Optimal Determination of Purchaser s Order Quantity and Producer s Process Mean

Transcription

1 Journal of Applied cience and Engineering, Vol. 15, No. 1, pp (2012) 41 Optimal Determination of Purchaser s Order Quantity and Producer s Process Mean Chung-Ho Chen 1 * and Chao-Yu Chou 2 1 Department of Management and Information Technology, outhern Taiwan University, Tainan, Taiwan 710,.O.C. 2 Department of Finance, National Taichung University of cience and Technology, Taichung, Taiwan,.O.C. Abstract Determination of optimum mean of the process characteristic is an important theme in quality improvement. Meanwhile, economic selection of order quantity is a key factor in inventory management since an inappropriate order quantity may result in unreasonably high inventory or stockout cost. In the present paper, we modify Chen and Liu s model [8] to simultaneously determine the optimal purchaser s order quantity and producer s process mean by incorporating an asymmetric loss function. The solution procedure for this modified model is developed and numerical eamples are given for illustration. Based on the sensitivity analyses, the common parameters having significant effects on the epected total profit of the system are the selling price per unit to the customer () and the mean demand of customer ( ). Therefore, accurate estimation of and in order to obtain the optimal control on the order quantity and the maimum epected total profit of the system would be highly desired. Key Words: Order Quantity, Process Mean, Asymmetric Quadratic Quality Loss Function, Uniform Distribution, Normal Distribution 1. Introduction *Corresponding author. chench@mail.stut.edu.tw Quality improvement of processes and/or products is always significant for enhancing customers satisfaction. One of important themes in quality improvement is the determination of optimum mean of the process characteristic, which usually has a major effect on the defective fraction of process, the inspection cost, the reprocessing cost, the scrap cost, and the epected total profit or cost associated with the process and products [1]. ecently, there are considerable attentions paid to this theme, e.g., Chen [2], Chen and Lai [3,4], and Chen and Khoo [5,6]. On the other hand, among various logistics costs in a company, inventory cost generally amounts to almost half of the company s total distribution dollar epense [7]. The economic selection of order quantity is a key factor in inventory management since an inappropriate order quantity may result in unreasonably high inventory or stockout cost. The economic determinations of order quantity and process mean look like two different problems for inventory management and quality improvement, respectively. However, for a modern supply chain system, the producer needs to determine the optimum process mean and the purchaser needs to consider the optimum order quantity. That is, a producer s decision on optimum process mean and a purchaser s decision on optimum order quantity could be simultaneously obtained such that the epected total profit of the system, including both the purchaser and the supplier, may be maimized. For previous years, several researchers focused their studies on modeling the profit of the producer and the purchaser in the supply chain system. For eample, Chen and Liu [8] presented an optimum profit model between

2 42 Chung-Ho Chen and Chao-Yu Chou the producer and the purchaser for the supply chain system with pure procurement policy from the regular supplier and with mied procurement policy from both the regular supplier and the spot market. In 2008, Chen and Liu [9] further proposed an optimal consignment policy considering a fied fee and a per-unit commission, which leads to a higher manufacturer s profit than the traditional production system and coordinates the retailer to obtain a higher supply chain profit. Li and Liu [10] considered the problem associated with the retailer for determining his optimal order quantity and the manufacturer for determining his optimal reserve capacity. Their model is able to increase profit for both sides of the supply chain. Darwish [11] proposed the optimum process mean setting for a single producer and a single purchaser in the filling industry, where the process mean, shipment quantity of product, and numbers of shipment are three decision variables to be determined. Chen and Huang [12] addressed that retailers purchase their products from options and online spot markets for hedging the risk of demand uncertainty. Arshinder and Deshmukh [13] presented a review of supply chain coordination and then proposed its research directions for further study. In Chen and Liu s model [8], they only considered the order quantity but did not account for the effect of product quality on the demand quantity of the end customers. Taguchi [14] refined the quality of product and presented the quadratic quality loss function for reducing total losses to the society. Taguchi s loss function, which eplicitly considers the quality loss due to process variability, can be used for the designs of control charts, sampling plans, process mean, manufacturing tolerances and specification limits [1519]. In the present paper, we modify Chen and Liu s model by incorporating Taguchi s quadratic loss function for determination of the producer s optimum process mean and the purchaser s optimum order quantity such that the epected total profit of the supply chain system may be maimized. The advantage of this modified model is to obtain the joint control on purchaser s order quantity and producer s process quality at the same time. The motivation behind this work stems from the fact that neglect of quality loss within the specification limits could overestimate the epected total profit of the system. The asymmetric loss function is employed in the present paper to well represent the real-world quality loss situations. In the net section, a brief review of Chen and Liu s model [8] is given. Then the modification of Chen and Liu s model is developed using an asymmetric loss function. Two numerical eamples are subsequently presented for illustration. ome final concluding remarks are then provided based on the analysis. 2. A Brief eview of Chen and Liu s Model Chen and Liu s pure procurement model [8] is actually based on the standard news-vendor model without spot markets. They consider a single period supplierbuyer relationship in which a regular supplier produces short life-cycle products and a buyer orders products from the regular supplier and then sells to the end customer. ome assumptions in their model are as follows: 1. A buyer purchases a finished product from a regular supplier and resells it at a price,, to the customer. 2. The regular supplier produces each unit at a cost, C. 3. The regular supplier and the buyer enter into a contract at a wholesale price, W. 4. The regular supplier sets the wholesale price to maimize his epected profit, while offering the buyer a specific order quantity, Q. 5. When realized demand eceeds procurement quantity, the unmet demand is lost; therefore, demand uncertainty eposes the buyer to the risk associated with mismatches between the procurement quantity and demand. 6. The procurement lead time is long relative to the selling season, such that the buyer cannot observe demand before placing an order. 7. The consumer demand, X, is uniformly distributed, i.e., X ~ U[ ( / 2), +( / 2)], where is the mean of X and is the variability of X. Let be the salvage value per unit of product. From Chen and Liu [8], the buyer s profit, denoted by,is given by (1) Then, the buyer s epected profit, ), may be epressed by

3 Optimal Determination of Purchaser s Order Quantity and Producer s Process Mean 43 (2) where f (t) is the density function of T. According to Chen and Liu [8], the optimum wholesale price and order quantity are as follows: distributed representing the process characteristic, i.e., Y ~ U[ y 3 y, y 3 y ],where y is the unknown mean of Y to be determined for maimization of profit and y is the known standard deviation of Y, Loss(Y) is the quality loss per unit of product, and mathematically, (3) and (4) Consequently, the buyer s epected profit and the supplier s epected profit, denoted by ), can be respectively obtained through (5) where k 1 is the quality loss coefficient for y<m, k 2 is the quality loss coefficient for y m, m is the target value of process characteristic, U is the upper specification limit of process characteristic which may be epressed by U = y + b y, L is the lower specification limit of process characteristic and can be epressed by L = y a y, where both a and b are constants, the random variable X represents the customers demand quantity and is assumed to be normally distributed, i.e., X ~ N (, 2 ),where is the known mean of X and is the known standard deviation of X. It can be shown that the purchaser s epected profit may be epressed by and (6) (8) 3. Development of the Modified Chen and Liu s Model Let customers demand quantity be normally distributed with a large variability and the quality characteristic of product have a finite value and be uniformly distributed. Assume that these two variables (i.e., demand quantity and product characteristic) are independent. In our modified model, an asymmetric loss function is incorporated to account for the quality loss of the conformance products. Hence, the purchaser s profit can be given by where f (, y) is the joint probability density function of X and Y.Notethatf (, y)=f () f (y)sincex and Y are independent. Eq. (8) can be rewritten as (7) (9) where the random variable Y isassumedtobeuniformly where

4 44 Chung-Ho Chen and Chao-Yu Chou Then the epected total profit of the system, including both purchaser and supplier, is (10) (14) For a given Q, we can obtain the optimum process mean y *. Differentiating Eq. (14) with respect to and setting equal to 0 gives where () is the cumulative distribution function of the standard normal random variable, and () isthe probability density function of the standard normal random variable. Consider the situation that conformance product is sold to the primary market and the non-conformance product is scrapped and sold to the secondary market. Then the supplier s profit is given by (11) where W is the selling price per unit of the conformance product, p is the discounted price per unit of the nonconformance product that is scrapped, f is the constant fied production cost, z is the variable production cost per unit of product, and i is the inspection cost per unit of product. According to the definition of epectation, the supplier s epected profit per unit may be epressed by (15) Q Q Let A [ ( ) ( )] and B = Q[1 Q ( )]. If the second derivative of Eq. (14) is neg- ative, that is, (12) Q ince the supplier needs to produce items PL ( yu) in order to satisfy the purchaser s order quantity, hence, the supplier s epected profit would be then * y is optimal for a given Q. In general, we have A > 0andB > 0. Therefore, the necessary condition to obtain the optimal solution is [k 1 (L m)+k 2 (U m)] > 0, that is, k 1 U m k 2 m L. By substituting L = y a y and U = y + b y into Eq. (15), we can solve Eq. (15) to obtain the optimum process mean for a given Q. Eq. (15) can be rewritten as (13) (16)

5 Optimal Determination of Purchaser s Order Quantity and Producer s Process Mean Numerical Eamples and ensitivity Analysis Eq. (16) has an optimum process mean y * b1 b1 4a c 2a as b 1 2 4a 1 c 1 > 0. Because b a 1 c 1 = 4[ kk 1 2( a b Qz ) 2 1 ( a b)( A B) ( k k )] y, we need the following necessary condition for obtaining the optimal solution, that is, The general procedure for obtaining the optimal solution of the aforementioned modified model in Eq. (14) is summarized as follows: tep 1. et the minimum order quantity Q min = 5 and the maimum order quantity Q ma = +5. tep 2. Let Q min Q Q ma. For a given Q, the optimum process mean may be obtained by under the necessary conditions k k 1 2 U m and m L tep 3. elect the maimum epected total profit of the system from tep 2 as the best policy. The corresponding combination of parameters ( y *, Q * ) having the maimum ETP is the optimum solution. The optimal solution of the modified Chen and Liu s model [8] depends on the several cost and profit parameters. The influences of these parameters need to be illustrated by conducting the sensitivity analysis of parameters in the net section. 4.1 Eample 1 Assume that the parameters in the system are as follows: a = 0.8, b = 0.6, k 1 =5,k 2 = 10, = 25, =5,W = 10, = 100, p =5, = 20, m =11, y = 0.9, i = 0.2, f = 2, and z = 0.5. By solving Eq. (14), we have the optimal order quantity Q * = 118 and the optimal process mean * y with the epected profit of the purchaser ) = , the epected profit of the supplier ) = , and the epected total profit of the society ETP = for the our model. Table 1 lists 50% range for parameter values and presents their effects on the order quantity, the process mean, the epected profit of the purchaser, the epected profit of the supplier, and the epected total profit of the system for the modified Chen and Liu s model. If the change percentage of the epected profit is greater than 20%, then the corresponding parameter is considered as a major effect on the epected profit. From Table 1, the following results can be observed: 1. The mean demand of the customer ( ), the variable production cost per unit (z), and the target value (m) have significant effects on the purchaser s order quantity. 2. The quality loss coefficients (k 1 and k 2 ), the coefficient of the lower specification limit (a), the coefficient of the upper specification limit (b), and the target value (m) have significant effects on the process mean. 3. The selling price per unit to the customer (), the standard deviation of the quality characteristic ( y ), the mean demand of the customers ( ), and the wholesale price per unit paid by the buyer to the regular supplier (W) have significant effects on the epected profit of the purchaser. 4. The mean demand of the customer ( ), the coefficient of the lower specification limit (a), the coefficient of the upper specification limit (b), the variable production cost per unit (z), the wholesale price per unit paid by the buyer to the regular supplier (W), the target value (m), the discounted price per unit for the nonconformance product scrapped ( p ), and the fied production cost (f) have significant effects on the epected profit of the supplier. 5. The selling price per unit for the end customer (), the mean demand of the customer ( ), the variable pro-

6 46 Chung-Ho Chen and Chao-Yu Chou Table 1. Effects of parameters on the optimal solution for the modified Chen and Liu s model Q y ) y Q y ) Q y ) Q y ) k 1 Q y ) k 2 Q y ) a Q y ) b Q y ) W Q y ) Table 1. Continued m Q y ) ) duction cost per unit (z), and the target value (m) have significant effects on the epected total profit of the system. 4.2 Eample 2 Consider the situation that the mean and standard deviation of the demand of customers are the same as those of Eample 1 in Chen and Liu s model [8]. Assume that the parameters are as follows: = 25, =5, = 100, = 40 3, and C = 2. By solving Eqs. (3)(6), we have the optimal order quantity Q * = 73 and the optimal wholesale price W * = with the epected profit of the purchaser ) , the epected profit of the supplier ) ,. and the epected total profit of the system ETP = for the Chen and Liu s model [8]. Table 2 lists 50% range for parameter values and presents the effects on the order quantity, the wholesale price, the epected profit of the purchaser, the epected profit of the supplier, and the epected total profit of the system for the Chen and Liu s model [8]. If the change percentage of the epected profit is greater than 20%, then the corresponding parameter would be a major effect on the epected profit. From Table 2, the following conclusions may be obtained: 1. The mean demand of customer ( ) has a significant effect on the purchaser s order quantity. 2. The standard deviation of demand of customer ( ) and the selling price per unit to the customer () have significant effects on the wholesale price. 3. The standard deviation of the demand of customer ( ) and the mean demand of customer ( ) have significant effects on the epected profit of the purchaser. ETP f Q y )

7 Optimal Determination of Purchaser s Order Quantity and Producer s Process Mean 47 Table 2. Effects of parameters on the optimal solution for Chen and Liu s model Q W ) ) 4. The standard deviation of the demand of customer ( ), the mean demand of the customer ( ), and the selling price per unit to customer () have significant effects on the epected profit of the supplier. 5. The standard deviation of the demand of customer ( ), the mean demand of customer ( ) and the selling price per unit to customer () have significant effects on the epected total profit of the system. 5. Conclusion In the present paper, we modify Chen and Liu s model [8] by involving an asymmetric quality loss function. By assuming that the demand quantity of the customer is normally distributed, the quality characteristic of the process is uniformly distributed, and both random variables are independent, the optimum process mean of ETP Q W ) ) ETP Q W ) C Q W ) Q W ) product and the order quantity can be simultaneously determined in this modified model by maimizing the epected total profit of the supply chain system, including both the purchaser and the producer. Based on the sensitivity analyses, the common parameters having significant effects on the epected total profit of the system are the selling price per unit to the customer () and the mean demand of customer ( ). Therefore, accurate estimation of and in order to obtain the optimal control on the order quantity and the maimum epected total profit of the system would be highly desired. The etension of this method to the modified Chen and Liu s model [8] with rectifying sampling inspection plan or mied procurement policy may be left for further study. eferences [1] Chen, C. H. and Chou, C. Y., Determining the Optimum Process Mean under a Lognormal Distribution, Quality & Quantity, Vol. 39, pp (2005). [2] Chen, C. H., The Optimum election of Imperfect Quality Economic Manufacturing Quantity and Process Mean by Considering Quadratic Quality Loss Function, Journal of the Chinese Institute of Industrial Engineers, Vol. 23, pp (2006). [3] Chen, C. H. and Lai, M. T., Determining the Optimum Process Mean Based on Quadratic Quality Loss Function and ectifying Inspection Plan, European Journal of Operational esearch, Vol. 182, pp (2007a). [4] Chen, C. H. and Lai, M. T., Economic Manufacturing Quantity, Optimum Process Mean, and Economic pecification Limits etting under the ectifying Inspection Plan, European Journal of Operational esearch, Vol. 183, pp (2007b). [5] Chen, C. H. and Khoo, M. B. C., Joint Determination of Optimum Process Mean and Economic pecification Limits for ectifying Inspection Plan with Inspection Error, Journal of the Chinese Institute of Industrial Engineers, Vol. 25, pp (2008). [6] Chen, C. H. and Khoo, M. B. C., Optimum Process Mean and Manufacturing Quantity ettings for erial Production ystem under the Quality Loss and ectifying Inspection Plan, Computers & Industrial Engineering, Vol. 57, pp (2009).

8 48 Chung-Ho Chen and Chao-Yu Chou [7] Pfohl, H. C., Cullmann, O. and tolzle, W., Inventory Management with tatistical Process Control: imulation and Evaluation, Journal of Business Logistics, Vol. 20, pp (1999). [8] Chen,. L. and Liu, C. L., Procurement trategies in the Presence of the pot Market-an Analytical Framework, Production Planning & Control, Vol. 18, pp (2007). [9] Chen,. L. and Liu, C. L., The Optimal Consignment Policy for the Manufacturer under upply Chain Coordination, International Journal of Production esearch, Vol. 46, pp (2008). [10] Li, J. and Liu, L., upply Chain Coordination with Manufacturer s Limited eserve Capacity: An Etended Newsboy Problem, International Journal of Production Economics, Vol. 112, pp (2008). [11] Darwish, M. A., Economic election of Process Mean for ingle-vendor ingle-buyer upply Chain, European Journal of Operational esearch, Vol. 199, pp (2009). [12] Chen,. L. and Huang,. C., Managing upply Chain isk with Options and Online pot Markets, Journal of tatistics & Management ystems, Vol. 13, pp (2010). [13] Arshinder, K. A. and Deshmukh,. G., upply Chain Coordination: Perspectives, Empirical tudies and esearch Directions, International Journal of Production Economics, Vol. 115, pp (2008). [14] Taguchi, G., Introduction to Quality Engineering, Asian Productivity Organization, Tokyo, Japan (1986). [15] Liu, H.., Chou, C. Y. and Chen, C. H., Minimum- Loss Design of X Charts for Correlated Data, Journal of Loss Prevention in the Process Industries, Vol. 15, pp (2002). [16] Chen, C. H. and Chou, C. Y., Economic Design of Dodge-omig LTPD ingle ampling Plans for Variables under Taguchi s Quality Loss Function, Total Quality Management, Vol. 12, pp. 511 (2001). [17] Chen, C. H. and Chou, C. Y., Determining the Optimum Process Mean under the Bivariate Quality Characteristics, International Journal of Advanced Manufacturing Technology, Vol. 21, pp (2003). [18] Chen, C. H. and Chou, C. Y., Tolerance Design for a ubsystem with Unequal pecification Limits Using Taguchi s Quadratic Loss Function, International Journal of Information and Management ciences, Vol. 14, pp (2003). [19] Chen, C. H. and Chou, C. Y., Determining a One- ided Optimum pecification Limit under the Linear Quality Loss Function, Quality & Quantity, Vol. 39, pp (2005). Manuscript eceived: Dec. 8, 2010 Accepted: Nov. 28, 2011