Economic Design of Control Charts Using Metaheuristic Approaches

Abstract

Statistical Process Control (SPC) is a collection of problem solving tools useful in achieving process stability and improving capability through the reduction of variability using statistical methods. It can help industries in reduction of cost, improvement of quality and pursuit of continuous improvement. Among all the SPC tools, the control chart is most widely used in practice. Out of all the control charts, chart is the simplest to use and hence most popularly used for monitoring and controlling processes in an industry.A process may go out-of-control due to shift in process mean and/or process variance. To detect both types of shifts, R chart is often used along with chart.
The design of chart refers to selection of three design variables such as sample size (n), sampling interval (h) and width of control limits (k). On the other hand, the joint design of and R charts involves four design variables i.e., sample size (n), sampling interval (h), and widths of control limits for both charts (i.e., k1 and k2). There are four types of control chart designs, namely (i) heuristic design, (ii) statistical design, (iii) economic design, and (iv) economic statistical design. In heuristic design, the values of design variables are selected using some thumb rules. In statistical design, the design variables are selected in such a way that the two statistical errors, namely Type-I error ( ), and Type-II error ( ) are kept at minimum values. In economic design, a cost function is constructed involving various costs like the cost of sampling and testing, the cost of false alarm, the cost to detect and eliminate the assignable cause(s), and the cost of producing non-conforming products when the process is operating out-of-control. The design parameters of the control chart are then selected so that this cost function is minimized. The design based on combined features of statistical design and economic design is termed as economic statistical design where the cost function is minimized while satisfying the statistical constraints. The effectiveness of economic design or economic statistical design depends on the accuracy of minimization of cost function. So, use of effectively designed control charts is highly essential for ensuring quality control at minimum cost.
Most of the researchers have used either approximate or traditional optimization techniques for minimizing the cost function. With time, more and more efficient optimization methods have been utilized for this purpose. There are a number of metaheuristic algorithms reported in literature for optimization in various types of design problems. Out of them one each from two different groups are selected for the present work i.e., simulated annealing (SA) and teaching-learning based optimization (TLBO). SA is a point to point based metaheuristic technique, whereas TLBO is population based technique. SA is one of the oldest metaheuristic algorithms and proved to be the most robust one, whereas TLBO is one of the most recent and promising techniques. The present work requires optimization techniques that can solve non-linear, non-differentiable, multi-variable, unconstrained as well as constrained type of objective function. Both the above techniques are capable of optimizing this type of objective function. However, from literature review it is observed that neither of these two metaheuristic approaches has been applied in economic or economic statistical design of any type of control chart. In this work, both these metaheuristic techniques (i.e., SA and TLBO) have been applied for minimization of cost function for economic as well as economic statistical design point of view for individual chart, and by taking and R charts jointly in case of continuous as well as discontinuous process. Thus, a total of the following eight distinct design cases have been considered for their optimization.
1. Economic design of chart for continuous process
2. Economic design of chart for discontinuous process
3. Economic statistical design of chart for continuous process
4. Economic statistical design of chart for discontinuous process
5. Joint economic design of and R charts for continuous process
6. Joint economic design of and R charts for discontinuous process
7. Joint economic statistical design of and R charts for continuous process
8. Joint economic statistical design of and R charts for discontinuous process
All the above designs are illustrated through numerical examples taken from literature using two metaheuristics i.e., SA and TLBO separately. These two independent techniques are used to validate their results with each other. Their results are found to be superior to that reported earlier in the literature. Thus, eight types of methodologies based on SA or TLBO approach are recommended in this thesis for designing control charts from economic point of view.
Sensitivity analysis has been carried out using fractional factorial design of experiments and analysis of variance for each of the eight design cases, to examine the effects of all the cost and process parameters on all the output responses such as sample size, sampling interval, width of control limits and expected loss costper unit time. The process parameters which significantly affect the output responses are identified in each of the eight design cases. These results are expected to be helpful for quality control personnel in identifying the significant factors and thereby taking utmost care in choosing their values while designing the control charts on economic basis.