Joint Design of Economic Manufacturing Quantity, Sampling Plan and Specification Limits

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1 c Heldermann Verlag Economic Quality Control ISSN Vol 17 (2002), No. 2, Joint Design of Economic Manufacturing Quantity, Sampling Plan and Specification Limits Chung-Ho Chen and Chao-Yu Chou Abstract: In this paper we propose an extension of the problem to determine the economic design of a Type I continuous sampling plan (CSP-1 plan) and the economic manufacturing quantity (EMQ) under imperfect quality investigated in Chen et al. (2000), where the economic objective equals the total inventory cost including the set-up cost, the holding cost and the expected quality loss. The extension refers to the economic specification limits. The problem can be formulated as a non-linear mixed-integer minimization problem. By solving it, the required level of product quality is obtained at minimum total inventory cost. A numerical example is given to illustrate the solution procedure. Keywords: Economic Manufacturing Quantity (EMQ), Type I Continuous Sampling Plan (CSP 1 Plan), Average Outgoing Quality (AOQ), Average Outgoing Quality Limit (AOQL), Average Fraction Inspected (AF I), Specification Limits, Taguchi s Quality Loss Function. 1 Introduction The general concept of specifications is that items must meet some limits for being conforming. The limits are usually selected according to technical criteria. However, in view of the economic character of almost any manufacturing process, it makes sense to select the specification limits using economic criteria. For this end, Taguchi s (1986) quadratic quality loss function for evaluating total losses to society seems to be appropriate. Following these ideas, Kapur and Wang (1987), Kapur (1988) and Kapur and Cho (1994) have addressed problems related to the application of the quality loss function in the economic design of specification limits. Traditionally when determining the economic manufacturing quantity (EMQ), it is assumed implicitly that items produced are of perfect quality. However, product quality is not always perfect and is usually a function of the production process. Porteus (1986) and Rosenblatt & Lee (1986) were the first to incorporate quality costs into the EMQ problem and found that the optimal production cycle is shorter than that of the traditional EMQ model. Continuous sampling plans were first proposed by Dodge (1943). Dodge s initial plan is called the CSP 1 plan which may be used as process control aimimg at improving outgoing product quality. Recently, Chen et al. (2000) and Chen and Chou (2001) have

2 146 Chung-Ho Chen and Chao-Yu Chou proposed to consider imperfect quality under the application of a CSP 1 plan. According to Chen et al. (2000), the specified quality level can be reached under minimum total inventory cost including quality cost. In this paper, the problem of a joint design of EMQ, CSP 1 plan and specification limits is addressed. Assuming that the quality characteristic follows a normal distribution and defining the total inventory cost as the sum of set-up cost, holding cost and expected quality loss, the determination procedure for a joint design of EMQ, CSP 1planand specification limits is an extension of Chen et al. s (2000) method. In the following sections, both Chen et al. s (2000) EMQ problem with imperfect quality and Kapur and Wang s problem for designing specification limits are briefly reviewed. Subsequently these two problems are integrated for jointly designing EMQ, CSP 1 plan and specification limits. A solution algorithm for the integrated problem is proposed and an example is provided for illustrating the algorithm. 2 Notation The following notations are used. Process and product parameters: Φ( ) Standard normal distribution function assumed for the quality characteristic under consideration. φ( ) Standard normal density function. µ Mean of the quality characteristic under consideration. σ 2 Variance of the quality characteristic under consideration. I Production rate in items per day. D Demand rate in items per unit time. k A constant called quality loss coefficient. O Demand rate in items per day. p Incoming quality. q =1 p Economic parameters: C i C r C S C S1 C S2 h Inspection cost per item. Replacement cost per item. Set-up cost for each production run. Cost of rework, repair, or replacement for a defective item which is not released to the market, i.e., internal failure cost. Cost of replacement and loss of goodwill for a defective item which is released to the market, i.e., external failure cost. Holding cost per item per unit time.

3 Economic Manufacturing Quantity, Sampling Plan and Specification Limits 147 Quality requirement: p L Required AOQL value. Cost functions: TC 0 TC 1 TC(Q, i, f, l) Total expected quality loss per item under 0% inspection. Total expected quality loss per item under 100% inspection. Total inventory cost per unit time. Quantities to be determined: Sampling plan f i Production Q Specification l Sampling frequency for short-run CSP 1plan. Clearance number of the 100% inspection stage for short-run CSP 1plan. Economic manufacturing quantity. Specification limit coefficient: if the quality characteristic falls within the limits defined by µ ± lσ 2, then the item is said to be conforming. Sampling plan characteristics: AOQ AOQ 1 AF I AF I 1 Average outgoing quality for long-run CSP 1plan. Average outgoing quality for short-run CSP 1 plan. Average fraction inspected for long-run CSP 1plan. Average fraction inspected for short-run CSP 1plan. 3 Chen et al. s Problem Chen et al. (2000) presented a solution to the problem of determining the economic manufacturing quantity when Yang s (1983, pp ) short-run CSP 1 plan is applied for controlling imperfect production quality. The total inventory cost includes the set-up cost, the holding cost, the inspection cost, and the internal and external failure cost. The problem investigated by Chen et al. (2000) can be formulated as follows: Minimize TC(Q, i, f) (1) Subject to max 0 p 1 AOQ 1 = p L (2) where

4 148 Chung-Ho Chen and Chao-Yu Chou where 0 Q D, Q IN (3) 0 i Q, i IN (4) 0 f 1 (5) D TC(Q, i, f) =C s Q + 1 ( 2 Q 1 O ) ] h + D [AF I 1 (C i + pc S1 )+AOQ 1 C S2 (6) I (1 f)pqi AOQ = (7) f +(1 f)q i AOQ 1 = AOQ + E[Z] ( ) V [W ]+E[W] 1 (8) 2Q (E[W ]) 2 with E[Z] = 1 f 1 (9) E[W ]= f +(1 f)q2 fpq i V [W ]= 1 pqi (2i +1) q 2i+1 p 2 q 2i + 1 fp AF I 1 =1 AOQ 1 p ( ) 1 fp 1 (10) (11) (12) The above problem consists of minimizing TC(Q, i, f) while guaranteeing the required quality given by p L. The decision variables of this problem are the economic manufacturing quantity Q, the clearance number i, andthesamplingfrequencyf. Wang and Chen (1998) presented a method to find the parameters for Yang s (1983) shortrun CSP 1 plan with specified average outgoing quality limit (AOQL) and production quantity. For the requirement given by (2), we can adopt Wang and Chen s (1998, pp ) method for obtaining the sampling plan given by the pair (i, f) meeting the requirement with respect to the maximum AOQ 1 for given value of Q. 4 Kapur and Wang s Problem Kapur and Wang (1987) addressed the problem of selecting economic specification limits using as criterion the quality loss function based on the assumption that 100% screening inspection by an on-line quality control system thought to be conducted as a short term approach to reduce variance of the product to be shipped to the customers. The strength of the Kapur and Wang s (1987) article is that it provides a fairly clear and thorough presentation of the topic of economic design of specifications as related

5 Economic Manufacturing Quantity, Sampling Plan and Specification Limits 149 to Taguchi s (1986) quality loss function. It covers the basis of quality loss function, underlying distributions, optimization problems and sensitivity analysis for the nominalthe-best, smaller-the-best, and larger-the-best cases. According to Equations (23)-(25) of Kapur and Wang (1987, p. 30), the total expected cost per item under 0% inspection for the nominal-the-best case is TC 0 = kσ 2 (13) and the total expected cost per item under 100% inspection for the nominal-the-best case is ) TC 1 = kσ (1 2 2l ( ) 2Φ(l) 1 φ(l) +2 1 Φ(l) C r + C i (14) Kapur and Wang s (1987) optimization problems consists of determining specification limits taking into account inspection costs, scrap/rework costs, and loss due to variance. 5 An Extended Problem The solution of the problem of jointly designing EMQ, CSP 1 plan and the specification limit l is based on the following three assumptions. 1. The quality characteristic has a normal distribution with expectation µ and variance σ The quality loss can be described by a quadratic loss function and the quality characteristic is of nominal-the-best type. 3. The process mean is centered at the target value. Following Chen et al. (2000), the problem is to minimize the total inventory cost under the required product quality. Combining Chen et al s. (2000) problem with that of Kapur and Wang (1987), an extended set of decision variables is obtained. These are the economic manufacturing quantity Q, the clearance number i, the sampling frequency f, and the standardized specification limit l. The modified problem for a joint determination of EMQ, CSP 1 plan and specification limits is formulated as follows. Minimize D TC(Q, i, f, l) =C s Q + 1 ( 2 Q 1 O ) h + D [AF I 1 TC 1 +(1 AF I 1 )TC 0 ] I D = C s Q + 1 ( 2 Q 1 O ) h + D [AF I 1 (TC 1 TC 0 )+TC 0 ] (15) I Subject to max 0 p 1 AOQ 1 = p L (16)

6 150 Chung-Ho Chen and Chao-Yu Chou where 0 Q D, Q IN (17) 0 i Q, i IN (18) 0 f 1 (19) l 0 (20) with AOQ 1 given by (8), AF I 1 by (12) and TC 0 = kσ 2 (21) ) TC 1 = kσ (1 2 2l ( ) 2Φ(l) 1 φ(l) +2 1 Φ(l) C r + C i (22) p =2(1 Φ(l)) (23) q =2Φ(l) 1 (24) 6 Solution Algorithm Chen et al. (2000) have developed an approach to obtain the optimal parameters for a CSP 1 plan and the economic manufacturing quantity Q for specified quality level and specification limits. The solution procedure for the modified problem given by (15) - (20) may be based on the former one and consists of the following steps. Step 1: Compute TC 0, the specification limit l, which minimizes TC 1, and the value of minimum TC 1. Step 2: For given p L select a value for Q and determine (compare Appendix of Wang and Chen (1998, pp )) the unique pair (i (Q),f (Q)) which minimizes TC(Q, i, f, l ). Step 3: Repeat Step 2 with Q replaced by Q + 1. Then repeat Step 2 with Q replaced by Q + 2, etc. Terminate the procedure when TC(Q, i, f, l) starts to increases with further increase in Q. Select as Q the last value before the increase starts. Step 4:

7 Economic Manufacturing Quantity, Sampling Plan and Specification Limits 151 ) The optimal solution is given by (Q,i (Q ),f (Q ),l. The minimum total inventory cost are given by TC ) (Q,i (Q ),f (Q ),l. 7 Numerical Example In this section, an example is given for illustrating the solution procedure. Consider the following set of input parameters: Process parameters µ σ 2 k I D O Economic parameters C i C r C S h Quality requirement p L 0.02 The problem is to determine the optimal parameters (Q,i,f,l ) such that the required level of quality p L is reached and at the same time the total inventory cost is minimum. For 0% inspection, the expected quality loss per item is given by TC 0 =1.25 (25) For 100% inspection, the minimum expected quality loss per item is attained for l =1.330 implying the lower specification limit LSL = and the upper specification limit USL = and TC 1 =1.146 (26) Applying the solution algorithm proposed in the previous section, we obtain the optimal policy: (Q,i,f,l ) = (2000, 86, , 1.330) (27) with the total inventory cost per unit time: TC(Q,i,f,l ) = (28) Thus, the result obtained is as follows: The economic manufacturing quantity is Q = 2000, the process shall be controlled with the CSP 1 plan with i =86andf =0.0354, and the specification limits shall be set at LSL =9.335 and USL =

8 152 Chung-Ho Chen and Chao-Yu Chou 8 Conclusion This article presents a solution for jointly designing the economic manufacturing quantity, a CSP 1 plan and specification limits aiming at minimizing the incurred cost subject to meeting some quality requirements. It is an extension of the paper of Chen et al. (2000) and may be continued by assuming a non-normal distribution for the quality characteristic on the one hand and applying other sampling plans than CSP 1 plan for controlling the out-going product quality. References [1] Chen, C.H., Chou, C.Y. and Cheng, T.S. (2000): CSP-1 plan applied in the economic manufacturing quantity model. Economic Quality Control 15, [2] Chen, C.H. and Chou, C.Y. (2001): Integrating EMQ model and product quality. Journal of the Chinese Institute of Engineers, Series A, 24, [3] Dodge, H.F. (1943): A sampling plan for continuous production. Annals of Mathematical Statistics 14, [4] Kapur, K.C. and Wang, C.J. (1987): Economic design of specifications based on Taguchi s concept of quality loss function, in Quality: Design, Planning, and Control. (Eds.: DeVor, R.E. and Kapoor, S.G.), The Winter Annual Meeting of the American Society of Mechanical Engineers, Boston, [5] Kapur, K.C. (1988), An approach for development of specifications for quality improvement, Quality Engineering, 1, pp [6] Kapur, K.C. and Cho, B.-R. (1994): Economic design and development of specifications. Quality Engineering 6, [7] Porteus, E.L. (1986): Optimal lot sizing, process quality improvement and set-up cost Reduction. Operations Research 34, [8] Rosenblatt, M.J. and Lee, H.L. (1986): Economic production cycles with imperfect production processes. IIE Transactions 17, [9] Taguchi, G. (1986): Introduction to Quality Engineering. Asian Productivity Organization, Tokyo, Japan. [10] Yang, G.L. (1983): A renewal-process approach to continuous sampling plans. Technometrics 25, [11] Wang R.C. and Chen C.H. (1998): Minimum AFI for short-run CSP-1 plan. Journal of Applied Statistics 25,

9 Economic Manufacturing Quantity, Sampling Plan and Specification Limits 153 Chung-Ho Chen Department of Industrial Management Southern Taiwan University of Technology 1 Nan-Tai Street, Yung-Kang City Tainan 710 Taiwan Chao-Yu Chou Department of Industrial Engineering and Management National Yunlin University of Science and Technology Touliu 640 Taiwan