Economic Order Quantity Model with Two Levels of Delayed Payment and Bad Debt

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1 Research Journal of Applied Sciences, Engineering and echnology 4(16): , 01 ISSN: Maxwell Scientific Organization, 01 Submitted: March 30, 01 Accepted: March 3, 01 Published: August 15, 01 Economic Order Quantity Model with wo Levels of elayed Payment and Bad ebt QIN Juanjuan Business School in ianjin University of Finance and Economics, ianjin, 300, hina Abstract: he purpose of this study is to determine the optimal retailer s replenishment policies considering the customers bad debt and delayed payment in the three-stage supply chain with the dominant retailer. he effect of bad debt is analyzed on the interest earned and interest charged to build the models of the retailer s decision in two cases. By analyzing the model, the retailer s optimal replenishment time and the optimal order quantity are obtained. Furthermore, analyze the effect of parameters on the retailer s optimal order policies. Finally, the numerical analysis is presented to demonstrate the conclusions. he results show that the delayed payment offered by the manufacturer becomes large, the retailer's optimal order cycle and the optimal order quantity increases or remains the same; When the delayed payment time offered by the retailer decreases, the retailer's optimal order cycle and the optimal order quantity increases or remains the same. When the fixed ordering cost is reduced, the retailer's optimal order cycle and the optimal order quantity decreases or remains the same. When the charged interest is greater than the earned interest, with the bad debt rate increasing, the retailer's optimal order cycle and optimal order quantity is converged to a certain value. Keywords: Bad debt, delayed payment, EOQ, the dominant retailer INROUION Nowadays, the retailers have more bargaining power than the manufacturers in some supply chains. For example, in hina maret, SUNING and GOME have almost controlled the home appliances maret; and in the medium-sized cities of the food and commodity marets, most of the maret share is dominated by the Wal-Mart and arrefour. herefore, the strong retailers often required the suppliers to provide a longer extension of the payment period, while the retailers usually provide an extension of pay period to its customers for encouraging the customers purchase behavior. Because the strong position of the retailers in the supply chain, the delayed payment period provided by the retailers is less than the credit period offered by the manufacturer. For the retailers facing the majority of customers for non-specific, there exist the un-receivable accounts that customers will have bad debt losses. he delayed payment has been focused on by many scholars. In the traditional Economic Order Quantity model (EOQ), the manufacturer receives payment in full of the purchase price immediately, without taing into the delayed payment period. EOQ model with the delayed payment is proposed by Goyal (1985), which is extended by hu et al. (1998) for considering the deteriorated products. he situation with the exponential rate of deterioration of the situation deteriorating items is analyzed by Aggarwal and Jaggi (1995). he order strategy is researched by Jamal et al. (1997) and hang and ye (001) allowing the shortage in the case of delayed payment and deteriorating conditions. eng (00) further considered the unit selling price is not the same as the unit product costs in the model. he economic production quantity model is analyzed by hung and Huang (003) considering the manufacturer to the retailer under the deferred payment policy. In addition, some authors too into account cash discounts of the delayed payment, for example, (Huang, 003, 005; hang, 00; Ouyang et al., 00). As the development of research, there are some papers discussing the two levels of delayed payment. Huang (003) first extended Goyal's model to analyze the two levels of the trade credit policy, the manufacturer s credit period available to retailers as M and the retailers provide the customers a credit period N, M>N. Qiu et al. (007) considers the manufacturer to give retailers the delayed payment on purchases, the retailers give the customers a fixed period of delayed payment of the threestage model of economic order quantity. Huang (007) analyzes the optimal inventory updating strategy under the two levels of the delayed payment time, but the delayed payment time which is given to the customers by the retailers is less than the delayed payment time which is given to the retailers by the manufacturer. Also, the retailer requires the customers to pay the deposit as the part of the retailing price to get the products. However, Huang (007) did not consider that when the retailers provide the customers a credit period N, the retailer does not receive the payment during the time 831

2 Res. J. Appl. Sci. Eng. echnol., 4(16): , 01 able 1: Notations : he annual demand rate c : he purchasing cost per unit Ie : he earned interest per year M : he delayed payment time provided by the manufacturer : he replenish cycle I(t) : he inventory level in time t,, 0#t# A : he order cost per order h : he unit holding cost I : he charged interest per year N : he delayed payment time provided by the retailer r : he bad debt rate, 0#r#1 interval [0, ]; the customers purchase the products at time point, which will pay the purchasing price at the time point +N. herefore, the retailer receives payment during the time interval [, +N]. Jinn-sair and hun- ao (009) amended the shortcomings of Huang (007) to relax the constraints I >I e and M>N, discussing the optimal product quantity in the two levels of the delayed payment time. he research on the delayed payment above did not consider the bad debt rate of the customer. When facing the non specific customers, the retailers will have the ris that the accounts receivable can not be recovered. So the retailers solving the economic order quantity, we must consider the losses caused by the bad debts. But the research above in the construction of retailer cost function, does not tae into consideration the bad debts to the function model. Qin (011) have done some research on the delayed payment time with the customer s bad debt, but which is on the situation with the low bad debt considering the retailing price larger than the production cost. But in this study, we extends the research to construct the models which is not relating to the high or low rate of bad debt to obtain the optimal decisions of the players; furthermore, I contrast the results to the EOQ model without the delayed payment and the bad debt. herefore, our study further expands the delayed payment research above, considering the retailer's dominance in the supply chain. here are two levels of the delayed payment time, considering the customers bad debt on interest income and interest payments. Further construct the cost function to obtain the retailers optimal order quantity and the optimal order cycle. he study is much closer to the actual situation and further enriches the academic research of the delayed payment time. MEHOOLOGY Notations and assumption: he notations are used in the study listed in the able 1. Additionally, the mathematical models proposed in this study are based on the following assumptions: he retailer has more power than the manufacturer in the three echo supply chain, therefore, we have M#N. he annual demand rate is nown and constant. he time horizon is infinite. he retailing price is equal to the production ost (Goyal, 1985; hu et al., 1998). In the extended research, we can relax this assumption. he shortages are not allowed because of the timebased competition in today s maret. he manufacturer is offered by the manufacturer a trade credit of M periods to settle the entire purchase cost. On the other hand, the customers are offered by the manufacturer a trade credit of N periods. However, the customers have the credit ris, resulting in the uncollectible account receivables. Uncollectible accounts expense is recorded in the period from N to +N, in which individual accounts receivable are determined to be worthless. he bad debt rate for the individual accounts is r. By the time +N, the manufacturer will request the write-off of the uncollectible accounts. Models: he inventory level I (t) is decreasing to meet the customers demand. If the customers demand rate is, the inventory level over time is: di() t dt, 0 t (1) And meeting the initial conditions I (t) 0, then we can obtain the solution of Eq. (1): I(t)!t +, 0#t# () he purchasing quantity of the retailer is: Q I(0) (3) he annual cost associated with the retailer is R() he ordering cost per year + the purchasing cost per year + the holding cost per year + the interest charged per year-the earned interest per year. In which: he ordering cost per year equals A/; the purchasing cost per year equals c; the holding cost per year (not including the interest) H is as follows: H h Itdt h () 0 he retailer's annual charged interest and annual earned interest are related to its action strategy. herefore, we discuss the problem in the following two cases. case 1: N#M#+N he interest charged per year: Assuming an unlimited supplying rate of the product, the inventory level is shown in Fig. 1a. Without considering the impact of bad debts, the 83

3 Res. J. Appl. Sci. Eng. echnol., 4(16): , 01 At time point M, the retailer needs financing to pay for not un-receivable money. After time point M, the retailer repays the financing money with the sales revenue, hence, the interest expense for the unit cycle Y is: ISA A ci Y N M ( + ) [( r + 1) ( 1 r)( M N) ] (4) If not considering bad debt, it means that r 0. (a) inventory level he interest earned per year: he retailer start to selling the products at the time point 0 and begins to receive the receivable money from its customers at time point N. At time point M, the retailer pays money to the manufacturer for covering the purchase cost. herefore, the retailers can invest the receiving money during the time interval [N, M] with the earned interest I e. Under the influence of bad debts, the earned interest per year is: (b) he accumulated sales revenue with no bad debt IS e MN cie( 1r)( M N) Ye when not consider the bad debt rate, it is r 0. (5) he total annual cost: Under the influence of bad debt, the retailer s total cost is: (c) he accumulated sales revenue with bad debt Fig. 1: he inventory level and the accumulated sales revenue with N < M # + N the retailer's total sales revenue is presented in Fig. 1b. aing into account the customer's bad debt losses, the retailer's total sales revenue are shown in Fig. 1c. he retailer s receivable accounts, after delayed payment time N providing to the customers, exists a certain bad debt rate r. ase : N < + N#M: In case, assuming the unlimited products supplying rate, the inventory level is shown in Fig. a. Without considering the impact of bad debts, the retailer's total sales revenue is presented in Fig. b. aing into account the customer's bad debt losses, the retailer's total sales revenue are shown in Fig. c. he retailer s receivable accounts, after delayed payment time N providing to the customers, exists a certain bad debt rate r. By the assumptions we can see that at time +N, the retailer has written off the bad debt losses of this cycle. At the time point M, the retailer pays the amount of financing with its own funds, its interest with zero. At time point 0, the retailer begins to sell the products; recover the receivable from the customers at time point N; pay the purchase cost to the 833

4 Res. J. Appl. Sci. Eng. echnol., 4(16): , 01 IS e A B MN ci e ( 1r)( M N ) Ye (7) he annual total cost is: A h ci e ( 1 r)( M N ) R ( ) + c + (8) MOEL ANALYSIS (a) inventory level ase: 1 N < M#+N: o obtain the optimal solution, tae the first order and the second order derivatives of the formula (6) with respect to : R1 ( ) A h ci [( 1 + r ) ci e( 1 r)( M N ) ( M N) ( 1 r) ] + R1 ( ) [ A c( M N) ( r)( I Ie) ] (b) he accumulated sales revenue with no bad debt If A+cd(M!N)(1!r)(I!I e ) > 0, we can obtain R1 ( ) 0. It is clearly that R1() is strictly > convex function in. herefore, we can obtain the R optimal order cycle * 11 based on ( ) 1 0. We can obtain: * A+ c( M N) ( 1 r)( I Ie) 11 h + ci ( 1+ r) (9) he corresponding optimal order quantity Q* 11 is: A c M N r I Ie Q11 * * + ( ) ( 1 )( ) 11 h+ ci ( 1+ r) (10) But in case 1, the retailer s optimal order cycle should meet the condition that M#*11+N, therefore, we can get the inequality as following: (c) he accumulated sales revenue with bad debt Fig. : he inventory level and the accumulated sales revenue with N<M#+N manufacturer at the time point M. herefore, theretailers can obtain the interest income at the time interval [N, M] with the earned interest Ie. With the bad debt, the earned interest per year is: 1 A+ c( M N) ( 1 r)( I Ie) h + ci ( + r) ( M N) A c( M N) [ ri + ( 1 r) Ie] h( M N) 0 [ 1 ] (11) If A+c(M!N) (1!r)(I!I e )#0, we can get ) 1 < 0 based on the formula (11). In case 1N#M#+N, we can get: 834

5 Res. J. Appl. Sci. Eng. echnol., 4(16): , 01 R1 ( ) A h ci [( 1 + r ) ci e( 1 r)( M N ) ( M N) ( 1 r) ] + R ( ) optimal order cycle based on 0 : * A h + cie( 1r) (1) ( M + N) > 1 + [( ) h ci 1 + r ci e( 1 r)( M N ) + 0 ( M N) ( 1 r) ] herefore, in case 1, when: A+c(M!N)(1!r) (I!I e )#0, the function R 1 () is increasing with. We can obtain that the optimal order cycle is * 1 M!N and the optimal order quantity is Q* 1 (M!N). Based on the analysis (1) and (), we can get the heorem 1. heorem 1: in case 1 with N#M#+N, he retailer s optimal order quantity is: * * Q A h+ cie( 1 r) (13) And the retailer s annual minimized cost is R(* ). But in case, the retailer optimal order cycle should meet the condition * +N#M, so we can get the following inequality: If ) 1 $0 then: )!A!(h + ci e (1!r))(M!N) #0 (14) * * 1 11 A+ c( M N) ( 1 r)( I Ie) h + ci ( 1 + r) Also, ) 1 - ) ci (M!N) (1!r)!cI (1+r)(M!N) # 0 (15) and and Q Q Q * * * If ) 1 <0, then: *1 *1-M-N Q* 1 Q* 1 (M!N) Proof: From the analysis above, it is easy to obtain the heorem 1. Based on which, we can get the retailer s annual minimized cost R1(*1). ase : N < + N#M: o obtain the optimal solution, tae the first order and the second order derivatives of the formula (8) with respect to, which is: R ( ) A h ci e( 1 r) Analysis of case 1 and case : Based on the analysis of case 1 and case, we can get the theorem : heorem : when M# N, If ) 1 $ 0, the retailer s optimal order cycle is * * 11; the optimal order quantity is: Proof: Q* Q* 11 * 11 If ) 1 < 0 and ) # 0, the retailer s optimal order cycle is * * ; the optimal order quantity is: Q* Q* * ; If ) 1 < 0 and ) > 0, the retailer s optimal order cycle is * M!N; the optimal order quantity is: Q * (M!N) R ( ) A 3 > 0 he function R () has the minimized value for R ( ) > 0. herefore, we can obtain the retailer If ) 1 $ 0, from formula (15), we can get: ) A!(h + cl e (1!r))(M!N) $0. In case, N#+N#M, for ) $0, we can obtain that: 835

6 Res. J. Appl. Sci. Eng. echnol., 4(16): , 01 R ( ) A h ci e( 1 r) ( M N) h ci e( 1 r) herefore, in case, if ) $0, the function R () is decreasing with. In case 1 N#+N#M for ) 1 $0, based on formula (11), we can obtain the optimal value of R1() is * 11. With ase 1 and ase analysis, when is satisfied N#+N#M, we can get: R ()$R (M!N) R 1 (M-N)$R 1 (*11). herefore, the heorem (1) has been proved. order quantity are increasing with M or unchanged; when the value of N is gradually decreases, the retailer's optimal order cycle and the optimal order quantity become larger or unchanged; when the value of A is decreasing, the retailer optimal order cycle and the order quantity is decreasing or unchanged; when I $I e, with the value of r increasing, the retailer optimal order cycle is increasing or decreasing to (M-N) and eep unchanged and the optimal order quantity is increasing or decreasing to (M- N) and eep unchanged. In the traditional EOQ model, not considering the delayed payment and the bad debt, the retailer and the customer both pay the money instantly and r 0. herefore, the traditional EOQ model is the special situation of case 1 with M N r 0. So it means Q* 0 A/ h + ci If ) 1 < 0, in case 1 N# M # +N, we can get that: R1 ( ) A h ci [( 1 + r ) ci e( 1 r)( M N) ( M N) ( 1 r) ] + ( M + N) > 1 [( ) h + ci 1 + r ci ( 1 ) e r ( M N ) ( M N) ( 1 r) ] + 0 heorem 3: when M$N and A+c(M!N) (1!r)(I!I e ) > 0 If I e (1!r) < I, then Q * > Q * 0 If I e (1!r) > I, then Q * < Q * 0 If I e (1!r) I, then Q * Q * 0 If I e $I, then Q * 11#Q * 0 Proof: When I e (1!r) < I : herefore, in case 1, the function R 1 () is decreasing with. In cased, for ) #0, from formula (14), we can get * the optimal solution of R (). Based on the analysis of case 1 and case, we can now that when meets N#M#+N, then: R 1 ()$R 1 (M!N) R (M!N)$R (* ) herefore, the theorem 1() is proved. If) 1 < 0 and ) #0 the retailer s optimal order cycle is * * and the optimal order quantity is Q* *. heorem 1(1) and () shows that in case 1 with N#M#+N, if ) 1 < 0, the function R 1 () is increasing with ; in case, N#+N#M, if ) > 0, R () is decreasing with the. herefore, when ) 1 < 0 and ) > 0, the retailer s optimal order cycle is * M!N and the optimal order quantity is Q* (M!N). From formula (11) and (14), we can get that with the credit period M increasing, the values of ) 1 and ) are decreased. Based on the heorem 1 and heorem, it shows that the retailer's optimal order cycle and optimal Q A > h + ci ( 1 r) A h + ci Q * * * 0 e Similarly, we can prove the () and (3). When A+c(M!N) (1!r)(I!I e )>0, then we get that: Q * 11 A + ci ( M N ) ( 1 r)( I I e) h + ci ( 1 + r) A + c ( M N ) ( 1 r)( I Ie) A * Q 0 h+ ci h+ ci heorem 3 shows that when the earned interest is higher than the charged interest, the retailers will tae advantage of the upstream extension of the payment period; the economic order quantity is less than the traditional economic order quantity. 836

7 Res. J. Appl. Sci. Eng. echnol., 4(16): , 01 NUMERIAL ANALYSIS o illustrate the feasibility of the models and theorems, we illustrate with the numerical analysis following Assume the retailer maes its decision about the order strategy of the products: A 00 dollars per time, 1500 units per year, c 5 dollars per year, h 15 dollars per year, I e 0.09, I 0.05, M 0/360 per year, N 15/360 per year, r alculate to obtain ) 1 > 0 and ) > 0 * , * the retailer minimixed cost is 40781; the optimal order cycle is * 0.119; the optimal order quantity. Q * Now, with the adjustment of model parameters M, N, A, the retailer's optimal decisions are shown in able, 3, 4 and 5. hrough the numerical analysis, it is easier to understand the influence of the parameters on the retailer's optimal order cycle and the optimal order quantities. able : he retailer s optimal decisions with the adjustment of parameter M he optimal he optimal order Minimized M order cycle quantity cost 35/ / / / able 3: he retailer s optimal decisions with the adjustment of parameter N he optimal he optimal Minimized N order cycle order quantity cost 15/ / / able 4: he retailer s optimal decisions with the adjustment of parameter A he optimal he optimal Minimized A order cycle order quantity cost able 5: he retailer s optimal decisions with the adjustment of parameter r he optimal he optimal Minimized r order cycle order quantity cost ONLUSION In the retailer dominant supply chain with two levels of delayed payment, the optimal decisions of the retailer are obtained considering the bad debt. he retailer's cost function is established in two cases. By calculating the functions, obtain the optimal order cycle and the optimal order quantity. he study found that when the delayed payment period offered by the manufacturer M becomes large, the retailer's optimal order cycle and the optimal order quantity increases or remains the same; when the retailer giving to the customers extended payment period N decreasing, the retailer's optimal order cycle and the optimal order quantity increases or remains the same. When the fixed ordering cost A is reduced, the retailer's optimal order cycle and the optimal order quantity decreases or remains the same. When the charged interest is greater than the earned interest, with the bad debt rate increasing, the retailer's optimal order cycle and optimal order quantity is converged, respectively, in (M-N) and (M-N). herefore, the results not only have a certain theoretical significance, but also can better guide decision-maing for the retailer. AKNOWLEGMEN he study is funded by ianjin ity High School Science and echnology Fund Planning Project (01017), Research evelopment Funding Project of ianjin University of Finance & Economics (Y1109) and National Natural and Science Funding ( , ). REFERENES Aggarwal, S.P. and.k. Jaggi, Ordering policies of deteriorating items under permissible delay in payments. J. Oper. Res. Soc., 46: hang,.., 00. Extended economic order quantity model under cash discount and payment delay. Int. J. Inform. Manage. Sci., 13(3): hang, H.J. and.y. ye, 001. An inventory model for deteriorating items with partial baclogging and permissible delay in payments. Int. J. Syst. Sci., 3: hu, P., K.H. hung and S.P. Lan, Economic order quantity of deteriorating items under permissible delay in payments. omput. Oper. Res., 5(10): hung, K.J. and Y.F. Huang, 003. he optimal cycle time for EPQ inventory model under permissible delay in payments. Int. J. Prod. Econ., 84:

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