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Two-echelon trade credit financing for perishable items in a supply chain when demand depends on both selling price and credit period
Computers & Industrial Engineering 57 (2009) 773–786
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Two-echelon trade credit financing for perishable items in a supply chain when demand depends on both selling price and credit period A. Thangam *, R. Uthayakumar Department of Mathematics, Gandhigram Rural University, Gandhigram, Dindigul 624 302, Tamilnadu, India
a r t i c l e
i n f o
Article history: Received 10 July 2008 Received in revised form 1 December 2008 Accepted 10 February 2009 Available online 20 February 2009 Keywords: Two-level trade credit financing EPQ model Supply chain Perishable items
a b s t r a c t A profitable decision policy between a supplier and the retailers can be characterized by an agreement on the trade credit scenario such as permissible delay in payments. In real life business, we observe that the demand is a function of both the selling price and credit period rather than the constant demand. Incorporating this demand function to the retailer of a supply chain, we develop an EPQ – based model for perishable items under two-echelon trade financing. The purpose of this paper is to maximize the profit by determining the optimal selling price, credit period and replenishment time. It is shown that the model developed by Jaggi et al. [Jaggi, J. K., Goyal, S. K., & Goel, S. K., 2008. Retailer’s optimal replenishment decisions with creditlinked demand under permissible delay in payments. European Journal of Operational Research, 190, 130–135] can be treated as a special case of this paper. Finally, through numerical examples, sensitivity analysis shows the influence of key model parameters. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Today’s research is interested in focusing on supply chain models which have real life applications. In real life business via share marketing, trade credit financing becomes a powerful tool to improve sales and profits in an industry. In practice, suppliers/retailers allow a fixed period to settle the payment without penalty for their retailers/customers to increase sales and reduce on-hand stock. This permissible delay in payments reduces the cost of holding stock because it reduces the amount of capital invested in stock for the duration of the permissible period. During the delay period (i.e. credit period) the retailer can accumulate revenue on sales and earn interest on that revenue via share market investment or banking business. Teng (2002) illustrated two more benefits of trade credit policy: (1) it attracts new customers who consider trade credit policy to be a type of price reduction; and (2) it should cause a reduction in sales outstanding, since some established customer will pay more promptly in order to take advantage of trade credit more frequently. However, the strategy of granting credit terms adds an additional dimension of default risk to the supplier (Teng, Chang, & Goyal, 2005). Over the years, the extensive use of trade credit has been addressed. Firstly, Goyal (1985) examined the effect of the trade credit period on the optimal inventory policy. Chand and Ward (1987) analyzed Goyal’s problem under assumptions of the classical economic order quantity model, obtaining different results. Chung (1998) developed an alternative approach to determine the eco* Corresponding author. Tel.: +91 451 2452371; fax: +91 451 2453071. E-mail address: [email protected] (A. Thangam). 0360-8352/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2009.02.005
nomic order quantity under the condition of permissible delay in payments. The perishability of goods is a realistic phenomenon. It is well known that certain products such as medicine, volatile liquids, blood bank, food stuff and many others, decrease under deterioration (vaporization, damage, spoilage, dryness and so on) during their normal storage period. So Aggarwal and Jaggi (1995) considered the inventory model with an exponential deterioration rate under the condition of permissible delay in payments. Chu, Chung, and Lan (1998) extended Goyal’s model to the case of deterioration. Jamal, Sarker, and Wang (1997), Chang and Dye (2001) further generalized the model with shortages. Many related articles can be found in Hwang and Shinn (1997), Jamal, Sarker, and Wang (2000), Arcelus, Shah Nita, and Srinivasan (2003), Abad and Jaggi (2003), Chang (2004), Chung, Goyal, and Huang (2005), Teng et al. (2005), Chung and Liao (2006) and their references. Jaber and Osman (2006) developed a two-level supply chain system in which the retailer’s permissible delay in payment offered by the supplier is considered as a decision variable in order to coordinate the order quantity between the two levels. All the aforementioned inventory models implicitly assumed one-level trade credit financing. But, in most business transactions, this assumption is unrealistic and usually the supplier offers a credit period to the retailer and the retailer, in turn, passes on this credit period to his/her customers. For example, in India, the TATA Company can delay the amount of purchasing cost until the end of the delay period offered by his supplier. The TATA Company also offers permissible delay payment period to his dealership. Recently, researchers developed inventory models under this two-echelon (or two-level) trade credit financing. Huang (2003) presented an inventory model assuming that the retailer also offers
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a credit period to his/her customer which is shorter than the credit period offered by the supplier, in order to stimulate the demand. Huang (2006) extended Huang’s (2003) model to investigate the retailer’s inventory policy under two levels of trade credit and limited storage space. Huang (2007) incorporated Huang (2003) to investigate the two-level trade credit policy in the EPQ frame work. Ho, Ouyang, and Su (2008) developed an integrated supplier-buyer inventory model with the assumption that demand is sensitive to retail price and the supplier adopts a two-part trade credit policy. Huang and Hsu (2008) have developed an inventory model under two-level trade credit policy by incorporating partial trade credit option at the customers of the retailer. Liao (2008) developed an EOQ model with non-instantaneous receipt and exponentially deteriorating items under two-level trade credit financing. Tsao (2009) developed an EOQ model under advance sales discount and twoechelon trade credits. Teng and Chang (2009) extended the Huang’s (2007) model by relaxing the assumption N < M. If we consider a supply chain problem for perishable items under two-echelon trade credit policy, it is very essential to consider the impact of both selling price and credit period on retailer’s demand. Because, the large piles of customer goods in a supermarket are often associated with a price cut to induce more sales, as well as profits. Also, the marginal effect of credit period on sales is proportional to the unrealized potential of the market demand. Thus the retailer’s demand becomes a function of both the selling price and credit period. Related to this context, recently, Jaggi, Goyal, and Goel (2008) have developed a simple EOQ model in which the retailer’s demand is linked to credit period alone. But, ignoring the impact of selling price on demand, they considered non-deteriorating items (i.e., unaffected by time) inventory system and replenishment is done instantaneously. In real life, their model is not quit applicable. Mostly, the selling items are perishable such as fruits, fresh fishes, gasoline, photographic films, etc. The effect of time is even more critical for the goods such as food stuff and cigarettes. As Liao (2008) illustrated, stocks are often replenished at certain production rate which is seldom infinite. Even for purchased items, when supply arrives at the warehouse, it may take days for receiving department to completely transfer the supply into storage room. The effect of these situations imposed us to establish an EPQ inventory model for perishable items when the retailer’s demand is a function both credit period and selling price under two-echelon trade credit financing. The main purpose of this paper is to amend the paper of Jaggi et al. (2008) with a view of making their model more relevant and so applicable to practice. Here we are taking into account the following factors: (1) The retailer’s demand is a function of both selling price and credit period; (2) the retailer’s trade credit period ðMÞ offered by the supplier is not necessarily longer than the customer’s trade credit period ðNÞ offered by the retailer; (3) replenishment rate is finite; (4) the selling items are perishable such as fruits, fresh fishes, gasoline, photographic films, etc.; and (5) a two-echelon trade credit financing is adopted instead of single level trade credit financing between supplier and retailer. Under these conditions, we determine optimal credit period ðN Þ, selling price ðs Þ, and replenishment time ðT Þ in order to maximize the retailer’s profit. The rest of the paper is organized as follows. In the next section, the assumptions and notations related to this study are presented. In Section 3, we formulate the model by considering the possible costs and revenues. Section 4 proves that the optimal replenishment policy not only exists but also is unique and we derive the optimality conditions to find optimal selling price. Section 5 shows that the inventory model of Jaggi et al. (2008) is a special case of our paper. In Section 6, several numerical examples are presented to illustrate the theory.
2. Notations and assumptions The following notations are adopted throughout this paper. Notations A h c s
ordering cost per cycle, holding cost per unit time excluding interest charges, unit purchasing cost per item, unit selling price per item of good quality (decision variable), M permissible delay period in payment for the retailer offered by the supplier, N permissible delay period in payment for the customer offered by the retailer (decision variable), kðs; NÞ the annual demand, as a function of both s and N, P annual replenishment rate, h constant deterioration rate, where 0 6 h < 1, the time at which the production stops in a cycle, t1 T cycle time in years (decision variable), IðtÞ the inventory level at time t where 0 6 t 6 T, interest earned per $ per year, Ie interest charged per $ per year by the supplier, Ik TPðs; T; NÞ the annual total profit. We use the notations kðs; NÞ and k interchangebly throughout this paper. In addition, the following assumptions are made in deriving the model. Assumptions
1. The demand, kðs; NÞ, is a marginally increasing function with respect to N and downward sloping function of price s. 2. The production rate is finite and P > k. 3. The gross profit ðs cÞkðs; NÞ is concave. 4. The time to deterioration of a product follows an exponential distribution with parameter h, i.e. the deterioration rate is a constant fraction of the on-hand inventory. 5. Before the settlement of an account, the retailer can use sales revenue to earn the interest with an annual rate Ie upto the end of period M. At time t ¼ M, the credit is settled and the retailer starts to pay the interest at rate Ik for the items in stock. 6. The retailer offers a credit period N for each of his customers to settle the account. 7. Time horizon is infinite. 8. Inventory holding cost is charged only on the amount of undecayed stock. 9. Shortages are not allowed and lead time is negligible.
3. Model formulation We first model the demand function as below. For a given selling price s > 0, the marginal effect of credit period on sales is proportional to the unrealized potential of market demand without any delay. Under this assumption, the demand kðs; NÞ can be defined in the following two ways. First, the demand kðs; NÞ is represented by the partial differential equation,
@kðs; NÞ ¼ r½aðsÞ kðs; NÞ @N
ð1Þ
where aðsÞ is the maximum demand over the planning horizon when the selling price is s, r is the saturation rate of demand and 0 6 r < 1.
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Secondly, kðs; NÞ can also be represented by the difference equation,
kðs; N þ 1Þ kðs; NÞ ¼ r½aðsÞ kðs; NÞ
ð2Þ
The solutions to Eqs. (1) and (2) can be found by using the initial condition that when N ¼ 0; kðs; NÞ is bðsÞ: That is, when there is no permissible delay period offered to the customers from the retailer, the demand is a function of selling price alone. This is a trivial case. We let bðsÞ such that bðsÞ < aðsÞ for any value of s > 0. Here both aðsÞ and bðsÞ are any non-negative, continuous, convex, decreasing functions of the selling price in ½0; su ], where su is an extremely large number. The solutions to Eqs. (1) and (2) are Type 1:
kðs; NÞ ¼ aðsÞ ½aðsÞ bðsÞerN
ð3Þ
and
By using the condition that I1 ðtÞ ¼ I2 ðtÞ at t ¼ t1 , we obtain
1 kðs; NÞ hT log 1 þ ðe 1Þ h P
t1 ðs; T; NÞ ¼
ð10Þ
We use the notations t 1 ðs; T; NÞ and t 1 interchangebly throughout this paper. 3.1. Determination of total profit function We now derive the equations for sales profit, ordering cost, holding cost, deterioration cost, interest payable and interest earned. These components are evaluated as in the following: (1) Sales profit (SP): The total sales profit is given by ðs cÞkðs; NÞ (2) Ordering cost (OC): The annual ordering cost is A=T (3) Holding cost (HC): Annual stock holding cost (excluding interest charges)
Z h T IðtÞdt T 0 h ¼ 2 ðP kðs; NÞÞ½eht1 þ ht1 1 h T þkðs; NÞ½ehðTt1 Þ hðT t1 Þ 1 ¼
Type 2:
kðs; NÞ ¼ aðsÞ½1 ð1 rÞN þ bðsÞð1 rÞN
ð4Þ
respectively. The rest of our theory is fittest for both the Type 1 and Type 2 demand functions. Due to the finite replenishment rate, the inventory level gradually increases from time t ¼ 0 and it reaches the maximum at t ¼ t1 . Production then stops at t ¼ t1 and the inventory gradually depletes to zero at the end of the cycle t ¼ T due to deterioration and consumption. The graphical representation of this inventory system is clearly depicted in Fig. 1. The objective is to determine the optimal values for s; T and N (namely, s ; T and N , respectively) such that the annual profit is maximized. From the above assumptions and notations, we know that the inventory level IðtÞ at time t satisfies the following differential equations:
dIðtÞ þ hIðtÞ ¼ P kðs; NÞ; dt
if 0 6 t 6 t 1
if t1 6 t 6 T
I1 ðtÞ if 0 6 t 6 t1 I2 ðtÞ if t1 6 t 6 T
ð6Þ
ð11Þ
ð7Þ
where
P kðs; NÞ ð1 eht Þ h kðs; NÞ hðTtÞ 1Þ I2 ðtÞ ¼ ðe h
I1 ðtÞ ¼
c Pt1 ckðs; NÞ T
ð5Þ
with the boundary conditions Ið0Þ ¼ 0 and IðTÞ ¼ 0. Consequently the solutions for Eqs. (5) and (6) are given by
IðtÞ ¼
¼
(4) Interest payable (IP): Based on the values of T; N and M, there are three cases to be considered: (i) N 6 M 6 T þ N, (ii) N 6 T þ N 6 M and (iii) M 6 N 6 T þ N: Case 1: N 6 M 6 T þ N. Here, the retailer sells the units up to the end of his permissible delay period M. The items in stock are charged at interest rate Ik by the supplier starting from time M to T þ N. As a result, the interest payable per unit time is
and
dIðtÞ þ hIðtÞ ¼ kðs; NÞ; dt
(3) Annual deteriorating cost (DC)
ð8Þ ð9Þ
Z cIk kðs; NÞ TþN hðTþNtÞ ðe 1Þdt hT M cIk kðs; NÞ hðTþNMÞ e hðT þ N MÞ 1 ¼ 2 h T
Case 2: N 6 T þ N 6 M. In this case, there is no interest payable by the retailer. Case 3: M 6 N 6 T þ N. The retailer pays interest on full replenished quantity for a period from M to N and on average stock held during the cycle length. As a result, the annual interest payable is
cIk T
Z
N
M
Pt1 dt þ
Z N
TþN
kðs; NÞ hðTþNtÞ 1dt ½e h
cIk ¼ 2 h2 Pt 1 ðN MÞ þ kðs; NÞðehT hT 1Þ h T
ð12Þ
(5) Interest earned (IE): Same as the interest payable, there are three cases to be considered. Case 1: N 6 M 6 T þ N, shown in Fig. 2 (as in Jaggi et al., 2008) During the period from N to M, the retailer earns interest Ie per dollar. Therefore the annual interest earned is
Fig. 1. Graphical representation of inventory system.
sIe kðs; NÞðM NÞ2 2T
ð13Þ
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A h ½ðP kðs; NÞÞ T Th2 þ ht1 1Þ þ kðs; NÞðehðTt1 Þ hðT t 1 Þ 1Þ
TP3 ðs; T; NÞ ¼ ððs cÞkðs; NÞÞ
ðeht1 cIk 2 ½h2 Pt 1 ðN MÞ þ kðs; NÞðehT hT 1Þ Th
ð19Þ
After some significant simplifications, we have
TP1 ðs; T; NÞ ¼ ðs cÞkðs; NÞ
Case 2: N 6 T þ N 6 M, shown in Fig. 3 (as in Jaggi et al., 2008)In this case, the retailer earns interest on average sales revenues received during the period from N to T þ N and on full sales revenue from a period from T þ N to M: As a result the annual interest earned is
" # sIe kðs; NÞT 2 þ kðs;NÞTðM T NÞ ¼ sIe kðs;NÞ½M N T=2 T 2
ð14Þ TP3 ðs; T; NÞ ¼ ðs cÞkðs; NÞ
cIk Th2
ð21Þ
A ðh þ chÞ Pt1 þ kðs; NÞ T h T
½h2 Pt 1 ðN MÞ þ kðs; NÞðehT hT 1Þ
ð22Þ
4. Solution procedure
The retailer’s annual profit is
ð15Þ
From the above arguments, the annual profit is given by
8 > < TP 1 ðs; T; NÞ if N 6 M 6 T þ N TPðs; T; NÞ ¼ TP 2 ðs; T; NÞ if N 6 T þ N 6 M > : TP 3 ðs; T; NÞ if M 6 N 6 T þ N
ð20Þ
þ sIe kðs; NÞ½M N T=2
Case 3: M 6 N 6 T þ N.In this case, there is no interest earned by the retailer.
TPðs; T; NÞ ¼ ðSPÞ ðOCÞ ðHCÞ ðDCÞ ðIPÞ þ ðIEÞ
cIk kðs; NÞ
½ehðTþNMÞ hðT þ N MÞ 1 Th2 sIe kðs; NÞðM NÞ2 þ 2T A ðh þ chÞ Pt1 kðs; NÞ TP2 ðs; T; NÞ ¼ ðs cÞkðs; NÞ þ T h T
Fig. 2. Retailer’s interest earned when N 6 M 6 T þ N.
A ðh þ chÞ Pt1 þ kðs; NÞ T h T
ð16Þ
To find the optimal solution, say ðs ; T ; N Þ, for TPðs; T; NÞ, the following procedures are considered. Definition 1. A function f defined on an open interval ða; bÞ is said to be concave if for x; y 2 ða; bÞ and each k; 0 6 k 6 1; we have
f ðkx þ ð1 kÞyÞ P kf ðxÞ þ ð1 kÞf ðyÞ
where A h TP 1 ðs;T;NÞ ¼ ðs cÞkðs;NÞ 2 ðP kðs;NÞÞðeht1 þ ht 1 1Þ T Th cPt 1 hðTt1 Þ þ ckðs;NÞ hðT t 1 Þ 1Þ þkðs;NÞðe T sIe kðs;NÞðM NÞ2 cIk kðs;NÞ hðTþNMÞ e hðT þ N MÞ 1 þ 2 2T Th ð17Þ
A h ðP kðs; NÞÞðeht1 þ ht 1 1Þ T Th2 cPt 1 þkðs; NÞðehðTt1 Þ hðT t 1 Þ 1Þ T ð18Þ þckðs; NÞ þ sIe kðs; NÞ½M N T=2
TP 2 ðs; T; NÞ ¼ ðs cÞkðs; NÞ
Intermediate value theorem (in real analysis) Let g be a continuous function on the closed interval [a, b] and let gðaÞgðbÞ < 0: Then there exists a number d 2 ða; bÞ such that gðdÞ ¼ 0. Lemma 1. If f ðtÞ is a continuous function on (a,b) and if increasing, then f is concave.
df dt
is non-
Proof. Please see Appendix A. h 4.1. Determination of the optimal replenishment cycle length T for any given s and N Case 1. N 6 M 6 T þ N For the given values of s and N, the first derivative of TP1 ðTjs; NÞ with respect to T is
dTP1 ðTjs; NÞ Z 1 ðTjs; NÞ ¼ dT T2 where
! sIe kðs;NÞðM NÞ2 Z 1 ðTjs;NÞ ¼ A 2 ðch þ hÞP @t 1 T t 1 ðs;T;NÞ h @T i cIk kðs;NÞ h hðTþNMÞ hðTþNMÞ hTe e þ hðN MÞ þ 1 h2 Fig. 3. Retailer’s interest earned when N 6 T þ N 6 M.
ð23Þ
The optimal value of T, say T 1 ðs; NÞ, can be found by solving the equation Z 1 ðTjs; NÞ ¼ 0. It is easy to obtain the following derivative
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" ! # dZ 1 ðTjs; NÞ ðch þ hÞP @ 2 t1 hðTþNMÞ þ cIk kðs; NÞTe < 0; ¼ T dT h @T 2
ðTjs;NÞ Hence, Z 3 ðTjs; NÞ is non-increasing on ð0; 1Þ and so dTP3dT is nonincreasing. From Lemma 1, TP 3 ðTjs; NÞ is a concave function on ð0; 1Þ: On the other hand, we have
2
since @@Tt21 > 0 (it is easy to verify). Hence Z 1 ðTjs; NÞ is non-increasing ðTjs;NÞ on ð0; 1Þ and so dTP1dT is non-increasing. From Lemma 1, TP1 ðTjs; NÞ is a concave function on ð0; 1Þ. On the other hand, we have
lim Z 1 ðTjs; NÞ ¼ 1 < 0
T!1
Based upon the above arguments, the intermediate value theorem yields that the optimal solution T 3 ðs; NÞ, not only exists but also is unique. Combining the above three cases we have the following Lemma.
and
!
Z 1 ð0Þ ¼ A
lim Z 3 ðTjs; NÞ ¼ 1 < 0 and Z 3 ð0Þ > 0:
T!1
sIe kðs;NÞðM NÞ2 cIk kðs;NÞ ½1 þ hðN MÞ ehðNMÞ 2 h2
2 A sIe kðs;NÞðMNÞ > 0: So 2 2 we restrict our attention to the condition of A sIe kðs;NÞðMNÞ > 0, 2 Since 1 þ hðN MÞ < ehðNMÞ ; Z 1 ð0Þ > 0 if
in the rest of our mathematical analysis. Hence, we have
Lemma 2. For the given values of s and N, (1) If A sIe kðs;NÞ ðM NÞ2 > 0, then T 1 ðs; NÞ is the unique optimal 2 solution to the profit function TP1 ðTjs; NÞ. (2) TP 2 ðTjs; NÞ has the unique optimal solution T 2 ðs; NÞ on the nonnegative interval ð0; 1Þ (3) TP 3 ðTjs; NÞ has the unique optimal solution T 3 ðs; NÞ on the nonnegative interval ð0; 1Þ
lim Z 1 ðTjs; NÞ ¼ 1 < 0 and Z 1 ð0Þ > 0:
T!1
Based upon the above arguments, the intermediate value theorem yields that the optimal solution T 1 ðs; NÞ, not only exists but also is unique. The similar procedure as described in Case 1 can be applied to the remaining two cases.
Let
sIe kðs;NÞðM NÞ2 ðch þ hÞP ðM NÞkðs;NÞehðMNÞ h 2 P þ kðs;NÞðehðMNÞ 1Þ
1 kðs;NÞ hðMNÞ 1Þ ðe log 1 þ h P
D1 ðs;NÞ ¼A
Case 2. N 6 T þ N 6 M For the given values of s and N, we have
ð26Þ and
dTP 2 ðTjs; NÞ Z 2 ðTjs; NÞ ¼ dT T2 where
! sIe kðs;NÞT 2 ðch þ hÞP @t 1 T t 1 ðs;T;NÞ Z 2 ðTjs;NÞ ¼ A h 2 @T
ð24Þ
The optimal value of T, say T 2 ðs; NÞ, can be found by solving the equation Z 2 ðTjs; NÞ ¼ 0. Now, we have
" # dZ 2 ðTjs; NÞ ðch þ hÞPT @ 2 t1 < 0; ¼ sIe kðs; NÞT dT h @T 2
ðTjs;NÞ Hence, Z 2 ðTjs; NÞ is non-increasing on ð0; 1Þ and so dTP2dT is nonincreasing. From Lemma 1, TP2 ðTjs; NÞ is a concave function on ð0; 1Þ. On the other hand, we have
lim Z 2 ðTjs; NÞ ¼ 1 < 0 and Z 2 ð0Þ > 0:
T!1
Based upon the above arguments, the intermediate value theorem yields that the optimal solution T 2 ðs; NÞ, not only exists but also is unique. Case 3. M 6 N 6 T þ N For the given values of s and N, we have
dTP 3 ðTjs; NÞ Z 3 ðTjs; NÞ ¼ dT T2 where
P @t 1 Z 3 ðTjs; NÞ ¼ A ½ðch þ hÞ þ cIk hðN MÞ T t 1 ðs; T; NÞ h @T cIk kðs; NÞ hT ½hTe ehT þ 1 ð25Þ h2 The optimal value of T, say T 3 ðs; NÞ, can be found by solving the equation Z 3 ðTjs; NÞ ¼ 0. Now, we have
dZ 3 ðTjs; NÞ P @ 2 t1 cIk kðs; NÞTehT < 0; ¼ ½ðh þ chÞ þ cIk hðN MÞT dT h @T 2
P ðM NÞkðs; NÞehðMNÞ D2 ðs; NÞ ¼ A ½ðch þ hÞ þ cIk hðN MÞ h P þ kðs; NÞðehðMNÞ 1Þ
1 kðs; NÞ hðMNÞ 1Þ log 1 þ ðe h P cIk kðs; NÞ ½hðM NÞehðMNÞ ehðMNÞ þ 1 h2 ð27Þ Using the values of D1 ðs; NÞ and D2 ðs; NÞ, we have the following Theorem. Theorem 1. For the fixed values of s and N, (1) If D1 ðs; NÞ P 0, then T ðs; NÞ ¼ T 1 ðs; NÞ. (2) If D1 ðs; NÞ 6 0, then T ðs; NÞ ¼ T 2 ðs; NÞ. (3) If D2 ðs; NÞ P 0 and M N < 0 then T ðs; NÞ ¼ T 3 ðs; NÞ.
Proof. Please refer Appendix B for details. h 4.2. Determination of optimal selling price s for any given value of T and N For any given values of T and N, the optimal value of s can be determined by solving the first order necessary condition (i.e., @TP ¼ 0) and examining the second order sufficient condition for @s 2
concavity (i.e.,@@sTP 2 < 0). Case 1. N 6 M 6 T þ N For the given values of T and N, the first order and second order partial derivatives of TP 1 ðs; T; NÞ with respect to s are given by
@TP 1 @k Ie ðM NÞ2 @k ðh þ chÞ @k P @t 1 ¼ ðs cÞ þ k þ s þk þ @s @s h @s T @s @s 2T cIk hðTþNMÞ @k 2 ½e hðT þ N MÞ 1 ð28Þ @s Th
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and 2
"
2
#
2
"
2
Algorithm 1. To find the optimal value of s and T for a given value of N, we adopt the following steps.
#
@ TP1 @ k @k Ie ðM NÞ @ k @k ðh þ chÞ þ þ ¼ ðs cÞ 2 þ 2 s 2 þ2 @s @s @s @s h @s2 2T " # 2 2 @ k P @ t1 cIk hðTþNMÞ @2k ½e hðT þ N MÞ 1 @s2 T @s2 @s2 Th2 ð29Þ Case 2. N 6 T þ N 6 M For the given values of T and N, the first order and second order partial derivatives of TP2 ðs; T; NÞ with respect to s are given by
@TP 2 @k @k ¼ ðs cÞ þ k þ Ie ðM N T=2Þ s þ k @s @s @s ðh þ chÞ @k P @t 1 þ h @s T @s
ð30Þ
and
" # @ 2 TP2 @2k @k þ Ie ðM N T=2Þ ¼ ðs cÞ 2 þ 2 @s @s @s2 " # " # @2k @k ðh þ chÞ @ 2 k P @ 2 t1 þ s 2 þ2 @s @s h @s2 T @s2
ð31Þ
Case 3. M 6 N 6 T þ N For the given values of T and N, the first order and second order partial derivatives of TP3 ðs; T; NÞ with respect to s are given by
@TP 3 @k ðh þ chÞ @k P @t1 ¼ ðs cÞ þ k þ @s h @s T @s @s cIk 2 @t 1 @k hT 2 h P ðN MÞ þ ðe hT 1Þ @s @s Th
ð32Þ
þ kðsÞ ¼ 0. 1. Let j ¼ 1. Find sl by solving the equation ðs cÞ @k @s 2. Find T ij ,(i ¼ 1 or 2 or 3) by using the value of sl and Theorem 1. Let sij ¼ sl . i ¼ 0 for s by using the value T ¼ T ij . Find ^si 3. Solve the equation @TP @s such that
@ 2 TP i j @s2 ðT¼T ij ;s¼^si Þ i equation @TP @T
< 0 and let si;jþ1 ¼ ^si .
4. Solve the ¼ 0 for T by using the value s ¼ si;jþ1 and let the solution be T i;jþ1 . 5. If jT ij T i;jþ1 j < and jsij si;jþ1 j < , where is any small positive number, then s ¼ si;jþ1 and T ¼ T i;jþ1 . Otherwise let j = j+1, go to step 3.
Algorithm 2. For finding the optimal solution ðs ; T ; N Þ, the following steps can be followed. 1. Let N ¼ 1 and TP i ðsð0Þ ; T ð0Þ ; 0Þ ¼ 0 for i ¼ 1; 2; 3. 2. Determine the optimal solution ðsðNÞ ; T ðNÞ Þ from Algorithm 1. 3. If N 6 M 6 T ðNÞ þ N then calculate TP 1 ðsðNÞ ; T ðNÞ ; NÞ; else go to step 5. 4. If TP1 ðsðN1Þ ; T ðN1Þ ; N 1Þ > TP 1 ðsðNÞ ; T ðNÞ ; NÞ then the optimum solution, say ðs ; T ; N Þ, is ðsðN1Þ ; T ðN1Þ ; N 1Þ and TP ¼ TP1 ðs ; T ; N Þ. Otherwise, let N ¼ N þ 1 and go to step 2; 5. If N 6 T ðNÞ þ N 6 M then calculate TP 2 ðsðNÞ ; T ðNÞ ; NÞ; else go to step 7. 6. If TP2 ðsðN1Þ ; T ðN1Þ ; N 1Þ > TP 2 ðsðNÞ ; T ðNÞ ; NÞ then the optimum solution, say ðs ; T ; N Þ, is ðsðN1Þ ; T ðN1Þ ; N 1Þ and TP ¼ TP2 ðs ; T ; N Þ. Otherwise, let N ¼ N þ 1 and go to step 2; 7. If M 6 N 6 T ðNÞ þ N then calculate TP3 ðsðNÞ ; T ðNÞ ; NÞ. 8. If TP3 ðsðN1Þ ; T ðN1Þ ; N 1Þ > TP 3 ðsðNÞ ; T ðNÞ ; NÞ then the optimum solution, say ðs ; T ; N Þ, is ðsðN1Þ ; T ðN1Þ ; N 1Þ and TP ¼ TP3 ðs ; T ; N Þ. Otherwise, let N ¼ N þ 1 and go to step 2.
and
" # " # @ 2 TP2 @2k @k ðh þ chÞ @ 2 k P @ 2 t 1 þ ¼ ðs cÞ 2 þ 2 @s @s h @s2 T @s2 @s2 " # cIk 2 @ 2 t 1 @ 2 k hT 2 h P 2 ðN MÞ þ 2 ðe hT 1Þ @s @s Th
5. Special case
ð33Þ
Note that TP i ðsjN; TÞ ði ¼ 1; 2; 3Þ are continuous functions of s over the compact set ½0; su , where su is an extremely large number. Hence TPi ðsjN; TÞ ði ¼ 1; 2; 3Þ has a maximum value. It is clear that TPi ðsjN; TÞ ði ¼ 1; 2; 3Þ is not maximum if s ¼ 0 or su . As a result, the optimal s must be an interior point between 0 and su . This implies that Eqs. (28), (30) and (32) have at least one solution. If the solution is unique, then it is the optimal solution (i.e. the Eqs. (28), (30) and (32) are necessary and sufficient conditions). Otherwise, we have to find the one at which the second order partial derivative is less than zero. Although we are not able to prove the uniqueness of the solution, the numerical examples 1 and 2 show that a unique solution exists for the considered optimization problem. We can easily calculate the Hessain matrix of TP as a negative definite although it is not straightforward to find a closed form solution when both the price and replenishment time are decision variables. Following the solution approach given in Chang, Teng, Ouyang, þ kðsÞ ¼ 0, say sl ; is the and Dye (2006), the solution of ðs cÞ @k @s lower bound for the optimal selling price s . The solution sl has been taken as a initial value for s in Algorithm 1. To find the optimal solution ðs ; T ; N Þ, we run Algorithms 1 and 2 simultaneously using Matlab 7.0. In order to find the optimal solutions, the following algorithms are used.
Here, we let h ! 0; P ! 1 and demand to be a function of credit period alone i.e., s is fixed. When the deterioration is ignored, there is no cost due to deteriorated units. If the selling price (s) is fixed then the demand is kðNÞ. By the above parametric considerations, we have limh!0þ t1 ¼ 0. Adopting above conditions, Eqs. (17)–(19) become as follows:
A hkT cIk kðT þ N MÞ2 T 2 2T 2 sIe kðM NÞ þ 2T A hkT þ sIe kðM N T=2Þ TP2 ðT; NÞ ¼ ðs cÞk T 2 A hkT cIk kðN M þ T=2Þ TP3 ðT; NÞ ¼ ðs cÞk T 2
TP1 ðT; NÞ ¼ ðs cÞk
ð34Þ ð35Þ ð36Þ
The Eqs. (34)–(36) are consistent with Eqs. (10), (12) and (14) in Jaggi et al. (2008) (where IC ¼ h). Consequently, we obtain that
sIe kðM NÞ2 hkT 2 2 2 cIk k ðT þ N MÞðT N þ MÞ 2 sIe kT 2 hkT 2 Z 2 ðT; NÞ ¼ A 2 2 2 hkT cIk kT 2 Z 3 ðT; NÞ ¼ A 2 2 Z 1 ðT; NÞ ¼ A
ð37Þ ð38Þ ð39Þ
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779
From Eqs. (37)–(39), we obtain the following,
kðM NÞ2 ½sIe þ h 2 2 kðM NÞ D2 ðNÞ ¼ A ½h þ cIk 2
D1 ðNÞ ¼ A
From the above, Theorem 1 becomes Theorem 2. For the fixed values N, (1) If D1 ðNÞ P 0, then T ðNÞ ¼ T 1 ðNÞ. (2) If D1 ðNÞ 6 0, then T ðNÞ ¼ T 2 ðNÞ. (3) If D2 ðNÞ P 0 and M N < 0 then T ðNÞ ¼ T 3 ðNÞ. These results of Theorem 2 are identical to Theorem 1 in Jaggi et al. (2008). Thus, Jaggi et al. (2008) can be treated as a special case of our paper. 6. Computational analyses The purposes of the numerical analysis are as follows:
Fig. 5. The optimal total profit for various values of s in type-1 and type-2 demands.
1. To obtain the optimal solutions for two types of demand functions. 2. To use sensitivity analysis to highlight the influence of model parameters.
is indeed the global optimum solution. Using the Algorithms 1 and 2, in Matlab 7.0, we obtained the following optimal results: optimal cycle length ðT Þ ¼ 17:92 days, optimal credit period offered by the retailer to his customers, N , is 17 days, optimal selling price s ¼ $48:77 per unit and the optimal total profit is $335.850.
6.1. Numerical examples
Example 2. Here, we consider the same data set as in numerical Example 1 but the demand function is type-2, i.e., kðs; NÞ ¼ aðsÞ½1 ð1 rÞN þ bðsÞð1 rÞN . The three dimensional graphs of total profit of the retailer are presented in Fig. 7. The graphs reveal that for any given N (e.g., N ¼ 8; 15; 26), there exists a corresponding optimal solution ðsðNÞ ; T ðNÞ Þ which maximizes the total profit. Furthermore, we run the numerical results with values of N ¼ 1; 2; . . . ; 100. The numerical results indicate that there is a unique integer N which maximizes the value of TPðNÞ ¼ TPðN; sðNÞ ; T ðNÞ Þ, as shown in Fig. 4b. Also, Fig. 5b shows that TPðsjN ; T Þ is strictly concave function of s. As a result, we are sure that the optimum obtained from the proposed algorithms is indeed the global optimum solution for the type-2 demand function also. Using the Algorithms 1 and 2, in Matlab 7.0, we obtained the following optimal results: optimal cycle length ðT Þ ¼ 17:90 days, optimal credit period offered by the retailer to his customers, N , is 15 days, optimal selling price s ¼ $48:70 per unit and the optimal total profit is $338.18.
Example 1. We consider the demand type-1, i.e., kðs; NÞ ¼ aðsÞ ½aðsÞ bðsÞerN . Let aðsÞ ¼ 80 1:21s; bðsÞ ¼ 30 1:21 s; r ¼ 0:35, production rate P ¼ 100; h ¼ 0:01; A ¼ $1000; M ¼ 30 days, h ¼ $4:5 per unit, c ¼ $30 per unit, Ie ¼ 25% per year, Ik ¼ 15% per year. The three dimensional graphs of total profit of the retailer are presented in Fig. 6. The graphs reveal that for any given N (e.g., N ¼ 8; 17; 26), there exists a corresponding optimal solution ðsðNÞ ; T ðNÞ Þ which maximizes the total profit. Furthermore, we run the numerical results with values of N= 1,2,. . .,80. The numerical results indicate that there is a unique integer N which maximizes the value of TPðNÞ ¼ TPðN; sðNÞ ; T ðNÞ Þ, as shown in Fig. 4a. Also, Fig. 5a shows that TPðsjN ; T Þ is strictly concave function of s. As a result, we are sure that the maximum obtained from the proposed algorithms
6.2. Effect of changing the inventory model parameters Here, we consider the demand function type-2 (one can also consider type-1). Let aðsÞ ¼ 80 1:21s; bðsÞ ¼ 30 1:21s; r ¼ 0:35, production rate P ¼ 100; h ¼ 0:01; A ¼ $1000; M ¼ 30 days, h ¼ $4:5 per unit, c ¼ $30 per unit, Ie ¼ 10% per year, Ik ¼ 15% per year. The sensitivity analysis is performed by varying different parameters and is given in Table 1. It is important to discuss the influence of key model parameters on the optimal solutions. The effect of changing the parameters are shown graphically in Figs. 8–14. Based on these figures, we have the following comments.
Fig. 4. The optimal total profit for various values of N in type-1 and type-2 demands.
1. It is observed that as r increases, N decreases and T marginally decreases; but TP increases. The optimal selling price remains at the threshold. It shows that retailer should offer lower credit period ðNÞ to customers when the rate of saturation of demand ðrÞ is higher (see Fig. 8).
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Fig. 6. The total profit for any given N (e.g., N = 8, 17, and 26) when demand is type-1.
Fig. 7. The total profit for any given N (e.g., N = 8, 15, and 26) when demand is type-2.
2. A higher value of M causes a higher value of TP , but lower value of s and T . It indicates the following managerial phenomena: when the supplier provides a longer credit period, the retailer
replenishes the goods more often. In other words, the retailer will shorten the cycle time and reduce the selling price in order to take advantage of the longer credit period (see Fig. 9).
A. Thangam, R. Uthayakumar / Computers & Industrial Engineering 57 (2009) 773–786 Table 1 Sensitivity analysis for various inventory model parameters. Parameter
N
T
s
TP
4.
r
0.32 0.35 0.38 0.41
18 17 16 14
18.542 18.537 18.534 18.532
48.869 48.869 48.869 48.857
337.025 337.578 338.023 338.464
M
30 35 40 45
17 17 17 17
18.537 18.529 18.521 18.513
48.870 48.848 48.825 48.802
337.578 338.995 340.414 341.833
5.
800 1000 1200 1400
17 17 17 17
16.610 18.537 20.275 21.867
48.776 48.871 48.957 49.036
343.808 337.578 331.913 326.674
6.
Ie
0.10 0.15 0.20 0.25
17 16 15 15
18.537 18.320 18.110 17.904
48.871 48.837 48.796 48.772
337.578 337.653 337.875 338.188
P
100 120 140 160
17 17 17 17
18.537 18.137 17.869 17.677
48.870 48.960 49.029 49.077
337.578 336.127 335.104 334.343
h
0.01 0.03 0.05 0.07
17 17 17 17
18.537 10.853 8.370 7.034
48.869 49.487 49.927 50.293
337.578 293.088 259.300 230.027
h
4.5 6 8 10
17 17 17 17
18.537 18.428 18.286 18.147
48.869 48.876 48.884 48.893
337.578 337.220 336.760 336.302
A
3. As ordering cost, A, increases, the replenishment cycle time T significantly increases; but optimal selling price marginally increases. Keeping the credit period ðN Þ at some threshold, the optimal profit decreases as A increases. It indicates the following managerial effect. If the ordering cost is higher, it is rea-
7.
781
sonable that the retailer lengths the cycle time to reduce the frequency of replenishment and he marginally increases the selling price (see Fig. 10). As Ie increases, N decreases; but T and s are marginally decreasing. Profit TP increases as interest earned rate of the retailer increases. It implies that if the retailer increases his interest earned rate, then he can shorten his trade credit period with marginally less selling price (see Fig. 11). As the value of h increases, TP and T decrease whereas s is increasing. That is, when the items are starting to deteriorate, it is optimal to rise marginally the selling price in order to manage the profit (see Fig. 12). When holding increases, it is seen that cycle length and profit decreases whereas the optimal selling price increases. So it is reasonable that when the holding cost increases the retailer will shorten the cycle time and increases the selling price in an effort to maintain his profit gained by keeping the threshold credit period ðN Þ (see Fig. 13). As production rate increases, TP decreases; but there are marginal increases in selling price, s , under the optimal threshold value of N . So it is not advisable to increase the production rate without the prior knowledege about the demands (see Fig. 14).
6.3. When customer’s payment exceeds the optimal credit period ðN Þ If a customer violates the payment condition, then the retailer can fix some another duration for payment with increased interest charges ð> Ik Þ. Using trade credit insurance, retailers can generally extend more credit to customers whilst reducing the risk of nonpayment, thereby enabling sales growth without a corresponding increase in risk. Insurance can also enable the retailers economy to secure more favorable financing terms, as insured accounts receivable may be used as collateral.
Fig. 8. Effect of changing saturation rate of retailer’s demand.
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Fig. 9. Effect of changing supplier’s credit period.
Fig. 10. Effect of changing ordering cost (A).
7. Conclusions and future research The increased prominence of financial markets and their widespread use in pricing and in credit period allocation provides an
opportunity for convergence between financial tools and operational problems. The latter is based mostly on a private valuation of profits and benefits associated with the trade credit policy. In this paper, we first formulated an EPQ – based inventory model
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783
Fig. 11. Effect of changing interest earned rate by the retailer.
Fig. 12. Effect of changing the deterioration rate.
for per ishable items under two-echelon trade credit policy with the assumptions that the market demand is sensitive to both the selling price and credit period offered by the retailer. After formulating the mathematical model, we then developed solution proce-
dures to determine the best payment method, the optimal selling price and optimal cycle length for the retailer. From the numerical Examples 1 and 2, we examined that the proposed algorithms provide global optimum solutions for both types of demand functions.
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Fig. 13. Effect of changing the holding cost.
Fig. 14. Effect of changing the replenishment rate (P).
The managerial implications of numerical results are clear and provide a suitable f ramework to assess the relative profitability. Finally, Jaggi et al. (2008) model becomes as a special case to this paper. In future research, our model can be extended to more general
supply chain networks, for example, multi-echelon or assembly supply chains. One can also extend this model for two-part trade credit term i.e., supplier offers two payment options: trade credit and early-payments with discount price to the retailer.
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Acknowledgements We sincerely express our gratitude to the two anonymous referees for their valuable comments and suggestions. This research work is fully supported by Senior Research Fellowship (Grant No: 09/ 715(0002)/2006 EMR-I) under Counsil of Scientific and Industrial Research (CSIR) – India. We also thank University Grants Commision, India for providing Special Assistance Program (UGC – SAP). Appendix A Proof of the Lemma 1. Proof. Given x; y with a < x < y < b, define a function g on [0, 1] by
gðtÞ ¼ f ðty þ ð1 tÞxÞ tf ðyÞ ð1 tÞf ðxÞ: Our goal is to show that g is non-negative on [0, 1]. Now g is continuous and gð0Þ ¼ gð1Þ ¼ 0. Moreover
dgðtÞ df ¼ ðy xÞ f ðyÞ þ f ðxÞ dt dt
785
Hence, if D1 ðs; NÞ 6 0, then T ðs; NÞ ¼ T 2 is the optimal solution of TPðTjs; NÞ. For the fixed values of s and N, there exists a unique value T (say T 3 ðs; NÞ) which maximizes TP3 ðTjs; NÞ and T 3 would satisfy the condition M N 6 0 6 T. Since Z 3 ðTjs; NÞ is non-increasing on ð0; 1Þ Z 3 ðM NÞ P Z 3 ðT 3 Þ ¼ 0 i.e. Z 3 ðM NÞ P 0. Now,
P ðM NÞkðs;NÞehðMNÞ Z 3 ðM NÞ ¼A ½ðch þ hÞ þ cIk hðN MÞ h P þ DðehðMNÞ 1Þ
1 kðs;NÞ hðMNÞ 1Þ log 1 þ ðe h P cIk kðs;NÞ ½hðM NÞehðMNÞ ehðMNÞ þ 1 h2
ð40Þ
So, D2 ðs; NÞ ¼ Z 3 ðM NÞ. Hence, if D2 ðs; NÞ P 0; T ðs; NÞ ¼ T 3 is the optimal solution of TPðTjs; NÞ. Combining the three possible cases, we obtain the Theorem 1. h References
For t þ h > t,
dgðt þ hÞ dgðtÞ df ðt þ hÞ df ðtÞ : ¼ ðy xÞ dt dt dt dt Since
df dt
is non-increasing,
df ðt þ hÞ df ðtÞ < 0: dt dt It implies that dgðtÞ is non-increasing on [0, 1]. Let c be a point where dt g assumes its minimum on [0, 1]. If c ¼ 1, then gðtÞ P gð1Þ ¼ 0 on [0, 1]. In this case, g is non-negative. Suppose that c 2 ð½0; 1Þ. Since P 0. But dg was nong has a local minimum at c, we have dgðcÞ dt dt P 0 on ½0; c. Consequently g is non-decreasing increasing and so dg dt on ½0; c and hence gðcÞ 6 gð0Þ ¼ 0 then the minimum of g on [0, 1] is non-negative and so g P 0 on [0, 1]. h
Appendix B Proof of the Theorem 1 Proof. For the given values of s and N; TP 2 ðs; T; NÞ and TP 3 ðs; T; NÞ are concave on T > 0. However, TP 1 ðs; T; NÞ is concave on T > 0 if ðM NÞ2 > 0. Thus there exists a unique value of T (say A sIe kðs;NÞ 2 T 1 ðs; NÞ) which maximizes TP1 ðTjs; NÞ and T 1 would satisfy the inequality 0 6 M N 6 T. Since Z 1 ðTjs; NÞ is non-increasing on ð0; 1Þ, we have that Z 1 ðM NÞ P Z 1 ðT 1 Þ ¼ 0: That is, Z 1 ðM NÞ P 0. Now,
sIe kðs;NÞðM NÞ2 ðch þ hÞP Z 1 ðM NÞ ¼ A h 2
hðMNÞ ðM NÞkðs;NÞe 1 kðs;NÞ hðMNÞ 1Þ ðe log 1 þ hðMNÞ P P þ kðs;NÞðe 1Þ h So, D1 ðs; NÞ ¼ Z 1 ðM NÞ. If D1 ðs; NÞ P 0, then T ðs; NÞ ¼ T 1 is the optimal solution of TPðTjs; NÞ. For the fixed values of s and N, there exists a unique value T (say T 2 ðs; NÞ) which maximizes TP 2 ðTjs; NÞ and T 2 ðs; NÞ would satisfy the condition 0 6 T 6 M N: Since Z 2 ðTjs; NÞ is non-increasing on ð0; 1Þ; Z 2 ðM NÞ 6 Z 2 ðT 2 Þ ¼ 0 i.e. Z 2 ðM NÞ 6 0. Now,
sIe kðs;NÞðM NÞ2 ðch þ hÞP Z 2 ðM NÞ ¼ A 2 h
hðMNÞ ðM NÞkðs;NÞe 1 kðs;NÞ hðMNÞ 1Þ log 1 þ ðe P þ kðs;NÞðehðMNÞ 1Þ h P ¼ D1 ðs;NÞ
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Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Two-echelon trade credit financing for perishable items in a supply chain when demand depends on both selling price and credit period A. Thangam *, R. Uthayakumar Department of Mathematics, Gandhigram Rural University, Gandhigram, Dindigul 624 302, Tamilnadu, India
a r t i c l e
i n f o
Article history: Received 10 July 2008 Received in revised form 1 December 2008 Accepted 10 February 2009 Available online 20 February 2009 Keywords: Two-level trade credit financing EPQ model Supply chain Perishable items
a b s t r a c t A profitable decision policy between a supplier and the retailers can be characterized by an agreement on the trade credit scenario such as permissible delay in payments. In real life business, we observe that the demand is a function of both the selling price and credit period rather than the constant demand. Incorporating this demand function to the retailer of a supply chain, we develop an EPQ – based model for perishable items under two-echelon trade financing. The purpose of this paper is to maximize the profit by determining the optimal selling price, credit period and replenishment time. It is shown that the model developed by Jaggi et al. [Jaggi, J. K., Goyal, S. K., & Goel, S. K., 2008. Retailer’s optimal replenishment decisions with creditlinked demand under permissible delay in payments. European Journal of Operational Research, 190, 130–135] can be treated as a special case of this paper. Finally, through numerical examples, sensitivity analysis shows the influence of key model parameters. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Today’s research is interested in focusing on supply chain models which have real life applications. In real life business via share marketing, trade credit financing becomes a powerful tool to improve sales and profits in an industry. In practice, suppliers/retailers allow a fixed period to settle the payment without penalty for their retailers/customers to increase sales and reduce on-hand stock. This permissible delay in payments reduces the cost of holding stock because it reduces the amount of capital invested in stock for the duration of the permissible period. During the delay period (i.e. credit period) the retailer can accumulate revenue on sales and earn interest on that revenue via share market investment or banking business. Teng (2002) illustrated two more benefits of trade credit policy: (1) it attracts new customers who consider trade credit policy to be a type of price reduction; and (2) it should cause a reduction in sales outstanding, since some established customer will pay more promptly in order to take advantage of trade credit more frequently. However, the strategy of granting credit terms adds an additional dimension of default risk to the supplier (Teng, Chang, & Goyal, 2005). Over the years, the extensive use of trade credit has been addressed. Firstly, Goyal (1985) examined the effect of the trade credit period on the optimal inventory policy. Chand and Ward (1987) analyzed Goyal’s problem under assumptions of the classical economic order quantity model, obtaining different results. Chung (1998) developed an alternative approach to determine the eco* Corresponding author. Tel.: +91 451 2452371; fax: +91 451 2453071. E-mail address: [email protected] (A. Thangam). 0360-8352/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2009.02.005
nomic order quantity under the condition of permissible delay in payments. The perishability of goods is a realistic phenomenon. It is well known that certain products such as medicine, volatile liquids, blood bank, food stuff and many others, decrease under deterioration (vaporization, damage, spoilage, dryness and so on) during their normal storage period. So Aggarwal and Jaggi (1995) considered the inventory model with an exponential deterioration rate under the condition of permissible delay in payments. Chu, Chung, and Lan (1998) extended Goyal’s model to the case of deterioration. Jamal, Sarker, and Wang (1997), Chang and Dye (2001) further generalized the model with shortages. Many related articles can be found in Hwang and Shinn (1997), Jamal, Sarker, and Wang (2000), Arcelus, Shah Nita, and Srinivasan (2003), Abad and Jaggi (2003), Chang (2004), Chung, Goyal, and Huang (2005), Teng et al. (2005), Chung and Liao (2006) and their references. Jaber and Osman (2006) developed a two-level supply chain system in which the retailer’s permissible delay in payment offered by the supplier is considered as a decision variable in order to coordinate the order quantity between the two levels. All the aforementioned inventory models implicitly assumed one-level trade credit financing. But, in most business transactions, this assumption is unrealistic and usually the supplier offers a credit period to the retailer and the retailer, in turn, passes on this credit period to his/her customers. For example, in India, the TATA Company can delay the amount of purchasing cost until the end of the delay period offered by his supplier. The TATA Company also offers permissible delay payment period to his dealership. Recently, researchers developed inventory models under this two-echelon (or two-level) trade credit financing. Huang (2003) presented an inventory model assuming that the retailer also offers
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a credit period to his/her customer which is shorter than the credit period offered by the supplier, in order to stimulate the demand. Huang (2006) extended Huang’s (2003) model to investigate the retailer’s inventory policy under two levels of trade credit and limited storage space. Huang (2007) incorporated Huang (2003) to investigate the two-level trade credit policy in the EPQ frame work. Ho, Ouyang, and Su (2008) developed an integrated supplier-buyer inventory model with the assumption that demand is sensitive to retail price and the supplier adopts a two-part trade credit policy. Huang and Hsu (2008) have developed an inventory model under two-level trade credit policy by incorporating partial trade credit option at the customers of the retailer. Liao (2008) developed an EOQ model with non-instantaneous receipt and exponentially deteriorating items under two-level trade credit financing. Tsao (2009) developed an EOQ model under advance sales discount and twoechelon trade credits. Teng and Chang (2009) extended the Huang’s (2007) model by relaxing the assumption N < M. If we consider a supply chain problem for perishable items under two-echelon trade credit policy, it is very essential to consider the impact of both selling price and credit period on retailer’s demand. Because, the large piles of customer goods in a supermarket are often associated with a price cut to induce more sales, as well as profits. Also, the marginal effect of credit period on sales is proportional to the unrealized potential of the market demand. Thus the retailer’s demand becomes a function of both the selling price and credit period. Related to this context, recently, Jaggi, Goyal, and Goel (2008) have developed a simple EOQ model in which the retailer’s demand is linked to credit period alone. But, ignoring the impact of selling price on demand, they considered non-deteriorating items (i.e., unaffected by time) inventory system and replenishment is done instantaneously. In real life, their model is not quit applicable. Mostly, the selling items are perishable such as fruits, fresh fishes, gasoline, photographic films, etc. The effect of time is even more critical for the goods such as food stuff and cigarettes. As Liao (2008) illustrated, stocks are often replenished at certain production rate which is seldom infinite. Even for purchased items, when supply arrives at the warehouse, it may take days for receiving department to completely transfer the supply into storage room. The effect of these situations imposed us to establish an EPQ inventory model for perishable items when the retailer’s demand is a function both credit period and selling price under two-echelon trade credit financing. The main purpose of this paper is to amend the paper of Jaggi et al. (2008) with a view of making their model more relevant and so applicable to practice. Here we are taking into account the following factors: (1) The retailer’s demand is a function of both selling price and credit period; (2) the retailer’s trade credit period ðMÞ offered by the supplier is not necessarily longer than the customer’s trade credit period ðNÞ offered by the retailer; (3) replenishment rate is finite; (4) the selling items are perishable such as fruits, fresh fishes, gasoline, photographic films, etc.; and (5) a two-echelon trade credit financing is adopted instead of single level trade credit financing between supplier and retailer. Under these conditions, we determine optimal credit period ðN Þ, selling price ðs Þ, and replenishment time ðT Þ in order to maximize the retailer’s profit. The rest of the paper is organized as follows. In the next section, the assumptions and notations related to this study are presented. In Section 3, we formulate the model by considering the possible costs and revenues. Section 4 proves that the optimal replenishment policy not only exists but also is unique and we derive the optimality conditions to find optimal selling price. Section 5 shows that the inventory model of Jaggi et al. (2008) is a special case of our paper. In Section 6, several numerical examples are presented to illustrate the theory.
2. Notations and assumptions The following notations are adopted throughout this paper. Notations A h c s
ordering cost per cycle, holding cost per unit time excluding interest charges, unit purchasing cost per item, unit selling price per item of good quality (decision variable), M permissible delay period in payment for the retailer offered by the supplier, N permissible delay period in payment for the customer offered by the retailer (decision variable), kðs; NÞ the annual demand, as a function of both s and N, P annual replenishment rate, h constant deterioration rate, where 0 6 h < 1, the time at which the production stops in a cycle, t1 T cycle time in years (decision variable), IðtÞ the inventory level at time t where 0 6 t 6 T, interest earned per $ per year, Ie interest charged per $ per year by the supplier, Ik TPðs; T; NÞ the annual total profit. We use the notations kðs; NÞ and k interchangebly throughout this paper. In addition, the following assumptions are made in deriving the model. Assumptions
1. The demand, kðs; NÞ, is a marginally increasing function with respect to N and downward sloping function of price s. 2. The production rate is finite and P > k. 3. The gross profit ðs cÞkðs; NÞ is concave. 4. The time to deterioration of a product follows an exponential distribution with parameter h, i.e. the deterioration rate is a constant fraction of the on-hand inventory. 5. Before the settlement of an account, the retailer can use sales revenue to earn the interest with an annual rate Ie upto the end of period M. At time t ¼ M, the credit is settled and the retailer starts to pay the interest at rate Ik for the items in stock. 6. The retailer offers a credit period N for each of his customers to settle the account. 7. Time horizon is infinite. 8. Inventory holding cost is charged only on the amount of undecayed stock. 9. Shortages are not allowed and lead time is negligible.
3. Model formulation We first model the demand function as below. For a given selling price s > 0, the marginal effect of credit period on sales is proportional to the unrealized potential of market demand without any delay. Under this assumption, the demand kðs; NÞ can be defined in the following two ways. First, the demand kðs; NÞ is represented by the partial differential equation,
@kðs; NÞ ¼ r½aðsÞ kðs; NÞ @N
ð1Þ
where aðsÞ is the maximum demand over the planning horizon when the selling price is s, r is the saturation rate of demand and 0 6 r < 1.
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Secondly, kðs; NÞ can also be represented by the difference equation,
kðs; N þ 1Þ kðs; NÞ ¼ r½aðsÞ kðs; NÞ
ð2Þ
The solutions to Eqs. (1) and (2) can be found by using the initial condition that when N ¼ 0; kðs; NÞ is bðsÞ: That is, when there is no permissible delay period offered to the customers from the retailer, the demand is a function of selling price alone. This is a trivial case. We let bðsÞ such that bðsÞ < aðsÞ for any value of s > 0. Here both aðsÞ and bðsÞ are any non-negative, continuous, convex, decreasing functions of the selling price in ½0; su ], where su is an extremely large number. The solutions to Eqs. (1) and (2) are Type 1:
kðs; NÞ ¼ aðsÞ ½aðsÞ bðsÞerN
ð3Þ
and
By using the condition that I1 ðtÞ ¼ I2 ðtÞ at t ¼ t1 , we obtain
1 kðs; NÞ hT log 1 þ ðe 1Þ h P
t1 ðs; T; NÞ ¼
ð10Þ
We use the notations t 1 ðs; T; NÞ and t 1 interchangebly throughout this paper. 3.1. Determination of total profit function We now derive the equations for sales profit, ordering cost, holding cost, deterioration cost, interest payable and interest earned. These components are evaluated as in the following: (1) Sales profit (SP): The total sales profit is given by ðs cÞkðs; NÞ (2) Ordering cost (OC): The annual ordering cost is A=T (3) Holding cost (HC): Annual stock holding cost (excluding interest charges)
Z h T IðtÞdt T 0 h ¼ 2 ðP kðs; NÞÞ½eht1 þ ht1 1 h T þkðs; NÞ½ehðTt1 Þ hðT t1 Þ 1 ¼
Type 2:
kðs; NÞ ¼ aðsÞ½1 ð1 rÞN þ bðsÞð1 rÞN
ð4Þ
respectively. The rest of our theory is fittest for both the Type 1 and Type 2 demand functions. Due to the finite replenishment rate, the inventory level gradually increases from time t ¼ 0 and it reaches the maximum at t ¼ t1 . Production then stops at t ¼ t1 and the inventory gradually depletes to zero at the end of the cycle t ¼ T due to deterioration and consumption. The graphical representation of this inventory system is clearly depicted in Fig. 1. The objective is to determine the optimal values for s; T and N (namely, s ; T and N , respectively) such that the annual profit is maximized. From the above assumptions and notations, we know that the inventory level IðtÞ at time t satisfies the following differential equations:
dIðtÞ þ hIðtÞ ¼ P kðs; NÞ; dt
if 0 6 t 6 t 1
if t1 6 t 6 T
I1 ðtÞ if 0 6 t 6 t1 I2 ðtÞ if t1 6 t 6 T
ð6Þ
ð11Þ
ð7Þ
where
P kðs; NÞ ð1 eht Þ h kðs; NÞ hðTtÞ 1Þ I2 ðtÞ ¼ ðe h
I1 ðtÞ ¼
c Pt1 ckðs; NÞ T
ð5Þ
with the boundary conditions Ið0Þ ¼ 0 and IðTÞ ¼ 0. Consequently the solutions for Eqs. (5) and (6) are given by
IðtÞ ¼
¼
(4) Interest payable (IP): Based on the values of T; N and M, there are three cases to be considered: (i) N 6 M 6 T þ N, (ii) N 6 T þ N 6 M and (iii) M 6 N 6 T þ N: Case 1: N 6 M 6 T þ N. Here, the retailer sells the units up to the end of his permissible delay period M. The items in stock are charged at interest rate Ik by the supplier starting from time M to T þ N. As a result, the interest payable per unit time is
and
dIðtÞ þ hIðtÞ ¼ kðs; NÞ; dt
(3) Annual deteriorating cost (DC)
ð8Þ ð9Þ
Z cIk kðs; NÞ TþN hðTþNtÞ ðe 1Þdt hT M cIk kðs; NÞ hðTþNMÞ e hðT þ N MÞ 1 ¼ 2 h T
Case 2: N 6 T þ N 6 M. In this case, there is no interest payable by the retailer. Case 3: M 6 N 6 T þ N. The retailer pays interest on full replenished quantity for a period from M to N and on average stock held during the cycle length. As a result, the annual interest payable is
cIk T
Z
N
M
Pt1 dt þ
Z N
TþN
kðs; NÞ hðTþNtÞ 1dt ½e h
cIk ¼ 2 h2 Pt 1 ðN MÞ þ kðs; NÞðehT hT 1Þ h T
ð12Þ
(5) Interest earned (IE): Same as the interest payable, there are three cases to be considered. Case 1: N 6 M 6 T þ N, shown in Fig. 2 (as in Jaggi et al., 2008) During the period from N to M, the retailer earns interest Ie per dollar. Therefore the annual interest earned is
Fig. 1. Graphical representation of inventory system.
sIe kðs; NÞðM NÞ2 2T
ð13Þ
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A h ½ðP kðs; NÞÞ T Th2 þ ht1 1Þ þ kðs; NÞðehðTt1 Þ hðT t 1 Þ 1Þ
TP3 ðs; T; NÞ ¼ ððs cÞkðs; NÞÞ
ðeht1 cIk 2 ½h2 Pt 1 ðN MÞ þ kðs; NÞðehT hT 1Þ Th
ð19Þ
After some significant simplifications, we have
TP1 ðs; T; NÞ ¼ ðs cÞkðs; NÞ
Case 2: N 6 T þ N 6 M, shown in Fig. 3 (as in Jaggi et al., 2008)In this case, the retailer earns interest on average sales revenues received during the period from N to T þ N and on full sales revenue from a period from T þ N to M: As a result the annual interest earned is
" # sIe kðs; NÞT 2 þ kðs;NÞTðM T NÞ ¼ sIe kðs;NÞ½M N T=2 T 2
ð14Þ TP3 ðs; T; NÞ ¼ ðs cÞkðs; NÞ
cIk Th2
ð21Þ
A ðh þ chÞ Pt1 þ kðs; NÞ T h T
½h2 Pt 1 ðN MÞ þ kðs; NÞðehT hT 1Þ
ð22Þ
4. Solution procedure
The retailer’s annual profit is
ð15Þ
From the above arguments, the annual profit is given by
8 > < TP 1 ðs; T; NÞ if N 6 M 6 T þ N TPðs; T; NÞ ¼ TP 2 ðs; T; NÞ if N 6 T þ N 6 M > : TP 3 ðs; T; NÞ if M 6 N 6 T þ N
ð20Þ
þ sIe kðs; NÞ½M N T=2
Case 3: M 6 N 6 T þ N.In this case, there is no interest earned by the retailer.
TPðs; T; NÞ ¼ ðSPÞ ðOCÞ ðHCÞ ðDCÞ ðIPÞ þ ðIEÞ
cIk kðs; NÞ
½ehðTþNMÞ hðT þ N MÞ 1 Th2 sIe kðs; NÞðM NÞ2 þ 2T A ðh þ chÞ Pt1 kðs; NÞ TP2 ðs; T; NÞ ¼ ðs cÞkðs; NÞ þ T h T
Fig. 2. Retailer’s interest earned when N 6 M 6 T þ N.
A ðh þ chÞ Pt1 þ kðs; NÞ T h T
ð16Þ
To find the optimal solution, say ðs ; T ; N Þ, for TPðs; T; NÞ, the following procedures are considered. Definition 1. A function f defined on an open interval ða; bÞ is said to be concave if for x; y 2 ða; bÞ and each k; 0 6 k 6 1; we have
f ðkx þ ð1 kÞyÞ P kf ðxÞ þ ð1 kÞf ðyÞ
where A h TP 1 ðs;T;NÞ ¼ ðs cÞkðs;NÞ 2 ðP kðs;NÞÞðeht1 þ ht 1 1Þ T Th cPt 1 hðTt1 Þ þ ckðs;NÞ hðT t 1 Þ 1Þ þkðs;NÞðe T sIe kðs;NÞðM NÞ2 cIk kðs;NÞ hðTþNMÞ e hðT þ N MÞ 1 þ 2 2T Th ð17Þ
A h ðP kðs; NÞÞðeht1 þ ht 1 1Þ T Th2 cPt 1 þkðs; NÞðehðTt1 Þ hðT t 1 Þ 1Þ T ð18Þ þckðs; NÞ þ sIe kðs; NÞ½M N T=2
TP 2 ðs; T; NÞ ¼ ðs cÞkðs; NÞ
Intermediate value theorem (in real analysis) Let g be a continuous function on the closed interval [a, b] and let gðaÞgðbÞ < 0: Then there exists a number d 2 ða; bÞ such that gðdÞ ¼ 0. Lemma 1. If f ðtÞ is a continuous function on (a,b) and if increasing, then f is concave.
df dt
is non-
Proof. Please see Appendix A. h 4.1. Determination of the optimal replenishment cycle length T for any given s and N Case 1. N 6 M 6 T þ N For the given values of s and N, the first derivative of TP1 ðTjs; NÞ with respect to T is
dTP1 ðTjs; NÞ Z 1 ðTjs; NÞ ¼ dT T2 where
! sIe kðs;NÞðM NÞ2 Z 1 ðTjs;NÞ ¼ A 2 ðch þ hÞP @t 1 T t 1 ðs;T;NÞ h @T i cIk kðs;NÞ h hðTþNMÞ hðTþNMÞ hTe e þ hðN MÞ þ 1 h2 Fig. 3. Retailer’s interest earned when N 6 T þ N 6 M.
ð23Þ
The optimal value of T, say T 1 ðs; NÞ, can be found by solving the equation Z 1 ðTjs; NÞ ¼ 0. It is easy to obtain the following derivative
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" ! # dZ 1 ðTjs; NÞ ðch þ hÞP @ 2 t1 hðTþNMÞ þ cIk kðs; NÞTe < 0; ¼ T dT h @T 2
ðTjs;NÞ Hence, Z 3 ðTjs; NÞ is non-increasing on ð0; 1Þ and so dTP3dT is nonincreasing. From Lemma 1, TP 3 ðTjs; NÞ is a concave function on ð0; 1Þ: On the other hand, we have
2
since @@Tt21 > 0 (it is easy to verify). Hence Z 1 ðTjs; NÞ is non-increasing ðTjs;NÞ on ð0; 1Þ and so dTP1dT is non-increasing. From Lemma 1, TP1 ðTjs; NÞ is a concave function on ð0; 1Þ. On the other hand, we have
lim Z 1 ðTjs; NÞ ¼ 1 < 0
T!1
Based upon the above arguments, the intermediate value theorem yields that the optimal solution T 3 ðs; NÞ, not only exists but also is unique. Combining the above three cases we have the following Lemma.
and
!
Z 1 ð0Þ ¼ A
lim Z 3 ðTjs; NÞ ¼ 1 < 0 and Z 3 ð0Þ > 0:
T!1
sIe kðs;NÞðM NÞ2 cIk kðs;NÞ ½1 þ hðN MÞ ehðNMÞ 2 h2
2 A sIe kðs;NÞðMNÞ > 0: So 2 2 we restrict our attention to the condition of A sIe kðs;NÞðMNÞ > 0, 2 Since 1 þ hðN MÞ < ehðNMÞ ; Z 1 ð0Þ > 0 if
in the rest of our mathematical analysis. Hence, we have
Lemma 2. For the given values of s and N, (1) If A sIe kðs;NÞ ðM NÞ2 > 0, then T 1 ðs; NÞ is the unique optimal 2 solution to the profit function TP1 ðTjs; NÞ. (2) TP 2 ðTjs; NÞ has the unique optimal solution T 2 ðs; NÞ on the nonnegative interval ð0; 1Þ (3) TP 3 ðTjs; NÞ has the unique optimal solution T 3 ðs; NÞ on the nonnegative interval ð0; 1Þ
lim Z 1 ðTjs; NÞ ¼ 1 < 0 and Z 1 ð0Þ > 0:
T!1
Based upon the above arguments, the intermediate value theorem yields that the optimal solution T 1 ðs; NÞ, not only exists but also is unique. The similar procedure as described in Case 1 can be applied to the remaining two cases.
Let
sIe kðs;NÞðM NÞ2 ðch þ hÞP ðM NÞkðs;NÞehðMNÞ h 2 P þ kðs;NÞðehðMNÞ 1Þ
1 kðs;NÞ hðMNÞ 1Þ ðe log 1 þ h P
D1 ðs;NÞ ¼A
Case 2. N 6 T þ N 6 M For the given values of s and N, we have
ð26Þ and
dTP 2 ðTjs; NÞ Z 2 ðTjs; NÞ ¼ dT T2 where
! sIe kðs;NÞT 2 ðch þ hÞP @t 1 T t 1 ðs;T;NÞ Z 2 ðTjs;NÞ ¼ A h 2 @T
ð24Þ
The optimal value of T, say T 2 ðs; NÞ, can be found by solving the equation Z 2 ðTjs; NÞ ¼ 0. Now, we have
" # dZ 2 ðTjs; NÞ ðch þ hÞPT @ 2 t1 < 0; ¼ sIe kðs; NÞT dT h @T 2
ðTjs;NÞ Hence, Z 2 ðTjs; NÞ is non-increasing on ð0; 1Þ and so dTP2dT is nonincreasing. From Lemma 1, TP2 ðTjs; NÞ is a concave function on ð0; 1Þ. On the other hand, we have
lim Z 2 ðTjs; NÞ ¼ 1 < 0 and Z 2 ð0Þ > 0:
T!1
Based upon the above arguments, the intermediate value theorem yields that the optimal solution T 2 ðs; NÞ, not only exists but also is unique. Case 3. M 6 N 6 T þ N For the given values of s and N, we have
dTP 3 ðTjs; NÞ Z 3 ðTjs; NÞ ¼ dT T2 where
P @t 1 Z 3 ðTjs; NÞ ¼ A ½ðch þ hÞ þ cIk hðN MÞ T t 1 ðs; T; NÞ h @T cIk kðs; NÞ hT ½hTe ehT þ 1 ð25Þ h2 The optimal value of T, say T 3 ðs; NÞ, can be found by solving the equation Z 3 ðTjs; NÞ ¼ 0. Now, we have
dZ 3 ðTjs; NÞ P @ 2 t1 cIk kðs; NÞTehT < 0; ¼ ½ðh þ chÞ þ cIk hðN MÞT dT h @T 2
P ðM NÞkðs; NÞehðMNÞ D2 ðs; NÞ ¼ A ½ðch þ hÞ þ cIk hðN MÞ h P þ kðs; NÞðehðMNÞ 1Þ
1 kðs; NÞ hðMNÞ 1Þ log 1 þ ðe h P cIk kðs; NÞ ½hðM NÞehðMNÞ ehðMNÞ þ 1 h2 ð27Þ Using the values of D1 ðs; NÞ and D2 ðs; NÞ, we have the following Theorem. Theorem 1. For the fixed values of s and N, (1) If D1 ðs; NÞ P 0, then T ðs; NÞ ¼ T 1 ðs; NÞ. (2) If D1 ðs; NÞ 6 0, then T ðs; NÞ ¼ T 2 ðs; NÞ. (3) If D2 ðs; NÞ P 0 and M N < 0 then T ðs; NÞ ¼ T 3 ðs; NÞ.
Proof. Please refer Appendix B for details. h 4.2. Determination of optimal selling price s for any given value of T and N For any given values of T and N, the optimal value of s can be determined by solving the first order necessary condition (i.e., @TP ¼ 0) and examining the second order sufficient condition for @s 2
concavity (i.e.,@@sTP 2 < 0). Case 1. N 6 M 6 T þ N For the given values of T and N, the first order and second order partial derivatives of TP 1 ðs; T; NÞ with respect to s are given by
@TP 1 @k Ie ðM NÞ2 @k ðh þ chÞ @k P @t 1 ¼ ðs cÞ þ k þ s þk þ @s @s h @s T @s @s 2T cIk hðTþNMÞ @k 2 ½e hðT þ N MÞ 1 ð28Þ @s Th
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and 2
"
2
#
2
"
2
Algorithm 1. To find the optimal value of s and T for a given value of N, we adopt the following steps.
#
@ TP1 @ k @k Ie ðM NÞ @ k @k ðh þ chÞ þ þ ¼ ðs cÞ 2 þ 2 s 2 þ2 @s @s @s @s h @s2 2T " # 2 2 @ k P @ t1 cIk hðTþNMÞ @2k ½e hðT þ N MÞ 1 @s2 T @s2 @s2 Th2 ð29Þ Case 2. N 6 T þ N 6 M For the given values of T and N, the first order and second order partial derivatives of TP2 ðs; T; NÞ with respect to s are given by
@TP 2 @k @k ¼ ðs cÞ þ k þ Ie ðM N T=2Þ s þ k @s @s @s ðh þ chÞ @k P @t 1 þ h @s T @s
ð30Þ
and
" # @ 2 TP2 @2k @k þ Ie ðM N T=2Þ ¼ ðs cÞ 2 þ 2 @s @s @s2 " # " # @2k @k ðh þ chÞ @ 2 k P @ 2 t1 þ s 2 þ2 @s @s h @s2 T @s2
ð31Þ
Case 3. M 6 N 6 T þ N For the given values of T and N, the first order and second order partial derivatives of TP3 ðs; T; NÞ with respect to s are given by
@TP 3 @k ðh þ chÞ @k P @t1 ¼ ðs cÞ þ k þ @s h @s T @s @s cIk 2 @t 1 @k hT 2 h P ðN MÞ þ ðe hT 1Þ @s @s Th
ð32Þ
þ kðsÞ ¼ 0. 1. Let j ¼ 1. Find sl by solving the equation ðs cÞ @k @s 2. Find T ij ,(i ¼ 1 or 2 or 3) by using the value of sl and Theorem 1. Let sij ¼ sl . i ¼ 0 for s by using the value T ¼ T ij . Find ^si 3. Solve the equation @TP @s such that
@ 2 TP i j @s2 ðT¼T ij ;s¼^si Þ i equation @TP @T
< 0 and let si;jþ1 ¼ ^si .
4. Solve the ¼ 0 for T by using the value s ¼ si;jþ1 and let the solution be T i;jþ1 . 5. If jT ij T i;jþ1 j < and jsij si;jþ1 j < , where is any small positive number, then s ¼ si;jþ1 and T ¼ T i;jþ1 . Otherwise let j = j+1, go to step 3.
Algorithm 2. For finding the optimal solution ðs ; T ; N Þ, the following steps can be followed. 1. Let N ¼ 1 and TP i ðsð0Þ ; T ð0Þ ; 0Þ ¼ 0 for i ¼ 1; 2; 3. 2. Determine the optimal solution ðsðNÞ ; T ðNÞ Þ from Algorithm 1. 3. If N 6 M 6 T ðNÞ þ N then calculate TP 1 ðsðNÞ ; T ðNÞ ; NÞ; else go to step 5. 4. If TP1 ðsðN1Þ ; T ðN1Þ ; N 1Þ > TP 1 ðsðNÞ ; T ðNÞ ; NÞ then the optimum solution, say ðs ; T ; N Þ, is ðsðN1Þ ; T ðN1Þ ; N 1Þ and TP ¼ TP1 ðs ; T ; N Þ. Otherwise, let N ¼ N þ 1 and go to step 2; 5. If N 6 T ðNÞ þ N 6 M then calculate TP 2 ðsðNÞ ; T ðNÞ ; NÞ; else go to step 7. 6. If TP2 ðsðN1Þ ; T ðN1Þ ; N 1Þ > TP 2 ðsðNÞ ; T ðNÞ ; NÞ then the optimum solution, say ðs ; T ; N Þ, is ðsðN1Þ ; T ðN1Þ ; N 1Þ and TP ¼ TP2 ðs ; T ; N Þ. Otherwise, let N ¼ N þ 1 and go to step 2; 7. If M 6 N 6 T ðNÞ þ N then calculate TP3 ðsðNÞ ; T ðNÞ ; NÞ. 8. If TP3 ðsðN1Þ ; T ðN1Þ ; N 1Þ > TP 3 ðsðNÞ ; T ðNÞ ; NÞ then the optimum solution, say ðs ; T ; N Þ, is ðsðN1Þ ; T ðN1Þ ; N 1Þ and TP ¼ TP3 ðs ; T ; N Þ. Otherwise, let N ¼ N þ 1 and go to step 2.
and
" # " # @ 2 TP2 @2k @k ðh þ chÞ @ 2 k P @ 2 t 1 þ ¼ ðs cÞ 2 þ 2 @s @s h @s2 T @s2 @s2 " # cIk 2 @ 2 t 1 @ 2 k hT 2 h P 2 ðN MÞ þ 2 ðe hT 1Þ @s @s Th
5. Special case
ð33Þ
Note that TP i ðsjN; TÞ ði ¼ 1; 2; 3Þ are continuous functions of s over the compact set ½0; su , where su is an extremely large number. Hence TPi ðsjN; TÞ ði ¼ 1; 2; 3Þ has a maximum value. It is clear that TPi ðsjN; TÞ ði ¼ 1; 2; 3Þ is not maximum if s ¼ 0 or su . As a result, the optimal s must be an interior point between 0 and su . This implies that Eqs. (28), (30) and (32) have at least one solution. If the solution is unique, then it is the optimal solution (i.e. the Eqs. (28), (30) and (32) are necessary and sufficient conditions). Otherwise, we have to find the one at which the second order partial derivative is less than zero. Although we are not able to prove the uniqueness of the solution, the numerical examples 1 and 2 show that a unique solution exists for the considered optimization problem. We can easily calculate the Hessain matrix of TP as a negative definite although it is not straightforward to find a closed form solution when both the price and replenishment time are decision variables. Following the solution approach given in Chang, Teng, Ouyang, þ kðsÞ ¼ 0, say sl ; is the and Dye (2006), the solution of ðs cÞ @k @s lower bound for the optimal selling price s . The solution sl has been taken as a initial value for s in Algorithm 1. To find the optimal solution ðs ; T ; N Þ, we run Algorithms 1 and 2 simultaneously using Matlab 7.0. In order to find the optimal solutions, the following algorithms are used.
Here, we let h ! 0; P ! 1 and demand to be a function of credit period alone i.e., s is fixed. When the deterioration is ignored, there is no cost due to deteriorated units. If the selling price (s) is fixed then the demand is kðNÞ. By the above parametric considerations, we have limh!0þ t1 ¼ 0. Adopting above conditions, Eqs. (17)–(19) become as follows:
A hkT cIk kðT þ N MÞ2 T 2 2T 2 sIe kðM NÞ þ 2T A hkT þ sIe kðM N T=2Þ TP2 ðT; NÞ ¼ ðs cÞk T 2 A hkT cIk kðN M þ T=2Þ TP3 ðT; NÞ ¼ ðs cÞk T 2
TP1 ðT; NÞ ¼ ðs cÞk
ð34Þ ð35Þ ð36Þ
The Eqs. (34)–(36) are consistent with Eqs. (10), (12) and (14) in Jaggi et al. (2008) (where IC ¼ h). Consequently, we obtain that
sIe kðM NÞ2 hkT 2 2 2 cIk k ðT þ N MÞðT N þ MÞ 2 sIe kT 2 hkT 2 Z 2 ðT; NÞ ¼ A 2 2 2 hkT cIk kT 2 Z 3 ðT; NÞ ¼ A 2 2 Z 1 ðT; NÞ ¼ A
ð37Þ ð38Þ ð39Þ
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779
From Eqs. (37)–(39), we obtain the following,
kðM NÞ2 ½sIe þ h 2 2 kðM NÞ D2 ðNÞ ¼ A ½h þ cIk 2
D1 ðNÞ ¼ A
From the above, Theorem 1 becomes Theorem 2. For the fixed values N, (1) If D1 ðNÞ P 0, then T ðNÞ ¼ T 1 ðNÞ. (2) If D1 ðNÞ 6 0, then T ðNÞ ¼ T 2 ðNÞ. (3) If D2 ðNÞ P 0 and M N < 0 then T ðNÞ ¼ T 3 ðNÞ. These results of Theorem 2 are identical to Theorem 1 in Jaggi et al. (2008). Thus, Jaggi et al. (2008) can be treated as a special case of our paper. 6. Computational analyses The purposes of the numerical analysis are as follows:
Fig. 5. The optimal total profit for various values of s in type-1 and type-2 demands.
1. To obtain the optimal solutions for two types of demand functions. 2. To use sensitivity analysis to highlight the influence of model parameters.
is indeed the global optimum solution. Using the Algorithms 1 and 2, in Matlab 7.0, we obtained the following optimal results: optimal cycle length ðT Þ ¼ 17:92 days, optimal credit period offered by the retailer to his customers, N , is 17 days, optimal selling price s ¼ $48:77 per unit and the optimal total profit is $335.850.
6.1. Numerical examples
Example 2. Here, we consider the same data set as in numerical Example 1 but the demand function is type-2, i.e., kðs; NÞ ¼ aðsÞ½1 ð1 rÞN þ bðsÞð1 rÞN . The three dimensional graphs of total profit of the retailer are presented in Fig. 7. The graphs reveal that for any given N (e.g., N ¼ 8; 15; 26), there exists a corresponding optimal solution ðsðNÞ ; T ðNÞ Þ which maximizes the total profit. Furthermore, we run the numerical results with values of N ¼ 1; 2; . . . ; 100. The numerical results indicate that there is a unique integer N which maximizes the value of TPðNÞ ¼ TPðN; sðNÞ ; T ðNÞ Þ, as shown in Fig. 4b. Also, Fig. 5b shows that TPðsjN ; T Þ is strictly concave function of s. As a result, we are sure that the optimum obtained from the proposed algorithms is indeed the global optimum solution for the type-2 demand function also. Using the Algorithms 1 and 2, in Matlab 7.0, we obtained the following optimal results: optimal cycle length ðT Þ ¼ 17:90 days, optimal credit period offered by the retailer to his customers, N , is 15 days, optimal selling price s ¼ $48:70 per unit and the optimal total profit is $338.18.
Example 1. We consider the demand type-1, i.e., kðs; NÞ ¼ aðsÞ ½aðsÞ bðsÞerN . Let aðsÞ ¼ 80 1:21s; bðsÞ ¼ 30 1:21 s; r ¼ 0:35, production rate P ¼ 100; h ¼ 0:01; A ¼ $1000; M ¼ 30 days, h ¼ $4:5 per unit, c ¼ $30 per unit, Ie ¼ 25% per year, Ik ¼ 15% per year. The three dimensional graphs of total profit of the retailer are presented in Fig. 6. The graphs reveal that for any given N (e.g., N ¼ 8; 17; 26), there exists a corresponding optimal solution ðsðNÞ ; T ðNÞ Þ which maximizes the total profit. Furthermore, we run the numerical results with values of N= 1,2,. . .,80. The numerical results indicate that there is a unique integer N which maximizes the value of TPðNÞ ¼ TPðN; sðNÞ ; T ðNÞ Þ, as shown in Fig. 4a. Also, Fig. 5a shows that TPðsjN ; T Þ is strictly concave function of s. As a result, we are sure that the maximum obtained from the proposed algorithms
6.2. Effect of changing the inventory model parameters Here, we consider the demand function type-2 (one can also consider type-1). Let aðsÞ ¼ 80 1:21s; bðsÞ ¼ 30 1:21s; r ¼ 0:35, production rate P ¼ 100; h ¼ 0:01; A ¼ $1000; M ¼ 30 days, h ¼ $4:5 per unit, c ¼ $30 per unit, Ie ¼ 10% per year, Ik ¼ 15% per year. The sensitivity analysis is performed by varying different parameters and is given in Table 1. It is important to discuss the influence of key model parameters on the optimal solutions. The effect of changing the parameters are shown graphically in Figs. 8–14. Based on these figures, we have the following comments.
Fig. 4. The optimal total profit for various values of N in type-1 and type-2 demands.
1. It is observed that as r increases, N decreases and T marginally decreases; but TP increases. The optimal selling price remains at the threshold. It shows that retailer should offer lower credit period ðNÞ to customers when the rate of saturation of demand ðrÞ is higher (see Fig. 8).
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Fig. 6. The total profit for any given N (e.g., N = 8, 17, and 26) when demand is type-1.
Fig. 7. The total profit for any given N (e.g., N = 8, 15, and 26) when demand is type-2.
2. A higher value of M causes a higher value of TP , but lower value of s and T . It indicates the following managerial phenomena: when the supplier provides a longer credit period, the retailer
replenishes the goods more often. In other words, the retailer will shorten the cycle time and reduce the selling price in order to take advantage of the longer credit period (see Fig. 9).
A. Thangam, R. Uthayakumar / Computers & Industrial Engineering 57 (2009) 773–786 Table 1 Sensitivity analysis for various inventory model parameters. Parameter
N
T
s
TP
4.
r
0.32 0.35 0.38 0.41
18 17 16 14
18.542 18.537 18.534 18.532
48.869 48.869 48.869 48.857
337.025 337.578 338.023 338.464
M
30 35 40 45
17 17 17 17
18.537 18.529 18.521 18.513
48.870 48.848 48.825 48.802
337.578 338.995 340.414 341.833
5.
800 1000 1200 1400
17 17 17 17
16.610 18.537 20.275 21.867
48.776 48.871 48.957 49.036
343.808 337.578 331.913 326.674
6.
Ie
0.10 0.15 0.20 0.25
17 16 15 15
18.537 18.320 18.110 17.904
48.871 48.837 48.796 48.772
337.578 337.653 337.875 338.188
P
100 120 140 160
17 17 17 17
18.537 18.137 17.869 17.677
48.870 48.960 49.029 49.077
337.578 336.127 335.104 334.343
h
0.01 0.03 0.05 0.07
17 17 17 17
18.537 10.853 8.370 7.034
48.869 49.487 49.927 50.293
337.578 293.088 259.300 230.027
h
4.5 6 8 10
17 17 17 17
18.537 18.428 18.286 18.147
48.869 48.876 48.884 48.893
337.578 337.220 336.760 336.302
A
3. As ordering cost, A, increases, the replenishment cycle time T significantly increases; but optimal selling price marginally increases. Keeping the credit period ðN Þ at some threshold, the optimal profit decreases as A increases. It indicates the following managerial effect. If the ordering cost is higher, it is rea-
7.
781
sonable that the retailer lengths the cycle time to reduce the frequency of replenishment and he marginally increases the selling price (see Fig. 10). As Ie increases, N decreases; but T and s are marginally decreasing. Profit TP increases as interest earned rate of the retailer increases. It implies that if the retailer increases his interest earned rate, then he can shorten his trade credit period with marginally less selling price (see Fig. 11). As the value of h increases, TP and T decrease whereas s is increasing. That is, when the items are starting to deteriorate, it is optimal to rise marginally the selling price in order to manage the profit (see Fig. 12). When holding increases, it is seen that cycle length and profit decreases whereas the optimal selling price increases. So it is reasonable that when the holding cost increases the retailer will shorten the cycle time and increases the selling price in an effort to maintain his profit gained by keeping the threshold credit period ðN Þ (see Fig. 13). As production rate increases, TP decreases; but there are marginal increases in selling price, s , under the optimal threshold value of N . So it is not advisable to increase the production rate without the prior knowledege about the demands (see Fig. 14).
6.3. When customer’s payment exceeds the optimal credit period ðN Þ If a customer violates the payment condition, then the retailer can fix some another duration for payment with increased interest charges ð> Ik Þ. Using trade credit insurance, retailers can generally extend more credit to customers whilst reducing the risk of nonpayment, thereby enabling sales growth without a corresponding increase in risk. Insurance can also enable the retailers economy to secure more favorable financing terms, as insured accounts receivable may be used as collateral.
Fig. 8. Effect of changing saturation rate of retailer’s demand.
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Fig. 9. Effect of changing supplier’s credit period.
Fig. 10. Effect of changing ordering cost (A).
7. Conclusions and future research The increased prominence of financial markets and their widespread use in pricing and in credit period allocation provides an
opportunity for convergence between financial tools and operational problems. The latter is based mostly on a private valuation of profits and benefits associated with the trade credit policy. In this paper, we first formulated an EPQ – based inventory model
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783
Fig. 11. Effect of changing interest earned rate by the retailer.
Fig. 12. Effect of changing the deterioration rate.
for per ishable items under two-echelon trade credit policy with the assumptions that the market demand is sensitive to both the selling price and credit period offered by the retailer. After formulating the mathematical model, we then developed solution proce-
dures to determine the best payment method, the optimal selling price and optimal cycle length for the retailer. From the numerical Examples 1 and 2, we examined that the proposed algorithms provide global optimum solutions for both types of demand functions.
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Fig. 13. Effect of changing the holding cost.
Fig. 14. Effect of changing the replenishment rate (P).
The managerial implications of numerical results are clear and provide a suitable f ramework to assess the relative profitability. Finally, Jaggi et al. (2008) model becomes as a special case to this paper. In future research, our model can be extended to more general
supply chain networks, for example, multi-echelon or assembly supply chains. One can also extend this model for two-part trade credit term i.e., supplier offers two payment options: trade credit and early-payments with discount price to the retailer.
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Acknowledgements We sincerely express our gratitude to the two anonymous referees for their valuable comments and suggestions. This research work is fully supported by Senior Research Fellowship (Grant No: 09/ 715(0002)/2006 EMR-I) under Counsil of Scientific and Industrial Research (CSIR) – India. We also thank University Grants Commision, India for providing Special Assistance Program (UGC – SAP). Appendix A Proof of the Lemma 1. Proof. Given x; y with a < x < y < b, define a function g on [0, 1] by
gðtÞ ¼ f ðty þ ð1 tÞxÞ tf ðyÞ ð1 tÞf ðxÞ: Our goal is to show that g is non-negative on [0, 1]. Now g is continuous and gð0Þ ¼ gð1Þ ¼ 0. Moreover
dgðtÞ df ¼ ðy xÞ f ðyÞ þ f ðxÞ dt dt
785
Hence, if D1 ðs; NÞ 6 0, then T ðs; NÞ ¼ T 2 is the optimal solution of TPðTjs; NÞ. For the fixed values of s and N, there exists a unique value T (say T 3 ðs; NÞ) which maximizes TP3 ðTjs; NÞ and T 3 would satisfy the condition M N 6 0 6 T. Since Z 3 ðTjs; NÞ is non-increasing on ð0; 1Þ Z 3 ðM NÞ P Z 3 ðT 3 Þ ¼ 0 i.e. Z 3 ðM NÞ P 0. Now,
P ðM NÞkðs;NÞehðMNÞ Z 3 ðM NÞ ¼A ½ðch þ hÞ þ cIk hðN MÞ h P þ DðehðMNÞ 1Þ
1 kðs;NÞ hðMNÞ 1Þ log 1 þ ðe h P cIk kðs;NÞ ½hðM NÞehðMNÞ ehðMNÞ þ 1 h2
ð40Þ
So, D2 ðs; NÞ ¼ Z 3 ðM NÞ. Hence, if D2 ðs; NÞ P 0; T ðs; NÞ ¼ T 3 is the optimal solution of TPðTjs; NÞ. Combining the three possible cases, we obtain the Theorem 1. h References
For t þ h > t,
dgðt þ hÞ dgðtÞ df ðt þ hÞ df ðtÞ : ¼ ðy xÞ dt dt dt dt Since
df dt
is non-increasing,
df ðt þ hÞ df ðtÞ < 0: dt dt It implies that dgðtÞ is non-increasing on [0, 1]. Let c be a point where dt g assumes its minimum on [0, 1]. If c ¼ 1, then gðtÞ P gð1Þ ¼ 0 on [0, 1]. In this case, g is non-negative. Suppose that c 2 ð½0; 1Þ. Since P 0. But dg was nong has a local minimum at c, we have dgðcÞ dt dt P 0 on ½0; c. Consequently g is non-decreasing increasing and so dg dt on ½0; c and hence gðcÞ 6 gð0Þ ¼ 0 then the minimum of g on [0, 1] is non-negative and so g P 0 on [0, 1]. h
Appendix B Proof of the Theorem 1 Proof. For the given values of s and N; TP 2 ðs; T; NÞ and TP 3 ðs; T; NÞ are concave on T > 0. However, TP 1 ðs; T; NÞ is concave on T > 0 if ðM NÞ2 > 0. Thus there exists a unique value of T (say A sIe kðs;NÞ 2 T 1 ðs; NÞ) which maximizes TP1 ðTjs; NÞ and T 1 would satisfy the inequality 0 6 M N 6 T. Since Z 1 ðTjs; NÞ is non-increasing on ð0; 1Þ, we have that Z 1 ðM NÞ P Z 1 ðT 1 Þ ¼ 0: That is, Z 1 ðM NÞ P 0. Now,
sIe kðs;NÞðM NÞ2 ðch þ hÞP Z 1 ðM NÞ ¼ A h 2
hðMNÞ ðM NÞkðs;NÞe 1 kðs;NÞ hðMNÞ 1Þ ðe log 1 þ hðMNÞ P P þ kðs;NÞðe 1Þ h So, D1 ðs; NÞ ¼ Z 1 ðM NÞ. If D1 ðs; NÞ P 0, then T ðs; NÞ ¼ T 1 is the optimal solution of TPðTjs; NÞ. For the fixed values of s and N, there exists a unique value T (say T 2 ðs; NÞ) which maximizes TP 2 ðTjs; NÞ and T 2 ðs; NÞ would satisfy the condition 0 6 T 6 M N: Since Z 2 ðTjs; NÞ is non-increasing on ð0; 1Þ; Z 2 ðM NÞ 6 Z 2 ðT 2 Þ ¼ 0 i.e. Z 2 ðM NÞ 6 0. Now,
sIe kðs;NÞðM NÞ2 ðch þ hÞP Z 2 ðM NÞ ¼ A 2 h
hðMNÞ ðM NÞkðs;NÞe 1 kðs;NÞ hðMNÞ 1Þ log 1 þ ðe P þ kðs;NÞðehðMNÞ 1Þ h P ¼ D1 ðs;NÞ
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