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1 Economic Order Quaniy under Condiions of Permissible Delay in Paymens Auhor(s): S. K. Goyal Source: The Journal of he Operaional Research Sociey, Vol. 36, No. 4 (Apr., 1985), pp Published by: Palgrave Macmillan Journals on behalf of he Operaional Research Sociey Sable URL: hp:// Accessed: 04/05/ :23 Your use of he JSTOR archive indicaes your accepance of JSTOR's Terms Condiions of Use, available a hp:// JSTOR's Terms Condiions of Use provides, in par, ha unless you have obained prior permission, you may no download an enire issue of a journal or muliple copies of aricles, you may use conen in he JSTOR archive only for your personal, non-commercial use. Please conac he publisher regarding any furher use of his work. Publisher conac informaion may be obained a hp:// Each copy of any par of a JSTOR ransmission mus conain he same copyrigh noice ha appears on he screen or prined page of such ransmission. JSTOR is a no-for-profi service ha helps scholars, researchers, sudens discover, use, build upon a wide range of conen in a rused digial archive. We use informaion echnology ools o increase produciviy faciliae new forms of scholarship. For more informaion abou JSTOR, please conac suppor@jsor.org. Operaional Research Sociey Palgrave Macmillan Journals are collaboraing wih JSTOR o digiize, preserve exend access o The Journal of he Operaional Research Sociey. hp://

2 J. Opl Res. Soc. Vol. 36, No. 4, pp , /85 $ Prined in Grea Briain. All righs reserved Copyrigh? 1985 Operaional Research Sociey Ld Economic Order Quaniy Under Condiions of Permissible Delay in Paymens S. K. GOYAL Deparmen of Quaniaive Mehods, Concordia Universiy, Monreal, Canada In his paper, mahemaical models have been derived for obaining he economic order quaniy for an iem for which he supplier permis a fixed delay in seling he amoun owed o him. An example has been solved o illusrae he mehod. Key word: invenory INTRODUCTION In deriving he economic order quaniy formula, i is acily assumed ha he supplier mus be paid for he iems as soon as he iems are received. However, in pracice a supplier will allow a cerain fixed period for seling he amoun owed o him for he iems supplied. Usually here is no charge if he ousing amoun is seled wihin he permied fixed selemen period. Beyond his period, ineres is charged. When a supplier allows a fixed ime period for seling he accoun, he is acually giving his cusomer a loan wihou ineres during his period. During he period before he accoun has o be seled, he cusomer can sell he iems coninue o accumulae revenue earn ineres insead of paying off he overdraf which is necessary if he supplier requires selemen of he accoun immediaely afer replenishmen. Therefore, i makes economic sense for he cusomer o delay he selemen of he replenishmen accoun up o he las momen of he permissible period allowed by he supplier. We would now develop he mahemaical models for deermining he economic order quaniy. MATHEMATICAL FORMULATION D = annual dem h = uni sock-holding cos per iem per year excluding ineres charges I,. = ineres charges per $ invesmen in socks per year Id= ineres which can be earned per $ in a year p = uni purchase price in $ S = cos of placing one order = permissible delay in seling accouns T = ime inerval beween successive orders Z(T) = oal annual variable cos. The following assumpions are made in deriving he model. (1) The dem for he iem is consan wih ime. (2) Shorages are no allowed. (3) During he ime he accoun is no seled, generaed sales revenue is deposied in an ineresbearing accoun. A he end of his period, he accoun is seled we sar paying for he ineres charges on he iems in sock. (4) Time period is infinie. The oal annual variable cos consiss of he following elemens. (1) Cos of placing orders = S/T. (2) Cos of sock holding (excluding ineres charges). The average sock equals DT/2 (see Figure 1), hence sock-holding cos per year is given by DTh/2. (3) Cos of ineres charges for he iems kep in sock. As iems are sold, before he replenishmen accoun is seled, he sales revenue is used o earn ineres. When he replen- 335

3 Journal of he Operaional Research Sociey Vol. 36, No. 4 0 IT Dr~~~ (T-) T ~~~~~~~~Time FIG. 1. Time-weighed invenory when T?. ishmen accoun is seled, he siuaion is reversed, effecively he iems sill in sock have o be financed a ineres rae I,. The sock level a he ime of seling he replenishmen accoun equals D(T - ) (see Figure 1) he ineres is payable during ime (T - ). Ineres payable in one cycle -p 2 ) c. Dp(T - )2j Dp TI Dp 2 I Ineres payable per year = _T 2 + Dp2I. p (4) Ineres earned during he permissible selemen period. The maximum accumulaed amoun earning ineres during he selemen period equals Dp if T > (see Figure 1) or DTp if T < (see Figure 2). Hence he ineres earned during he permissible selemen period for he wo cases is obained as follows: Case I: T ), shown in Figure I Ineres earned in one cycle = Dp 2Id 2 Ineres earned in one year = 2d Case II: T <, shown in Figure 2 Ineres earned in one cycle = + DTp(-T))Id =DTpId( - Ineres earned in one year = DpId(-T 0)~~~~(T 0) C-) }D'D LST) FIG. 2. Time-weighed invenory when T <. 336

4 S. K. Goyal-Economic Order Quaniy Under Condiions of Permissible Delay in Paymens Noe ha he ineres earned should be subraced from oher variable coss in order o ge he oal variable cos per year. Now we shall develop he mahemaical models for deermining economic order quaniies for he wo cases. or DETERMINATION OF ECONOMIC ORDER QUANTITY WHEN T In his case he oal variables cos per year is given by S DTh DTpIC Dp I2I Dp 2Id Z(T)=- ~+- + c-dpi - T T 2 2 2T 2 (5+ Dp2(Ic - I)) D Z(T) = (2S + ( + (h + pi) - Dp. (1) 2T ~~2 In order o minimize Z(T), se [dz(t)/dt] = 0 obain he economic value of T =T T.* = /( (2+Dp2(j-c)) Dh ).(2) (2) 1D (h + ph) I may be poined ou ha he effec of use of money, which means > 0 (Ic -Id) 0 for normal condiions, is o increase he apparen order placemen cos. The economic order quaniy Q (T ) he minimum annual variable cos Z(T ) are given by Q(T*) =DT*= D(2S + Dp2(Ic - Id)) 1( ) 1 ' j (3) (h +pij) Z(T*) =,/D(2S + Dp2(I -Id)) (h + - DpIc. (3a) As a resul of permissible delay in selemen of he replenishmen accoun, he order quaniy as obained by (3) is generally higher han he order quaniy given by he classical economic order quaniy model given by Q* 2DS/(h+ph) (3b) The exen of such a change in he order quaniy depends enirely on he parameers of he problem siuaion. When Ic = Id, hen he economic order quaniy given by (3) is exacly equal o he order quaniy obained from he classical economic order quaniy model. On he oher h, he highes value of Q(T ) is obained when he funds canno be invesed (Id = 0). Under such condiions, he economic order is given by D(2S + Dp2j,) (h + pi) In he rare case when I - Id < 0, he economic order quaniy obained from (3) is lower han he economic order quaniy obained from he classical economic order quaniy model. DETERMINATION OF ECONOMIC ORDER QUANTITY WHEN T < In his case, no ineres charges are paid for he iems kep in sock. The oal variable cos in his case is given by Z() S +DTh DI( T) Z(T) =-?+ (h +pld)-dpld. (4) 337

5 Journal of he Operaional Research Sociey Vol. 36, No. 4 Hence he economic order inerval he order quaniy are given by T * = A ;(5) 2 D h +pld)(5 2SD Q(T* ) = DT* = -/ (6) 2 2 ~ (h +pld)(6 Noe ha for I, ) Id Q(Tfl) > Q* I may be poined ou ha he oal annual variable cos, Z(), a T= is obained on subsiuing T = in (1) or in (4). S D h Dp ld Z() =+ 2-2 (7) 2 2 In order o obain he economic operaing policy, follow hese seps: Q() = D. (8) Sep 1: Deermine T1 from (2). If T1 ), obain Z(T ) from (1). Sep 2: Deermine Tf from (5). If T2 <, evaluae Z(Tf ) from (4). Sep 3: If T1 < T2 ), hen evaluae Z() from (7). Sep 4: Compare Z(T *), Z(Tf ) Z(). Selec he replenishmen inerval he order quaniy associaed wih he leas annual cos value evaluaed in seps 1 2 or 3. CONCLUDING REMARKS As a resul of permissible delay in seling he replenishmen accoun, he economic replenishmen inerval order quaniy generally increases marginally, alhough he annual cos decreases considerably. The saving in cos as a resul of permissible delay in seling he replenishmen accoun largely comes from he abiliy o delay paymen wihou paying any ineres. As a resul of increased order quaniy under condiions of permissible delay in paymens, we need o order less ofen. The economic order quaniy model given in his paper is an exension of he classical economic order quaniy model hence can be used in a similar manner for deermining ordering policy for iems having probabilisic dem. Acknowledgemen-I am very much indebed o he wo referees for heir valuable commens suggesions. 338