Economic Design of Control Chart Using Differential Evolution

Transcription

1 Economic Design of Control Chart Using Differential Evolution Rukmini V. Kasarapu 1, Vijaya Babu Vommi 2 1 Assistant Professor, Department of Mechanical Engineering, Anil Neerukonda Institute of Technology and Sciences, Visakhapatnam, India 2 Professor, Department of Mechanical Engineering, Andhra University College of Engineering, Visakhapatnam, India Abstract The process control charts are a powerful tool of Statistical Process Control (SPC) to improve a firm s quality and productivity. Control charts are expensive and increase the cost of controlling the process. Therefore, an appropriate design of the process control chart in an economic point of view is necessary before the chart is used. The economic design of a control chart refers to the determination of optimal control chart parameters viz. the sample size, the sampling interval, and the control limits coefficient to minimize the total cost of the control chart. In the present work, control chart is considered for the economic design. Differential Evolution (DE) algorithm is used to determine the economically optimal parameters of control chart. The design has been conducted for two different process models, to which the control chart is applied. The comparison of results obtained by the present algorithm with the published results shows that the proposed DE algorithm is simple, effective, and efficient for the economic design of control charts. Keywords Differential Evolution, Economic Control Chart Design, Statistical Process Control, Control Chart. I. INTRODUCTION Control charts have been established as major tool for on-line quality control in industry. In particular, the control chart has been used extensively for providing information helpful in bringing a process into control, maintaining surveillance of the process, and improving process capability. The economic design of control chart has been studied by various authors. Duncan [6] proposed an economic model for the selection of control chart parameters that has become a standard in research. Goel et al. [7] presented an algorithm which gives exact optimal solution and sensitivity analysis of the loss-cost function for chart design. Chiu and Wetherill [2] proposed a very simple semi-economic procedure for the design of a control plan using chart for practical application. Montgomery [9] provided a computer program that determines the optimal design parameters for economically-based charts. Panagos et al. [10] developed economic models of the control chart for two different manufacturing process models and explored the sensitivity of the control chart parameters to the choice of model. Collani and Chung [5, 3] presented a simple procedure for selecting the optimal economic design of control chart that can be applied at the workshop level. Chung [4] determined optimal design parameters of control chart for small process shifts. Celano and Fichera [1] developed an evolutionary algorithm for multiobjective economic design of control chart using genetic algorithm. Yu and Low [16] described a very simple procedure to determine the design parameters of control chart in Duncan s [6] economic model. Vommi and Seetala [13, 14] dealt with robust economic design of control chart by using genetic algorithm search tool. Kasarapu and Vommi [11] developed an evolutionary algorithm for economic design of joint and R control charts using differential evolution. The usual approach to the economic design of control charts is to specify a model for the process, make appropriate assumptions about the relevant costs, specify a mathematical model that relates the control chart design parameters to the costs, and then optimize this model. In this paper, Panagos et al. [10] notation, cost models and cost and risk factors are considered for the design. II. DISCUSSION OF ASSUMPTIONS In this paper, two different process models are considered to which the cost model of control chart is applied. The process is assumed to start in a state of statistical control with mean μ 0 and standard deviation σ. The measurable quality characteristic of the process is assumed to be normally distributed, N (μ 0, σ 2 ). The control chart is used to maintain current surveillance of a process that is randomly subjected to a single assignable cause of magnitude δ, which takes the form of a shift in the process mean from μ 0 to μ 0 ±δσ, with an intensity of occurrences per hour. That is, assuming that the process begins in the in-control state, the time interval that the process remains in control is an exponential random variable with mean 1/ hours. Samples of size n are taken every h hours and the sample means ( ) are plotted on control chart with centre line μ 0 and control limits μ 0 ±kσ/ n, where k is control limit coefficient. If a single sample point falls outside the control limits, the process is assumed to be out of control, and a search for the assignable cause is begun. 541

2 In the first model considered, it is assumed that process continues during the search for the assignable cause (continuous model). The second process model considered assumes that the process is shut down during the search for an assignable cause (discontinuous model). The parameters μ 0, δ, and σ are assumed known, whereas n, k, and h are to be determined. Both models minimize the sum of the expected costs of sampling, while searching for an assignable cause, and operating in the out-of-control state, expressed on a per cycle basis. The cycle is defined as the time period during which the process begins in the in-control state, shifts to the out-of-control state, and as a result of an action signal on the control chart, returns to the initial incontrol state. The cost of returning the process to the incontrol state after the completion of a cycle is not included in the cost model. The cost of repairing the assignable cause, however, is included. Given the occurrence of the assignable cause between the j th and (j+1) st samples, the expected time of occurrence of the shift within this interval is = ( ) ( ) =. (1) When the assignable cause occurs, the probability that it will be detected on any subsequent sample is. (2) Where is the standard normal density. The quantity 1-β is the power (P) of the chart, and β is the type II error probability. The probability of a false alarm is. (3) Furthermore, the expected number of false alarms that occur before a shift is α times the expected number of samples taken before the shift or = =. (4) These approximations are used in the two cost models for simplification. III. COST MODELS OF CONTROL CHART A. The Continuous Process The total cycle time for the continuous process consists of one in-control period and three out-of-control periods. The in-control portion has expected length of 1/ and includes, the fraction of the time between the j th and (j+1) st samples when the process is in control. The outof-control portion consists of: 542 (a) The time until the signal is detected. The probability that a signal will be detected when an assignable cause occurs is a geometric random variable with mean 1/P. Since the length of time between samples is h, then is the expected time period that the process will be out of control before a signal is detected. Thus, using Eq. (1),. (5) (b) The time to test and interpret the results. This time is equal to a constant e times the sample size n. (c) The time to find and repair an assignable cause, D. Thus the total cycle time for the continuous process is. (6) The expected net income earned during one cycle is the net hourly income while the process is in control (V 0 ) times the expected length of time the process is in control (1/) plus the net hourly income while the process is out of control (V 1 ) times the expected length of time the process is out of control less process costs for the cycle. The cost of taking a sample is b+cn (b, c are fixed and variable costs of sampling respectively), and the expected number of samples taken during a cycle is the expected length of the cycle [E(t)] divided by the time between samples (h). The cost of investigating a false alarm is T, and the expected number of false alarms encountered during a cycle is α times the expected number of samples taken before the shift. The cost of finding and repairing an assignable cause is W, and this will occur once during a cycle. Thus the expected net income per cycle is * + and The expected net hourly income is. (7) =. (8) Let, (9), (10). (11) Then the cost function can be rewritten as Where the loss-cost function is (12)

3 B. The Discontinuous Process α. (13) The cycle time for the discontinuous process consists of one in-control period, two out-of-control periods, and three shut down periods. The expected length of time the process is in control is 1/. The expected length of time the process is out of control consists of: the time until the signal is detected * + and the time required to test and interpret the results (en). The portion of time the process is stopped consists of: (a) The time to search for and repair an assignable cause, D. (b) The time to restart the process or the setup time S 1. This time does not include the time required after the process is restarted to bring it back into statistical control. (c) The time to search for a false alarm and restart the process. This is the sum of the expected time to search for a false alarm (D 1 ) and the setup time (S 1 ) multiplied by the expected number of false alarms per cycle,. Thus the total cycle time for the non-continuous process is. (14) The expected net income for one cycle is the expected hourly income while the process is in control times the expected length of time the process is in control [V 0 (1/)] plus the expected out-of-control income times the expected length of time the process is out of control ( ( ) ) less process costs. The process costs are (a) The cost of taking a sample times the expected number of samples taken while the process is in operation,, -. (b) The cost of investigating a false alarm times the expected number of false alarms that occur before the shift in the process, α. (c) The cost of finding and repairing an assignable cause, W. (d) The setup cost, S. Therefore, the expected net income per cycle is ( ) [ ( ( ) α The expected net hourly income is ) ]. (15). (16) 543 Let α (17) (18) M= (19) Then the loss cost function for this model is ( ) α. (20) IV. APPLICATION OF DIFFERENTIAL EVOLUTION TO CONTROL CHART Differential Evolution is a population-based, directsearch algorithm for globally optimizing the complicated objective functions. For the present economic design, Neoteric Differential Evolution algorithm suggested by Vitaliy Feoktistov [15] has been used. Storn and Price [12] first proposed classical Differential Evolution algorithm which forms the base for the present Neoteric Differential Evolution. In Differential Evolution, the individuals of population contain design parameters and represent potential optimal solutions. The population is initialized by randomly generating individuals within the lower and higher boundary limits of the design parameters. Each individual of the initial population is evaluated by the cost function. In order to obtain next generation from the initial population, any one individual is chosen as the current best individual. Then, the initial population is subjected to repeated generations of differentiation, crossover and selection. Differentiation and crossover operations are used to create one trial or child individual for each target or parent individual. In order to perform the differentiation, a set of individuals, mutually different and also different from the current target individual, are randomly chosen from the current population. The search strategies of differentiation are designed on the basis of these individuals. In the crossover, by recombining the trial and target individuals, the trial individual inherits parameters of the target individual with certain probability. Next, boundary limits of the trial individual parameters are verified. If any parameter exceeds the limits, the parameter is reset by re-initialization. This trial individual is evaluated by the cost function. Afterwards, selection is fulfilled by comparing the cost function values of target and trial individuals. If the trial individual has an equal or lower cost to the target individual, it replaces its target individual in the population. If the trial individual has higher cost than the target one then the target individual is retained. Then, if the new trial individual of the population is better than the current best individual, the current best individual s index is updated. Figure 1 shows how the Differential Evolution is applied for economic design of control chart.

4 In the present work, an individual of the population represents a set of design parameters of control chart, namely n, h, and k. To define the limits of search space, feasible values are taken as lower and higher boundary limits of design parameters by considering the published economic designs on control chart. Table1 contains the boundary constraints taken on the parameters of the control chart. Once the search space has been defined, the next step is to find the best parameters of the evolutionary algorithm. Parametric tuning has been carried out to find the effective parameters for the algorithm viz. the population size, constant of differentiation, and constant of crossover. A few loss-cost function evaluations have been made using different combinations of control parameters, generations and search strategies. Different population sizes in multiples of 10, number of generations in multiples of 50 and search strategies of differentiation as suggested in Vitaliy Feoktistov [15] have been tested. For refining the selection of constant of differentiation, F and constant of crossover, Cr, different values in multiples of 0.05 have been chosen in the intervals F (0, 1] and Cr [0, 1] respectively. By using the feedback provided by these function evaluations and practical guidelines given in Vitaliy Feoktistov [15], the parameters for the Differential Evolution Algorithm have been finalized. Table2 shows the parameters of DE employed in the present work to obtain the optimum control chart design parameters. As the performance of any evolutionary algorithm is best represented by the probability distribution of the best objective value ( Kuo et al. [8]), it is required to run algorithm for a number of times, in order to check its consistency in providing the best solution. While obtaining robust chart designs using genetic algorithm, Vommi and Seetala [14] has run the algorithm for 300 times and obtained the statistics of the best objective values. In the present case, to check the consistency of the algorithm in providing the best solutions, the algorithm has been run for 300 times for 10 sets of randomly selected cost and risk factors for each continuous and discontinuous process models. All the 300 runs yielded the same best design parameters reported in this paper for each of the selected cost and process parameters sets. Since there is no variation in the best values of the objective function, single solution corresponding to each input data set for control chart has been tabulated. The economic designs of Panagos et al. [10] are considered for comparison of optimum designs obtained by the present algorithm for the same cost and process parameters for each continuous and discontinuous process models. A comparison of the results is presented in Tables 3 and 4 for each continuous and discontinuous process models respectively. 544

5 Start Set appropriate lower and higher boundary limits for each design parameter (n, h, and k) of the control chart Generate a predetermined number of sets of design parameters randomly within the respective low and high bounds Evaluate these sets of design parameters by the cost function. Choose one of the sets as the current best set of design parameters Initialize index of current generation Are predetermined generations over? Choose n, h, k values of the current best set of design parameters Update index of current generation Initialize index of current target set of design parameters Stop Are all target sets of design parameters over? Randomly choose 3 sets of n, h, and k values for differentiation operation Apply differentiation and crossover operations to create a new trial set of n, h, and k values Check whether any design parameter of the trial set exceeds the boundary limits. Reset the out-of-bounds design parameter Evaluate the trial set of design parameters by the cost function Select better set of design parameters by comparing the cost function values of current target and trial set of design parameters Update index of the current best set of design parameters if the trial set of design parameters is better than the current best set Update index of current target set of design parameters Figure 1. Procedure for Economic Design of Control Chart Using Differential Evolution 545

6 TABLE I BOUNDARY LIMITS OF CONTROL CHART PARAMETERS USED IN PRESENT DIFFERENTIAL EVOLUTION ALGORITHM Control Chart Design Parameters Low High Boundary Limits n 2 33 h k TABLE III PARAMETERS USED IN PRESENT DIFFERENTIAL EVOLUTION ALGORITHM Description of the Differential Evolution Algorithm Parameters Magnitude/ Method No. of design parameters in a set 3 Population size 30 Search strategy Rand3 Constant of Differentiation, F 0.85 Type of Crossover Combinatorial Constant of Crossover, Cr 0.50 Selection scheme Elitist Number of generations 300 V. RESULTS AND DISCUSSION Differential Evolution algorithm has been applied in the economic design of chart by utilizing the cost and risk factors of Panagos et al. [10] for continuous and discontinuous process models. Given cost and risk factors and other process parameters, the present work finds the sample size, the interval between samples and the control limits coefficient for chart that minimize the expected loss-cost per hour. For two process models in Panagos et al. [10] paper, optimum designs were obtained by the method described by Montgomery [9], which, in turn, was based on the approximate procedure developed by Chiu and Wetherill [2]. In Tables 3 and 4, optimum economic designs obtained by the present Differential Evolution algorithm for these cost and risk factors are compared with those reported by Panagos et al. [10] for continuous and discontinuous process models respectively. Comparison of the results shows that the costs found by Differential Evolution algorithm are lower than those of Panagos et al. [10] in many cases of the cost and process parameters considered. The cost reduction of 0 to 3.31% in continuous process model and 0 to 3.61% in discontinuous process model has been achieved. Also, it has been observed that the algorithm could provide the same best solutions even after a number of times the algorithm was run with different initial solutions. 546 The present Neoteric Differential Evolution algorithm efficiently creates new sets of design parameters. Comparison of results indicates that the algorithm is effective and accurate. Figure1 shows the simplicity of the present algorithm. That is, it shows that the algorithm is easy to understand and also easy to program. To conclude, the proposed Neoteric Differential Evolution algorithm is simple, efficient and effective. And also, it is both easy to implement and easy to use in the economic design of control charts. REFERENCES [1] Celano, G. and Fichera, S. (1999). Multiobjective Economic Design of an control Chart. Computers & Industrial Engineering, 37, [2] Chiu, W. K. and Wetherill, G. B. (1974). A Simplified Scheme for the Economic Design of Charts. Journal of Quality Technology, 6(2), [3] Chung, K. J. (1990). A Simplified Procedure for the Economic Design of Charts. International Journal of Production Research, 28(7), [4] Chung, K. J. (1992). Determination of Optimal Design Parameters of an Control Chart. The Journal of the Operational Research Society, 43(12). [5] Collani, E. V. (1986). A Simple Procedure to Determine the Economic Design of an Control Chart. Journal of Quality Technology, 18(3), [6] Duncan, A. J. (1956). The Economic Design of charts Used to Maintain Current Control of a Process. Journal of the American Statistical Association, 51(274), [7] Goel, A. L., Jain, S. C. and Wu, S. M. (1968). An Algorithm for the Determination of the Economic Design of Charts Based on Duncan s Model. Journal of the American Statistical Association, 63(321), [8] Kuo, W., Prasad, V.R. and Hwang, C.L. (2001). Optimal Reliability Design, Cambridge: Cambridge University Press. [9] Montgomery, D.C. (1982). Economic Design of an Control Chart. Journal of Quality Technology, 14(1), [10] Panagos, M. R., Heikes, R. G. and Montgomery, D. C. (1985). Economic Design of Control Charts for Two Manufacturing Process Models. Naval Research Logistics Quarterly, 32, [11] Rukmini V. Kasarapu and Vijaya Babu Vommi (2011). Economic Design of Joint and R Control Charts Using Differential Evolution. Jordan Journal of Mechanical and Industrial Engineering, 5(2), [12] Storn, R. and Price, K. (1995). Differential Evolution - A Simple and Efficient Adaptive Scheme for Global Optimization Over Continuous Spaces. Technical Report TR , International Computer Science Institute, Berkeley, CA. [13] Vijaya Babu Vommi and Murty, S.N. Seetala (2007a). A New Approach to Robust Economic Design of Control Charts. Applied Soft Computing, 7, [14] Vijaya Babu Vommi and Murty, S.N. Seetala (2007b). A Simple Approach for Robust Economic Design of Control Charts. Computers and Operations Research, 34, [15] Vitaliy Feoktistov (2006). Differential Evolution In Search of Solutions, USA: Springer Publications. [16] Yu, F. J. and Low, C. (2005). An Algorithm for the Determination of Optimal Design Parameters of Control Charts. International Journal of Advanced Manufacturing Technology, 26,

7 TABLE IIIII COMPARISON OF OPTIMUM DESIGNS FOR CHART FOR THE CONTINUOUS PROCESS MODEL COST AND RISK FACTORS PANAGOS et al. RESULTS DIFFERENTIAL EVOLUTION RESULTS * The numbers are same as used by Panagos et al. [10]. % OF REDUCTION IN LOSS- COST e.g., M δ e D b c W T n h k L 1 n h k L 1 No. *

8 TABLE IVV COMPARISON OF OPTIMUM DESIGNS FOR CHART FOR THE DISCONTINUOUS PROCESS MODEL e.g., No. * COST AND RISK FACTORS PANAGOS et al. RESULTS DIFFERENTIAL EVOLUTION RESULTS M δ e D b c W T V 0 S S 1 D 1 n h k L 2 n h k L * The numbers are same as used by Panagos et al. [10]. % OF REDU CTIO N IN LOSS- COST