Research Article Inventory and Credit Decisions under Day-Terms Credit Linked Demand and Allowance for Bad Debts

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1 Avances in Decision Sciences, Aricle ID 67856, pages hp://x.oi.org/.55/24/67856 Research Aricle Invenory an Crei Decisions uner Day-erms Crei Linke Deman an Allowance for Ba Debs K. K. Aggarwal an Arun Kumar yagi Deparmen of Operaional Research, Faculy of Mahemaical Sciences, Universiy of Delhi, Delhi 7, Inia Corresponence shoul be aresse o Arun Kumar yagi; arunyagi.u@gmail.com Receive 28 July 24; Accepe 26 Ocober 24; Publishe 8 December 24 Acaemic Eior: Jen-Chih Yao Copyrigh 24 K. K. Aggarwal an A. K. yagi. his is an open access aricle isribue uner he Creaive Commons Aribuion License, which permis unresrice use, isribuion, an reproucion in any meium, provie he original work is properly cie. In orer o simulae eman of heir prouc, firms generally give crei perio o heir cusomers. However, selling on crei exposes he firms o he aiional imension of ba ebs expense (i.e., cusomer s efaul). Moreover, crei perio hrough is influence on eman becomes a eerminan of invenory ecisions an invenory sol on crei ges convere o accouns receivable inicaing he ineracion beween he wo. Since invenory an crei ecisions are inerrelae, invenory ecisions mus be eermine joinly wih crei ecisions. Consequenly, in his paper, a mahemaical moel is evelope o eermine invenory an crei ecisions joinly. he eman rae is assume o be a logisic funcion of crei perio. he accouns receivable carrying cos along wih an explici consieraion of ba eb expense which have been ofen ignore in previous moels are incorporae in he presen moel. he iscoune cash flow approach (DCF) is use o evelop he moel an he objecive is o maximize he presen value of he firm s ne profi per uni ime. Finally, numerical example an sensiiviy analysis have been one o illusrae he effeciveness of he proposeoel.. Inroucion he basic purpose of a firm is maximizaion of is presen value an in orer o achieve his goal proper invenory managemen is an imporan aspec. he basic objecive of any invenory conrol sysem is o saisfy he fuure eman in a bes possible manner. he classical EOQ moel assumes haemancannobeinfluencebyheecisionmaker. However, ecision maker can influence he eman by giving crei perio o is cusomers. rae crei is use by he firms as a markeing sraegy o simulae eman by aracing he cusomers who consier i o be a ype of price reucion. Moreover, o realize sales from cusomers who o no have money for insan paymen he firm mus wai unil hey resell he goos before oing he paymen. Many cusomers woul like o verify he qualiy of firm s prouc prior o making he paymen. In such circumsances, firm allows sales on crei so ha i can sell more goos in comparison o when i relies only on cash sales. eng [] also illusrae wo benefis of rae crei policy o he supplier: () i shoul arac new cusomers who consier i o be a ype of price reucion; (2) i shoul cause a reucion in he sales ousaning, since some esablishe cusomers will pay more promply in orer o ake avanage of permissible elay more frequenly. he wo common forms of rae crei are ay-erms an aeerms. In ay-erms crei paymen has o be one wihin a fixe ime perio afer a purchase an in ae-erms crei he firm specifies a ue ae on which paymen has o be one (Kingsman [2], Carlson an Rousseau [3], Robb, an Silver [4]). hus, in ay-erms crei each cusomer ges same amoun of crei perio irrespecive of is purchase ae, while in ae-erms crei policy he crei perio availe by he cusomers is he ifference beween ue ae (i.e., maximum crei perio) an he ime a which cusomer has purchase he goos. he response of eman rae wih respec o crei perio will have a significan impac on he ecision making. In a crei elasic marke, crei perio is suppose o increase eman of he prouc by inucing exising cusomer o orer larger quaniy an aracing new cusomers. he ne effec of hese will be reflece in he eman rae. Since in realiy he marke consiss of cusomers having ifferen

2 2 Avances in Decision Sciences financial capaciies so heir propensiy owars availing crei perio woul also be ifferen. For example, a low value of crei perio has uiliy only for cusomers having low income while a larger value of crei perio woul also be worh full o high income cusomers along wih he low income cusomers. herefore, a firs eman increases slowly wih he increase in crei perio bu as crei perio increases furher he eman rae increases subsanially in view of he fac ha crei perio has now value o a large number of cusomers. Bu he eman rae canno increase inefiniely wih he increase in crei perio as here is a limi o his growh ue o he finie marke poenial. Consequenly, he eman rae will evenually reach is sauraion level afer a cerain value of crei perio. Hence, in general, as crei perio increases, he eman rae a firs increases slowly, hen here is a subsanial increase in is value, an finally reaches is sauraion level ue o finie marke poenial. he growh curve of eman rae uner such circumsances can be represene by he sigmoi or S- shapepaern(e.g.,logisicfuncion),whenemanraeis ploe agains crei perio (Figure ). he logisic curve consiss of hree ifferen paerns, AB, BC,anCD.FromAoBhecurvegrauallyrises,fromB o C i is almos an exponenial increase, an from C o D i ges flaene. I implies ha a smaller value of crei perio he eman effec of crei perio is less (AB), afer ha as crei perio increases he eman effec of crei perio increases significanly (BC) an hen graually sauraes o is maximum level wih a furher increase in crei perio (CD). he crei policy hrough is impac on eman becomes a eerminan of invenory policy. Moreover, when invenoriesaresoloncrei,heygeconvereinoaccouns receivable. he close ineracion among hese componens implies ha he opimal policy for one componen canno be eermine wihou he simulaneous consieraion of oher componen. he invenory policy ses he iming an quaniy of cash ouflows, while crei policy ses he iming an quaniy of cash inflows. Because of he link among heir associae cash flows, he invenory an crei ecisions will have an effec on each oher. Since invenory an crei ecisions are inerrelae, invenory ecisions mus be eermine joinly wih he crei ecisions. However, invenory ecisions are normally eermine inepenenly of he crei ecisions. Furhermore, when a firm allows sales on crei i resuls in increase revenue as well as causing cerain coss o he firm. Selling on crei will be economical for he firm if he revenue generae ue o crei sales is sufficien o compensaehecosofgivingraecrei.goossoloncrei ge convere o accouns receivables whose accumulaion woul resul in accouns receivable carrying cos o he firm. he coss associae wih carrying accouns receivables are he cos of financing accouns receivable, aminisraive coss in running a crei eparmen, elinquency or collecion coss, an cos of efaul by he cusomers, ha is, ba eb losses. Ou of he oal cos in graning crei perio he mos significan cos is uncollecible accouns or ba ebs. his is ue o he fac ha selling on crei exposes he firm o he aiional imension of efaul risk (i.e., nonpaymen) Deman rae A B Crei perio C Figure : (Logisic curve). from he cusomers as some cusomers are eiher unable or unwilling o pay. If a cusomer eclares bankrupcy, no paymen may be forhcoming. If he cusomer leaves he ciy orhesae,imaybeoocoslyoracehimaneman paymen. In hese cases, he firm is force o accep a baeb loss on he accoun. Mos firms expec o incur baeb losses in he normal course of business. hese losses are properly viewe as a cos of aminisering a crei policy. If he firm can ienify such nonpaying cusomers in avance hen i woul refuse o rae wih hem excep on a cash basis. Since he firm canno ienify such cusomers beforehan so his loss may be a necessary loss of business. I is he cos which mus be susaine in orer o obain he profi from he paying cusomers. Hence, he ba-eb expenses nee o be given an explici consieraion in ecision making. Alhough he firm canno isinguish beween he paying an nonpaying cusomers in avance, someimes from pas experiences he firm knows wih a reasonable egree of cerainy ha a number represening a cerain porion of accouns receivable will never be collece, an hence ha much amoun mus be wrien off from he profi. Since he maching principle of accrual accouning requires he recogniion of ba ebs expense a he same ime as he relae revenue; herefore, in orer o recor aneasure he accouns uncollecible, he firm usually uses allowance meho for ba eb losses as i confirms he maching principle of accrual accouning. In his meho, he firm makes an esimae of he porion of crei sales ha will ulimaely be uncollecible from as ye unienifie cusomers. Consequenly, he firms usually express he amoun of ba ebs as a percenage of oal crei sales, which relies on he hisorical relaionship beween crei sales an uncollecible ebs, ha is, which epens on he pas experience of he firm. Owning o he fac ha crei perio has an influence on he eman K. K. Aggarwal an S. P. Aggarwal [5] evelope an EOQ moel wih crei linke eman when firmpurchasesiemsoncashinaninflaionaryconiion. Jaggi e al. [6] expane on his heme an evelope an EOQ moel wih rae crei linke eman uner he wo sage rae crei financing. However, in hese moels D

3 Avances in Decision Sciences 3 he crei perio is assume o be given an only orering ecisions were eermine. Oher relevan aricles wih crei linke eman ealing wih invenory or invenory an crei ecisions are Su e al. [7], Jaggi e al. [8], hangam an Uhayakumar [9], Maii [], Ho [], Annaurai an Uhayakumar [2], Giri an Maii [3], an Shah e al. [4]. All he above menione aricles have ignore he cos of graning crei perio o cusomers an none of hem have assume eman rae o be a logisic funcion of crei perio. In aiion, average cos approach was use in eveloping all he aforemenioneoels wih he excepion of K. K. Aggarwal an S. P. Aggarwal [5], Jaggi e al. [6], an Maii []. In average cos approach he ime value of money is no explicily aken ino accoun an here is no isincion beween ou-of-pocke holing cos an opporuniy coss ue o invesmen of funs in invenory as well as in accouns receivable. Moreover, he effec of elaye revenue realizaion because of crei perio given o cusomers canno be capure by he average cos approach. I is wiely accepe ha he ne presen value or DCF approach leas o a beer ecision (Kim an Feis [5]). he DCF approach allows proper recogniion of he financial implicaion of he opporuniy cos an ou-of-pocke coss associae wih he economic sysem. I also permis an explici recogniion of he exac iming of each cash-flow associae wih he economic sysem an consiers he ime value of money. Consequenly, in his paper, a mahemaical moel is evelope wihin he iscoune cash flow approach o eermine opimal invenory an crei perio ecisions joinly for a firm. he firm purchases a single prouc on cash an sells on ay-erms crei o is cusomers. he eman rae has been moele as a logisic funcion of crei perio. he cos of carrying accouns receivable along wih he explici consieraion of he allowance for ba eb expense is consiere. he objecive is o maximize he presen value of firm s ne profi per uni ime. A hypoheical numerical example, sensiiviy analysis, an observaions are presene o illusrae he effeciveness of he proposeoel. 2. oaions an Assumpions 2.. oaions. he noaions use in his paper are as follows: Q = Orering quaniy per orer = Invenory cycle ime (in years) = Crei perio given by he firm (in years) () = Deman rae (in unis/year) f = Proporion of crei sales ha becomes ba ebs O = Orering cos per orer (in ollars) C =Unipurchasecos(inollars) P = Uni selling price (in ollars) k =Raeofineresoriscounraeperyear(in percen per year) I = Ou-of-pocke invenory carrying cos per year, as apercenageofunipurchasecos R = Ou-of-pocke receivable carrying cos per year, asapercenageofunisellingprice I() =Invenorylevelaanyime R() = Accouns receivable level a any ime Z(, ) = e profi per year as a funcion of ecision variables an Assumpions. he moel has been evelope uner he following assumpions. () Invenory sysem involves one ype of iem. (2) Firm purchases on cash an sells on crei. (3) he firm offers same amoun of crei perio o each of is crei cusomers. hus, firm follows a ay-erms crei policy, ha is, ne, whereis he crei perio. (4) he eman rae is a funcion of he crei perio offere by he firm. Uner he assumpion ha eman rae firs increases slowly, hen increases subsanially, an afer ha sauraes as crei perio increases heemanraefuncionwihouheloss of generaliy can be escribe by he logisic funcion () = /( + ae b ),wherea >, b >,are consans an is he maximum achievable eman rae,hais,upperbounonheemanrae.figure 2 epics several logisic curves for various values of growh parameer b wih =, an a =5are epice. (5) he effec of crei policy on eman is observe insananeously wihou any elay. (6) A cerain porion of accouns receivable of he firm remains uncollecible; ha is, he firm incurs ba eb losswhoseproporionouofheoalcreisalesis known o he firm. he proporion of crei sales ha becomes ba-eb loss can be esimae from he pas aa of crei sales an accouns uncollecible. (7) All he nonefauling cusomers sele heir accoun imely. (8) Replenishmen rae is infinie. (9) Shorages are no allowe. () Lea ime is zero or negligible. () Discoune cash flow approach is use o reflec he ime value of money. 3. Mahemaical Moeling A he sar of he cycle, he invenory level is raise o Q unis; aferwars as ime progresses, he level of invenory ecreases o fulfill eman up o an becomes zero a (Figure 3). here is a cash ouflow of orering an purchase cos a he sar of cycle an he revenue from crei sales aking place uring he replenishmen inerval will be receive by he firm from o + (Figure 4).

4 4 Avances in Decision Sciences Deman rae (()) 9 8 b=3 7 b=8 6 b= Crei perio () Figure 2: Deman rae versus crei perio. Q I() Since replenishmen rae is infinie an shorages are no allowe, so he iniial invenory level, I() (i.e., he orer quaniy, Q) is I () =Q= ( m )= m. (2) +ae b +ae b An he variaion of invenory level wih respec o ime can be escribe by he following ifferenial equaion: I () = ( ). (3) +ae b Wih he bounary coniions: I()=Q= /( + ae b ) an I() =, consequenly, he soluion of (3) is given by I () = m ( ),. (4) +ae b Furhermore, he level of accouns receivable a any ime epens upon he sales rae an he rae of revenue collecion a ha poin of ime. As he collecion of revenue corresponing o he sales which ake place uring he invenory cycle, ha is, from o willoccurfrom o +, herefore, calculaion for he level of accouns receivable will epen upon he following cases, namely, () (Figure 5) an (2) (Figure 6), which are explaine in he subsequen secions. By using DCF approach, he various componens of he profi funcions are as follows: Figure 3: Invenory level versus ime. he presen value of revenue per year from sales = ( f) P + ( ( +ae b )e k ) = ( f) Pe b k(+) (e k ), (a + e b )k Orering cos + purchase cos Revenue he presen value of he orering cos per year = O, Figure 4 + he presen value of purchasing cos per year = CQ = C ( C )= m +ae b +ae, b he presen value of invenory carrying cos per year For he firm, he presen value of is ne profi per year Z(, ) can be expresse as Z (, ) = Revenue from sales Orering cos Purchase cos Invenory carrying cos Accouns receivable carrying cos. () = IC I () e k = IC ( ( ) +ae b )e k = ICe b (e k +k ). (a + e b )k 2 he calculaion of accouns receivable carrying cos will epen upon he following cases. (5)

5 Avances in Decision Sciences 5 R() R 2 () R 3 () R () + Figure 5: Accouns receivable versus ime. R 2 () = ( + ) b +ae (Sales rae Collecion rae) ) b = ( +ae + ( m +ae b ) +ae b =( +ae b ),, (8) R() R () R 2 () R 3 () Figure 6: Accouns receivable versus ime. + Case ( ). aking ino accoun he crei sales ha occur uring he replenishmen inerval, we observe ha he behavior of he level of accouns receivable will be as follows: a he sar of he cycle, he level of accouns receivable is zero; aferwars as ime progresses i accumulaes up o ue o crei sales. From o, wo hings happen simulaneously; ha is, he accouns receivable are creae as a resul of crei sales an also he accouns receivables are collece by he firm. Since he rae of crei sales is equal o he rae of collecion of accouns receivable, he ne level of accouns receivable remains consan for.from o +he level of accouns receivable ecreases ue o heir collecionan becomes zero a +. he accouns receivable creae from o +will be accoune in nex cycle. Le R () ={R () = accouns receivable level a any ime, for, R 2 () = accouns receivable level a any ime, for, R 3 () = accouns receivable level a any ime, for +}. We have R () =+ ( m )= m +ae b +ae, b (6), (7) + R 3 () = ( ) b +ae (9) =( m ) (+ ), +ae +, b he presen value of accoun receivables carrying cos per year = RP ( R () e k + + R 2 () e k + R 3 () e k ) = RP ( ( m +ae b )e k + ( +ae b )e k + + ( +ae ) (+ ) b e k ) = RP (e (b k) (e k k ) (a + e b )k 2 + e b (e k e k ) (a + e b )k + e k(+) ( + e k (k )) ( + ae b )k 2 ). () Using (5) an (), he presen value of firm s ne profi per year, Z (, ) is Z (, ) =[ ( f) Pe b k(+) (e k ) (a + e b )k O C +ae b ICe b (e k +k ) (a + e b )k 2 RP (e (b k) (e k k ) (a + e b )k 2

6 6 Avances in Decision Sciences + e b (e k e k ) (a + e b )k + e k(+) ( + e k (k )) ( + ae b )k 2 )]. () he necessary coniions for he maximizaion of Z (, ) are which gives Z (, ) =, Z (, ) = (2) [ Pe b k(+) (e k ) (f ) (e b k+a(k b)) (a + e b ) 2 k Cabe b ( + ae b ) 2 ICabe b k ( + e k (k )) (a + e b ) 2 k 2 RPe (b k) (e b k 2 +a(k 2 +b(e k k ))) (a + e b ) 2 k 2 + (RP e b k(+) (e b (e k +e k (k )) +a(e k (+b) ((a+e b ) 2 k) ) (RP e b k(+) (e b (e k )k +a(b k +e k (k b ))) +e k (k+b(k )))) ((a+e b ) 2 k 2 ) )]=, [( e b k(+) (f ) P (e k k ) (a + e b )k 2 ) +( O 2 ) (ICe b k (e k k ) (a + e b )k 2 2 ) ( PRe (b k) ( e k + k) (a + e b )k 2 2 ) ( RPe b k(+) (e k (+k) e k ) (a + e b )k 2 ) +( RPe b k(+) ( + e k (k )) (+k) )] (a + e b )k 2 2 =. (3) he soluion of above equaions gives he opimal values of an for he maximizaion of Z (, ) provie ha hey saisfy he sufficiency coniions given by 2 Z 2, 2 Z 2, ( 2 Z 2 )( 2 Z 2 ) ( 2 Z ) 2. (4) However, i is ifficul o solve he necessary coniions analyically in a close form an also o check he valiiy of sufficien coniions analyically. Consequenly, numerical approach is use o obain he soluion. Case 2 ( ). aking ino accoun he crei sales ha occur uring he replenishmen inerval, we observe ha he behavior of he level of accouns receivable will be as follows: a he sar of he cycle, he level of accouns receivable is zero; aferwars as ime progresses i accumulaes up o ue o crei sales an remains a his level up o. From o +he level of accouns receivable ecreases ue o heir cash realizaion an becomes zero a +. he accouns receivable creae from o +will be accoune in nex cycle. Le R () ={R () = accouns receivable level a any ime, We have R () =+ for, R 2 () = accouns receivable level a any ime, for, R 3 () = accouns receivable level a any ime, ( )= +ae b for +}. (5) m +ae b, R 2 () = ( m )= m +ae b +ae, b + R 3 () = (, (6), (7) +ae )= (+ ) b +ae, b +, (8)

7 Avances in Decision Sciences 7 he presen value of accoun receivables carrying cos per year = RP ( R () e k + + R 2 () e k + R 3 () e k ) = RP ( ( m +ae b )e k + ( +ae b )e k + + ( (+ ) +ae b )e k ) = RPe b k (e k k ) (a + e b )k 2 + RPe b (e k e k ) (a + e b )k + RPe k(+) ( + e k (k )). ( + ae b )k 2 (9) Using (5) an (9), he presen value of firm s ne profi per year, Z 2 (, ) is Z 2 (, ) =[ ( f) Pe b k(+) (e k ) O (a + e b )k C m +ae ICe b (e k +k ) b (a + e b )k 2 RPe b k (e k k ) (a + e b )k 2 RPe b (e k e k ) (a + e b )k RPe k(+) ( + e k (k )) ]. ( + ae b )k 2 (2) he necessary coniions for he maximizaion of Z 2 (, ) are Z 2 (, ) which gives =, Z 2 (, ) = (2) [ Pe b k(+) (e k ) (f ) (e b k+a(k b)) (a + e b ) 2 k Cabe b ( + ae b ) 2 ICabe b k ( + e k (k )) (a + e b ) 2 k 2 RPabe b k (e k k ) (a + e b ) 2 k 2 RPe b k(+) (abe k +e k (e b k+a(k b))) (a + e b ) 2 k + (RP e b k(+) (e b k+a(k b)) (+e k (k )) ((a+e b ) 2 k 2 ) )]=, [( Pe b k(+) (f ) (e k k ) (a + e b )k 2 ) +( O 2 ) (ICe b k (e k k ) (a + e b )k 2 2 ) + RPe b k (e k k k 2 2 ) (a + e b )k RPe b k a+e b RPe b k(+) (e k k ) (a+e b ) k 2 2 ]=. (22) he soluion of above equaions gives he opimal values of an for he maximizaion of Z 2 (, ) provie ha hey saisfy he sufficiency coniions given by 2 Z 2 2, 2 Z 2 2, ( 2 Z 2 2 )( 2 Z 2 2 ) ( 2 Z 2 ) 2. (23) However, in his case also, i is ifficul o solve he necessary coniions analyically in a close form an also o check he valiiy of sufficien coniions analyically. Consequenly, like he previous case, numerical approach is use o obain he soluion. Furhermore, combining boh of he cases ha is, () an (2),wegehefirm sneprofiperyear,z(, ) as Z (, ) =( Z (, ), ). (24) Z 2 (, ), Our problem is o fin he values of an which maximize Z(, ). o solve he moel we solve boh of he cases separaely an hen combine he resuls o obain he opimal soluion. Due o highly complex an nonlinear form, iisifficulosolvehemoelanalyicallyinacloseform. However, he moel can be solve numerically using LIGO which uilizes generalize reuce graien algorihm.

8 8 Avances in Decision Sciences Z = Z 2 = Z ($).5 Z 2 ($).5 (year) (year).8.5 (year) (year).8 Figure 7: Graph of profi funcion ( ). Figure 8: Graph of profi funcion ( ). 4. umerical Example For numerical illusraion, he values of he moel parameers are aken as follows: =, unis/year, a =5,b =5,O = $/orer, C =$2/uni,P = $25/uni, I =3%,R =2%, k =5%,anf =4%(.4). Solvinghemoelaccoringohesoluionproceure escribe above, we ge Z ($) Z = Z Z (, ) = $, = (.4633 (years) = (ays)), = (.4633 (years) = (ays)), (, ) = $, = ( (years) = (ays)), = ( (years) = (ays)). Clearly, Max{Z,Z 2 } is Z 2. herefore, (25) = 2 = (ays), = 2 = (ays), (26) Z(, )=Z 2 = $. he opimal orering quaniy is Q = /( + ae b ) = (Unis). he surface graphs (Figures 7, 8, an9) ofheneprofi funcion Z(, ) wih respec o ecision variables an have been ploe using MALAB. he graphs clearly show ha a (, ) he value of Z(, ) is maximum. hus, for he given values of parameers in he numerical example, = (ays) an = (ays) is he opimal soluion. 5. Sensiiviy Analysis In his secion we have suie he effec of changes in he value of inpu parameers I, R, f,,ank on he opimal soluion. We consier he aa as given in he numerical example. I is assume ha all oher parameers are known an saionary in he ime perios uner consieraion. he sensiiviy analysis is performe by changing one parameer.8.6 (year) (year) Figure 9: Graph of profi funcion (combine graph of boh cases). a a ime an keeping he remaining parameers unchange. Following ables show he changes in opimal soluion for ifferen values of he parameers I, R, f,,ank. We obaine he following observaions ananagerial insighs from he resuls of numerical exercise. (i) Fromable,iisobservehaashecosofcarrying invenory increases he opimal value of crei perio, cycle lengh, an orering quaniy ecreases. his is rue since as he cos of carrying invenory increases he firm woul like o carry invenory in a lesser amoun as well as for shorer ime uraionaninoingsohefirmwoulreuceiseman by ecreasing he crei perio given o cusomer. he resul is consisen wih he properies of EOQ moels an also shows ha in a crei elasic marke he ecision of graning creiperioocusomersisinfluencebyinvenorycarrying cos. herefore, invenory carrying cos shoul be aken ino accoun by he firm while eciing he level of invesmen in accouns receivable when eman is influence by crei perio. Resuls sugges ha he firm shoul follow a sringen crei policy in presence of high invenory carrying cos. I can be ha he opimal value of profi ecreases as invenory carrying cos increases, which is quie obvious an confirms our expecaions. (ii) able 2 shows ha as accouns receivable carrying cos increases he opimal value of crei perio ecreases. his

9 Avances in Decision Sciences 9 I able : Sensiiviy analysis wih respec o I. (ays) (ays) Q (unis) Z(, ) $ R able 2: Sensiiviy analysis wih respec o R. (ays) (ays) Q (unis) Z(, ) $ is rue because as per economic raionale he firm shoul carry accouns receivable in a lesser amoun as well as for a shorer uraion as he cos of carrying accouns receivable increases. In oing so he firm shoul reuce he crei perio given o cusomers so as o reuce he eman. A change in eman woul cause a simulaneous change in invenory ecisions as per he srucure an parameers of he moel. he resul of increasing accouns receivable carrying cos on invenory ecisions inicaes ha he firm shoul lenghen is replenishmen inerval o save on orering cos in orer o maximize is profi. he above analysis confirms ha invenory ecisions are influence by accouns receivable carrying cos as well as by crei ecisions when eman is epenen on crei perio. herefore, in orer o correcly analyze invenory an crei ecisions wih crei linke eman funcion he accouns receivable carrying cos mus also be inegrae in he overall cos-benefi srucure of heecisionmoel.heresulssuggeshahefirmshoul inves less in accouns receivable an hence shoul follow a sringen crei policy in he siuaion of high value of accouns receivable carrying cos. Moreover, i can also be observe from able 2 ha he opimal value of crei perio becomes zero as accouns receivable carrying cos become very high. his suggess ha if he value of accouns receivable carrying cos is very high he firm shoul go for all cash sales program. Furhermore, he opimal oal profi ecreases as accouns receivable carrying cos increases, which is quie obvious an confirms our expecaions. (iii) able 3 shows ha as he value of allowance for oubful accouns increases he opimal crei perio ecreases. his is quie reasonable because if he firm is f able 3: Sensiiviy analysis wih respec o f. (ays) (ays) Q (unis) Z(, ) $ expecing o have high rae of efaul (i.e., ba accouns) i woul like o reuce is oal crei sales in orer o reuce is ba eb expenses an for oing ha i shoul reuce he amoun of crei perio given o cusomers. Consequenly, here will be a reucion in accouns receivable carrying cos as lesser amoun of receivables has o be kep for shorer uraion. he saving in accouns receivable carrying cos will conribue in offseing he cos of ba eb expenses. Furher, reucion in crei perio leas o a change in eman causing a simulaneous change in opimal invenory ecisions as per he srucure an parameers of he moel. he resuls show ha when higher proporion of sales becomes ba accouns he firm shoul increase is invenory cycle lengh for maximizing he profi. he increase cycle lengh resuls in less orering cos. he saving in orering cos will conribue in offseing he cos of ba eb expenses. he above analysis an inerpreaion shows ha he invenory ecisions are influence by he value of he allowance for oubful accouns. his fac becomes even more significan a higher value of ba eb proporion. hus, he allowance for oubful accouns mus be consiere an nees an explici consieraion for making invenory-crei ecisions in an invenory sysem wih crei linke eman. he resuls imply ha he firm shoul evise a sringen crei policy when i is expecing o have higher incience of ba accouns an he esimaion of allowance for oubful accouns shoul be one properly. Furhermore, he resuls inicae ha he cusomer selecion crieria have imporan implicaions for invenory-crei ecisions. herefore, he characer, capaciy, an capial of he cusomers have implicaions no only forcreiecisionsbualsoforinvenoryecisionsan herefore, mus be aken ino accoun in invenory conrol. Moreover, he opimal oal profi ecreases when he allowance for ba eb expense increases, which is quie obvious an confirms our expecaions. (iv) From able 4 i can be seen ha as he value of maximum achievable eman rae increases he opimal value of crei perio increases. his is quie reasonable becauseifhereisahugemarkepoenial,hefirmwoul efiniely like o give more crei perio in orer o capure largershareofi.alhoughiwillcauseanincreaseinba eb expenses, he loss ue o ba eb will be offse by he profi from he paying cusomers. hus, if here is a huge marke poenial for crei eman hen he firm shoul

10 Avances in Decision Sciences able 4: Sensiiviy analysis wih respec o. (ays) (ays) Q (unis) Z(, ) $ followamoreliberalcreipolicy.anincreaseincrei perio resuls in increase in eman causing a simulaneous change in invenory ecisions. I can be seen ha firm shoul orer more frequenly as well as orering more quaniy if hereisahugecreisalespoenialinhemarke.his is because he higher profis generae even afer ajusing for ba ebs will offse he increase in orering, invenory holing, an accouns receivable carrying coss as can be seen from he increase in he value of opimal profis. hese resuls are quie obvious an confirm our expecaions. (v) able 5 showshaheopimalvaluesofcreiperio an orering quaniy ecreases as iscoun rae increases. hisisruebecausehefirmwoullikeohavelessinvesmen in invenory an accouns receivable as he opporuniy cos of funs increases. In orer o o so he firm has o reuce is eman for which i has o ecrease he amoun of crei perio given o cusomers. A change in eman causes a simulaneous change in invenory ecisions accoring o he srucure an parameers of he moel. hus, he resuls show ha invenory-crei ecisions are influence by he ime value of money. Furhermore, a a very high value of iscoun rae he crei perio becomes zero. his suggess ha in he even of very high opporuniy cos of funs he firm shoul go for all cash sales by invesing less in invenory (i.e., smaller orering quaniy) for a shorer uraion (i.e., smaller replenishmen inerval). his is in confirmaion wih he properies of EOQ moel in he presen value framework. Also,hevalueofheopimalprofiecreasesasiscounrae increases, which is quie obvious an confirms he economic logic. 6. Conclusions Offering crei perio is a common business pracice followe by he firm an hrough is influence on eman has a significan impac on invenory managemen. In aiion, selling on crei also exposes he firm o he aiional imension of ba eb loss from he cusomers. Consequenly, in his paper an invenory-crei perio ecision moel has been evelope wihin he DCF framework wih an explici consieraion of allowance for ba ebs along wih accouns receivable carrying cos. he eman rae is aken o be logisic funcion of crei perio given o cusomers. he objecive is o maximize he presen value of firm s ne profi per uni ime by opimizing he values of crei perio an invenory cycle lengh joinly. Subsequenly, a soluion proceure is escribe o obain he opimal ecisions rules. umerical example is presene o illusrae k able 5: Sensiiviy analysis wih respec o k. (ays) (ays) Q (unis) Z(, ) $ he moel. Finally, sensiiviy analysis has been one followe by he iscussion on he resuls. he resuls are foun o be consisen wih he economic logic an confirm he moel. he proposeoel can be use as a framework for he coorinaion an analysis of invenory an crei perio ecisions in carrying ou working capial planning aciviies of he firm. Conflic of Ineress here is no conflic of ineress regaring financial gains for all he auhors. Acknowlegmens he commercial ieniies menione in he paper such as LIGO an MALAB sofware are use only for he numerical analysis of he moel iscusse in he paper which is only for he acaemic purpose. he auhors of his paper o no have any irec financial relaion wih he commercial ieniies menione in he paper. References [] J.-. eng, On he economic orer quaniy uner coniions of permissible elay in paymens, he Operaional Research Sociey,vol.53,no.8,pp.95 98,22. [2] B. G. Kingsman, he effec of paymen rules on orering an sockholing in purchasing, he Operaional Research Sociey, vol. 34, no., pp , 983. [3]M.L.CarlsonanJ.J.Rousseau, EOQunerae-erms supplier crei, he Operaional Research Sociey,vol. 4,no.5,pp.45 46,989. [4] D. J. Robb an E. A. Silver, Invenory managemen uner ae-erms supplier rae crei wih sochasic eman an leaime, he Operaional Research Sociey, vol. 57, no. 6, pp , 26. [5]K.K.AggarwalanS.P.Aggarwal, Invenoryconrolwih inflaion an rae crei, in Operaions Research-heory an Pracice, pp , Spaniel Publicaions, ew Delhi, Inia, 995.

11 Avances in Decision Sciences [6] C.K.Jaggi,K.K.Aggarwal,anS.K.Goel, Reailer'sopimal orering policy uner wo sage crei financing, Avance Moeling an Opimizaion,vol.9,no.,pp.67 8,27. [7] C.-H. Su, Y.-L. Ouyang, C.-H. Ho, an C.-. Chang, Reailer s invenory policy an supplier s elivery policy uner wolevel rae crei sraegy, Asia-Pacific Operaional Research, vol. 24, no. 5, pp , 27. [8]C.K.Jaggi,S.K.Goyal,anS.K.Goel, Reailer sopimal replenishmen ecisions wih crei-linke eman uner permissible elay in paymens, European Operaional Research,vol.9,no.,pp.3 35,28. [9] A. hangam an R. Uhayakumar, wo-echelon rae crei financing for perishable iems in a supply chain when eman epensonbohsellingpriceancreiperio, Compuers an Inusrial Engineering,vol.57,no.3,pp ,29. [] M. K. Maii, A fuzzy geneic algorihm wih varying populaion size o solve an invenory moel wih crei-linke promoional eman in an imprecise planning horizon, European Operaional Research, vol. 23, no., pp. 96 6, 2. [] C. H. Ho, he opimal inegrae invenory policy wih pricean-crei-linke eman uner wo-level rae crei, Compuers & Inusrial Engineering,vol.6,no.,pp.7 26,2. [2] K. Annaurai an R. Uhayakumar, wo-echelon invenory moel for eerioraing iems wih crei perio epenen eman incluing shorages uner rae crei, Opimizaion Leers,vol.7,no.6,pp ,23. [3] B. C. Giri an. Maii, Supply chain moel wih pricean rae crei-sensiive eman uner wo-level permissible elay in paymens, Inernaional Sysems Science, vol. 44, no. 5, pp , 23. [4].H.Shah,D.B.Shah,anD.G.Pael, Opimalcreiperio an purchase quaniy for crei epenen rene eman, OPSEARCH,24. [5] S. H. Kim an W. R. Feis, Examinaion of he equivalen relaionship beween he wo crei policy ecision approaches: he opporuniy cos an PV approaches, he Financial Review,vol.3,no.4,pp.7 737,995.

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