Constrained optimization of a newsboy problem with return policy

Transcription

1 Appl. Math. Inf. Sci. 6 No. S pp. 635S-64S () Applied Mathematics & Information Sciences An International NSP Natural Sciences Publishing or. onstrained optimization of a newsboy problem with return policy Po-hung Yang, Suzanne Pai, Ling Yang Hui-Ming Wee Department of Marketing Logistics Management, St. John s University, amsui, aipei 535, aiwan Department of Industrial Systems Engineering, hung Yuan hristian University, hungli, 33 aiwan Address: weehm@cycu.edu.tw Received Jul. 5, ; Revised Dec. 4, ; Accepted Dec. 4, Abstract: A newsboy model in which a vendor has a limited budget to procure the required items is developed. It is assumed that the manufacturer will either sell the items to the vendor outright or offer the items to the vendor with return policy. In the latter case, the manufacturer buys back from the vendor the unsold items at the end of the selling season. his study considers one item with budget constraints. he vendor has the option of purchasing the goods outright, obtaining the goods with return policy, or a combination of both. By using Kuhn-ucker onditions inductive method, our analysis proposes an optimal inventory policy theorem. he purpose of this study is to investigate how the vendor should replenish the items with return policy under limited budget changing prices. We show that there exists a set of conditions under which the vendor s optimal strategy will change based on the amount of available budget for product procurement. If the purchase price with return policy is greater than the price of outright purchase the vendor has a small amount of available budget, only outright purchase will be optimal. However, if the budget reaches a certain threshold amount, mixed strategies where items are obtained by outright purchase with return policy are used. We also show that when the purchase price with return policy is not greater than the outright purchase, the vendor will only obtain the items with return policy. Keywords: Return Policy, Kuhn-ucker onditions, Ordering policy. Introduction his study investigates a single period problem in which a vendor has the option of purchasing the item outright /or obtaining the item through a return-policy agreement with the manufacturer. he vendor has limited budget. A return policy allows a vendor to return the unsold products for a full or partial refund. his will entice the vendor to order a larger quantity, resulting in an increase in the joint profit. Some products like the catalogue or style goods are examples where return policies are used [,]. he catalogue goods are sold to customers through catalogue advertisement with fixed price during a particular selling season. Pasternack [3] modeled a return policy derived a global optimization in a single period with uncertain dem. He demonstrated that a return policy where a manufacturer offers the vendors partial credits for all unsold products could achieve channel coordination. Padmanabhan Png [4] illustrated that the useful return policy can increase a manufacturer s profit increase the vendor competition. Emmons Gilbert [] studied the effect of return policy on both the manufacturer the vendor. Such policy is to maximize the manufacturer s profit by inducing the vendor to place larger order when dem is uncertain. he importance of the single period problem increases due to the shortening of the product life cycle in recent years. Many extensions of the single period problem have been studied [5]. wo major extensions are the unconstrained, single-item single-period problem, the constrained, multiitem single-period problem. Hadley Whitin [6]

2 636 Po-hang Yang et al: onstrained Optimization of a Newsboy derived a constrained multi-item problem in a single period. Jucker Rosenblatt [7] considered an unconstrained model with three types of quantity discounts: all-units quantity discount, incremental quantity discounts arload-lot discounts. Gerchak Parlar [8] developed an unconstrained model in which the vendor decides the price order. Lau Lau [9] modeled a newsboy problem with price-dependent distribution dem. Khouja [] developed a newsboy model in which multiple discounts are used to sell excess inventory. Khouja Mehrez [] extended Khouja s model [] to the multi-item case. his model dealt with a newsboy selling many items under a budget constraint. Lau Lau [] derived a capacitated multiple-product single period inventory model. Pasternack [3] developed a capacitated single-item newsboy model with revenue sharing. Vlachos Dekker [4] derived order quantity for single period products with return policy. Arcelus et al. [5] evaluated manufacturer s buyback policy under price-dependent stochastic dem. hiu et al. [6] addressed returns supply contract for coordinating supply chains with price-dependent dems. his study considers one item with budget constraints. he vendor has the option of purchasing the goods outright, obtaining the goods with return policy, or a combination of both. By using Kuhn-ucker onditions, our analysis proposes an optimal inventory policy theorem. We discuss the conditions for obtaining goods through outright purchase, return-policy purchase, or a combinatory purchase of both. In Section we present a general model a theorem for three cases. In Section 3 four numerical examples are given to illustrate the theorem. he first three examples demonstrate the various optimal strategies with changing budget. he last example demonstrates the strategies when the purchase price through return policy changes for a fixed limited budget. he concluding remark is given in the last section. Mathematical Modeling Analysis he mathematical model is developed based on the following assumptions: (a) he dem is uncertain. (b) An item with single order period, short selling season long production lead-time is considered (an example of this type of product is the catalogue or style product). (c) A vendor has the option of purchasing the item outright /or obtaining the item through a returnpolicy agreement with the manufacturer. (d) he products purchased through return policy begin selling only after the outright purchase items are sold out. he decision variables are: vendor s lot size obtained from the manufacturer through outright purchase vendor s lot size obtained from the manufacturer through return policy he known parameters are: f(x) probability density function of uncertain dem x x) cumulative distribution function of the probability density function f(x) vendor s purchase price through outright purchase vendor s purchase price through return policy P unit retail price S vendor s shortage cost per unit if the item is out of stock R vendor s return price per unit if the item is unsold total amount of funds the vendor has for obtaining the item through either outright purchase or return policy, or both. EP vendor s expected profit he vendor s expected profit can be expressed as EP P( S R xf ( x) dx f ( x) dx R ( ) f ( x) dx) ( x ) f ( x) dx ( x) f ( x) dx () he first two terms in the right side of () are the expected sales revenue. he second two terms are the return revenue for the unsold units. he last three terms are the expected shortage cost the purchase costs. he problem faced by the vendor is a nonlinear programming with constraints as follows: Maximize EP Subject to: Looking at the partial second derivatives for EP, one has: EP f )( P S R) Rf ( ), () (

3 Po-hang Yang et al: onstrained Optimization of a Newsboy 637 EP f ( )( P S R), (3) EP f ( )( P S R), (4) Hence, if P S R, EP is concave. he following Kuhn-ucker conditions are required for optimality:. u ( ). u 3. u 3 P S ( u) u )( P S R) 4. ) R 5. P S u ) u )( P S R) ( 3, u, u3,, 6. u hree cases of solution are discussed. he first case is,. he second case is,. he last case is,. ase :, ( u, ) From Kuhn-ucker conditions 4 5, one has P S u ) )( P S) (5) ( P S ( u) u3 )( P S R) (6) From (5), it is u [ P S )( P S)] (7) hen, P S ). (8) P S Substituting (7) into (6), one has u 3 )[( P S)( ) R] ( P S) ( ). (9) If ( P S)( ) R, then ( P S)( ) F ( ) (ontradiction) ( P S)( ) R () If ( P S)( ) R, then ( P S)( ) F ( ) (ontradiction) ( P S)( ) R () If, then u3 ) R (ontradiction) () If, then ( P S)( ) ) (3) ( P S)( R ) One can see that conditions (8) (3) must be satisfied simultaneously for the case of,. ase :, ( u 3, ) Derived from Kuhn-ucker conditions 4 5, one has u P S )( P S R) (4) u ( P S)( ) )( )( P S R) (5) From (4), one has P S ), (6) P S R where R. From (5), if, one has P S F ( ) P S R From (5), if, one has P S ) P S R (ontradiction) (7) (8) From (6) (8), one can see that conditions P S ) must be satisfied P S R simultaneously for the case of,. ase3:, ( u, 3 ) From Kuhn-ucker conditions 4 5, one has P S ( u) )( P S R) ) R (9)

4 638 Po-hang Yang et al: onstrained Optimization of a Newsboy P S u ) )( P S R) () ( he optimal solution of must satisfy (9), () simultaneously. If the solution of is positive, after substituting ( )/ into, the first derivatives of EP with respect to is greater than zero when =, that is ( )[ )( P S R) ( P S)]. () If, one has P S F ( ) (ontradiction) P S R () If, one has P S ) P S R (3) If the solution of is positive, then, after substituting ( )/ into, the first derivatives of EP with respect to is greater than zero when =, that is F ( )[( P S)( ) R] ( P S)( ) (4) If, one has ( P S)(( ) ) ( P S)( R (5) ) If ( P S)( ) R, one has ( P S)( ) ) (ontradiction) ( P S)( ) R (6) If ( P S)( ) R, one has ( P S)( ) ) (7) ( P S)( R ) Solution for > > is located at the intersection of (3) {(5) or (7)}. One can find that the two conditions ( P S)(( ) ) must be satisfied for ( P S)( R ) the case of,. he theorems resulted from the above discussion can be stated as follows: For P S R, one has heorem (i) if, ( P S)( ) ) ( P S)( ) R P S ). P S heorem (ii) if P S ). P S R heorem (iii) if ( P S)( ) ). ( P S)( R ) 3 Numerical Example he preceding theorems are illustrated by the following examples: Example his example illustrates heorem (i). Suppose f(x)= U(,6), P =, S =, R = 6, = = 8, when the available fund is unlimited, the solutions are = 36, = 43 (see Appendix A). From ( P S)( ) ) ( P S)( ) R P S ), one has P S respectively. he intersection of is 43. herefore, when 43, the solution is (i.e., / ). Numerical data for heorem (i) are given in able illustrated in Fig.. Example his example illustrates heorem (i) (iii). Suppose f(x)= U(,6), P =, S =, R = 6, = = 5, when the available fund is unlimited (see Appendix A), the solutions are = 3, =

5 Po-hang Yang et al: onstrained Optimization of a Newsboy 639 From ( P S)( ) ) ( P S)( ) R P S ), one has P S 4 43 respectively. herefore, when 4, the solution is (i.e., ) ; when 4 475, the solution is. In the case when 46, the solution of is 36 6 respectively. It is noted that when the available fund is small ( 4 ), the optimal strategy is to buy only; when the available fund is abundant ( 4 475), it is better to buy both. Numerical data for heorem (i) (iii) are given in able illustrated in Fig.. Example 3 his example illustrates heorem (ii). Suppose f(x)= U(,6), P =, S =, R = 6, = = 8, when the available fund is unlimited (see Appendix A), the solution is =, = From P S ), one has 44. P S R herefore, when 44, the solution is = > (i.e., / ). Example 4 his example illustrates the sensitivity analysis of the purchase price through return policy when the available limited fund is fixed at 4. Letting f(x)= U(,6), P =, S =, R = 6 =, the result of the sensitivity analysis is given in able 3 illustrated in Fig. 3 When , the inventory fund needed when the available fund is unlimited is less than 4. herefore, when is not greater than (i.e., ), the optimal strategy is to buy only; when increases slightly (i.e., 5 ), the optimal strategy is to buy both. However, when increases significantly ( 5 ), it is more economical to buy only. able : Some numerical data for heorem (i) with =8 Remark Unlimited budget , P S 35 9 ) P S (i.e., 43 ), ( P S)( ) ) R ( P S)( ) (i.e., 543) Figure : Vendor s lot sizes with various budgets when a higher price =8 able : Some numerical data for heorem (i) (iii) with =5 Remark Unlimited budget , ( P S)( ) ) ( P S)( ) R (i.e., 4 ) 4 333, P S ) 35 9 P S 3 5 (i.e., 43 ) 5 8 ( P S)( ) ) ( P S)( ) R (i.e., 4 ) Figure : Vendor s lot sizes with various budgets when a lower price =5 able 3: Sensitivity analysis of the purchase price with return policy when 4 Remark >576 Unlimited budget 7 57, P S 333 ) P S R (i.e., , )

6 64 Po-hang Yang et al: onstrained Optimization of a Newsboy ( P S)( ) ) ( P S)( ) R (i.e., 5 ), P S ) P S (i.e., ), ( P S)( ) ) ( P S)( ) R (i.e., 5 ) Appendix A Maximize EP Subject to: where EP is stated in () Solution: Equating the first derivatives of EP with respect to, one has P S )( P S R) ) R (A) Figure 3: Vendor s lot sizes with various when 4 4 oncluding Remark his study analyzes the vendor s replenishment strategy in a newsboy model with limited budget. We show that there exists a set of conditions under which the vendor s optimal strategy will change based on the amount of available budget for product procurement. If the purchase price with return policy is greater than the price of outright purchase the vendor has a small amount of available budget, only outright purchase will be optimal. However, if the budget reaches a certain threshold amount, mixed strategies where items are obtained by outright purchase with return policy are used. We also show that when the purchase price with return policy is not greater than the outright purchase, the vendor will only obtain the items with return policy. When the available budget is fixed, three conditions to obtain the optimal strategies are derived. he first condition is when the purchase price with return policy is not greater than the price with outright purchase; it is better to buy the items with return policy only. he second condition is when the purchase price with return policy is greater than the price of outright purchase over a certain threshold value; it is better to choose outright purchase only. he last condition is when the purchase price with return policy is between the two conditions, it is better to choose a mixture of items with outright purchase return policy. Acknowledgements: he authors would like to thank the editor anonymous referees for their helpful comments. his paper is supported in part by the NS of RO under grant NS 99-4-H-9-4-MY P S )( P S R) (A) Let the solution of the two simultaneous equations (A) (A) be denoted as ( # #, ). If # #, then the optimal solution, denoted # # by ( ), is. If # #, then the optimal solution is derived by the following steps: (i) Let (ii) After substituting into (A), one has P S F ( ) (A3) P S R If # #, the following steps derive the optimal solution: (iii) Let (iv) After substituting into (A), one has P S F ( ) (A4) P S References [] H. Emmons S. M. Gilbert, he role of returns policies in pricing inventory decisions for catalogue goods, Management Science. 44 (998), [] M. K. Mantrala K. Raman, Dem uncertainty supplier s returns policies for a multi-store style-good retailer. European Journal of Operational Research. 5 (999), [3] B. A. Pasternack, Optimal pricing return policies for perishable commodities. Marketing Science. 4 (985), [4] V. Padmanabhan I. P. L. Png, Returns policies: make money by making good. Sloan Management Review. 37 (995), [5] M. Khouja, he single-period (news-vendor) problem: literature review suggestions for future research. he International Journal of Management Science. 7 (999), [6] G. Hadley. M. Whitin, Analysis of Inventory Systems. liffs, New Jersey, Prentice-Hall, Englewood, 963. [7] J. V. Jucker M. J. Rosenblatt, Single-period inventory models with dem uncertainty quantity discounts: behavioral implications a new solution

7 Po-hang Yang et al: onstrained Optimization of a Newsboy 64 procedure. Naval Research Logistics. 3 (985), [8] Y. Gerchak M. Parlar, A single period inventory problem with partially controlled dem, omputers Operations Research. 4 (987), 9. [9] A. H.-L. Lau H.-S. Lau, he newsboy problem with price dependent distributions. IIE ransactions., (988), [] M. Khouja, he newsboy problem under progressive multiple discounts. European Journal of Operational Research. 84 (995), [] M. Khouja A. Mehrez, A multi-product constrained newsboy problem with progressive multiple discounts. omputers & Industrial Engineering. 3, (996), 95. [] H.-S. Lau A. H.-L. Lau, he newsst problem: A capacitated multi-product single-period inventory problem. European Journal of Operational Research. 94 (996), 9 4. [3] B. A. Pasternack, he capacitated newsboy problem with revenue sharing. Journal of Applied Mathematics Decision Sciences. 5 (), 33. [4] D. Vlachos R. Dekker, Return hling options order quantities for single period products. European Journal of Operational Research. 5 (3), [5] F. J. Arcelus, S. Kumar G. Srinivasan, Evaluating manufacturer s buyback policies in a single-period twoechelon framework under price-dependent stochastic dem. Omega. 36 (8), [6]. H. hiu,. M. hoi. S. ang, Price, rebate, returns supply contracts for coordinating supply chains with price-dependent dems. Production Operations Management. (), 8 9. Po-hung Yang is a professor in Marketing Logistics Management Department at St. John s University. He received a BSc in ontrol Engineering MSc from National hiao ung University, PhD from hung Yuan hristian University. His research interests are in the field of supply chain management, engineering economics, production inventory control. He has had publications in EUR J OPER RES, J OPER RES SO, PROD PLAN ONROL, OMPU IND ENG, OMPU OPER RES, MAH OMPU MODEL, ESWA OMEGA etc. Suzanne Pai is a senior lecturer in Marketing Logistics Management Department at St. John s University. She received MS in Industrial Management from National aiwan University of Science echnology. Her research interests are in the field of quality management, production inventory control. She has published in omputers & Industrial Engineering, International Journal of Systems Science, Expert Systems with Applications. Logistics Management at St. John s University, aiwan. She received MS in Management Science from National hiao ung University (aiwan), PhD in Management Science from National aiwan University of Science echnology. Her research interests are in the field of quality management, supply chain management, production inventory control. She has published in journals such as International Journal of Advanced Manufacturing echnology, uality echnology uantitative Management, uality Reliability Engineering International, omputers & Industrial Engineering, International Journal of Systems Science, Expert Systems with Applications. Hui-Ming Wee is a Professor of Industrial Engineering at hung Yuan hristian University in aiwan. He received his BSc in Electrical Electronic Engineering from Strathclyde University (UK), a MEng in Industrial Engineering Management from Asian Institute of echnology (AI) a PhD in Industrial Engineering from level State University, Ohio (USA). His research interests are in the field of production/inventory control, optimization supply chain management. He has published widely in the area of his research, edited four books serves on the Advisory Board for a number of international journals. Further details can be found at Department of Industrial Systems Engineering, hung Yuan hristian University, hungli, aiwan (, hung Pei Rd., hung Li, aiwan 33, RO) weehm@cycu.edu.tw Ling Yang is an associate professor in the Department of Marketing