Optimum Profit Model for Determining Purchaser s Order Quantity and Producer s Order Quantity and Producer s Process Mean and Warranty Period

Transcription

1 International Journal of Operations Research International Journal of Operations Research Vol. 7, No. 3, 4-4 (200) Optimum Profit Model for Determining Purchaser s Order uantit and Producer s Order uantit and Producer s Process Chung-Ho Chen Department of Management and Information Technolog, outhern Taiwan Universit Nan-Tai treet, Yung-Kang Cit Tainan 70, Taiwan Received eptember 200; Revised November 200; Accepted November 200 Abstract In this paper, the author presents a modified Chen and Liu s (2007) model for determining the optimum order quantit, process mean, and warrant period of product between the producer and the purchaser. Assume that the demand quantit of the end of customer and the qualit characteristic of product are independent normall distributed. Taguchi s smmetric quadratic qualit loss function is applied in measuring the product qualit. The numerical result shows that the standard deviation of the qualit characteristic,, the mean demand of the end of customer,, the parameter of demand,, the parameter of selling price, a, and the sale price per unit for the end of customer, R, have a significant effect on the epected total profit of the societ. Kewords Order quantit, process mean, quadratic qualit loss function, warrant period.. INTRODUCTION Product qualit improvement is useful for increasing the customer s satisfaction. Hence, the maimum epected profit model between the producer and the purchaser is an important problem for the suppl chain sstem. The producer needs to determinate the optimum product qualit and the purchaser needs to consider the order quantit of product. Hence, the market needs to solve the problem o f how to get a trade-off between them. Recentl, man researchers have addressed this work. Chen and Liu (2007) presented the optimum profit model between the producer and the purchaser for the suppl chain sstem with pure procurement polic from the regular supplier and mied procurement polic from the regular supplier and the spot market. In 2008, the further proposed an optimal consignment polic considering a fied 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 (2008) considered the problem about the retailer determining his optimal order quantit and the manufacturer determining his optimal reserve capacit. Their model can make both sides of the suppl chain to increase their profit. The optimum process mean setting has a major effect on the defective fraction of product, the inspection/ reprocessing cost, and the epected total profit/cost. There are considerable attentions paid to this work b appling Taguchi s (986) quadratic qualit loss function, e.g., Chen (2006), Chen and Lai (2007a, 2007b), and Chen and Khoo (2008, 2009). Corresponding author s chench@mail.stut.edu.tw 83-73X Copright 200 ORTW

2 5 There are some works that assured a post-sale warrant cost where imperfection is prevalent in the production sstem. The works of Djamaludin et al. (994), Yeh and Lo (998), Yeh, et al. (2000), Wang and heu (2000, 2003), Wang (2004), and Yeh and Chen (2006) can be quoted along this line. Wang and heu (2000) suggested a speed solution procedure for Djamaludin et al. s (994) model. ubsequentl, Wang and heu (2003) investigated the lot size problem for repairable products sold under the free-repair warrant polic. The searched the optimal lot size for minimizing the epected total cost including the setup cost, the inventor holding cost, and the warrant cost. Yeh and Chen (2006) eamined the problem of free minimal repair warrant product in a deteriorating production sstem. The developed an approimate solution for obtaining the optimal lot size and the corresponding inspection polic. In the above mentioned models, an assumption is made where the warrant period of the product is a known quantit. In 2004, Ladan and hore considered the problem of determining the optimal warrant period of product with lower specification limit. The products are assumed that sale-price per manufactured item increases linearl with the warrant period. Ladan and hore (2007) further considered the joint problem of sale price and warrant period setting. In Chen and Liu s (2007) model, the neglected the effect of product qualit on the demand quantit of the end of customer and onl considered the order quantit satisfing the uniform distribution. In 2009, Chen proposed a modified Chen and Liu s model based on the demand quantit of the end of customer correlated with producer s product qualit. Taguchi s smmetric quadratic qualit loss function is applied for measuring the product qualit. The bivariate normal distribution is adopted for formulating the purchaser s epected profit model. The optimum purchaser s order quantit and supplier s process mean are jointl determined. Although the economic order quantit, process mean setting, and warrant period of product setting are three different problems that occur in the inventor management, qualit control, and warrant polic. If the are combined into an integrated model, then one can obtain the optimum decision parameters with maimum epected total profit of the societ. In this paper, the author proposes a modified Chen and Liu s (2007) model for determining the optimum order quantit, process mean, and warrant period of product between the producer and the purchaser. Assume that the demand quantit of the end of customer and the qualit characteristic of product are independent normall distributed. Ladan and hore s (2007) model is integrated into the modified Chen and Liu s (2007) model for obtaining the optimum purchaser s order quantit, the epected lifetime of product, and the warrant period of product b maimizing the epected total profit of the societ. The advantage of this modified Chen and Liu s (2007) model is to obtain the joint control of production quantit, process qualit level, and warrant period. Taguchi s smmetric quadratic qualit loss function is applied in measuring the product qualit. Finall, the sensitivit analsis of parameters will be provided for illustration. 2. LITERATURE REVIEW 2. Chen and Liu s (2007) Model Chen and Liu s (2007) pure procurement model is actuall based on the standard news-vendor model without spot markets. The consider a single period supplier-buer relationship in which a regular supplier produces short-life ccle products and a buer orders products from the regular supplier and then sells to the end customer. Assumptions:. A buer purchases a finished product from a regular supplier and resells it at a price, R, to the end customer. 2. The regular supplier produces each unit at a cost, C. 3. The regular supplier and the buer 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 buer a specific order quantit,. 5. When realized demand eceeds procurement quantit, unmet demand is lost; therefore, demand uncertaint eposes the buer to risks associated with mismatches between the procurement quantit

3 6 and demand. 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., X ~ U / 2, / 2 is the epected demand of customer and is the customer demand variabilit. The buer s profit is given b, where ( ), R RX W X X () R W, X where R is the sales price per unit; is the salvage value per unit; is the quantit procured b the buer from the regular supplier; X is the stochastic demand; W is the wholesale price per unit, paid b the buer to the regular supplier. The buer s epected profit can be epressed as E R R W f d R / 2 / 2 W f d (2) where f( ) is the probabilit distribution of X. de( Let the partial derivative of the buer s epected profit function with respect to be zero, i.e., d R ) 0. We have the optimal order quantit 2 R W R X (3) The regular supplier maimizes his epected profit and determines the wholesale price per unit based on the buer s order quantit. Hence, the regular supplier s epected profit is epressed as W C E (4) de( Let the partial derivative of the regular supplier s epected profit function with respect to W be zero, i.e., dw ) 0. The optimal W value of equation (4) ields W R R 2 C 2 4 (5) E and ubstituting equation (5) into equation (3), the optimal value can be rewritten as R C X (6) 2 2 2R From equations (2), (4), (5), and (6), the buer s and supplier s epected profits can be epressed as R 2 X 2 ( R )[( ) ( X ) ] 2 (7) 2 X

4 7 ( R ) X R 2C [ ] E (8) 2 4 X 2.2 Ladan and hore s (2007) MODEL ome assumptions of Ladan and hore s (2007) model are as follows: The lifetime of product, Y, is eponential with known upper specification limit (U) and unknown lower specification limit (L) which is also defined as the warrant period of product. The selling price of product, r r a b L s, is positive proportional to L. Let, where a > 0 and b >0. r The selling price is for the lifetime of product between L and U. The selling price is r rl for the lifetime of product below L, where 0 r L. The selling price is r ru for the lifetime of product beond U, where 0 r U. The epected profit per item is L U s L s s U 0 L U (9) EP r r f ( ) d r f ( ) d r r f ( ) d ( a b L)[( r ) P( Y L) ( r ) P( Y U ) r ] L U U where f () is the probabilit densit function of Y and P( Y.) is the cumulative distribution function of Y. According to Cobb-Douglas tpe demand function, the demand quantit of product,, is negativel proportional to the selling price, r s, and positivel proportional to the warrant period of product lifetime, L. Hence, Eq. (0) shows the demand quantit of product r r L (0) where > 0, r > 0, and < 0. The epected total profit of product for Ladan and hore s (2007) model is as follows: TEP [ R C ] K r a b L L rl P Y L ru P Y U ru C K ( ) {[( ) ( ) ( ) ( ) ] } () where C is the production cost per item and K is the fied cost associated with manufacturing and selling. Ladan and hore (2007) adopted the response modeling methodolog for obtaining the optimum warrant period of product with the maimum epected total profit. 3. MODIFIED CHEN AND LIU s (2007) MODEL Assume that demand quantit of the end of customer and the qualit characteristic of product are independent normall distributed. The modified Chen and Liu s (2007) model is as follows: The purchaser s profit is given b

5 8 R RX W ( X ) X Loss( Y ), if X, L Y U (2) R W Loss( Y ), if X, L Y U ; is the unknown mean of Y; where Y is the normal qualit characteristic of the product, 2 Y N, ~ is the known standard deviation of Y; Loss(Y ) is the qualit loss per unit, LossY ky 2 ; k is the qualit loss coefficient; 0 is the known standard deviation of X. Hence, the purchaser s epected profit is is the target value of product.; 2 X N, ~ 0 ; is the known mean of X; R E( ) U U,, R W f dd R W f dd L L U U, +, Loss f dd Loss f dd L L (3) where f(, ) is the joint probabilit densit function of X and Y. The purchaser s epected profit can be rewritten as E R U {( R )[ ( ) ( )] ( W ) ( )} [ ( ) L U ( )] [ ( ) ( )] Loss f d ( R W ) [ ( )] L U L U [ ( ) ( )] ( ) Loss f d L (4) where U L Loss f d U 2 2 L L k{[( 0) ] [( 2 0 L) ( ) U ( 2 0 U) ( )]} () is the cumulative distribution function of the standard normal random variable; () is the probabilit densit function of the standard normal random variable. (5) Consider the conforming product sold to the primar market and the non-conforming product scrapped and sold to the secondar market. The supplier s profit is given b, W z cy i L Y U p z cy i, Y L or Y U where W is the selling price per unit for the conforming product; p (6) is the discounted price per unit for the non-conforming product scrapped; z is the constant product cost per unit; c is the variable product cost per unit; i is the inspection cost per unit. Hence, the supplier s epected profit per unit is

6 9 E P( L Y U )[ W ( z c i)] (7) P( L Y U ) p[ P( L Y U )] P ( L Y U ) The supplier needs to produce items in order to satisf the buer s order quantit. P( L Y U) Hence, the supplier s epected profit is E E( P( L Y U ) ) p L Y U PL Y U W z c i P U L z c i p W U L (8) The lifetime of product is one of the qualit characteristics. One usuall assumes that the qualit characteristic of product is normall distributed. Hence, the normal lifetime distribution is considered in the Ladan and hore s (2007) model. Let rl ru, r W, and r rl p, where r a b L and 0r L. B substituting some parameters of Ladan and hore s (2007) model into the modified Chen and Liu s (2007) model, the supplier s epected profit can be rewritten as E( ) Where U L z c i p W U L (9) r L r (20) W r a b L (2) p r r (22) L The epected total profit of the societ including the buer and the supplier is R ETP E( ) E( ) (23) It is difficult to show that the Hessian s matri is a negative definite matri for Eq. (23) with optimum solution. One cannot obtain a closed-form solution. For the given parameters, we can adopt the direct search method for obtaining the optimal warrant period of product, L, the optimal process mean,, and the optimal order quantit,. The procedure of approimate optimum solution for the above integrated model (23) is as follows:

7 0 tep. et the maimum warrant period L, Lma U. tep 2. et the minimum searched warrant period L, Lmin 0.0, and let the epected lifetime searched value L min Compute ETP using Eq. (23). tep 3. Let Compute ETP using Eq. (24). Repeat this step until U tep 4. Let L L min 0. 0 and L Compute ETP using Eq. (23). Repeat steps 3 and 4 until L U tep 5. elect the maimum epected total profit of the societ from above teps -4 as the best polic. The corresponding parameters of L,, and values having the maimum ETP is the optimum solution. The optimal solution of integrated model depends on the several cost and profit parameters. The influences of them need to be illustrated b considering the sensitivit analsis of parameters. 4. NUMERICAL EXAMPLE AND ENITIVITY ANALYI Assume that some parameters are as follows: a = 0, b =2, W r a bl, R 3W, L U 0. 2W, rl 0.4, 00, p 0. 4W, 20, 0, 0. 8, i 0. 05, 2 k 0, c 5, 00, β -0,. r 0,. and U = 3. B solving Eq. (23), one obtain the optimal order quantit, the optimal epected lifetime of product 75, the optimal warrant period of.42 product L 0.6 with the epected profit of buer R, the epected profit of supplier E( ) and the epected total profit of the societ E( ) 25.0 ETP Table lists the 20% change for parameter values and presents the effect on the order quantit, the epected lifetime, the warrant period, the epected profit of buer, the epected profit of supplier, and the epected total profit of the societ. If the change percentage of the epected profit is larger than 0 %, then the parameter has a major effect on the epected profit. From Table, one has the following conclusions: () The standard deviation of the qualit characteristic,, the mean demand of the end of customer,, the upper specification limit of product lifetime, U, the variable product cost per unit, c, the parameter of demand,, the parameter of selling price,, and the sale price per unit for the end of customer, R, have a significant effect on the purchaser s order quantit. (2) The standard deviation of the qualit characteristic,, the mean demand of the end of customer,, the upper specification limit of product lifetime, U, the variable product cost per unit, c, the parameter of demand,, the parameter of warrant period, r, the parameter of selling price, a, the parameter of selling price, b, and the sale price per unit for the end of customer, R, have a significant effect on the epected lifetime and warrant period of product. (3) The standard deviation of the qualit characteristic,, the mean demand of the end of customer,, the upper specification limit of product lifetime, U, the variable product cost per unit, c, the parameter of demand,, the parameter of selling price, a, and the sale price per unit for the end of customer, R, have a significant effect on the epected profit of buer. (4) The standard deviation of the qualit characteristic,, the mean demand of the end of customer,, the upper specification limit of product lifetime, U, the variable product cost per unit, c, the parameter of demand,, the parameter of selling price, a, and the sale price per unit for the end of customer, R, have a significant effect on the epected profit of supplier.

8 (5) The standard deviation of the qualit characteristic,, the mean demand of the end of customer,, the parameter of demand,, the parameter of selling price, a, and the sale price per unit for the end of customer, R, have a significant effect on the epected profit of epected total profit of the societ. 5. DICUION AND CONCLUION The parameters having a significant effect on the combination (L, ) of parameters are the standard deviation of the qualit characteristic,, the mean demand of the end of customer,, the upper specification limit of product lifetime, U, the variable product cost per unit, c, the parameter of demand,, the parameter of selling price, a, and the sale price per unit for the end of customer, R. The parameters having a significant effect on the order quantit are the standard deviation of the qualit characteristic,, the mean demand of the end of customer,, the upper specification limit of product lifetime, U, the variable product cost per unit, c, the parameter of demand,, the parameter of selling price,, and the sale price per unit for the end of customer, R. The parameters having a significant effect on the epected total profit of the societ are the standard deviation of the qualit characteristic,, the mean demand of the end of customer,, the parameter of demand,, the parameter of selling price, a, and the sale price per unit for the end of customer, R. Hence, one needs to have an eact estimation of,,, a, and R in order to obtain the optimal control of the product qualit and the maimum epected total profit of the societ. In this paper, the author has presented an integrated Ladan and hore s (2007) model into the modified Chen and Liu s (2007) model with qualit loss and warrant period of product. The warrant period of product, the epected lifetime of product, and the order quantit are simultaneousl determined in the modified Chen and Liu s (2007) model. The etension to an integrated model with rectifing sampling inspection plan ma be left for further stud. REFERENCE. Chen, C.H. (2006). The Optimum election of Imperfect ualit Economic Manufacturing uantit and Process Mean b Considering uadratic ualit Loss Function. Journal of The Chinese Institute of Industrial Engineers, 23: Chen, C.H. (2009). Optimum Profit Model Between Producer and Purchaser Having Optimum Order uantit and Process Mean. 2th International mposium of ualit Management, Taipei, Taiwan. 3. Chen, C.H. and Khoo, M.B.C. (2008). Joint Determination of Optimum Process Mean and Economic pecification Limits for Rectifing Inspection Plan with Inspection Error. Journal of The Chinese Institute of Industrial Engineers, 25: Chen,C.H. and Khoo, M.B.C. (2009). Optimum Process Mean and Manufacturing uantit ettings for erial Production stem Under the ualit Loss and Rectifing Inspection Plan. Computers & Industrial Engineering, 57: Chen, C.H. and Lai, M.T. (2007a). Economic Manufacturing uantit, Optimum Process Mean, and Economic pecification Limits etting Under the Rectifing Inspection Plan. European Journal of Operational Research, 83: Chen, C.H. and Lai, M.T. (2007b). Determining the Optimum Process Mean Based on uadratic ualit Loss Function and Rectifing Inspection Plan. European Journal of Operational Research, 82: Chen, C.L. and Liu, C.L. Liu (2007). Procurement trategies in the Presence of the pot Market-an Analtical Framework. Production Planning & Control, 8: Chen, C.L. and Liu, C.L. (2008). The Optimal Consignment Polic for the Manufacturer Under uppl Chain Coordination. International Journal of Production Research, 46: Dajmaludim, V.D., Murth, N.P., and Wilson, R.J. (994). ualit Control Through Lot izing for Items old with Warrant. International Journal of Production Economics, 33: Ladan,.P. and hore, H. (2004). Optimal Warrant Period When ale-price Increases with the Lower pecification Limit. In Frontiers in tatistical ualit Control 7 (Eds.: H.J. Lenz and P.TH. Wilrich), Phsica- Verlag, Ladan,.P. and hore, H. (2007). Profit Maimizing Warrant Period with ales Epressed b a Demand Function. ualit and Reliabilit Engineering International, 23:29-30.

9 2 2. Li, J. and Liu, L. (2008). uppl Chain Coordination with Manufacturer s Limited Reserve Capacit: an Etended Newsbo Problem. International Journal of Production Economics, 2: Taguchi, G. (986). Introduction to ualit Engineering, Asian Productivit Organization, Toko, Japan. 4. Wang, C.H. (2004). The Impact of a Free-Repair Warrant Polic on EM Model for Imperfect Production stems. Computers & Operations Research, 3: Wang, C.H. and heu,.h. (2000). Fast Approach to the Optimal Production/PM Polic. Computer and Mathematics with Applications, 40: Wang, C.H. and heu,.h. (2003). Optimal Lot izing for Products old Under Free-Repair Warrant. European Journal of Operation Research, 49: Yeh, R.H. and Chen, T.H. (2006). Optimal Lot ize and Inspection Polic for Products old with Warrant. European Journal of Operational Research, 74: Yeh, R.H. and Lo, H.C. (998). ualit Control Products Under Free-Repair Warrant. International Journal of Operations and uantitative Management, 4: Yeh, R.H., Ho, W.T., and Tseng,.T. (2000). Optimal Production Length for Products old with Warrant. European Journal of Operational Research, 20:

10 3 k Table. The effect of parameters on the optimal solution R L E E ETP R L E E ETP X R L E E ETP X R L E E ETP U R L E E ETP z R L E E ETP a R L E E ETP b R L E E ETP

11 4 Table. (Continued) R L E E ETP R L E E ETP r R L E E ETP c R L E E ETP i R L E E ETP R R L E E ETP 2.4W W R L E E ETP 0.6W W P L R E E ETP 0.32W W