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 yctsao@ttu.edu.tw 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
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
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
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 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)
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
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.
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 65 65 65 65 65 5, 5.5. 5 8 5.5 5. 5 6 7.5. 5.5 5. 5 5 5.5... 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 99.88. Usng heorem for tem,.86 8.9 and 96.88, the optmal replenshment cycle tme s.96 and the total cost s 86.55. Usng heorem for tem, 9.5 and 9.9, the optmal replenshment cycle tme s.779 and the total cost s 68.5. Usng heorem for tem 5, 9.55 and 99.6, the optmal replenshment cycle tme s.597797 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, 78.57. 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
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.. VC9.5.9797 VC99.9.565 VC7.7.86 VC7..668 VC86.. VC878.77.5 VC58.97.6 VC65..6796 VC68.7.5 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 5 65 65 5 65 able Effects of M and A 5 7 VC56.5.779 VC86.. VC88.97.58 VC697.7.668 VC5.6.689 VC6.77.68 VC867..978 VC6798.6.9968 VC5.57.97676 5. 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
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