematical Problems in Engineering

Volume 2018, Article ID 6516879, 10 pages

https://doi.org/10.1155/2018/6516879

## Joint and Control Charts Optimal Design Using Genetic Algorithm

Department of Industrial Engineering, German Jordanian University, Mushaqar, Amman 11180, Jordan

Correspondence should be addressed to Saleem Z. Ramadan; oj.ude.ujg@nadamar.meelas

Received 11 February 2018; Revised 23 June 2018; Accepted 3 July 2018; Published 12 July 2018

Academic Editor: Ali Ramazani

Copyright © 2018 Saleem Z. Ramadan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A simple and flexible model for economic statistical design of joint and control charts was proposed. The design problem was approached by constrained fuzzy multiobjective modeling for three objectives: joint* power*, joint* Type I* error, and joint total control cost. Fuzzy membership functions were created to measure the satisfaction levels of the objectives, and the overall satisfaction level of the design was calculated using a weighted-average method. A genetic algorithm was designed to solve this model. The strength of this model lies in its effectiveness in detecting the assignable causes through the joint design and in its simplicity and flexibility in dealing with uncertainties in the design.

#### 1. Introduction

Nonconforming products are often caused by assignable causes in the production process. These nonconformities cost companies a lot of money due to warranty, rework, and scrap costs among other costs. Statistical process control (SPC) is aimed at identifying and isolating assignable causes in a timely manner. These assignable causes may cause a shift in the process mean or adrift in the process variation or both and hence lead to an increase in the number of nonconforming products. Control charts, as an on-line SPC tool, are the foremost used tool for this purpose as they are very effective in this respect if they were designed well. Unfortunately, no single control chart can be effective in detecting the assignable causes for both the shift in the process mean and the drift in the process variation. Usually, control chart is used to detect the shifts in the process mean along with one of , , or control charts to detect the drift in the process variation.

Since the mean and variation are considered statistically independent for a normally distributed process, the and the control charts can be designed simultaneously to be effective in detecting the assignable causes for the shifts in the mean and the drifts in the variation. Simultaneous design of these control charts involves the determination of the sample size, frequency of sampling, and width of control limits that achieve a certain statistical and economical requirements such as* Type I* and* Type II* errors and total control cost. Unfortunately, achieving high statistical standards is usually accompanied with low economic standards and vice versa. This dilemma calls for a design that compromises between the statistical requirements and the economic requirements when the control charts are designed.

The statistical design (SD) is oriented to maximize the power of the control chart under an upper limit constraint for* Type I* error. References [1–5] among other references designed their control charts based on the SD. The drawback of this design is that it does not account for the economic consequences of the design because the main focus of this design is to enhance the statistical properties of the SPC and this usually happens at the expense of the economic aspect.

The economical design (ED) is oriented to minimize the total cost of the SPC. References [6–14] among other references designed their control charts based on ED. Reference [15] showed that ED usually gives poor statistical properties and model parameters estimation.

To overcome the disadvantages of the SD and the ED, [16] proposed a design method in which the total cost of the SPC is minimized under minimum* power* and maximum* Type I* error constraints, known as economic statistical design (SED). Many authors followed the steps of [17] and designed their control charts based on this design. Reference [18] used Markov chain approach to design the control charts using SED. Reference [19] used data development analysis in multiobjective optimization schema to design the control charts based on SED. References [20–23] utilized Taguchi loss function in the SED framework to design their control charts. References [24, 25] investigated the optimal SED design of the control chart when the underlying process is a nonnormal process. Joint design of control charts was also discussed in the literature; [26] used genetic algorithm to achieve the best design in the joint and control charts. Reference [14] designed joint and control charts using differential evolution. Reference [27] investigated the joint design of and control charts in preventive maintenance context.

In this paper, a multiobjective model for designing the joint and control charts was presented to find a compromise between* Type I* error,* Type II* error, and total control cost of the SPC objectives. The proposed model utilized fuzzy logic to calculate the scores of each objective through a membership function that reflects the satisfaction level for that objective. The overall satisfaction level of the design was calculated as the weighted average of the three satisfaction levels. The model maximized the total satisfaction level under total cost per hour, sample size, frequency of sampling, and width of control limits constraints. A genetic algorithm was designed to optimize the model such that the sample size, frequency of sampling, and width of control limits for and control charts are the decision variables.

To the author’s knowledge, the combination of the economic statistical joint design of the and control charts in the context of fuzzy logic multiobjective constrained optimization using genetic algorithm was not discussed in a single model in the literature before. The model developed in this paper dealt with designing the and control charts simultaneously to enhance the effectiveness of the two control charts in detecting the assignable causes for the shifts in the mean and the drifts in the variation. Moreover, this model also used fuzzy logic in the objective function of the model to give the designer the required flexibility to express his/her point of view about the relative importance of the statistical and economic requirements of the design throughout the weights in the objective function and the decay shape of the membership functions. The strength of this model is its effectiveness in detecting the assignable causes for the shifts in the mean and the drifts in the variation due to the usage of the joint design of the and control charts as well as its simplicity and flexibility in dealing with uncertainties in the design process due to the usage of the fuzzy membership functions.

The rest of the paper was organized as follows: Section 2 presented the assumptions and notations used in the paper, Section 3 showed the model derivation and formulation, Section 4 discussed the fuzzy logic, Section 5 discussed the genetic algorithm used, Sections 6 and 7 presented the experiments and discussion for verification purposes, Section 8 presented a comparative analysis between the joint design of the and control charts and the joint design of the and control charts, and Section 9 contained the conclusions.

#### 2. Assumptions and Notations

In this paper, the assignable causes were modeled as a Poisson process with a rate of and a mean exponential time between assignable causes of . Moreover, it was assumed that the quality cycle followed a renewal reward process and the quality control process is not self-correcting. Furthermore, the variability in the quality characteristic output of the process followed a normal distribution and the process started with known mean and variance. These assumptions are widely used in the literature such as [28, 29].

The total cost of the quality cycle in this paper was adopted from the cost schema in [17]. In this scheme, the total cost of quality cycle consisted of three stochastic elements: the cost of producing nonconforming item, the cost of investigating false alarms and repairing the process, and the cost of sampling and testing. Furthermore, the total time of quality cycle was also divided into two stochastic elements: the time that the process spends in statistical control state and the time that the process spends out of statistical control state.

Figure 1, adapted from [30], shows the quality cycle. The figure shows that the cycle is divided into two major periods, the in-control period and the out-of-control period. The figure shows the adjusted average time to signal (*AATS*) as the average time from the occurrence of an assignable cause to the time of detecting it. Also, the figure shows that the average time of the cycle (*ATC*) is the sum of the in-control period and the* AATS*. Moreover, the time of sampling and interpreting the results is the difference between the sum of* AATS* and the time to find and repair an assignable cause in one hand and the out-of-control period on the other hand.