370 European J. Industrial Engineering, Vol. 7, No. 3, 013 Retailer s optimal order and credit policies when a supplier offers either a cash discount or a delay payment linked to order quantity Chih-e Yang Department of Industrial Management, Ching Yun University, Jung-Li, aoyuan 30, aiwan E-mail: ctyang@cyu.edu.tw Qinhua Pan Department of Economies and Finance, School of Economics and Management, ongji University, Shanghai, 0009, China E-mail: panqh490@tongji.edu.cn Corresponding author Liang-Yuh Ouyang Department of Management Sciences, amkang University, amsui, aipei 51, aiwan, E-mail: liangyuh@mail.tku.edu.tw Jinn-sair eng Department of Marketing and Management Sciences, Cotsakos College of Business, he William Paterson University of New Jersey, Wayne, New Jersey 07470, USA E-mail: engj@wpunj.edu Abstract: o increase sales and reduce default risks, a supplier may offer its retailers either: 1 a cash discount a fixed credit period M if the order quantity is greater than or equal to a predetermined quantity W. Likewise, a retailer in turn offers its customers a credit period N, which has a positive impact on its demand but a negative impact on its default risks. In this paper, we establish an inventory model for a retailer in a supply chain when a supplier offers either a cash discount or a delay payment linked to order quantity; meanwhile it offers its customers a permissible delay in payments. Copyright 013 Inderscience Enterprises Ltd.
Retailer s optimal order and credit policies 371 hen, we derive several theoretical results to determine the optimal solution under various situations and develop an algorithm to solve this complex inventory problem. Finally, several numerical examples are given to illustrate the theoretical results and provide some managerial insights. [Received 9 April 011; Revised 14 August 011; Accepted 1 October 011] Keywords: inventory; finance; default risks; cash discount; delay payment linked to order quantity. Reference to this paper should be made as follows: Yang, C-., Pan, Q., Ouyang, L-Y. and eng, J-. (013) Retailer s optimal order and credit policies when a supplier offers either a cash discount or a delay payment linked to order quantity, European J. Industrial Engineering, Vol. 7, No. 3, pp.370 39. Biographical notes: Chih-e Yang is an Assistant Professor in the Department of Industrial Management at Ching Yun University in aiwan. He earned his PhD from the Graduate Institute of Management Sciences at amkang University in aiwan. His research interests are in the field of production/inventory control, and supply chain management. He has published articles in International Journal of Production Economics, Computers and Industrial Engineering, International Journal of Information and Management Sciences, Asia-Pacific Journal of Operational Research, European Journal of Operational Research, OP, Central European Journal of Operations Research and so on. Qinhua Pan received her Master in Business Administration in 1997 at La robe University in Australia, and PhD in Management of Science and Engineering in 011 at ongji University in China. She is an Associate Professor at School of Economics and Management in ongji University. Her research interests include financial management, industry economics, and supply chain management. She has published research articles in Omega, and other Chinese journals. Liang-Yuh Ouyang is a Professor in the Department of Management Sciences at amkang University in aiwan. He earned his MS in Mathematics and PhD in Management Sciences from amkang University. His research interests are in the field of production/inventory control, probability and statistics. He has publications in Journal of the Operational Research Society, Computers and Operations Research, European Journal of Operational Research, Computers and Industrial Engineering, International Journal of Production Economics, IEEE ransactions on Reliability, Production Planning and Control, International Journal of Systems Science, Mathematical and Computer Modelling, Applied Mathematical Modelling, Applied Mathematical and Computation, and Journal of Global Optimisation. Jinn-sair eng received his MS in Applied Mathematics from National sing Hua University in aiwan, and PhD in Industrial Administration from Carnegie Mellon University in USA. He joined the Department of Marketing and Management Sciences at Cotsakos College of Business in William Peterson University of New Jersey in 199. His research interests include supply chain management and marketing research. He has published research articles in Management Sciences, Marketing Science, Omega, Computers and Operations Research, European Journal of Operational Research, International Journal of Production Economics, Journal of the Operational Research Society, European Journal of Industrial Engineering, and others.
37 C-. Yang et al. 1 Introduction In practice, the supplier usually allows the retailer a fixed period M for settling the account (i.e., an up-stream trade credit). Likewise, the retailer in turn offers its customers a credit period N (i.e., a down-stream trade credit). o the seller, the permissible delay in payments produces two benefits: 1 it attracts new buyers who consider it to be a type of price reduction it may be applied as an alternative to price discount because it does not provoke competitors to reduce their prices and thus introduce lasting price reductions. On the other hand, the policy of granting trade credit adds not only an additional cost but also an additional dimension of default risk (i.e., the event in which the buyer will be unable to payoff its debt obligations). ime changes everything and creates uncertainty. Hence, the longer the trade credit period offered by the seller, the higher the default risk will be. For example, a 30-year mortgage has a higher default risk than a 15-year mortgage. o mitigate the impact of default risk, the lenders charge a higher 30-year mortgage rate than a 15-year mortgage rate. As a result, the retailer s down-stream trade credit has a positive impact on its customers demand but a negative impact on its default risks. In today s global competition, to attract more buyers the retailer is exposed to the default risk in all forms of trade credits such as allowing its customers to pay by instalments or credit cards. In addition, the triangular debt defaults (i.e., debt default among inter-enterprises) in the construction industry in China were totalled 1,500 billion Yuan in 006 (e.g., see Pan and Zhang, 009, 010). Furthermore, one of the main reasons for the financial meltdown in 008 was because the banks did not pay attention to their mortgagees default risks. herefore, the default risk is an important and relevant issue for the retailer to increase its sales by offering its trade credit financing. In this paper, we extend the previous EOQ models with trade credit financing to reflect the real-life situations by incorporating the following concepts: 1 the supplier offers the retailer an up-stream trade credit which is either a cash discount or a permissible delay linked to order quantity the retailer provides his/her customers a down-stream permissible delay which has a positive impact on its demand but a negative impact on its default risks. hen we derive several theoretical results to determine the optimal credit period and order quantity for the retailer under various situations. Furthermore, we develop an algorithm to obtain the optimal solution to this complex inventory problem. Finally, several numerical examples are given to illustrate the theoretical results and provide the managerial insights. he remainder of this paper is organised as follows. We briefly survey the related inventory literature with trade credit financing in Section. In Section 3, we define the assumptions and notation, which are used throughout the entire paper. hen we establish the mathematical model to maximise the annual total profit for the retailer in Section 4. In Section 5, we develop some important theorems to characterise the optimal solutions. We then provide a simple algorithm to find the optimal replenishment cycle time and order quantity in Section 6. Several numerical examples are provided in Section 7 to illustrate the theoretical results and managerial insights. Finally, the conclusions and suggestions for the future research are given in Section 8.
Retailer s optimal order and credit policies 373 Literature review At the earliest, Goyal (1985) developed an EOQ model under an up-stream trade credit. Aggarwal and Jaggi (1995) extended Goyal s (1985) model to consider the deteriorating items. Jamal et al. (1997) further generalised Aggarwal and Jaggi s (1995) model to allow for shortages. eng (00) amended Goyal s (1985) model by considering the difference between unit price and unit cost, and found that it makes economic sense for a well-established retailer to order less quantity and take the benefits of payment delay more frequently. Chang et al. (003) developed an EOQ model for deteriorating items under supplier credits linked to ordering quantity. Huang (003) extended Goyal s (1985) model to develop an EOQ model with up-stream and down-stream trade credits in which the length of down-stream trade credit period is less than or equal to the length of the up-stream trade credit period. eng and Goyal (007) complemented the shortcoming of Huang s (003) model and proposed a generalised formulation. Other interesting and relevant papers related to the delay in payments such as Hwang and Shinn (1997), Chang and Dye (001), Ouyang et al. (005), eng et al. (005, 007, 01), Goyal et al. (007), Mohan et al. (008), eng and Chang (009) and others. o avoid default risks, the supplier frequently offers its retailers a cash discount. Chang (00) amended Goyal s (1985) model to the case in which the supplier offers a cash discount. Ouyang et al. (005) extended Chang s (00) model to consider the deteriorating items. Furthermore, Ouyang et al. (007) established an EOQ model with limited storage capacity where the supplier provides cash discount and permissible delay in payments for the retailer. Recently, Yang (010) uses an alternate approach-discount cash flow (DCF) analysis to establish an inventory model for deteriorating items in which the supplier provides the retailer either a conditionally permissible delay or a cash discount. Many related articles can be found in eng (006), Chang et al. (010) and their references. All articles described above are only from the perspective of the buyer whereas in practice the length of the credit period is set by the seller. So far, how to determine the optimal length of the credit period for the seller has received a very little attention by the researchers. Abad and Jaggi (003) determined the seller s and the buyer s policies under non-cooperative as well as cooperative relationships. However, their model did not reflect any benefit for the seller to offer a trade credit to the buyer. Lately, Jaggi et al. (008) developed the optimal credit as well as replenishment policy jointly for the seller when credit period has a positive impact on demand. However, they did not consider the fact that the longer the credit period, the higher the default risk. 3 Notation and assumptions he following notation is used throughout this paper: A he retailer s ordering cost per order. c he retailer s purchasing cost per unit. p he retailer s selling price per unit, with p > c. h he retailer s unit holding cost per year excluding interest charge.
374 C-. Yang et al. I e he retailer s interest earned per dollar per year. I c he retailer s interest charged per dollar per year. M he up-stream trade credit period in years offered by the supplier. r he cash discount rate offered by the supplier, 0 < r < 1. W he minimum order quantity at which the up-stream trade credit is permitted. N he down-stream trade credit period in years offered by the retailer (a decision variable). D(N) he annual market demand which is a strictly increasing function of N. he retailer s replenishment cycle time in years (a decision variable). W he time period that W units are depleted to zero due to consumption demand rate, i.e., W = W / D(N). Q he retailer s order quantity = D(N). P(N, ) he retailer s total profit per year, which is a function of N and. N he retailer s optimal credit period to its customers. he retailer s optimal replenishment cycle time. Q he retailer s optimal order quantity = D (N ). In addition, the following assumptions are used throughout this paper: 1 In today s global competition, many retailers have no pricing power. As a result, the selling price is hardly changed for many retailers. herefore, we may assume without loss of generality that the selling price is constant within a year. he demand rate D(N) depends on the down-stream credit period N and is given by D(N) = Ke, where K and a are positive constants and a 1. 3 he retailer has two choices: either to take a permissible delay linked to order quantity a cash discount. If the retailer s order quantity is greater than or equal to W, then the delayed payment is permitted and the retailer must payoff the outstanding balance by time M. On the other hand, if the retailer wants to enjoy the giveback of cash discount, then the outstanding balance must be paid immediately. 4 During the credit period M, sales revenue is deposited in an interest bearing account. At the end of the permissible delay M, the retailer pays off all units sold, and starts paying for the interest charges on the items in stocks. 5 o reflect the fact that the longer the credit period the higher the default risk, we assume that the rate of not receiving debt obligations giving the credit period N is F(N) = 1 e bn, where b is a positive constant and 0 < b 1. Likewise, the reader can obtain more general results by assuming the rate of default risk is a function of N without a specific form. 6 Replenishment is instantaneous and shortages are not allowed.
Retailer s optimal order and credit policies 375 4 Model formulation In this section, we first present the EOQ model in which the retailer takes a delay in payments linked to order quantity and then discuss the other situation in which the retailer takes a cash discount. 4.1 aking a permissible delay linked to order quantity By Assumption 3, we know that the retailer takes either: 1 the permissible delay by ordering more than or equal to W the cash discount by paying the discount price immediately. If the retailer takes the permissible delay, then the retailer s order quantity Q W (i.e., W = W / D(N)). Based on the values of, W, M and N, we know the replenishment cycle has the following three alternative cases: 1 W / D(N) and N M + N W / D(N) and + N M 3 W / D(N) and N M. Let us discuss them accordingly. Case 1 W / D(N) and N M + N In this case, the retailer pays off all units sold by M N at time M, keeps the profits, and starts paying for the interest charges on the items sold after M N, which is shown in Figure 1. Consequently, the retailer s annual total profit consists of the following elements: a he selling revenue per year (denoted by SR) is the product of the unit selling price, the units sold, and the rate of receiving debt obligations as shown below: [ 1 F( N) ] SR = pd( N) = pd( N) e (1) b C d Purchasing cost per year (denoted by PC) is PC = cd( N). () he ordering cost per year (denoted by OC) is OC = A /. (3) he stock holding cost per year (denoted by HC) is given by HC = hd( N) /. (4) e Interest earned and charged. In this case, the retailer can not payoff the supplier by M because the supplier credit period M is shorter than the customer last payment time + N. As a result, the interest charged per year is given by
376 C-. Yang et al. cicd( N) ( ) + N M. (5) On the other hand, the retailer starts selling products at time 0, but getting the money at time N. Consequently, the retailer accumulates revenue in an account that earns I e per dollar per year starting from N through M. hus, the interest earned per year is pied( N) ( M ) N. (6) herefore, the retailer s annual total profit for Case 1 (denoted by P 1 (N, )) as follow: A hd( N) cicd( N) P1 ( N, ) = pd( N) e cd( N) ( + N M ) pied( N) + M N (7) bn ( h cic ) ( cic pie) ( M N) A Ke = + pe c + ci. c M N Figure 1 W / D(N) and N M + N Case W / D(N) and + N M In this case, the retailer receives the total revenue at time + N and is able to pay the supplier the total purchase cost at time M. Hence, there is no interest charge. he graphical representation of this case is shown in Figure. As a result, the annual interest earned is pied( N) pied( N) + pied( N)( M N) = pied( N)( M N). (8) herefore, the retailer s annual total profit for Case (denoted by P (N, )) as follow: ( + ) A h pie D( N) P ( N, ) = pd( N) e cd( N) + pied( N)( M N) ( h+ pie ) A = Ke pe c + pie ( M N). (9)
Retailer s optimal order and credit policies 377 Figure W / D(N) and + N M Case 3 W / D(N) and N M Since the down-stream trade credit N is greater than or equal to the up-stream trade credit M, there is no interest earned for the retailer. In addition, the retailer must finance all items ordered at time M at an interest charged I c per dollar per year, and start to payoff the loan after time N as shown in Figure 3. Hence, the interest charged per year is given by ci D( N) c [ ( N M) ] +. (10) herefore, the retailer s annual total profit for Case 3 (denoted by P 3 (N, )) is A hd( N) cicd( N) P3 ( N, ) = pd( N) e cd( N) ( N M) + h cic [ ( N M) + ] A = Ke pe c [ ] (11) Figure 3 W / D(N) and N M
378 C-. Yang et al. 4. aking a cash discount In this situation, the retailer takes a cash discount by paying the supplier the full payment amount of (1 r)cq at the time 0. Hence, there is no interest earned and the interest charged per year is (1 r)ci c D(N)(N + ) / as shown in Figure 4. herefore, the retailer s annual total profit (denoted by P 4 (N, )) is A hd( N) (1 rcidn ) c P4 ( N, ) = pd( N) e (1 r) cd( N) ( N + ) (1) h (1 rci ) c ( N+ ) A = Ke pe (1 r) c. Figure 4 Graphical representation when taking a cash discount 5 heoretical results Now, we shall determine the retailer s optimal credit period and replenishment time to maximise its annual total profit for each case. 5.1 aking a permissible delay linked to order quantity Case 1 W / D(N) and N M + N aking the first-order derivatives of P 1 (N, ) with respect to N and, setting the results to be zero, we get and bn a h+ cic ( a b) pe ac + acic ( M N) a( cic pie) ( M N) ( cic pie) ( M N) cic + = 0, ( h+ ci ) Ke ( ci pi ) Ke ( M N) A c c e + = 0. (14) (13)
Retailer s optimal order and credit policies 379 It seems to be intractable to find a closed-form solution for N due to the complexity of (13). However, for any fixed N, it is clear from (14) that the unique closed-form solution of (denoted by 1 ) is given as 1 = ( c e) ( + ) A+ ci pi Ke ( M N). h ci Ke c (15) o ensure 1 W / D(N) (i.e., 1 W / (Ke )) and 1 + N M (i.e., 1 M N), we substitute equation (15) into these two inequalities and have and {( h+ ci ) ( ) c W Ke + pie cic M N } A Ke Δ1( N), A h+ pi Ke ( M N) Δ ( N). (17) e It is obvious that Δ 1 (N) Δ (N) = (h + ci c )Ke {[W / (Ke )] (M N) } which implies that Δ 1 (N) Δ (N) if and only if W (M N)Ke. Consequently, we have the following result. (16) heorem 1: For any given N, the optimal value of (denoted by the retailer s annual total profit P 1 (N, ) in (7) is as follows: 1 ) which maximises a When W (M N)Ke, if A Δ 1 (N), then 1 1 ; = otherwise, 1 = W /( Ke ). b When W < (M N)Ke, if A Δ (N), then Proof: See Appendix A. 1 1 ; = otherwise, 1 = M N. Next, for any given 1, we study the condition under which the optimal down-stream trade credit period (denoted by N 1 ) also exists and is unique. Substituting = 1 into (13), and then we let bn a h+ cic F1 ( N) = ( a b) pe ac + acic ( M N) a( ci pi )( M N) ( ci pi )( M N) +. For any given 1 c e c e ci c 1 1 1, taking the first-order derivative of F 1(N) with respect to N, we obtain ( ci pi )[ am ( N) 1] df1 ( N) c e = ( a b) bpe acic + dn 1 aci bn c 1 ( M N) pi [ am ( N) 1] ci = ( a b) bpe e c 1 1 1 It is obvious that if a(m N) 1 (i.e., N M 1 / a), then df 1 (N) / dn < 0. hus, F 1 (N) is a strictly decreasing function in N [N L, M 1 / a], where (18) (19)
380 C-. Yang et al. { ( K ) 1 1 } NL = Max ln W / a, M. Furthermore, from (18) we have and bn 1 L a h+ cic F NL = a b pe ac + acic M N a ci pi ci pi + ( )( M N ) ( )( M N ) 1 c e L c e ci c 1 1 ( 1/ ) c 1 c e 1 = + a1 bm a a h+ ci ci pi F ( M 1/ a) ( a b) pe ac. It is obvious that F 1 (N L ) F 1 (M 1 / a) and then we have the following results. Lemma 1: For any given a b 1, If F 1 (N L ) 0 F 1 (M 1 / a), then the solution of N (denoted by N 1 ) which satisfies (13) not only exists but is unique. Otherwise, the solution of N which satisfies (13) does not exists. Proof: See Appendix B. heorem : For any given 1, L L, a b c 1 1 If F 1 (N L ) 0 F 1 (M 1 / a), then the retailer s annual total profit P ( N, ) is concave and reaches its global maximum at the point N (13). 1 N 1, = where N 1 satisfies 1 1 If F 1 (N L ) F 1 (M 1 / a) > 0, then the retailer s annual total profit P ( N, ) reaches its global maximum at the point N1 = M 1/ a. 1 1 If 0 > F 1 (N L ) F 1 (M 1 / a), then the retailer s annual total profit P ( N, ) reaches its global maximum at the point 1 =. N N L Proof: See Appendix C. Case W / D(N) and + N M he necessary conditions for the total profit per year in (9) to be maximised are P ( N, ) P = 0 and ( N, ) = 0, simultaneously. hat is, N bn a h+ pie ( a b) pe ac + apie( M N) pie = 0. (0)
and Retailer s optimal order and credit policies 381 ( + ) A h pie Ke = 0. (1) Similar to the approach used in Case 1, for any fixed N, we let Δ 3 (N) (h + pi e )W / (Ke ), and then have the optimal value of (denoted by ) that maximises the retailer s annual total profit P (N, ) in (9) is M N, A>Δ( N), =, Δ( N) A Δ3( N), W ( Ke ), A<Δ3( N), where = ( + ) A/ h pie Ke. Next, substituting = into (0), and then we let bn a h+ pie F N = a b pe ac + apie ( M N ) pie, () where N [ln[ W / ( K )] / a, M ]. aking the first-order derivative of F (N) with respect to N (ln[ W / ( K )] / a, M ), we obtain df ( N ) = ( a b) bpe apie < 0. dn hus, F (N) is a strictly decreasing function in Furthermore, from (), we have N [ln[ W / ( K )] / a, M ]. and 1 W ( a b) pw ( a+ b) a h+ pie F ln = e ac a K K ( M ) 1 W + apie M ln pi e, a K b M a h pi e = ( ). e F a b pe ac pi It is obvious that results. F (ln[ W / ( K )] / a) > F ( M ), and then we have the following Lemma : For any given, if F(ln[ W / ( K )] / a) 0 F( M ), then the solution of N (denoted by N ) which satisfies (0) not only exists but is unique. Otherwise, the solution of N which satisfies (0) does not exists.
38 C-. Yang et al. Proof: Using the similar analogous as in the proof of Lemma 1, we can easily prove Lemma. hus, the proof is omitted. heorem 3: For any given, a b if F W K a F M (, ) (ln[ / ] / ) 0 ( ), then the retailer s annual total profit P N is concave and reaches its global maximum at the point where N satisfies (0) if F W K a F M (, ) N = N, (ln[ / ] / ) > ( ) > 0, then the retailer s annual total profit P N reaches its global maximum at the point = N M c (, ) if 0 > F (ln[ W /( K )]/ a) > F ( M ), then the retailer s annual total profit P N reaches its global maximum at the point = N ln[ W / ( K )] / a. Proof: Using the similar analogous as in the proof of heorem, we can prove heorem 3. Hence, the proof is omitted. Case 3 W / D(N) and N M he necessary conditions for the retailer s annual total profit in (11) to be maximised are P3 ( N, ) P3 = 0 and ( N, ) = 0, simultaneously. hat is, N and [ ] ah acic N M + ( a b) pe ac cic = 0, (3) ( + ) A h cic Ke = 0. (4) Similar to the approach used in Case 1, for fixed N, let Δ 4 (N) (h + ci c )W / (Ke ), we have the optimal value of (denoted by 3 ) that maximises the retailer s annual total profit is 3 3, A Δ4( N), = ( W Ke ), A<Δ4( N), where = ( + ) 3 A h cic Ke. Next, substituting = 3 into (3), and then we let bn ah aci 3 c ( N M) + 3 F3 ( N) = ( a b) pe ac cic, (5) where N [N U, ) and NU = Max{ln[ W / ( K3 )] / a, M}. aking the first-order derivative of F 3 (N) with respect to N (N U, ), we obtain
Retailer s optimal order and credit policies 383 df3 ( N ) = ( a b) bpe acic < 0. dn hus, F 3 (N) is a strictly decreasing function in N [N U, ). Furthermore, from (5), we have ( ) + bn ah aci 3 c NU M 3 U F3 ( NU ) = ( a b) pe ac cic, and lim F3 ( N) = < 0. hen we have the following results. N Lemma 3: For any given 3, if F 3(N U ) 0, then the solution of N (denoted by N 3 ) which satisfies (3) not only exists but is unique. Otherwise, the solution of N which satisfies (3) does not exists. Proof: Similar to the proof of Lemma 1, we can prove Lemma 3 easily. So the proof is omitted. heorem 4: For any given 3, a 3 3 if F 3 (N U ) 0, then the retailer s annual total profit P ( N, ) is concave and reaches its global maximum at the point N satisfies (3) 3 N 3, = where N 3 is the point which b if F 3 (N U ) < 0, then the retailer s annual total profit P3( N, 3 ) reaches its global maximum at the point N3 = N U. Proof: One can prove heorem 4 easily by following the same analogous in the proof of heorem. Hence, the proof is omitted. 5. aking a cash discount he necessary conditions for the retailer s annual total profit in (1) to be maximised are P4 ( N, ) P4 = 0 and ( N, ) = 0, simultaneously. hat is, N and ah (1 raci ) c ( N+ ) ( a b) pe (1 r) ac (1 r) cic = 0, (6) [ + (1 ) ] A h r cic Ke = 0. (7) Similar to the approach used above, for fixed N, we have the optimal value of (denoted by that maximises the retailer s annual total profit is 4 ) {[ (1 ) c ] } 4 = A/ h+ r ci Ke.
384 C-. Yang et al. Next, substituting = 4 into (6), and then we let (1 ) bn ah raci 4 c N + 4 F4 ( N) = ( a b) pe (1 r) ac (1 r) cic, (8) where N [0, ). aking the first-order derivative of F 4 (N) with respect to N (0, ), we obtain df4 ( N ) = ( a b) bpe (1 r) acic < 0. dn hus, F 4 (N) is a strictly decreasing function in N [0, ). From (8), we have F (0) = ( a bp ) (1 rac ) { ah [ + (1 rci ) ] }/ (1 rci ), and lim F4 ( N) = < 0. 4 c 4 Consequently, we have the following results. c N Lemma 4: For any given 4, if F 4(0) 0, then the solution of N (denoted by N 4 ) which satisfies (6) not only exists but is unique. Otherwise, the solution of N which satisfies (6) does not exists. Proof: he proof is similar to that in Lemma 1. Hence, the proof is omitted. heorem 5: For any given 4, a 4 4 if F 4 (0) 0, then the retailer s annual total profit P ( N, ) is concave and reaches its global maximum at the point N 4 N 4, = where N 4 satisfies (6) b if F 4 (0) < 0, then the retailer s annual total profit P4( N, 4 ) reaches its global maximum at the point N 4 = 0. Proof: Again, the proof is similar to that in heorem. So, the proof is omitted. 6 Solution procedures o obtain the optimal solution (N, ) under various situations, we develop two solution approaches to solve the problem. Firstly, we establish a simple algorithm by using the characteristics of heorems 1 to 5 as following: Algorithm 1 Step 1: Start with j = 0 and the initial value of N ij = 0, i = 1,, 3, 4. Step : Determine the values i ij =, i = 1,, 3, 4 which maximise the corresponding total profits per year for a given N ij. Step 3: Determine the values Ni( j+ 1) = Ni, i = 1,, 3, 4 which maximise the corresponding total profits per year by the obtained values ij. Step 4: If the difference between N ij and N i(j+1) is enough small (i.e., N ij N i(j+1) 10 5 ), then set N = N ij and = ij. hus, (N, ) is the optimal solution. Otherwise, set j = j + 1 and go back to Step.
Retailer s optimal order and credit policies 385 Step 5: Find i= 1,,3,4 i i i Max P ( N, ), optimal solution. i i i i= 1,,3,4 and if P( N, ) = Max P ( N, ), (N, ) is the he convergence of the procedure can easily be proved by adopting a similar graphical technique used in Hadley and Whitin (1963). Once the optimal solution (N, ) is obtained, and then the optimal order quantity Q = Ke can also be determined. he second approach is to use any standard non-linear programming software such as LINGO, or Evolutionary Solver for Microsoft Excel to solve the following four sub-cases in which each objective function and constraints are non-linear. A hd( N) P-1 Max P1 ( N, ) = pdne cdn cidn c pidn e ( + N M) + ( M N). subject to W / D( N), N M + N, 0 and N 0. bn A h+ pie D( N) P- Max P ( N, ) = pdne cdn + pied( N)( M N). subject to W / D( N), + N M, 0 and N 0. A hd( N) P-3 Max P3 ( N, ) = pd( N) e cd( N) cicd( N) [( N M ) + ]. subject to W / D( N), N M, 0 and N 0. A hd( N) P-3 Max P4 ( N, ) = pd( N) e (1 r) cd( N) (1 rcidn ) c ( N + ). subject to 0 and N 0. he non-linear problem in each sub-case should have a unique global maximum. Consequently, we can solve sub-problems P-1, P-, P-3 and P-4 and then compare them to find the optimal solution. We have used both LINGO and Evolutionary Solver for Microsoft Excel (two different kinds of software) to solve the four sub-problems in the second solution procedure separately, and found both obtain the same optimal solution. In addition, both solution procedures get the same optimal solution, too. 7 Numerical examples he numerical examples given below are to illustrate above solution procedure.
386 C-. Yang et al. able 1 he solution procedure of Example 1 aking a permissible delay link to order quantity (D(N) W) aking a cash discount Case 1. N M + N Case. + N M Case 3. N M j N 1j P 1j 1 j N j P j j N 3j P 3j 3 j N 4j P 4j 4 0 0 1.5 4,759. 0 0 0.5 4,596.0 0 0 1.5 4,750.0 0 0 0.64766 4,908.0 1 0.05 0.97350 5,81.8 1 0 0.5 4,596.0 1 0.35133 0.6819 16,715.3 1 0.43506 0.186 1,374.0 0.05 0.97350 5,81.8 0.4957 0.18693 5,453.4 0.49997 0.18557 5,555.5 3 0.50837 0.18111 6,314.0 3 0.50504 0.1834 5,896.0 4 0.5098 0.18070 6,376.3 4 0.50540 0.18307 5,90.4 5 0.50935 0.18067 6,380.8 5 0.50543 0.18306 5,9. 6 0.50935 0.18067 6,381.1 6 0.50543 0.18306 5,9.3 7 0.50935 0.18067 6,381.1 7 0.50543 0.18306 5,9.3 Note: Italic type expresses the optimal solution of Example 1.
Retailer s optimal order and credit policies 387 able Optimal solutions under different parametric values r K A N Q P(N, ) 0.01 3,000 150 50 350 4,000 150 50 350 5,000 150 50 350 0.0 3,000 150 50 350 4,000 150 50 350 5,000 150 50 350 0.03 3,000 150 50 350 4,000 150 50 350 5,000 150 50 350 N = N 3 = 0.5154 N = N 3 = 0.50459 N = N 3 = 0.4979 N = N 3 = 0.51611 N = N 3 = 0.50935 N = N 3 = 0.50371 N = N 3 = 0.51851 N = N 3 = 0.5154 N = N 3 = 0.50757 N = N 3 = 0.5154 N = N 3 = 0.50459 N = N 3 = 0.4979 N = N 3 = 0.51611 N = N 3 = 0.50935 N = N 3 = 0.50371 N = N 3 = 0.51851 N = N 3 = 0.5154 N = N 3 = 0.50757 N = N 4 = 0.5069 N = N 4 = 0.5188 N = N 4 = 0.50633 N = N 4 = 0.5419 N = N 4 = 0.51755 N = N 4 = 0.5101 N = N 4 = 0.5665 N = N 4 = 0.5069 N = N 4 = 0.51580 = 3 = 0.16031 6,37.99 0,170.8 = 3 = 0.1111 7,894.80 19,4.7 = 3 = 0.5399 9,186.73 18,463.0 = 3 = 0.13760 7,67.53 7,479.6 = 3 = 0.18067 9,5.19 6,381.1 = 3 = 0.1680 10,76.40 5,495.5 = 3 = 0.134 8,174.9 34,850.8 = 3 = 0.16031 10,396.60 33,618.0 = 3 = 0.1905 1,149.60 3,63.3 = 3 = 0.16031 6,37.99 0,170.8 = 3 = 0.1111 7,894.80 19,4.7 = 3 = 0.5399 9,186.73 18,463.0 = 3 = 0.13760 7,67.53 7,479.6 = 3 = 0.18067 9,5.19 6,381.1 = 3 = 0.1680 10,76.40 5,495.5 = 3 = 0.134 8,174.9 34,850.8 = 3 = 0.16031 10,396.60 33,618.0 = 3 = 0.1905 1,149.60 3,63.3 = 4 = 0.15787 6,398.37 0,871.7 = 4 = 0.078 8,100.6 19,910.3 = 4 = 0.4996 9,49.10 19,136.3 = 4 = 0.13553 7,453.3 8,43.5 = 4 = 0.17789 9,463.68 7,307.4 = 4 = 0.134 11,043.50 6,407.5 = 4 = 0.1050 8,38.31 36,038.7 = 4 = 0.15787 10,663.90 34,786.1 = 4 = 0.18909 1,464.60 33,775.4
388 C-. Yang et al. Example 1: Let us consider an inventory system with the following data: A = 50, p =.4, c = 1, h = 0., M = 1/4 = 0.5, I c = 0.1, I e = 0.08, r = 0.0, W = 5,000, K = 4,000, a = 5, and b = 0.8 in appropriate units. Under the given values of the parameters and according to the algorithm in the above section, the computational results can be found as shown in able 1. able 1 reveals that the retailer s optimal credit period is N = N 3 = 0.50935, and its optimal replenishment cycle time is = 3 = 0.18067. Hence, the retailer s optimal order quantity is Q = D( N ) = 9,5.19 units, and its optimal total profit per year is 3 3 3 3 3 P( N, ) = P ( N, ) = 6,381.1. In this case, the retailer orders more than W units to take the permissible delay by the supplier. Example : We now study the effects of changes in parameters r, K and A on optimal credit period N, optimal replenishment cycle time, optimal ordering quantity Q, and total profit per year P(N, ). We use the values of r, K and A as r {0.01, 0.0, 0.03}, K {3,000, 4,000, 5,000}, and A {150, 50, 350}, respectively. Using the proposed algorithm above, we obtain the computational results for different values of r, K and A as shown in able. able reveals the following managerial insights: 1 If the supplier s cash discount rate r is high enough (for example, r = 0.03 in this numerical example), then the retailer will pay immediately to profit from the cash discount. Otherwise, the retailer will take the permissible delay instead. he larger the demand parameter K, the larger the down-stream trade credit N, and as well as the larger the order quantity Q and the total profit P(N, ). 3 If the order cost A increases, then the retailer reduces the length of down-stream trade credit while orders more quantity, and receives less total profit. 8 Conclusions In this paper, we have established an inventory model to incorporate the facts that: 1 both the supplier and the retailer often offer trade credits to their buyers the down-stream trade credit period is a decision variable and has impacts both on demand and default risk 3 the retailer can take either a permissible delay linked to order quantity or a cash discount. In theoretical analysis, we have derived several results as shown in heorems 1 to 5 to determine the optimal solution under various situations, and then developed an algorithm to solve this complex inventory problem. Finally, numerical examples have been provided to illustrate how the retailer makes its three choices: 1 the optimal order quantity the choice between a cash discount or a permissible delay
Retailer s optimal order and credit policies 389 3 the optimal down-stream credit period. For further research, this paper can be extended in several ways. For instance, we may add the constant deterioration rate for the deteriorating items. Also, we could generalise the model to allow for shortages. Finally, we can consider the effect of inflation rates on the economic order quantity. Acknowledgements he authors greatly appreciate Editor Lars Moench, and three anonymous referees for their valuable and helpful suggestions to improve the quality of the paper. References Abad, P.L. and Jaggi, C.K. (003) A joint approach for setting unit price and the length of the credit period for a seller when end demand is price sensitive, International Journal of Production Economics, Vol. 83, No., pp.115 1. Aggarwal, S.P. and Jaggi, C.K. (1995) Ordering policies of deteriorating items under permissible delay in payments, Journal of the Operational Research Society, Vol. 46, No. 5, pp.658 66. Chang, C.. (00) Extended economic order quantity model under cash discount and payment delay, International Journal of Information and Management Sciences, Vol. 13, No. 3, pp.57 69. Chang, C.., Ouyang, L.Y. and eng, J.. (003) An EOQ model for deteriorating items under supplier credits linked to ordering quantity, Applied Mathematical Modelling, Vol. 7, No. 1, pp.983 996. Chang, C.., Ouyang, L.Y., eng, J.. and Cheng M.C. (010) Optimal ordering policies for deteriorating items using a discounted cash-flow analysis when a trade credit is linked to order quantity, Computers & Industrial Engineering, Vol. 59, No. 4, pp.770 777. Chang, H.J. and Dye, C.Y. (001) An inventory model for deteriorating items with partial backlogging and permissible delay in payments, International Journal of Systems Science, Vol. 3, No. 3, pp.345 35. Goyal, S.K. (1985) EOQ under conditions of permissible delay in payments, Journal of the Operational Research Society, Vol. 36, No. 4, pp.335 338. Goyal, S.K., eng, J.. and Chang, C.. (007) Optimal ordering policies when the supplier provides a progressive interest-payable scheme, European Journal of Operational Research, Vol. 179, No., pp.404 413. Hadley, G. and Whitin,. (1963) Analysis of Inventory Systems, pp.169 17, Prentice-Hall, Englewood Cliff NJ. Huang, Y.F. (003) Optimal retailer s ordering policies in the EOQ model under trade credit financing, Journal of the Operational Research Society, Vol. 54, No. 9, pp.1011 1015. Hwang, H. and Shinn, S.W. (1997) Retailer s pricing and lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payments, Computers and Operations Research, Vol. 4, No. 6, pp.538 547. Jaggi, C.K., Goyal, S.K. and Goel, S.K. (008) Retailer s optimal replenishment decisions with credit-linked demand under permissible delay in payments, European Journal of Operational Research, Vol. 190, No. 1, pp.130 135.
390 C-. Yang et al. Jamal, A., Sarker, B. and Wang, S. (1997) An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, Journal of the Operational Research Society, Vol. 48, No. 8, pp.86 833. Kosmala, W.A.J. (1999) Advance Calculus: A Friendly Approach, Prentice-Hall, Upper Saddle River, New Jersey. Mohan, S., Mohan G. and Chandrasekhar A. (008) Multi-item, economic order quantity model with permissible delay in payments and a budget constraint, European Journal of Industrial Engineering, Vol., No. 4, pp.446 460. Ouyang, L.Y., Chang, C.. and eng, J.. (005) An EOQ model for deteriorating items under trade credits, Journal of the Operational Research Society, Vol. 56, No. 6, pp.719 76. Ouyang, L.Y., Wu, K.S. and Yang, C.. (007) An EOQ model with limited storage capacity under trade credits, Asia-Pacific Journal of Operational Research, Vol. 4, No. 4, pp.575 59. Pan, Q.H. and Zhang, X.F. (009) he model of operational risk measurement for regional industrial cluster, China Economic & rade Herald, Vol. 5, No. 17, pp.55 56. Pan, Q.H. and Zhang, X.F. (010) Financing innovation when Chinese construction enterprises overseas expansion, Journal of Enterprise Economy, Vol. 30, No. 1, pp.164 166. eng, J.. (00) On the economic order quantity under conditions of permissible delay in payments, Journal of the Operational Research Society, Vol. 53, No. 8, pp.915 918. eng, J.. (006) Discount cash-flow analysis on inventory control under various supplier s trade credits, International Journal of Operations Research, Vol. 3, No. 1, pp.3 9. eng, J.. and Chang, C.. (009) Optimal manufacturer s replenishment policies in the EPQ model under two levels of trade credit policy, European Journal of Operational Research, Vol. 195, No., pp.358 363. eng, J.. and Goyal, S.K. (007) Optimal ordering policies for a retailer in a supply chain with up-stream and down-stream trade credits, Journal of the Operational Research Society, Vol. 58, No. 9, pp.15 155. eng, J.., Chang, C.. and Goyal, S.K. (005) Optimal pricing and ordering policy under permissible delay in payments, International Journal of Production Economics, Vol. 97, No., pp.11 19. eng, J.., Chang, C.., Chern, M.S. and Chan, Y.L. (007) Retailer s optimal ordering policies with trade credit financing, International Journal of Systems Science, Vol. 38, No. 3, pp.69 78. eng, J.., Min, J. and Pan, Q.H. (01) Economic order quantity model with trade credit financing and non-decreasing demand, Omega, Vol. 40, No. 3, pp.38 335. Yang, C.. (010) he optimal order and payment policies for deteriorating items in discount cash flows analysis under the alternatives of conditionally permissible delay in payments and cash discount, OP, Vol. 18, No., pp.49 443. Appendix A Proof of heorem 1 Proof of part (a): For any given N, we take the first-order and second-order derivatives of P 1 (N, ) in (7) with respect to, and have dp 1( N, ) A+ cic pie Ke M N h+ cic Ke =, (A1) d
Retailer s optimal order and credit policies 391 and dpn 1(, ) A+ cic pie Ke ( M N) = 3 d. (A) When W (M N)Ke, if A Δ 1 (N), then we have ( c e) { ( c W Ke ) ( e c) } A+ ci pi Ke ( M N) Ke h+ ci + pi ci M N + ( ci ) ( c pie Ke M N = h + cic Ke W Ke ) > 0. (A3) Consequently, 1 in (15) is well-defined, and P 1 (N, ) is a strictly concave function of by using (A) and (A3). In addition, it is clear from (A3) that dp1 ( N, ) d = W A+ ci pi Ke ( M N) h+ ci Ke = > 0, (A4) c e c W and ( + ) dp1 ( N lim, ) h cic Ke = < 0. (A5) d herefore, there exists a unique Otherwise, if A < Δ 1 (N), then 1 1 = ( > W /( Ke )) such that dp 1 (N, 1 ) / d = 0. { Ke } ( c ) W ( ) dp1 ( N, ) h+ ci Ke < < 0, for ( W ( ), Ke ). d Hence, for any given N, P 1 (N, ) is a strictly decreasing function in [W / (Ke ), ]. herefore, 1 = W /( Ke ) is the optimal value which maximises P 1 (N, ). Proof of part (b): For the case W < (M N)Ke, similar to the arguments as above, we can show that when A Δ (N), = and conversely, when A < Δ (N), = M N his completes the proof. 1 1 1. Appendix B Proof of Lemma 1 Proof of part (a): For any given 1, it can be found that F 1(N) is a strictly decreasing function in N [N L, M 1 / a] from (19). Furthermore, from (18), we have F 1 (N L ) F 1 (M 1 / a). Hence, if F 1 (N L ) 0 F 1 (M 1 / a), then there exists a unique N 1 [N L, M 1 / a] such that F 1 (N 1 ) = 0 by intermediate value theorem (see, for example, Kosmala, 1999).
39 C-. Yang et al. Proof of part (b): If F 1 (N L ) F 1 (M 1 / a) or 0 > F 1 (N L ) F 1 (M 1 / a), then we cannot find a value of N [N L, M 1 / a] such that F 1 (N) = 0. his completes the proof. Appendix C Proof of heorem Proof of part (a): For given 1, if F 1(N L ) 0 F 1 (M 1 / a), then we see that N 1 is the unique solution of (13) from Lemma 1(a). aking the second derivative of F 1 (N) with respect to N and then finding the value of the function at the point N = N 1, we obtain (, N) ( c e) ( ) 1 1 1 1 1 = Ke ( a b) bpe ci c + dn N= N 1 1 acic 1 ( M N1) pie a( M N1) 1 1 ci 1 c Ke a b bpe 1 1 1 dp ci pi a M N 1 = ( ) < 0. 1 1 hus, N 1 is the global maximum point of P (, N ). Proof of part (b): If F 1 (N L ) F 1 (M 1 / a) > 0, then F 1 (N) > 0, for all N [N L, M 1 / a]. hus, we have dp (, N) 1 1 dn = F1 ( N) > 0, N NL, M 1/ a. which implies P1( 1, N ) is a strictly increasing function of N [N L, M 1 / a]. As a result, P1( 1, N ) has a maximum value at boundary point N1 = M 1/ a. Proof of part (c): If 0 > F 1 (N L ) F 1 (M 1 / a), then F 1 (N) < 0, for all N [N L, M 1 / a]. hus, we have dp (, N) 1 1 dn = F1 ( N) < 0, N NL, M 1/ a. which implies P1( 1, N ) is a strictly decreasing function of N [N L, M 1 / a]. As a result, P1( 1, N ) has a maximum value at boundary point N1 = N L. his completes the proof.