The Joint Determination of Optimum Process Mean and Economic Order Quantity

Transcription

1 Tamkang Journal of cience and Engineering, Vol. 14, No. 4, pp (2011) 303 The Joint Determination of Optimum Process Mean and Economic Order Quantit Chung-Ho Chen Department of Management and Information Technolog, outhern Taiwan Universit, Tainan 710, Taiwan,.O.C. Abstract In Chen and Liu s [1] model with traditional production sstem, the neglected the effect of product qualit on the retailer s order quantit. Their model onl considered the order quantit obeing the uniform distribution. In fact, the retailer s order quantit is concerned with product qualit. Chen and Liu s [1] model with simple manufacturing cost did not consider the used cost of customers. Hence, the modified Chen and Liu s [1] model should be addressed for determining the optimal process parameter. In this stud, the author proposes a modified Chen and Liu s [1] model with qualit loss and single sampling inspection plan. Assume that the retailer s order quantit is concerned with the manufacturer s product qualit and the qualit characteristic of product is normall distributed. Taguchi s smmetric quadratic qualit loss function is applied in evaluating the product qualit. The optimal retailer s order quantit and the manufacturer s process mean will be simultaneousl determined b maximizing the expected total profit of societ including the manufacturer and the retailer. Ke Words: Economic Order Quantit, Process Mean, Taguchi s Quadratic Qualit Loss Function 1. Introduction The suppl chain sstem is a major topic for the manufacturing industries in order to obtain the maximum expected total profit of societ including the manufacturer and the retailer. The manufacturer s objective needs to consider the sale revenue, the manufacturing cost, the inspection cost, and the inventor cost for having the maximum expected profit. The retailer s objective needs to consider the order quantit, the holding cost, the goodwill loss of cost, and the used cost of customer for having the maximum expected profit. Hence, the market needs to solve the problem of how to get a trade-off between them. Goal [2] first proposed an integrated inventor model with the total costs from the producers and the purchasers and obtained the optimum production quantit and order ccle. Lu [3] considered *Corresponding author. chench@mail.stut.edu.tw the problem of an integrated inventor model with a single producer and multi-purchasers and presented a heuristic method for obtaining the model s optimal solution. Hill [4] modified the Goal s [2] and Lu s [3] models for determining the optimum production quantit and schedule under the minimum expected total cost. Goal and Nebebe [5] proposed the optimum production and transportation polic with a minimum cost between the producers and the purchasers. Goal [6] further formulated the modified Hill s [4] model for obtaining the optimum transportation quantit under the minimum time cost per unit. eifert et al. [7] explained the resulting procurement change and proposed the benefits of using spot markets from a suppl chain perspective. The developed the mathematical model that determines the optimal order quantit to purchase via forward contracts and spot markets. Haksoz and eshadri [8] discussed about suppl chain sstem with spot market and presented a literature review about recent work. Chen

2 304 Chung-Ho Chen and Liu [9] presented the optimum profit model between the producers and the purchasers for the suppl chain sstem with pure procurement polic from the regular supplier and mixed procurement polic from the regular supplier and the spot market. Chen and Liu [1] further proposed an optimal consignment polic considering a fixed fee and a per-unit commission. Their model determines a higher manufacturer s profit than the traditional production sstem and coordinates the retailer to obtain a large suppl chain profit. Li and Liu [10] considered the problem about the retailer determining his optimal order quantit and the manufacturer determining his optimal reserve capacit. Their model can make two sides of the suppl chain increase profit. Darwish [11] proposed the optimum process mean setting for a single purchaser and a single purchaser in the filling industr. The optimum process mean, shipment quantit of product, and numbers of shipment need to be determined in his model. Chen and Huang [12] addressed that the retailers purchase their products from options and online spot markets for hedging the risk of demand uncertaint. Arshinder et al. [13] proposed a review of suppl chain coordination and presented its research directions for further stud. In 1997, ana et al. [14,15] proposed a volume flexible inventor model considering the demand rate of defective items sold at a reduced price as a function of the reduction rate of the selling price. The further addressed the imperfect production sstem including the unit production cost taken to be a function of the finite production rate. In 2010, ana [16,17] considered the production-inventor model with imperfect production sstem for defective items restored to their original qualit b rework. His model also discusses and provides an optimal solution for product reliabilit parameter and dnamic production rate. ana [18] discussed that the enterprises initiatives affect the time varing demand for a multi-item economic order quantit problem. ana [19] further presented a economic order quantit model when demand is price demand and partial backorder is permitted. In Chen and Liu s [1] model with traditional production sstem, the neglected the effect of product qualit on the retailer s order quantit. The onl considered the order quantit obeing the uniform distribution. In fact, the retailer s order quantit is concerned with the product qualit. Chen and Liu s [1] model with simple manufacturing cost did not consider the used cost of customers in the traditional production sstem. Hence, the modified Chen and Liu s [1] model needs to be addressed for determining the optimum process parameters. Economic selection of process mean is an important problem for modern statistical process control. It will affect the expected profit/cost per item. ecentl, man researchers have addressed this work. Both 100% inspection and sampling inspection are considered for different models. Taguchi [20] presented the quadratic qualit loss function for redefining the product qualit. Hence, the optimum product qualit should be the qualit characteristic with minimum bias and variance. ecentl, his qualit loss function has been successfull applied in the problem of optimum process mean setting. In this paper, the author proposes a modified Chen and Liu s [1] model with qualit loss and adopting the single sampling inspection plan for determining the lot qualit of manufacturer. Assume that the retailer s order quantit is concerned with the manufacturer s product qualit and the qualit characteristic of product is normall distributed. Taguchi s [20] smmetric quadratic qualit loss function will be applied in evaluating the product qualit. The optimal retailer s order quantit and the manufacturer s process mean will be simultaneousl determined b maximizing the expected total profit of societ including the manufacturer and the retailer. The motivation behind this work stems from the fact that the neglect of the qualit loss within the specification limits should have the overestimated expected total profit of societ. 2. Literature eview --- Chen and Liu s [1] Traditional stem Model Chen and Liu s [1] traditional sstem model is actuall based on the standard news-vendor model without spot markets. The consider a single period manufacturer-retailer relationship in which a regular manufacturer produces short-life ccle products and a retailer orders products from the regular manufacturer and then sells to the end customer. ome assumptions are as follows: 1. A retailer purchases a finished product from a regular supplier and resells it at a price,, totheendcustomer.

3 The Joint Determination of Optimum Process Mean and Economic Order Quantit The regular manufacturer produces each unit at a cost, C. 3. The regular manufacturer and the retailer enter into a contract at a wholesale price, W. 4. The regular manufacturer sets the wholesale price to maximize his expected profit while offering the buer a specific order quantit, Q. 5. When realized demand exceeds procurement quantit, unmet demand occurs a goodwill lost,, for the retailer; therefore, demand uncertaint exposes the retailer to risks associated with mismatches between the procurement quantit and demand. When realized demand is less than procurement quantit, the retailer occurs a carring cost, H, for the retailer. 6. The procurement lead time is long relative to the selling season, so that the buer cannot observe demand before placing the order. 7. The consumer demand, X, is an uniform distribution, i.e., on the retailer s order quantit Q *. Hence, the regular manufacturer s expected profit is expressed as (4) where C is the regular manufacturer s production cost per unit. Let the partial derivative of the regular manufacturer s expected profit function with respect to W be zero, i.e., de ( ) 0. The optimal W value of equation dwp (4) ields (5) ubstituting equation (5) into equation (3), the optimal Q value can be rewritten as (6) 8. where x is the mean of X and x is the variabilit of X. The retailer s profit is given b From equations (2), (4), (5), and (6), the retailer s and manufacturer s expected profit can be expressed as (1) The retailer s expected profit can be expressed as (7) (2) (8) where f (x) is the densit function of X. Let the partial derivative of the retailer s expected profit function with respect to Q be zero, i.e., de ( ) = dqp 0. From Appendix of Chen and Liu (2008), the optimal order quantit (3) The regular manufacturer maximizes his expected profit and determines the wholesale price per unit based 3. Modified Chen and Liu s [1] Traditional stem Model Chen and Liu s [1] traditional sstem model with constant manufacturing cost per unit for the manufacturer is too simple. In fact, the manufacturing cost per unit should include the constant and variable production costs. The variable production cost is proportional to the value of qualit characteristic. Chen and Liu s [1] model also did not consider the used cost of customers. The neglect of the qualit loss within the specification limits

4 306 Chung-Ho Chen should have the overestimated expected profit per item for the retailer. Assume that the qualit characteristic Y ~ N ( m, s 2 ) and X Y ~ N(l1 + l2, s2), where l1, l2, and s2 are constants. Hence, we have X ~ N(l1 + l2m, æ l 2s 2 ( x - l 1 ) + m s 2, l 22s 2 + s 2 ) and Y X ~ N ç ç l 22s 2 + s 2 è s 2s 2 ö. Hence, the author proposes the following l 22s 2 + s 2 ø modified Chen and Liu s [1] model. The retailer s profit is given b (9) where Y is the normal qualit characteristic of product, Y ~ N (m, s 2 ); my is the unknown mean of Y; sy is the known standard deviation of Y; Loss(Y) is the qualit loss per unit, Loss(Y) = k(y - 0)2; k is the qualit loss coefficient; 0 is the target value of product. From Appendix, the retailer s expected profit is (10) where (11) (12) (13)

5 The Joint Determination of Optimum Process Mean and Economic Order Quantit 307 L is the lower specification limit of product; U is the upper specification limit of product. The expected total profit of societ including the retailer and the manufacturer is (19) (14) where k = 1 + 2, 2 2 2, A = ; B ; C 0 ; ( ) is the cumulative distribution function of standard normal random variable; ( ) is the probabilit densit function of standard normal random variable. Assume that the retailer s order quantit is equal to the lot size of single sampling inspection plan. A single sampling inspection plan is adopted for determining the lot qualit of manufacturer. If the lot is accepted, then the selling price of product per unit is W. If the lot is rejected, then a product is scrapped and sold at a price p. The manufacturer s profit under the normal qualit characteristic is given b k 2 (15) where n is the sample size; c is the variable production cost per unit; i is the inspection cost per unit; d 0 is the acceptance number; D is the number of non-conformance in the sample. The manufacturer s expected profit is where (16) (17) (18) The trade-off model between Eqs. (10) and (16) needs to obtain the optimal retailer s order quantit (Q * ) and the optimal manufacturer s process mean ( * ) with the maximum expected profits for the retailer and the manufacturer. Let L < < U. One can adopt direct * search method for obtaining the optimal with the maximum expected total profit of societ for Eq. (19) * * with the given order quantit. The combination ( Q, ) with maximum expected total profit of societ is the optimal solution. 4. Numerical Example and ensitivit Analsis Assume that some parameters are as follows: = 70, W = 10, =3,H =2, 1 =1, 2 =2,n = 16, d 0 =0, =1, 0 = 10, = 0.5, i = 0.05, k =1,c = 0.5, L=8, U = 12, and p = 2. B setting 8 < < 12 and solving Eq. (19) b direct search method, one obtains the optimal process mean * The corresponding optimal order quantit Q * = 21 with retailer s expected profit E( P ) = , manufacturer s expected profit E( P ) , and expected total profit of societ ETP(Q, ) = Table 1 lists 20% change for parameter value and presents the effect on the process mean, the order quantit, the retailer s expected profit, the manufacturer s expected profit, and the expected total profit of societ. If the change percentage of expected profit is larger than 10%, then the parameter has a major effect on the expected profit. From Table 1, one has the following conclusions: 1. The order quantit is constant and the process mean increases as the sale price per unit,, increases. The sale price per unit has a major effect on the retailer s and societ s expected profits. 2. The order quantit and the process mean are constants as the goodwill loss per unit of stockout,,increases. The goodwill loss per unit of stockout has a slight effect on the retailer s, manufacturer s and societ s expected profits.

6 308 Chung-Ho Chen Table 1. The effect of parameters for optimal solution Q ETP(Q, ) Per P Q ETP(Q, ) Per P W Q ETP(Q, ) Per P k Q ETP(Q, ) Per Q ETP(Q, ) Per P P Q ETP(Q, ) Per P Q ETP(Q, ) Per Q ETP(Q, ) Per c Q ETP(Q, ) Per i Q ETP(Q, ) Per P Q ETP(Q, ) Per n Q ETP(Q, ) Per P P P P P P d 0 Q ETP(Q, ) Per P ETP( Q, ) Note: Per 100% P P P P P P P P P P P P P

7 The Joint Determination of Optimum Process Mean and Economic Order Quantit The order quantit is constant and the process mean increases as the wholesale price, W, increases. The wholesale price has a major effect on the manufacturer s expected profit. 4. The order quantit is constant and the process mean decreases as the qualit loss coefficient, k, increases. The qualit loss coefficient has a slight effect on the retailer s, manufacturer s and societ s expected profits. 5. The order quantit is constant and the process mean decreases as the standard deviation of qualit characteristic of product,, increases. The standard deviation of qualit characteristic of product has a major effect on the retailer s, manufacturer s, and societ s expected profits. 6. The order quantit is constant and the process mean increases as the standard deviation of conditional random variable X Y,, increases. The standard deviation of conditional random variable X Y has a major effect on the retailer s, manufacturer s, and societ s expected profits. 7. The order quantit is constant and the process mean decreases as the intercept of mean for conditional random variable X Y, 1, increases. The intercept of mean for conditional random variable X Y has a slight effect on the retailer s, manufacturer s, and societ s expected profits. 8. The order quantit increases and the process mean decreases as the slope of mean for conditional random variable X Y, 2, increases. The slope of mean for conditional random variable X Y has a slight effect on the retailer s, manufacturer s, and societ s expected profits. 9. The order quantit and the process mean are constants as the variable production cost per unit, c, increases. The variable production cost per unit has a major effect on the manufacturer s expected profit. 10. The order quantit and the process mean are constants as the inspection cost per unit, i, increases. The inspection cost per unit has a slight effect on the retailer s, manufacturer s and societ s expected profits. 11. The order quantit is constant and the process mean increases as the discounted price per unit for the nonconformance product scrapped, p, increases. The discounted price per unit for the non-conformance product scrapped has a slight effect on the retailer s, manufacturer s, and societ s expected profits. 12. The order quantit is constant and the process mean increases as the sample size, n, increases. The sample size has a slight effect on the retailer s, manufacturer s, and societ s expected profits. 13. The order quantit and the process mean decrease as the acceptance number, d 0, increases. The acceptance number has a slight effect on the retailer s, manufacturer s, and societ s expected profits. 5. Conclusion In this paper, the author has presented a modified Chen and Liu s [1] traditional sstem model with qualit loss of product. Assume that the retailer s order quantit is concerned with the manufacturer s product qualit and the qualit characteristic of product is normall distributed. The qualit of lot for manufacturer is decided b adopting a single sampling inspection plan. The process mean of qualit characteristic and the order quantit of retailer are simultaneousl determined in the modified model. From the above numerical results, one has the following conclusion: (1) The sale price per unit, the standard deviation of qualit characteristic of product, and the standard deviation of conditional random variable X Y have a major effect on the expected profit of the retailer; (2) The wholesale price, the standard deviation of qualit characteristic of product, the standard deviation of conditional random variable X Y, and the variable production cost per unit have a major effect on the expected profit of the manufacturer; (3) The sale price per unit, the standard deviation of qualit characteristic of product, and the standard deviation of conditional random variable X Y have a major effect on the expected total profit of the societ. The extension to modified Chen and Liu s (2008) model with qualit investment or mixed procurement model ma be left for further stud. Appendix. Derivative of Expected Profit for etailer The retailer s profit is given b (A1)

8 310 Chung-Ho Chen Hence, the expected profit of retailer can be divided b the following four parts. Let A = l 22s 2 l 22s 2 + s s 2s 2 l 22s 2 + s 2,B = 2 m s 2 - l 1l 22s 2 l 22s 2 + s 2-0, and C0 =. Eq. (A4) can be rewritten as (A2) (A3) (A5) (A4) where mk = l1 + l2 m and s k = l 22s 2 + s 2.

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