Manufacturers are often faced with the problem of selecting the optimum process mean. Wen and Mergen (1999) used the unbalanced step loss function for measuring the cost of the non-conforming item and adopted a trade-off model for determining the optimum process mean. They assumed that the quality characteristic is normally distributed, the process variance constant and the process mean is unknown. This paper presents the modified Wen and Mergen model with a step loss function and piecewise function using an exponential distribution. The proposed model is a generalization of Wen and Mergens model. Keywords: step-loss function, piecewise linear loss function, exponential distributio

Owido, Patrick K.

Vishwanathan, Krishnan

English

International Institute for Science, Technology and Education (IISTE): E-Journals

European Journal of Business and Management                                                                                                                               www.iiste.org ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online) Vol.6, No.6, 2014  155 Towards Determining the Optimum Process Mean using an Exponential Distribution  Patrick K. Owido*    Krishnan Vishwanathan Jomo Kenyatta University of Agriculture and Technology, P.O. Box 62000-00200 Nairobi, Kenya * Email of corresponding author: powido1@gmail.com  Abstract Manufacturers are often faced with the problem of selecting the optimum process mean. Wen and Mergen (1999) used the unbalanced step loss function for measuring the cost of the non-conforming item and adopted a trade-off model for determining the optimum process mean. They assumed that the quality characteristic is normally distributed, the process variance constant and the process mean is unknown. This paper presents the modified Wen and Mergen model with a step loss function and piecewise function using an exponential distribution. The proposed model is a generalization of Wen and Mergens model. Keywords: step-loss function, piecewise linear loss function, exponential distribution  1. Introduction The selection of economic process mean is an important topic for statistical process control. The setting of the optimum process mean will directly affect the total cost to the society including the inspection cost, scrap or rework cost and the loss to the customer. Considerable researches in this area include Li (1997), Wu and Tang (1998), Lin and Chirng (1999), Wen and Mergen (1999), Li and Cherng (2000) Maghsoodloo and Li (2000), Philips and Cho (2000) and Li and Wu (2001). Wen and Mergen (1999) used the unbalanced step loss function for measuring the cost of the non-conforming item. The normal quality characteristic, the constant process variance, and the unknown process mean are assumed in their model. They selected the optimum process mean based on minimizing the costs of falling below the lower specification limit (TL) and exceeding the upper specification limit (TU). Cho and Leonard (1997) presented that the piecewise linear quality loss function for product is roughly proportional to the deviation of the quality characteristic from its specification limits. The linear loss function is usually applied in the filling/canning problem for determining the optimum manufacturing target Carlsson (1984), Golhar and Pollock (1998), Misiorek and Barnett (2000) and Lee et. Al (2001). The lognormal distribution is usually adopted for describing the lifetimes of mechanical and electrical systems and other survival data. It is apparent that the exponential distribution is an important competitor to the lognormal, gamma or weibull distributions as models for non-negative phenomena. This paper further presents the modified Wen and Mergen’s (1999) model with exponential distribution . the step loss and the piecewise linear loss function of product are considered in the modified model.  2. Literature Review Wen And Mergen’s Model With Normal Distribution By minimizing the unbalanced costs of out-of-specification, Wen and Mergen obtained the optimum process mean. There are three assumptions in their model. 1. The quality characteristic, X, is normally distributed with an unknown mean µ and a known variance σ2 2. The quality characteristic nominal-is-best. 3. The target value, T, is the middle value of the specification, i.e., T=(Tl + Tu) /2 According to Wen and Mergen , the expected total loss per item is  =   	∞ +   	∞ =   1 −    +                  (1) Where Tu = the upper specification limit Tl= the lower specification limit CT = total loss per item due to exceeding the Tu and Tl Du = the monetary loss per item of exceeding Tu Dl = the monetary loss per item of staying below Tl 	 =  exp "−  # $            (-∞ <  < ∞                                                      (2) Φ (z) = the cumulative distribution function for the standard normal random variable with probability density function  	 =  √ ' −()      ,        (−∞ < * < ∞                                                                            (3) In order to determine the optimum process mean µ, Wen and Mergen took the derivative of Equation 1. Since European Journal of Business and Management                                                                                                                               www.iiste.org ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online) Vol.6, No.6, 2014  156 the second order derivative of equation 1 is positive, we set the first-order derivative equal to zero. The optimum µ is, +∗ = - − )./ 01 2.2/                                                                                                               (4) 2. Modified Wen And Mergen’s Model With Exponential Distribution Assume that the quality characteristic X follows the exponential distribution. The probability density function of X is as follows 	 = 3'4#         ,        x  ≥ 0 ,               (5) Where λ is the parameter of the exponential distribution. The cumulative distribution function, the expected value, and the variance of the exponential distribution, respectively, are 5; 3 =  1 −  '4#      ,  x ≥ 0               (6) 7 = 4                                                                                                                                                      (7) 89:  =  ';)<') − 1=                                                                                                                      (8) We now formulate the modified Wen and Mergen model with the step loss function of exponential characteristic for determining the optimum process mean. Step Loss Function The expected total loss per item of the modified Wen and Mergen model with the step loss function is - =    	 +  ∞  	 =  > '4. +  1 − '4/                                             (9) Where Tu = the upper specification limit Tl= the lower specification limit Du = the monetary loss per item of exceeding Tu Dl = the monetary loss per item of staying below Tl If we assume a Weibull function then - =    	 +    	 =  > ?'[A.BCD ]FG +  ?1 − '[A/BCD ]FG ∞                                   (10) After taking the first order derivative we get the equation as follows 	 ′ =  HIF J K'A.BCD FL [- − MH] − N'[O/I]FQ[- − MH]R              (11) Equation 11 is not a closed function. In order to determine the optimum µ, using an exponential function we take the derivative of equation 9. Since the second-order derivative of Equation 9 is positive, we set the first-order derivative equal to zero. The optimum parameter lambda is; 3 =   ./ 01 2. ∗.2/∗ /                                                                                                                            (12) Hence, the optimum process mean E(X)= µ = 4             (13)   3. Piecewise Linear Loss Function Cho and Leonard (1997) presented the piecewise linear quality loss function for the nominal-is best quality characteristic as follows: S = T 0                V	   - ≤  ≤ -- − ,         V	   ≤ - − -     V	    >  - Z                                                     (14) Where Dl = the quality loss coefficient when the quality characteristic is less than Tl Du= the quality loss coefficient when the quality characteristic exceeds the Tu The expected total loss per item of the modified Wen and Mergen model with the piecewise linear loss function is.  =   ∞  −  - 	 +   >  - −  	     2. ∗[B\A.4 + 2/∗[B\A/4 +   ∗ - − 2/4 = 0                                                                              (15)                                                              Since equation 15 is not closed, one can adopt the simple interval search method for obtaining the optimum parameter λ*  (the optimum process mean)  E(X)= µ = 4  European Journal of Business and Management                                                                                                                               www.iiste.org ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online) Vol.6, No.6, 2014  157 4. Numerical Example Assume that the quality characteristic follows an exponential distribution. Let the lower specification limit, Tl =2 and the upper specification limit, Tu =4. The monetary loss per item of falling below Tl is Dl = 1.5. The monetary loss per item of exceeding Tu is Du =1. We would like to determine the optimum process mean for minimizing the expected total loss per item. Step Loss Function For Product By solving equation (12) we have  λ* =0.144. Hence, the optimum process mean E(X) = 6.952 Piecewise Linear Quality Loss Function For Product By solving equation (15) we have λ*= 0.50209. Hence the optimum process mean E(X) = 2 Step Loss Sensitivity Analysis Figure 1: Graphical relationship between optimum values of λ* and the upper specification limit Tu. As Tu increases λ* increases initially at an increasing rate then gradually decreasing and then constant.    Figure 2: Graphical relationship between optimum values of λ* and the upper specification limit Tl. λ* decreases at a decreasing rate leveling off at higher values of Tl.    Figure 3: Graphical relationship between optimum values of λ* and the monetary loss per item Du. λ* increases at a decreasing rate with increases in Du.  00.10.23 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4λ*Tu012345670.3 0.4 0.5 0.6 0.7 0.8 0.9 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6λ*TlEuropean Journal of Business and Management                                                                                                                               www.iiste.org ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online) Vol.6, No.6, 2014  158   Figure 4: Graphical relationship between optimum values of λ* and the monetary loss per item Dl .  λ* decreases at a decreasing rate with increases in Dl.   Piecewise Sensitivity Analysis Figure 5: Graphical relationship between optimum values of λ* and the upper specification limit Tu. λ* decreases in varying rates with increases in Tu.   Figure 6: Graphical relationship between optimum values of λ* and the monetary loss per item Du. λ* increases with increases with increases in Du and decreasing gradually.  00.10.20.30.40.50.60.70.80.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3λ*Du00.20.40.60.811.21.41.61 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19λ*Dl0.440.460.480.50.520.543.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5λ*TuEuropean Journal of Business and Management                                                                                                                               www.iiste.org ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online) Vol.6, No.6, 2014  159   Figure 7: Graphical relationship between optimum values of λ* and the upper specification limit Tu. λ* decreases at a decreasing rate with increases in Tu.    Figure 6: Graphical relationship between optimum values of λ* and monetary loss per item Dl. λ* decreases steadily with increases in Dl.      00.10.20.30.40.50.60.70.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5λ*Du00.10.20.30.40.50.60.71.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5λ*Tu00.10.20.30.40.50.60.71 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2λ*DlEuropean Journal of Business and Management                                                                                                                               www.iiste.org ISSN 2222-1905 (Paper) ISSN 2222-2839 (Online) Vol.6, No.6, 2014  160 References 1. Carlsson, O. (1984). Determining the most profitable process level for a production process under different sales conditions. Journal of Quality technology 16:44-49 2. Cho, B.-R. and Leonard, M.S. (1997). Identification and extensions of quasiconvex quality loss functions. International Journal of Reliability, Quality and Safety Engineering 4:191-204 3. Golhar, D.Y. and Pollock, S.M. (1998). Determination of the optimal process mean and the upper limit of the canning problem. Journal of Quality Technology 20: 188-192. 4. Lee, M.K., Hong, S.H. and Elsayed, E.A. (2001). The optimum target value under single and two-stage screenings. Journal of Quality Technology 33:506-514. 5. Li, M.-H.C. (1997). Optimal setting of the process mean for assymetrical quadratic quality loss function. Proceedings of the Chinese Institute of Industrial Engineers Conference, pp  415-419. 6. Li, M.-H.C. and Cherng H.S. (2000). Unbalanced tolerance design with asymmetric truncated linear loss function. The 14th Asia Quality Symposium, pp. 162-165. 7. Li,M.-H.C. and Chirng H.S. (1999). Optimal setting of the process mean for assymetrical linear quality loss function. 1999 Conference of Technology and Applications of Quality Management for Twenty-first Century 2-6-2-11 8. Li,M.-H.C. and Wu, F.-W. (2001). A general model of unbalanced tolerance design and manufacturing setting with asymmetric quadratic loss function. Proceeding of Conference of the Chinese Society for Quality, pp. 403-409 9. Misiorek, V.I. and Barnett, N.S. (2000). Mean selection for filling processes under weights and measures requirements . Journal of Quality Technology 32:111-121. 10. Wen D. and Mergen, A.E. (1999). Running a process with poor capability, Quality Engineering II: 505-509 11. S. Maghsoodloo, and M.-H.C. Li, : Optimal assymetrical tolerance design, IEE Transactions, 32, pp.1127-1137, 2000 12. Golhar, D.Y., Pollock, S.M., 1988. Determination of the optimal process mean and the upper limit for a canning problem. Journal of Quality Technology 20, 188-192. 13. Phillips, D.M., Cho,R.B., 2000. Anon linear model for determining the most economical process mean under a Beta distribution. International Journal of Reliability, Quality and Safety Engineering 7(1), 61-74.       

Towards Determining the Optimum Process Mean using an Exponential Distribution

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Towards Determining the Optimum Process Mean using an Exponential Distribution

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