Published online: 18 Jun 2013.

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1 This rticle ws downloded by: [Dokuz Eylul University ] On: 3 My 014, At: :48 Publisher: Tylor & Frncis Inform Ltd Registered in Englnd nd Wles Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journl of Informtion nd Optimiztion Sciences Publiction detils, including instructions for uthors nd subscription informtion: Economic order quntity with prtil bckorders under supplier credit Ling-Yuh Ouyng, Chin-Chun Wu & Ki-Wyne Chung b Deprtment of Mngement Sciences 1, Tmkng University, Tmsuim, 51, Tipei, Tiwn R.O.C. b Deprtment of Sttistics 1, Tmkng University, Tmsuim, 51, Tipei, Tiwn R.O.C. Published online: 18 Jun 013. To cite this rticle: Ling-Yuh Ouyng, Chin-Chun Wu & Ki-Wyne Chung (003) Economic order quntity with prtil bckorders under supplier credit, Journl of Informtion nd Optimiztion Sciences, 4:, 55-67, DOI: / To link to this rticle: PLEASE SCROLL DOWN FOR ARTICLE Tylor & Frncis mkes every effort to ensure the ccurcy of ll the informtion (the Content ) contined in the publictions on our pltform. However, Tylor & Frncis, our gents, nd our licensors mke no representtions or wrrnties whtsoever s to the ccurcy, completeness, or suitbility for ny purpose of the Content. Any opinions nd views expressed in this publiction re the opinions nd views of the uthors, nd re not the views of or endorsed by Tylor & Frncis. The ccurcy of the Content should not be relied upon nd should be independently verified with primry sources of informtion. Tylor nd Frncis shll not be lible for ny losses, ctions, clims, proceedings, demnds, costs, expenses, dmges, nd other libilities whtsoever or howsoever cused rising directly or indirectly in connection with, in reltion to or rising out of the use of the Content. This rticle my be used for reserch, teching, nd privte study purposes. Any substntil or systemtic reproduction, redistribution, reselling, lon, sub-licensing, systemtic supply, or distribution in ny form to nyone is expressly forbidden. Terms & Conditions of ccess nd use cn be found t

2 Economic order quntity with prtil bckorders under supplier credit Ling-Yuh Ouyng Deprtment of Mngement Sciences 1 Emil : lingyuh@mil.tku.edu.tw Chin-Chun Wu Deprtment of Sttistics 1 Ki-Wyne Chung Deprtment of Mngement Sciences 1 Deprtment of Interntionl Trde 1 Tmkng University Downloded by [Dokuz Eylul University ] t :48 3 My 014 Tmsuim Tipei Tiwn 51 R.O.C. Kungwu Institute of Technology Tipei Tiwn 11 R.O.C. Abstrct This pper dels with the problem of economic order quntity with prtil bckorders under supplier credit. In prctice, the supplier llows certin fixed credit period to settle the ccount for stimulting retiler s demnd. In ddition, in some situtions, the supplier lso my offer csh discount to encourge retiler to py for his purchses quickly. This study develops n inventory model with credit period nd csh discount simultneously. We lso consider both bckorders nd lost sles during the shortge period. The objective of this model is to determine the optiml replenishment policies so tht the totl cost per unit time is minimized. Numericl exmples re presented to illustrte the proposed model nd the sensitivity nlysis of the optiml solution with respect to prmeters of the system is lso included. Key Words : Inventory, supplier credit, bckorders. Journl of Informtion & Optimiztion Sciences Vol. 4 (003), No., pp c Tru Publictions /03 $

3 56 L. Y. OUYANG ET AL 1. Introduction Downloded by [Dokuz Eylul University ] t :48 3 My 014 In the clssicl economic order quntity (EOQ) model, it is often ssumed tht the retiler s pyment is mde to the supplier immeditely fter receiving the items. Trde credit plys n importnt role in rel business trnsctions. In empiricl observtion, we cn find tht the supplier my llow certin fixed credit period to settle the ccount for stimulting retiler s demnd. The retiler pys no interest during the fixed credit period, but if the pyments re delyed beyond the credit period, interest will be chrged. Goyl [1] first considered n EOQ inventory model under the condition of permissible dely in pyments. Aggrwl nd Jggi [] developed the inventory model with constnt deteriortion rte under the condition of permissible dely in pyments. In the most recent investigtions, the studies cn be referred to Jml et l. [3], Chung [4], Srker et l. [5, 6] nd Lio et l. [7]. In the bove models, the effects of the csh discount re not tken into ccount. The supplier s regulr credit terms include credit period nd csh discount. In prctice, the supplier my employ csh discount to encourge retiler to py for his purchses quickly. For exmple, the supplier offer 1% discount off the price of the merchndise if the pyment is mde within 10 dys ; otherwise the full price of the merchndise is due within 30 dys, nd the credit terms denote s 1/10, n/30. In this study, we will extend Goyl s [3] work by considering shortge nd csh discount to fit tody s business trnsctions. Our objective is to determine the optiml inventory policies so tht the totl cost per unit time is minimized. The reminder of this pper is orgnized s follows. In the next section, the nottions nd ssumptions re presented. The model formultion nd solution procedure re shown in Section 3 nd Section 4. In Section 5, numericl exmples illustrte the model nd the sensitivity nlysis of the optiml solution with respect to prmeters of the system is lso included. Finlly, Section 6 contins concluding remrks.. Nottions nd ssumptions To develop the mthemticl model, the following nottions nd ssumptions re used. Nottions: c 1 = setup cost ($/setup). c = holding cost excluding interest chrge ($/unit/unit time).

4 ECONOMIC ORDER QUANTITY 57 Downloded by [Dokuz Eylul University ] t :48 3 My 014 c 3 = shortge cost ($/unit/unit time). c 4 = opportunity cost due to lost sles ($/unit). c = item cost ($/unit). R = demnd rte (units/unit time). Q = order quntity (units/cycle). I(t) = the inventory level t time t. I e = the interest erned per dollr per unit time. I p = the interest pid per dollr per unit time. r = csh discount rte, 0 < r < 1. M 1 = the period of csh discount. M = the lst time of permissible dely in settling the ccounts, M > M 1. T = length of the replenishment cycle. T 1 = length of the positive stock period, 0 < T 1 T. (T 1, T ) = the optiml vlue of (T 1, T) in Cse 1.1. (T b 1, Tb ) = the optiml vlue of (T 1, T) in Cse 1.. (T c 1, Tc ) = the optiml vlue of (T 1, T) in Cse.1. (T d 1, Td ) = the optiml vlue of (T 1, T) in Cse.. Assumptions: 1. The replenishment occurs instntneously t n infinite rte.. Supplier offers csh discount if pyment is mde within M 1 ; otherwise the full pyment is due within M. 3. Shortge re llowed. The frction of the demnd during the shortge period tht will be bckordered is δ, 0 < δ Model formultion Using the nottions nd ssumptions mentioned bove, the behvior of the inventory system is shown in Figure 1. In inventory system, the replenishment cycle [0, T] is divided into two mjor phses : inventory depletion period (Phse 1) nd shortges period (Phse ). In Phse 1, the inventory is depleted due to demnd. In

5 Downloded by [Dokuz Eylul University ] t :48 3 My L. Y. OUYANG ET AL Phse, the inventory is depleted due to demnd nd prtil bckorders re replenished t the beginning of the next cycle. The rte of chnge of inventory t time t [0, T], di(t), is given by dt di(t) R, 0 t T 1 = (1) dt δr, T 1 < t T with boundry condition I(T 1 ) = 0. Figure 1 Inventory system

6 ECONOMIC ORDER QUANTITY 59 The solution of (1) my be represented by R(T 1 t), 0 t T 1 I(t) = δr(t 1 t), T 1 t T. () Thus, the order quntity per cycle is Q = I(0) + δr(t T 1 ) = R(T 1 + δt δt 1 ). (3) Since the supplier offers premium of the csh discount, there re two pyment policies for the retiler: (1) pyment is pid t time M 1 to get the csh discount (Cse 1); () the pyment is pid t time M not to get the csh discount (Cse ). We discuss these two cses s follows. Downloded by [Dokuz Eylul University ] t :48 3 My 014 Cse 1. Pyment is pid t time M 1 Cse 1.1 M 1 < T 1 In this cse, the length of the positive stock period is lrger thn the period of csh discount (see Figure 1 ()). Consequently, the totl cost is consisted of the sum of the setup cost, holding cost, shortge cost, opportunity cost due to lost sles nd interest pyble minus the interest erned nd csh discount. The components re clculted s follows. () The setup cost per setup is fixed t c 1. (b) The holding cost during the intervl [0, T 1 ] is given by HC = c T1 0 I(t)dt = c RT 1. (4) (c) The shortge cost during the intervl [T 1, T] is given by SC = c 3 T I(t)dt = c 3δR(T T 1 ) T 1. (5) (d) The opportunity cost due to lost sles during the intervl [T 1, T] is given by OC = c 4 T T 1 R(1 δ)dt = c 4 R(1 δ)(t T 1 ). (6)

7 60 L. Y. OUYANG ET AL (e) The interest pyble per cycle is given by IP = ci p T1 I(t)dt = ci pr(t 1 M 1 ) M 1 (f) The interest erned per cycle is given by. (7) IE = c I e M1 0 Rt dt = c I erm 1. (8) (g) Since the pyment is pid t time M 1, the retiler cn get r csh discount off the price of the merchndise is given by CD = r cq Downloded by [Dokuz Eylul University ] t :48 3 My 014 = rcr(t 1 + δt δt 1 ). (9) Therefore, the totl cost per unit time is clculted s TC 1 (T 1, T) = c 1 + HC + SC + OC + IP IE CD T { = 1 c 1 + c RT1 + c 3δR(T T 1 ) + c 4 R(1 δ)(t T 1 ) T + ci pr(t 1 M 1 ) c I erm 1 rcr(t 1 + δt δt 1 ) }. (10) Our objective is to determine the optiml vlues of T 1 nd T (denoted by T1 nd T, respectively) which minimize the totl cost per unit time TC 1 (T 1, T). The necessry condition for TC 1 (T1, T ) to be minimum is the point (T1, T ) stisfies : TC 1 (T 1, T) T 1 = 0 nd TC 1 (T 1, T) T = 0, (11) simultneously. By exmining the second-order sufficient conditions (SOSC) for minimum vlue, it cn be shown tht SOSC re stisfied since the Hessin Mtrix is positive definite t point (T 1, T ) (see Appendix for the proof).

8 ECONOMIC ORDER QUANTITY 61 Cse 1. T 1 M 1 In this cse, the length of the positive stock period is not greter thn the period of csh discount (see Figure 1 (b)). The setup cost, holding cost, shortge cost, opportunity cost due to lost sles nd csh discount re identicl to Cse 1.1. However, since T 1 M 1, the retiler pys no interest nd erns the interest during the period [0, M 1 ]. The interest erned in this cse is [ T1 ] IE = c I e R t dt + R T 1 (M 1 T 1 ) = c R I e (M 1T 1 T1 ). (1) 0 Therefore, the totl cost per unit time cn be obtined s TC (T 1, T) Downloded by [Dokuz Eylul University ] t :48 3 My 014 = c 1 + HC + SC + OC IE CD T { = 1 c 1 + c RT1 + c 3δR(T T 1 ) T + c 4 R(1 δ)(t T 1 ) cri } e (M 1T 1 T1 ) rcr(t 1 + δt δt 1 ). (13) The necessry condition for TC (T b 1, Tb ) to be minimum is the point (T b 1, Tb ) stisfies TC (T 1, T) T 1 = 0 nd TC (T 1, T) T = 0, (14) simultneously. It cn be shown tht the SOSC for the minimum problem re lso stisfied. The proof is similr to Cse 1.1. Cse. Pyment is pid t time M Cse.1 M < T 1 In this cse, the length of the positive stock period is greter thn the lst credit period (see Figure 1 (c)). The setup cost, holding cost, shortge cost nd opportunity cost due to lost sles re identicl to Cse 1.1. However, the retiler hs no csh discount in this cse nd the interest pyble is given by IP = c I p T1 I(t)dt = c I pr(t 1 M ) M. (15)

9 6 L. Y. OUYANG ET AL The interest erned per cycle is given by IE = c I e M 0 R t dt = ci erm Therefore, the totl cost per unit time is clculted s TC 3 (T 1, T). (16) Downloded by [Dokuz Eylul University ] t :48 3 My 014 = c 1 + HC + SC + OC + IP IE T { = 1 c 1 + c RT1 + c 3δR(T T 1 ) T + ci pr(t 1 M ) ci erm + c 4 R(1 δ)(t T 1 ) }. (17) The necessry condition for TC 3 (T c 1, Tc ) to be minimum is the point (T c 1, Tc ) stisfies TC 3 (T 1, T) T 1 = 0 nd TC 3 (T 1, T) T = 0, (18) simultneously. It cn be shown tht the SOSC for the minimum problem re lso stisfied. The proof is similr to Cse 1.1. Cse. T 1 M In this cse, the length of the positive stock period is not lrger thn the lst credit period (see Figure 1 (d)). The setup cost, holding cost, shortge cost nd opportunity cost due to lost sles re identicl to Cse 1.1. However, the retiler hs no csh discount nd no interest pyble in this cse, nd the interest erned is given by IE = ci e [ T1 0 ] R t dt + R T 1 (M T 1 ) Therefore, the totl cost per unit time cn be obtined s = c R I e (M T 1 T 1 ). (19) TC 4 (T 1, T) = c 1 + HC + SC + OC + IP IE T

10 ECONOMIC ORDER QUANTITY 63 { = 1 c 1 + c RT1 + c 3δR(T T 1 ) + c 4 R(1 δ)(t T 1 ) T c R I } e (M T 1 T1 ). (0) The necessry condition for TC 4 (T d 1, Td ) to be minimum is the point (T d 1, Td ) stisfies TC 4 (T 1, T) T 1 = 0 nd TC 4 (T 1, T) T = 0, (1) simultneously. It cn be shown tht the SOSC for the minimum problem re lso stisfied. The proof is similr to Cse 1.1. Downloded by [Dokuz Eylul University ] t :48 3 My 014 Remrk. When the csh discount is neglected nd shortge is not llowed, our model is reduced to tht of Goyl [1]. 4. Solution procedure The optiml replenishment policies nd minimum totl cost per unit time cn be obtined by using the following lgorithm : Step 1. Determine T 1 nd T from (11). If M 1 < T 1, obtin TC 1 (T 1, T ) from (10) ; otherwise (T 1, T ) is infesible. Step. Determine T b 1 nd T b from (14). If M 1 T b 1, obtin TC (T b 1, Tb ) from (13) ; otherwise (T b 1, Tb ) is infesible. Step 3. Determine T c 1 nd Tc from (18). If M < T c 1, obtin TC 3 (T c 1, Tc ) from (17) ; otherwise (T c 1, Tc ) is infesible. Step 4. Determine T d 1 nd T d from (1). If M T d 1, obtin TC 4 (T d 1, Td ) from (0) ; otherwise (T d 1, Td ) is infesible. Step 5. By compring TC 1 (T1, T ), TC (T1 b, Tb ), TC 3 (T1 c, Tc ) nd TC 4 (T1 d, Td ), select the optiml replenishment cycle nd optiml positive stock period (denoted by T nd T1, respectively) with the lest totl cost per unit time (denoted by TC ). Once the optiml vlue T nd T1 re obtined, the optiml order quntity, Q, cn be obtined from (3). 5. Nemericl exmples In order to illustrte the bove solution procedure, let us consider n inventory system with the following dt : R = 1000 units/yer, c 1 = $00 per setup, c = $ /unit/yer, c 3 = $8 /unit/yer, c 4 = $ /unit,

11 64 L. Y. OUYANG ET AL c = $0 /unit, δ = 0.8, I e = 0.13 /yer nd I p = 0.15 /yer. The optiml solutions for different prmeters vlues of r, M 1 nd M re shown in Tble 1. For instnce, when r = 0.01, M 1 = 5 dys nd M = 30 dys, the optiml vlues T 1 = 91 dys, T = 134 dys, Q = 344 units, TC = $100.40, nd the optiml pyment time is M 1. It implies tht the retiler should py the pyment t 5 th dy to get the csh discount. Tble 1 Optiml solutions Downloded by [Dokuz Eylul University ] t :48 3 My 014 r M 1 M T1 T Q TC Pyment time M M M M M M M M M M M M M M M M M M M M M M M M M M M

12 ECONOMIC ORDER QUANTITY 65 From Tble 1, we cn find tht the results re consistent with the economic incentives. For fixed r nd M 1, the lrger the vlue of M is, the smller the optiml totl cost per unit time would be. For fixed r nd M, the lrger the vlue of M 1 is, the smller the optiml totl cost per unit time would be s the optiml pyment time is M 1 ; however, if the optiml pyment time is M, the optiml totl cost per unit time is independent of the vlue of M 1. When r nd M 1 increse, the retiler will be more possible to py his purchses quickly to get the csh discount. The phenomenon will reduce the optiml totl cost per unit time. 6. Concluding remrks Downloded by [Dokuz Eylul University ] t :48 3 My 014 In this rticle, we consider n inventory model with llowble shortge under supplier credit. Prticulrly, we consider the impcts of the csh discount. In prctice, the supplier offers csh discount to encourge retiler to py for his purchses quickly. This ssumption is more relistic in tody s business trnsctions. When the csh discount is neglected nd shortge is not llowed, our model is reduced to tht of Goyl [1]. In further reserch, we would like to consider the vrible demnd rte (e.g., time-vrying demnd or stock-dependent demnd) or finite rte of replenishment. Appendix To show the solution (T 1, T ) stisfies the second-order sufficient conditions for the minimum problem. We first obtin the Hessin Mtrix H s follows : TC 1 (T 1, T) TC 1 (T 1, T) T1 T 1 T H = TC 1 (T 1, T) T T 1 TC 1 (T 1, T) T. From (11), we hve TC 1 (T 1, T) (T T 1 1,T ) = 0 nd TC 1(T 1, T) T (T 1,T ) = 0. Tking the second-order prtil derivtives of TC 1 (T 1, T) with respect to T 1 nd T, nd then substituting (T 1, T) by (T1, T ), one hs TC 1 (T 1, T) T 1 (T 1,T ) = (c + ci p + c 3 δ)r T > 0 (A1)

13 66 L. Y. OUYANG ET AL TC 1 (T 1, T) T TC 1 (T 1, T) T 1 T (T 1,T ) = c 3δR T TC 1 (T 1, T) T T (T 1,T ) = c 3δR T > 0 (A) (T 1,T ) = TC 1 (T 1, T) T T 1 (T 1,T ) = c 3δR T 1 T TC 1 (T 1, T) T 1 (T 1,T ) Downloded by [Dokuz Eylul University ] t :48 3 My 014 = c 3δR T. (A3) It is not difficult to infer tht ( ) ( TC 1 (T 1, T) ) ( TC 1 (T 1, T) T1 T ) TC 1 (T 1, T) T 1 T (T 1,T ) = c 3δR (c + ci p ) (T ) > 0. (A4) Thus, from (A1) nd (A4), it cn be concluded tht the Hessin Mtrix H is positive definite t point (T 1, T ). Acknowledgements. The lst uthor s reserch ws supported by the Kungwu Institute of Technology. References [1] S. K. Goyl, Economic order quntity under conditions of permissible dely in pyments, Journl of the Opertionl Reserch Society, Vol. 36 (1985), pp [] S. P. Aggrwl nd C. K. Jggi, Ordering policies of deteriorting items under permissible dely in pyments, Journl of the Opertionl Reserch Society, Vol. 46 (1995), pp [3] A. M. M. Jml, B. R. Srker nd S. Wng, An ordering policy for deteriorting items with llowble shortge nd permissible dely in pyment, Journl of the Opertionl Reserch Society, Vol. 48 (1997), pp [4] K. J. Chung, A theorem on the determintion of economic order quntity under conditions of permissible dely in pyments, Computers nd Opertions Reserch, Vol. 5 (1998), pp

14 Downloded by [Dokuz Eylul University ] t :48 3 My 014 ECONOMIC ORDER QUANTITY 67 [5] B. R. Srker, A. M. M. Jml nd S. Wng, Supply chin models for perishble products under infltion nd permissible dely in pyment, Computers nd Opertions Reserch, Vol. 7 (1999), pp [6] B. R. Srker, A. M. M. Jml nd S. Wng, Optiml pyment time under permissible dely in pyment for products with deteriortion, Production Plnning nd Control, Vol. 11 (000), pp [7] H. C. Lio, C. H. Tsi nd C. T. Su, An inventory model with deteriorting items under infltion when dely in pyment is permissible, Interntionl Journl of Production Economics, Vol. 63 (000), pp Received Mrch, 001