Strojniški vestnik - Journal of Mechanical Engineering 54(2008)12, 855-865 UDC 658.562 Paper received: 28.01.2008 Paper accepted: 25.09.2008 Economic Design of Control Charts Rok Zupančič1* - Alojzij Sluga2 1 Elan, d.o.o., Begunje, Slovenia 2 Faculty of Mechanical Engineering, University of Ljubljana, Slovenia Control charts are widely used in industry for monitoring and controlling manufacturing processes. They should be designed economically in order to achieve minimum quality control costs. In this paper, an economic design of Shewhart control charts for process mean is proposed that takes into account various parameters. Standards for sample size within statistical process control do not exist due to high diversity of modern production. In the proposed economic model process-mean shift is assumed as random variable. This is a better approximation of the real world, than the models that assume process-mean shift as a constant value. Probability density function is used for description of process-mean shift. The optimum sample size is computed on base of loss function, regarding to constraints of particular production process. The comparison of optimum sample sizes assuming process-mean shift as a constant value versus random variable is presented. © 2008 Journal of Mechanical Engineering. All rights reserved. Keywords: control charts, quality control, economic design, statistical process control 0 INTRODUCTION A quality of a product is usually understood as a capability of fulfilling needs of a customer. A manufacturer desires to produce such products that match the desired specification and also fulfil customer needs. Hence, quality control is an essential part of the manufacturing process. In modern production the product's specifications are not controlled directly. The quality control is carried out by controlling the manufacturing process. A process quality is usually deducted from the manufactured products. In this case, not all of the products can be inspected directly, but only statistically. The quality control is therefore called Statistical Process Control (SPC). A control chart is a primary tool for SPC. A purpose of the control chart is to detect possible changes of the process, and inform the process operator about it. Many types of control charts have been developed in the past. Shewhart control charts, Cumulative sum (Cusum) control chart, Exponential weighted moving average (EWMA) control chart, and others are presented elsewhere for instance in [1]. Control chart is a quality control technique. Some quality improvement techniques are discussed in [2]. Shewhart control charts are widely used in industry. The design of the control charts has economic consequences since the cost of sampling and testing, the investigating out-of-control signals and possibly correcting probable causes, as well as costs of allowing nonconforming products to reach the customer are all affected by choice of the control charts parameters [3]. The most important parameter that affects the cost of quality control and indirectly the cost of production is sample size. In literature, description of standards that deal with sample size for acceptance sampling can be found in [3]. Due to their generality of use, these standards are very robust. The standards for SPC, which deal with sample size, do not exist. The reason is a standard's uselessness due to its generality and high diversity of modern production. It is not the same if low- or high-priced products are manufactured. Due to diversity of quality control costs, optimum sample size should be calculated for every production process respectively. In industrial environment, no statistical expertise is available. Normally, a sample size is defined by a rule of thumb, lacking the optimality in economical sense. Due to fierce competition on the global market, an economic control charts design can be a critical issue for a manufacturing enterprise. Some approaches to the economic control charts design are given in literature [4] to [7]. In all of these approaches, the process-mean shift (PMS) is assumed constant. In [4], Duncan's loss-cost function is used, which is originally proposed in [8]. In [5], simplified theoretical backgrounds and directions of economic design of control charts are proposed. In [6], economic design of *Corr. Author's Address: Elan, d.o.o., Begunje 1, SI-4275 Begunje na Gorenjskem, Slovenia, rok.zupancic@elan.si control charts of cumulative count of conforming products is presented. In [7], an economic design of control charts is presented for variable sampling size and sampling interval. In approaches [4] to [7], a PMS parameter is supposed to occupy an exact constant value. PMS is actually not known in advance. If constant value for PMS is assumed, the economic model is valid only for that PMS. PMS directly affects the optimum sample size calculation. In this paper, an economic design of Shewhart control chart for the mean value of the process (x chart) is discussed. The approach does not assume an exact value for PMS. We show that by finding its probability density function of PMS, we can compute the optimum sample size (n*). In Section 1, some theoretical backgrounds and parameters of the Shewhart control chart are discussed. In Section 2, the sampling policy and the economic model are proposed. In Section 3, the optimisation is presented in form of optimum sample size as a function of the PMS parameter. These curves are created for different economic parameters (costs). In Section 4, discussion and conclusions are presented. 1 BACKGROUNDS OF SHEWHART CONTROL CHART Shewhart control chart method is about 80 years old. It is a method for statistical process control, in sense of process-mean and process variability supervision. The statistical parameters are deduced from a statistical sample. The accuracy depends on the sample size (n). In this paper, we are concentrating on x control chart for PMS detection. The control chart is a kind of hypothesis testing hence the process is in a state of statistical control. The hypothesis testing is done with sample statistics. The sample statistics is assumed to follow a normal distribution with mean (m) and sample standard deviation (o). A sample standard deviation depends on standard deviation of population (a) and sample size (n). The null hypothesis states that the mean of a measured process variable has a desired value. The alternative hypothesis is that this mean value may be changed by ka. That means that the new mean value is n+ko. The centre line of the control chart is kept at the mean of the process (m). The upper control limit (UCL) and the lower control limit (LCL) are: UCL = u + La/Jn (1) LCL = U- La/4n (2) where L is the control limit coefficient (usually 3 is used). The interval of normal distribution between m-3o and m+3o covers 99.73% of the entire population. Statistically, this means that 99.73% of sample values are supposed to fall between the control limits. In spite of no change in a process, 0.27% of the sample values are to exceed the control limits. This is treated as type I error (false-positive). The probability of type I error (a) is defined as: a = 20(-L) (3) where PDFk (■)• dk+J ARLÇy PDFk (■)• dk\N (17) Eq. (17) does not have k among its arguments. PMS-multiplier (k) is handled by PDFk. Since the zero case (k=0) is not The optimum sample size is the sample size that minimizes LT. All of the discussed parameters have impact on the optimum sample size. In Fig. 4 we can see loss function (LT) as a function of the sample size for different modes (M). The following constants were assumed: CC1=10, CnC=100, Cfa=Crac=2 (process capability), Nl=100, o=0.2. Äp=500, Cp=1 Fig. 4. Loss function (LT) vs. the sample size (n), where Cc1=10, Cnc=l00, Cfa=Cra Nl=l00, ak=0.2; the ordinate is in a logarithmic scale =200, Xp=500, Cp=1, From these curves the optimum sample size for a chosen mode Mk can be found. The curve (Mk=0.25) is monotonically increasing. In this case, we can conclude that control of the process is economically unacceptable since the optimum sample size is 0. The curve (Mk=0.5) has the minimum at approximately n = 1, but it is not explicitly significant. If greater sample size is chosen, error gets smaller. Curves, with larger Mk, have a significantly expressed optimum sample size. Using expression LT, the curves for the optimum sample size in dependency of Mk were plotted for various parameters. The optimum sample size and lot size can be calculated numerically, using LT. The lot size defines the sampling frequency. In the presented case, the sampling frequency is 1 sample per 100 manufactured products, which means vs=0.01. The optimum sample size, where lot size is 100, can seen from diagrams, presented in Figs. 5a, b and 5c. By changing Cfa and Crac, the optimum sample size curves change. When Cfa and Crac increase, the optimum sample sizes lowers and peaks of the optimum sample size curves move to the right. Cfa and Crac have impact on the optimum sample size, where Mk is small. For large values of Mk the cost constants Cfa and Crac have no significant impact. i' 1 ■ \ % .7 < / . < £¡3 •' i 11 LV. t ■ ' f 1/ J V- A- ----Co1 = 1 ----Ccl =2 - -a- - Cc, = 5 Ccl = 10 Ccl = 50 Ccl = 100 Fig. 5a. Optimum sample size (n*) vs. modeMk for different Cc1, where, Cnc=100, Cfa=Crac=100, X„=500, Cv=l, N^lOO, ok=0.2 Fig. 5b. Optimum sample size (n*) vs. modeMk for different Ccl, where, Cnc=100,Cfa=Crac=200, Ap=500, Cp=1, Ni=100, ak=0.2 Fig. 5c. Optimum sample size (n*) vs. modeMkfor different Cc1, where, Cnc=100, Cfa=Crac=500, Ap=500, Cp=1, Ni=100, ak=0.2 In Figs. 5a, b and c the optimum sample size in dependency of the absolute value of mode (Mk) are presented. In some cases when the optimum sample size is 0, that means SPC is not recommended, because it is too expensive. In those cases, the total loss function LT is monotonically increasing, as we can see in Fig. 4 (Mk = 0.25). 4 DISCUSSION AND CONCLUSIONS In this paper, a new approach for the economic design of x charts is given. Optimum intensity of quality inspection, that means sample size and sampling frequency, can be found by the optimisation of the proposed loss function. A sampling frequency is defined by lot size. The majority of the overall loses in quality control are contributed by the state, where the process is out of control and the process-mean shift is not detected. The duration of this state, which is estimated by ARL1, depends on the sample size, PMS and the control limits. Due to different PMS values appearing in practice, we describe PMS in terms of probability. In our approach PMS is not assumed to be a constant value, but it is described by the probability density function PDFk. The proposed case study is obtained with bimodal probability density function within the proposed economic model. The curves for the optimum sample size for different cost constants are presented. In other approaches, proposed in [4] to [7], a constant value for PMS is used, which may be questionable in real production. The advantage of our approach is that PMS, estimated with PDFk, is closer to reality. In real world, PMS varies with time. Our approach takes this fact into account and describes PMS as a random variable. For PDFk an arbitrary probability density function can be used. Zero case (k=0), that means PMS is 0, is excluded from the integral area in Expressions (17) and (18). If there are any constraints for sample size or lot size, given by the observed production process, the optimum sample size (n*) and the optimum lot size can be found iteratively. For the loss functions in Fig. 4 and optimum sample size in Figs. 5a, b and c the value of Cp is 1. If the process capability ratio is greater (Cp>1), the optimum sample sizes lowers and peaks of their curves move to the right. For greater Mk (Mk>4) no significant difference of optimum sample size for different Cc1 was observed. In our case study, ak = 0.2, but normally it depends on the nature of the process shift. In Fig. 6, the curves of the optimum sample size for various ak values are presented. Proposed PDFk is taken into consideration. When a variance of PDFk approaches 0 (ak ^ 0), Mk approaches a constant value. Due to this fact, the curve of the optimum sample size taking into account PDFk differs from the curve of the optimum sample size, where PMS is 14 12 10 8 6 4 2 0 t A / 1 / I ! y / .r - ok = 0 ----crk = 0.2 —■— <^k = 0.6 0 0.5 1 1.5 2.5 3 M„ 3.5 4.5 Fig. 6. Comparison of optimum sample sizes (n*) for different (ak), where Cc1=2, Cnc=100, Cfa=Crac=200, Xp=500, Cp=1, Nl=100 supposed to be a constant value. The difference is significant for small values of Mk, (Mk<1.5). If the ak increases (ak=0.6), the optimum sample size approaches a constant value. In spite of that, the optimum sample size cannot be generalised. In Fig. 6 comparison of optimum sample size curves for different ak is presented. The optimum sample sizes, calculated with the proposed economic model (where ak=0.2 and ak=0.6) are compared with the optimum sample sizes, where PMS is assumed as a constant value. In this paper, the economic model for Shewhart control charts for process mean design was proposed. An economic design of control charts is a complex operation. In industrial environment, a statistical and quality control expertise is not available. With intention to support this expertise, a web service based operation support will be available soon and proposed in [11]. The web service for this kind of operation support will help operators in the industrial environment not only to control the quality of processes, but also to design the optimum lot sizes and sample sizes. Economic design of control charts is a key operations support for the manufacturing enterprises in order to achieve a competitive position on the global market. 5 ACKNOWLEDGEMENT This paper was partially supported by the Ministry of Higher Education, Science and Technology Republic of Slovenia, Grant No. 3211-06-000521 and inspired by 6 FP NoE VRL-KCiP - "Virtual Research Lab for Knowledge Community in Production", Contract No. FP6-507487/2. 6 GLOSSARY AND NOTATION ARL0 Average Run Length of false alarms ARL1 Average Run Length of process- out-of-control detection Cc1 Cost per unit (product) controlled Cfa Cost per false alarm C ^nc Cost per nonconforming unit Cp Process capability ratio C ^rac Cost to locate and repair an assignable cause of the process Cusum Cumulative sum EWMA Exponential weighted moving average k Multiplier of a process standard deviation ko Shift of the process mean value L LCL LSL Lt lxx m Mk n n* N PDFk PMS P nc Control limits coefficient Lower control limit Lower specification limit Total loss function Loss function Sample statistic mean Mode of PDFk Sample size Optimum sample size Lot size Probability density function of k Process-mean shift Fraction of nonconforming products SPC Statistical Process Control UCL Upper control limit USL Upper specification limit a Probability of type I error P Probability of type II error Process failure rate i Process mean v, Sampling frequency a Process standard deviation ak Standard deviation of PDFk a Sample standard deviation