EFFECTS OF JOINT REPLENISHMENT POLICY ON COMPANY COST UNDER PERMISSIBLE DELAY IN PAYMENTS

Transcription

1 Mathematcal and Computatonal Applcatons, Vol. 5, No., pp. 8-58,. Assocaton for Scentfc Research EFFECS OF JOIN REPLENISHMEN POLICY ON COMPANY COS UNDER PERMISSIBLE DELAY IN PAYMENS Yu-Chung sao, Mng-Yu Wang, Pe-Lng Lee Department of Busness Management, atung Unversty, ape, awan Abstract - In today s severely compettve busness envronment, reducng replenshment costs has become one of the most mportant objectves for companes. hs study deals wth the replenshment problem under the condton of permssble delay n payments. o better reflect real-world busness stuatons, we extend the tradtonal EOQ model by consderng the stuatons of permssble delay n payments and mult-tem replenshment. hs study presents both sngle-tem and jont mult-tem replenshment models, and develop theorems to solve the problems. he objectve of the sngle-tem replenshment polcy s to determne the optmal replenshment cycle tme for each tem whle mnmzng the total cost. he objectve of the jont mult-tem replenshment polcy s to determne a common optmal replenshment cycle tme for all tems. Usng computatonal analyss, we llustrate the soluton procedures and draw conclusons. he results of ths study can serve as a reference for busness managers or admnstrators. Key Words - EOQ, delay n payments, mult-tem, jont replenshment. INRODUCION In practce, supplers often provde forward fnancng to retalers. In ths stuaton, the suppler allows the retaler a credt perod n whch to settle the amount owed for goods already suppled. Snce the publcaton of Goyal s [] paper almost 5 years ago, over 5 papers have appeared n the lterature dealng wth varety of trade credt stuatons ncludng prcng-dependent demand (Abad and Jagg [], Sheen and sao [], sao and Sheen []), shortages allowed (Jamal et al. [5], Ouyang et al. [6]), partal bacloggng and deteroraton (Aggarwal and Jagg [7], Hwang and Shnn [8]), and varable cost (sao and Sheen [9]) etc. hese studes ndcate that the ssue of trade credt s a very popular feld of research. It s essental to consder trade credt when formulatng a decson-mang model. Jont mult-tem replenshment strateges are already wdely appled n the real world. Examples of ths type of strategy nclude the supplyng of parts for automotve

2 Y.C. sao, M.Y. Wang and P.L. Lee 9 assembly (Hahm and Yano []) and refrgerated goods to supermarets (Hammer []). In the automotve ndustry, a suppler normally produces several dfferent tems for a sngle customer and puts together a combned shpment for that customer. In the grocery supply ndustry, dfferent types of refrgerated goods (e.g., General Mlls yogurt and Land O Laes butter) can be shpped n the same truc to the same supermaret (Hammer []). Other researches such as Goyal [], Kao [], Graves [], Ben-Khedher and Yano [5], van Ejs [6], Rempala [7], and Chen and Chen [8] have proposed models and algorthms for solvng mult-tem replenshment problems for dfferent stuatons. However, the trade credt papers above only consder sngle-tem problems, and gnore the effect of jont mult-tem replenshment. In practce, trade credt and mult-tem replenshment coexst. herefore, none of these studes can be an approprate reference. o address ths problem, ths paper formulates a model that combnes the credt perod wth the jont mult-tem replenshment polcy. As a result, ths s the frst study to consder the jont replenshment problem n a trade credt stuaton. he objectve of ths study s to determne the optmal replenshment polcy whle stll mnmzng total cost. We present both sngle-tem and jont mult-tem replenshment models and develop theorems to solve these reploenshment problems. hs study also compares, for the frst tme, the performance of these two polces under delay n payments. Usng computatonal analyss, we llustrate the soluton procedures and draw conclusons. Results show that the jont mult-tem replenshment polcy s better than the sngle-tem replenshment polcy. We also provde useful references for manageral decson-mang and admnstraton based on mathematcal modelng. hs study uses the followng notatons. I s the annual nterest charged per dollar, I e s the annual nterest earned per dollar, M s the credt perod, s the replenshment cycle tme for tem n the sngle-tem replenshment polcy, s the replenshment cycle tme n the jont replenshment polcy, A s the major orderng cost per order, a s the mnor orderng cost for tem, p s the sellng prce per unt for tem, c s the purchasng prce per unt for tem, P d s the demand rate for tem, h s the nventory holdng cost per unt for tem I, s the number of tems, θ s the set of unts whose replenshment cycle s longer or equal to the credt perod, and φ s the set of unts whose replenshment cycle s shorter than the credt perod. he mathematcal model s developed under the followng assumptons:. he demand rates for tems are constant wth tme.. Replenshments occur nstantaneously.. Shortages are not allowed.. he sellng prce s hgher than the purchasng prce. 5. he unt retal prce of the products sold durng the credt perod s deposted n an

3 5 Effects of Jont Replenshment Polcy on Company Cost nterest bearng account wth rate I e. At the end of ths perod, the credt s settled and the retaler starts payng nterest charges for the tems n stoc wth rate I P ( I P > I e ).. HE SINGLE-IEM REPLINISHMEN MODEL In the sngle-tem replenshment model, the total annual cost VC( ) has two dfferent functons as follows: M A+ a h d ci pd piedm VC + + f M, () θ A+ a h d VC + p Ied M f < M φ, () +, and θ φ {,,,...,} VC VC VC Case :When +. () M, n whch belongs to θ, the frst and second-order dervatves of VC( ) wth respect to are ( ) A+ a + dm c I p p Ie d h + I pc VC +, () ( ) A+ a + d M c I p p I e VC. (5) Case :When VC( ) wth respect to are < M, n whch belongs toφ, the frst and second-order dervatves of ( + ) ( + ) d h p I A a VC, (6) e ( A+ a ) VC >. (7) Equaton ( 7 ) mples that VC ( ) s convex on >. A+ a + d M ( c I p p I e ), then Equaton ( 5 ) mples that convex on > f >. Furthermore, Equatons ( ) and VC( ) s convex on > f >. At. Hence, VC M VC M VC s contnuous and well-defned. Let VC s mply that M, we fnd

4 Y.C. sao, M.Y. Wang and P.L. Lee 5 o mnmze VC( ), solve Equatons replenshment cycle tme for tem n ( A+ a ) ( + ) d h I p e the optmal order quantty for tem n If >, Equaton < M, 6 to obtan the optmal, (8) < M s Q( ) d ( + ) d A a h + I p e 5 mples that ( ) to obtan the optmal replenshment cycle tme for tem n ( + ) A+ a + d M c I p p I e d h I c p VC s convex on >. Solve Equaton. M,, (9) and the optmal order quantty for tem n Q d d A+ a + d M c I p I h + I c p e p. M s Based on the equatons above, we derve and deduce heorem and heorem to determne the optmal replenshment cycle tme for sngle-tem replenshment when < and respectvely. heorem : If Proof: Because <, then VC has the mnmum value VC s convex on > and s decreasng on (, and ncreasng on mnmum value at. < M. herefore, VC ( ), M on(, M ]. On the other hand, f <, Equaton( ) mples. As a result, VC( ) has a that VC ( ) and VC( ) s ncreasng on >. hus, ncreasng on [ M, ). So VC ( M) VC ( M), Equatons ( ) and ( ) mply that mnmum value at heorem : If VC s VC has a mnmum value at M. From on >. herefore,. + + e, then, let A a d M ( h p I ) VC has the (a) When >, the optmal replenshment cycle tme s (b) When <, the optmal replenshment cycle tme s..

5 5 Effects of Jont Replenshment Polcy on Company Cost (c) When, the optmal replenshment nterval s M. Proof: (a) If >, Equatons (9) and (8) mply that > M and > M. Accordng to the convextes and the defntons of VC( ) for Case and VC( ) for Case, we fnd that VC( ) s decreasng on [ M, ] and VC( ) s decreasng on (, ] value at and M. hs means that VC has the mnmum VC has the mnmum value at M. herefore, from VC ( M ) VC ( M ) VC ( ) VC has the mnmum value at. he proofs n (b) and (c) are smlar to that n (a)., we now that. HE JOIN MULI-IEM REPLENISHMEN MODEL In the jont mult-tem replenshment model, the total annual cost VC two dfferent functons as follows: ( M) A a h d c I pd p IedM has VC f M, () VC A a hd VC + pd Ie M f M. () + < Case : When M VC wth respect to are, the frst and second-order dervatves of VC A+ a + d M c I p I c I d + h d K p e p I ( ) +, () VC A+ a + dm c I p p Ie. () Case : When < M, the frst and second-order dervatves of VC ( ) wth respect to VC are A e a h d p I d, () A a VC + >. (5)

6 Y.C. sao, M.Y. Wang and P.L. Lee 5 Equaton ( 5) mples that ( p e), Equaton A+ a + d M c I p I convex on > when VC s convex on >. mples that >. Furthermore, Equatons ( ) and > when >. At M. Hence, VC VC s convex on VC M VC M o mnmze VC( ), solve Equatons replenshment cycle tme n < M, s contnuous and well-defned. Let VC s mply that, we fnd to obtan the optmal A+ a h d + I p d e, (6) the optmal order quantty n < M s Q( ) d d (A+ a ). h + I p e If >, Equaton ( ) mples that VC( ) s convex on >. Solve Equaton to obtan the optmal replenshment cycle tme n > M, A+ a + M I c d I p d, (7) p e I c d + h d p and the optmal order quantty n M s d[a+ a+ M ( I p cd Ie pd )] Q d I c + h p Based on the equatons above, we derve and deduce heorem and heorem to determne the optmal replenshment cycle tme for jont mult-tem replenshment when < and respectvely. heorem : If <, then VC has the mnmum value.. Proof: VC( ) s convex on > and < M. herefore, VC ( ) s

7 5 Effects of Jont Replenshment Polcy on Company Cost decreasng on (, and ncreasng on mnmum value at on( ], M. hus, VC has a, M. On the other hand, f <, Equaton( ) mples that VC and VC( ) s ncreasng on >. herefore, VC ncreasng on [ M, ). hus, VC ( M) VC ( M), Equatons ( ) and ( ) mply that VC mnmum value at on >. herefore, s VC has a mnmum value at M. From. has the heorem : If, let A+ a M h d+ Ie pd, then (a) When >, the optmal replenshment cycle tme s (b) When <, the optmal replenshment cycle tme s.. (c) When, the optmal replenshment nterval s M. Proof: (a) If >, Equatons (7) and (6) mply that convextes and the defntons of show that that VC( ) s decreasng on (, M ]. hs means that VC has the mnmum value at M. herefore, from we now that VC are smlar to that n (a). M and M. he VC for Case and [ M, ] and and VC ( ) has the mnmum value at has the mnmum value at VC for Case VC s decreasng on VC ( M ) VC ( M ) VC ( ),. he proofs n (b) and (c). COMPUAIONAL ANALYSIS hs secton dscusses fve tems and summarzes the parameter values n able.

8 Y.C. sao, M.Y. Wang and P.L. Lee 55 Item Item Item Item Item 5 able values of parameters M A d c Ip p Ie h a , he Sngle-tem Replenshment Model v.s. he Jont Mult-tem Replenshment Model Usng heorem for tem,.7 and 97., the optmal replenshment cycle tme s and the total cost s.9. Usng.9 heorem for tem, 6.7 and 98.7, the optmal replenshment cycle tme s and the total cost s Usng heorem for tem, and 96.88, the optmal replenshment cycle tme s.96 and the total cost s Usng heorem for tem, 9.5 and 9.9, the optmal replenshment cycle tme s.779 and the total cost s Usng heorem for tem 5, 9.55 and 99.6, the optmal replenshment cycle tme s and the total cost s 7.. he sum of these tems, or total cost, s.5. For the jont mult-tem replenshment model, we get 9.68, Usng heorem, the optmal replenshment cycle tme s. and the total cost s 86.. he results above show that the jont mult-tem replenshment model s better than the sngle-tem replenshment model n reducng total cost... Effects of dfferent parameter values able presents the effects of I and that when I e ncreases, the optmal replenshment cycle tme p I e on total cost and decson, showng and total cost VC

9 56 Effects of Jont Replenshment Polcy on Company Cost wll decrease. When I p ncreases, the optmal replenshment cycle tme decrease, but the total cost VC wll ncrease. wll Ip..5.7 able Effects of I p and I e Ie.8.. VC VC VC VC VC86.. VC VC VC VC able presents the effects of and M on total cost and decson, showng that when A ncreases, the optmal replenshment cycle tme and total cost VC wll ncrease. When M ncreases, the optmal replenshment cycle tme wll ncrease, but the total cost VC wll decrease. M able Effects of M and A 5 7 VC VC86.. VC VC VC VC VC VC VC CONCLUSION hs paper consders replenshment problems under the permssble delay n payments. We present both sngle-tem and jont mult-tem replenshment models, and develop theorems to solve these problems. he objectve of ths study s to determne the optmal replenshment polcy whle mnmzng the total cost. Usng computatonal examples, we llustrate the soluton procedures and show that the jont mult-tem replenshment polcy s better than the sngle-tem replenshment polcy. Numercal

10 Y.C. sao, M.Y. Wang and P.L. Lee 57 analyss revealsthe effects of nterest charged, nterest earned, orderng cost, and credt perod on the total cost and replenshment decson. hs study provdes a useful reference for manageral decson-mang and admnstraton. Acnowledgment-he author expresses hs grattude to the edtor and the anonymous revewers for ther detaled comments and valuable suggestons to mprove the exposton of ths paper. hs paper was supported by atung Unversty under the grant B98-B REFERENCES. Goyal SK. Economcs order quantty under condtons of permssble delay n payments. Journal of the Operatonal Research Socety 6, 5-8, Abad PL, Jagg CK. A jont approach for settng unt prce and the length of the credt perod for a seller when end demand s prce senstve. Internatonal Journal of Producton Economcs 8,5-,.. Sheen GJ, sao YC. Channel coordnaton, trade credt and quantty dscounts for freght cost. ransportaton Research Part E: Logstcs and ransportaton Revew, -8, 7.. sao YC, Sheen GJ. Dynamc prcng, promoton and replenshment polces for a deteroratng tem under permssble delay n payments. Computers & Operatons Research 5, 56-58, Jamal AMM, Sarer BR, Wang S. An orderng polcy for deteroratng tems wth allowable shortage and permssble delay n payment. Journal of the Operatonal Research Socety 8, 86-8, Ouyang LY, eng J, Goyal SK, Yang C. An economc order quantty model for deteroratng tems wth partally permssble delay n payments lned to order quantty, European Journal of Operatonal Research 9, 8-, Aggarwal SP, Jagg CK. Orderng polces of deteroratng tems under permssble delay n payments. Journal of the Operatonal Research Socety 6, , Hwang H, Shnn SW. Retaler s prcng and lot szng polcy for exponentally deteroratng products under the condton of permssble delay n payment. Computers and Operatons Research, 59-57, 997.

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