Economic Design of X-bar Chart Using Genetic Algorithm

Control chart is a key tool in Statistical Process Control. This chart is one type of statistical tool which is used to monitor the quality of a process. It gives a visual representation of the status of the process indication whether the process is under control or not. It is used for finding any variation present in any process. Control charts display the variation in a process, so that anyone can easily determine whether the process is within control or it is out of control. For the design of X-bar control chart we need to find the optimal values of sample size, sampling frequency and width of control limit. In our work, we made a computer program in MATLAB based on Genetic Algorithm for finding the optimal values of above three parameters so that the total expected cost is minimized. Our result showed that Genetic Algorithm provides better result as compared to others reported in the literature

A Quality Systems Economic-Risk Design Theoretical Framework

Quality systems, including control charts theory and sampling plans, have become essential tools to develop business processes. Since 1928, research has been conducted in developing the economic-risk designs for specific types of control charts or sampling plans. However, there has been no theoretical or applied research attempts to combine these related theories into a synthesized theoretical framework of quality systems economic-risk design. This research proposes to develop a theoretical framework of quality systems economic-risk design from qualitative research synthesis of the economic-risk design of sampling plan models and control charts models. This theoretical framework will be useful in guiding future research into economic risk quality systems design theory and application

Diseño óptimo de planes de muestreo polietápicos para la monitorización de la capacidad de un proceso

[ES] El presente trabajo tiene como objetivo el planteamiento y el desarrollo de una metodología para la obtención de planes de muestreo en varias etapas para la valoración de la capacidad de un proceso, medida esta a través de la familia de índices de capacidad de Vännman Cp(u,v) y, como caso particular, del índice de capacidad Cpk. Para ello, en primer lugar se estudia la distribución en el muestreo del índice de capacidad y se implementa el cálculo de probabilidades relacionadas con dicho estadístico. Esto hace posible, posteriormente, el diseño y obtención de planes de muestreo óptimos (desde el punto de vista del tamaño muestral) para la capacidad de un proceso utilizando un muestreo polietápico. Se obtienen y comparan entre sí diseños en una, dos y tres etapas.[EN] The objective of this final master's work is to propose and develop a methodology to obtain sampling plans in several stages for the assessment of the capacity of a process, measured through the family of indices of Vännman capacity Cp (u, v) and, as a particular case, of the capacity index Cpk. To do this, first the distribution in the sampling of the capacity index is studied and the calculation of probabilities related to said statistic is implemented. This makes it possible, later, to design and obtain optimal sampling plans (from the point of view of the sample size) for the capacity of a process using multistage sampling. One-, two-, and three-stage designs are obtained and compared with each other.Arce Romero, EJ. (2021). Diseño óptimo de planes de muestreo polietápicos para la monitorización de la capacidad de un proceso. Universitat Politècnica de València. http://hdl.handle.net/10251/173991TFG

Economic Design of Control Charts Using Metaheuristic Approaches

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