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A note on “Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime”
European Journal of Operational Research 239 (2014) 868–871
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Short Communication
A note on ‘‘Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime’’ Chung-Yuan Dye a, Chih-Te Yang b,⇑, Fang-Cheng Kung a a b
Graduate School of Business and Administration, Shu-Te University, Yen-Chao, Kaohsiung 824, Taiwan Department of Industrial Management, Chien Hsin University of Science and Technology, Taoyuan 320, Taiwan
a r t i c l e
i n f o
Article history: Received 22 March 2014 Accepted 25 June 2014 Available online 3 July 2014 Keywords: Supply chain management Inventory Maximum lifetime Trade credit
a b s t r a c t In 2014, Wang et al. (2014) extended the model of Lou and Wang (2012) to incorporate the credit period dependent demand and default risk for deteriorating items with maximum lifetime. However, the rates of demand, default risk and deterioration in the model of Wang et al. (2014) are assumed to be specific functions of credit period which limits the contributions. In this note, we first generalize the theoretical results of Wang et al. (2014) under some certain conditions. Furthermore, we also present some structural results instead of a numerical analysis on variation of optimal replenishment and trade credit strategies with respect to key parameters. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Trade credit is an increasingly important payment behavior in real business transactions. Since the trade credit management needs to balance the trade-off between increased sales and the risk of granting credit, Teng and Lou (2012) first incorporated the creditlinked demand and default risks to establish an EOQ model for the retailer in a supply chain with up-stream and down-stream trade credits. In an article published in Journal of the Operational Research Society, Lou and Wang (2012) extended the model of Teng and Lou (2012) by introducing the credit-linked demand and default risks. Later, Wang, Teng, and Lou (2014) extended the model of Lou and Wang (2012) for deterioration with a maximum lifetime. Base on the model of Teng and Lou (2012), Wu, Ouyang, Cárdenas-Barrón, and Goyal (2014) subsequently built an EOQ model for deteriorating items with expiration dates under two-level trade credit financing. However, the rates of demand and default risk in the abovementioned studies are assumed to be specific functions of the credit period. In this short note, we first generalize the theoretical results of Lou and Wang (2012) and Wang et al. (2014) under some certain conditions. Next, a sensitivity analysis with respect to key parameters are further performed in rigorous discussions. 2. The model For easy tractability with Wang et al. (2014), we use the same notations and assumptions. Further, in order to generalize the ⇑ Corresponding author. Tel.: +886 3 4581196x6123. E-mail address: [email protected] (C.-T. Yang). http://dx.doi.org/10.1016/j.ejor.2014.06.037 0377-2217/Ó 2014 Elsevier B.V. All rights reserved.
models of Lou and Wang (2012) and Wang et al. (2014), we make assumptions on the rates of demand and default risk. Let n be the credit period offered by the retailer to its customers, the demand is given by a demand function DðnÞ, which is assumed to be nonnegative, continuous and twice differentiable. Because the trade credit allows customers to enjoy the benefits of delayed payments, lengthening the credit period will stimulate sales. The longer the credit period, the higher the demand; hence, the demand strictly increases in the credit period, that is D0 ðnÞ > 0. Meanwhile, since a longer credit period may tie up the retailer’s capital and then increase the risk of default, the rate of default risk giving the credit period n is assumed to be FðnÞ, where Fð0Þ ¼ 0; 0 6 FðnÞ < 1 and F 0 ðnÞ > 0. Hence, the total profit per unit time constructed by Wang et al. (2014) can be reviewed as follows:
o T
Pðn; TÞ ¼ pDðnÞ½1 FðnÞ
hDðnÞ T
Z 0
T
Z
cDðnÞ T
Z
T
edðtÞ dt
0
T
edðuÞdðtÞ dudt;
ð1Þ
t
Rz where dðzÞ ¼ 0 hðuÞdu and hðtÞ is the time-varying deterioration rate at time t with 0 6 hðtÞ 6 1. Then the problem can be formulated as max0 n, we can observe that dDðnÞfp½1FðnÞ ¼ D0 ðnÞ dn fp½1 FðnÞ cg DðnÞF 0 ðnÞ < 0, which represents the gross sale revenue after the default risk decreases strictly as n increases. By ðn;TÞ , and using the result, we can then obtain that dPdn < 0 for n > n
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C.-Y. Dye et al. / European Journal of Operational Research 239 (2014) 868–871
is the upper bound for the optimal value n . Hence, we can so n from further consideration, and the probexclude the region n > n lem can be then rewritten as max0 0, then a unique n 2 ½0; n such that Pðn; TÞ is maximized.
Theorem 2. For a given n, a unique T 2 ð0; m exists such that Pðn; TÞ is maximized. Proof. Assuming there is an interior solution, the first order condition is
RT dPðn; TÞ o cDðnÞ 0 edðTÞ edðtÞ dt ¼ 2 dT T2 T n R o RT RT T dðTÞdðtÞ hDðnÞ T 0 e dt 0 t edðuÞdðtÞ dudt ¼ 0: ð3Þ T2
Proof. If DðnÞ and 1 FðnÞ are both log-concave in n, we have 2 2 ½F 0 ðnÞ D00 ðnÞDðnÞ D0 ðnÞ 6 0 and F 00 ðnÞ P 1FðnÞ , respectively. Fur-
Substituting (3) into the second order derivative and simplifying yields
½1FðnÞ thermore, dDðnÞdn > 0 implies that D0 ðnÞ½1 FðnÞ DðnÞ 0 F ðnÞ > 0. Assuming there is an interior solution, the first order condition is
d Pðn; TÞ
dPðn; TÞ ¼ p D0 ðnÞ½1 FðnÞ DðnÞF 0 ðnÞ dn Z T Z T Z T D0 ðnÞ c edðtÞ dt þ h edðuÞdðtÞ dudt ¼ 0: ð2Þ T 0 0 t Substituting these results and (2) into the second order derivative, it follows that 2
d Pðn; TÞ ¼ p½1 FðnÞD00 ðnÞ 2pD0 ðnÞF 0 ðnÞ pDðnÞF 00 ðnÞ dn2 Z T Z T Z T D00 ðnÞ c edðtÞ dt þ h edðuÞdðtÞ dudt T 0 0 t 0 2 F ðnÞ 6 p½1 FðnÞD00 ðnÞ 2pD0 ðnÞF 0 ðnÞ þ pDðnÞ 1 FðnÞ D00 ðnÞ 0 D ðnÞ½1 FðnÞ DðnÞF 0 ðnÞ p 0 D ðnÞ pF 0 ðnÞ ¼ DðnÞF 0 ðnÞ D0 ðnÞ½1 FðnÞ 1 FðnÞ 2 o pF 0 ðnÞ n 00 D ðnÞDðnÞ D0 ðnÞ < 0: þ 0 D ðnÞ
2
dT 2
¼
n o RT DðnÞ cedðTÞ hðTÞ þ h þ hhðTÞ 0 edðTÞdðtÞ dt T
:
ð4Þ
2
ðn;TÞ Since hðTÞ > 0 for all T 2 ð0; mÞ, we then have d P < 0. Because it dT 2 is strictly concave at any interior critical point, Pðn; TÞ is strictly pseudoconcave in T. Furthermore, since a strictly pseudoconcave function can be either unimodal or monotone, we discuss the two situations separately below. By L’Hospital’s rule, we have
R T dðTÞ e edðtÞ dt 0
lim
T
T!0
2
¼ lim T!0
hðTÞedðTÞ 1 ¼ hð0Þ 2 2
and
lim
T
RT 0
edðTÞdðtÞ dt
RT RT
T
T!0
0 2
t
edðuÞdðtÞ dudt
¼ lim
1þhðTÞ
T!0
RT
edðTÞdðtÞ dt 1 ¼ : 2 2
0
Combining these result yields
lim T!0
If
dPðn; TÞ o cDðnÞhð0Þ hDðnÞ ¼ lim 2 ¼ 1: T!0 T dT 2 2
dPðn;TÞ dT
T¼m
ðn;TÞ < 0, since limT!0 dPdT > 0; Pðn; TÞ is unimodal in T.
Therefore, the solution to (2) is the unique interior maximum. Con dPðn;TÞ ðn;TÞ P 0, since limT!0 dPdT > 0; Pðn; TÞ is strictly dT
According to Theorems 3.4.7 in Cambini and Martein (2009), since
versely, if
Pðn; TÞ is strictly concave at any interior critical point, Pðn; TÞ is
monotonic increasing in T, and it is straightforward to see that T ¼ m. Combining the two arguments stated above, we can conclude that there exists a unique global maximum for Pðn; TÞ. This completes the proof of Theorem 2. h
strictly pseudoconcave in n. Since a strictly pseudoconcave function can be either unimodal or monotone, we discuss the two situations separately below. If dPðn;TÞ ðn;TÞ > 0, since dPdn < 0, Pðn; TÞ is unimodal in n. dn n>n
n¼0
Therefore, the solution to (2) is the unique interior maximum. ðn;TÞ ðn;TÞ 6 0, since dPdn < 0, Pðn; TÞ is strictly Conversely, if dPdn n¼0
n>n
monotonic decreasing in n, and it is straightforward to see that n ¼ 0. Combining the two arguments stated above, we can conclude that there exists a unique global maximum for Pðn; TÞ. This completes the proof of Theorem 1. h
Remark 1. If dPðn;TÞ dn
dDðnÞ½1FðnÞ dn
6 0, it is obvious to see from (2) that
< 0 for a given T. Therefore, Pðn; TÞ is maximized at n ¼ 0. ½1FðnÞ The intuition of the condition dDðnÞdn > 0 in Theorem 1 is sim1FðnÞ ple. Once dDðnÞ½dn 6 0, it represents that the marginal revenue after the default risk is less than or equal to zero. It is no benefit for the retailer to offer the customer a credit period. And thus, the retailer should shorten its credit period as much as possible. We next state our second structural result as the following theorem, which proves that for arbitrary deterioration rate, the optimal replenishment strategy is unique.
T¼m
From the analysis carried out so far, it follows that, if dDðnÞ½1FðnÞ dn
6 0, we can obtain from Remark 1 that max0
½1FðnÞ > 0, since guaranteed by Theorem 2. On the other hand, if dDðnÞdn the optimal value of T can be uniquely determined for a given n, (3) implicitly defines T as a function of n. Let P ðn; TðnÞÞ denote the maximum value function of P for a given n, then the problem becomes max06n6n P ðn; TðnÞÞ. By Berge’s Theorem of the Maximum, since the maximum value function P ðn; TðnÞÞ is continuous , the existence of a n in in the closed and bounded interval ½0; n which maximizes P ðn; TðnÞÞ is guaranteed by Weierstrass’ ½0; n > 0 and P ð0; Tð0ÞÞ > 0, theorem. In addition, if dP ðn;TðnÞÞ dn n¼0
; Tðn ÞÞ < 0, there exists at least one n 2 ð0; n Þ such because P ðn
¼ 0. However, since we cannot show the concavity that dP ðn;TðnÞÞ dn property of P ðn; TðnÞÞ, one can solve P ðn; TðnÞÞ many times from distinct starting values, and take the highest total profit obtained as the global maximum to avoid saddle points or local maximum.
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C.-Y. Dye et al. / European Journal of Operational Research 239 (2014) 868–871
(4) Taking the mixed derivative of Pðn; TÞ with respect to n, T; c and h yields
3. Optimal replenishment time and trade credit strategies Although we know the optimal replenishment time and trade credit strategies exist from the above discussions, we do not have any further information about the system behavior. In this section we study how the optimal replenishment time and trade credit strategies vary with respect to the key parameters by using the property of supermodular function. In the following, we first introduce the definition of supermodular function.
@ 2 Pðn; TÞ DðnÞ ¼ @T@c
hR
T 0
RT
0
2
edðtÞ dt TedðTÞ T2 edðtÞ dt
i
¼
hR i T DðnÞ 0 edðtÞ edðTÞ dt T2
< 0;
D ðnÞ 0 @ Pðn; TÞ ¼ < 0; @n@c nR TR o RT T T dðuÞdðtÞ dudt T 0 edðTÞdðtÞ dt @ 2 Pðn; TÞ DðnÞ 0 t e ¼ <0 @T@h T2
Definition 1. A function F : X ! R is supermodular if
F ðx _ x0 Þ þ F ðx ^ x0 Þ P F ðxÞ þ F ðx0 Þ; 8x; x0 2 A;
and
where
D ðnÞ @ 2 Pðn; TÞ ¼ @n@h
0
x _ x0 ¼ max x1 ; x01 ; max x2 ; x02 ; . . . ; max xN ; x0N and
x ^ a ¼ min x1 ; x01 ; min x2 ; x02 ; . . . ; min xN ; x0N : 0
2
ðxÞ If F is a smooth function, F is supermodular if and only if @@xF@x P 0. i j Furthermore, if the inequality is strict, then F is strictly supermodular. We now state our third structural result as the following proposition.
Proposition 1. Suppose that the conditions in Theorem 1 are met, then: (1) The optimal value of T decreases as n increases. (2) The optimal value of n increases, but the optimal value of decreases as p increases. (3) The optimal value of n decreases, but the optimal value of increases as o increases. (4) For a given T, the optimal value of n decreases as c or increases. Likewise, for a given n, the optimal value of decreases as c or h increases.
0
@ Pðn; TÞ cD ðnÞ ¼ @n@T þ
RT 0
T h T
@ 2 Pðn;TÞ @n@T
:
< 0. According to Lemma 1 in Amir et al.
@ Pðn;TÞ @n@T
< 0 implies that Pðn; TÞ is supermodular in ðn; TÞ. (2005), Hence the optimal value of T is a decreasing function of n. (2) Taking the mixed derivatives of Pðn; TÞ with respect to n; T and p gives @
Pðn;TÞ @T@p
¼ 0 and @
2
Pðn;TÞ @n@p
¼ D0 ðnÞ½1 FðnÞ DðnÞF 0 ðnÞ > 0. 2
Pðn;TÞ < 0, we obtain that Combining these with the fact that @ @n@T Pðn; TÞ is supermodular in ðn; T; pÞ. Therefore, the optimal value of n increases, but the optimal value of T decreases as p increases. (3) Taking the mixed derivative of Pðn; TÞ with respect to n, T
and o yields
@ 2 Pðn;TÞ @T@o @ 2 Pðn;TÞ @n@T
¼
1 T2
> 0 and
@ 2 Pðn;TÞ @n@o
Finally, the rates of default risk and deterioration defined in the model of Wang et al. (2014) are used to study the impacts of default risk and deterioration on the optimal replenishment time and trade credit strategies. We then state the structural results on the variation of optimal replenishment time and trade credit strategies with respect to b and m as the following proposition. 1 Proposition 2. If DðnÞ ¼ ke ; FðnÞ ¼ 1 ebn and hðtÞ ¼ 1þmt with a > b, then
(1) The optimal value of n decreases, but T increases as b increases. (2) For a given T, the optimal value of n increases as m increases. Likewise, for a given n, the optimal value of T increases as m increases.
Proof. d2 ln DðnÞ dn2
¼0
d ln ½1FðnÞ dn2
2
2
respectively. Using the definition of supermodular function, we obtain that Pðn; TÞ is supermodular in ðn; c; hÞ and ðT; c; hÞ, respectively. Therefore, for a given T, the optimal value of n decreases as c or h increases. Likewise, for a given n, the optimal value of T decreases as c or h increases. h
2
Since D0 ðnÞ > 0 and ddðzÞ ¼ hðzÞ > 0 for all z 2 ð0; mÞ, it is straightfor R Tdz dðuÞdðtÞ RT ward to see that t e du < 0 edðTÞdðtÞ du < T edðTÞdðtÞ , which implies that
edðuÞdðtÞ dudt < 0; T
½1FðnÞ (1) If a > b; dDðnÞdn ¼ ða bÞKeðabÞn > 0. Because
2
T2
t
an
edðtÞ edðTÞ dt
T nR R o RT T T 0 hD ðnÞ 0 t edðuÞdðtÞ dudt T 0 edðTÞdðtÞ dt
0
T
Proof. (1) Taking the mixed derivative of Pðn; TÞ with respect to n and T we get 2
RT RT
¼ 0 . Combining these with
the fact that < 0, we then obtain that Pðn; TÞ is supermodular in ðn; T; oÞ. Hence the optimal value of n decreases, but the optimal value of T increases as o increases.
¼ 0, DðnÞ and 1 FðnÞ are both log-concave and in n. Therefore, based on Proposition 1.1, we can conclude that
@ 2 Pðn;TÞ @n@T
< 0. Further, since
@ 2 Pðn;TÞ @n@b
¼ pK ½1 þ ða bÞn
2
Pðn;TÞ ¼ 0, Pðn; TÞ is supermodular in eðabÞn < 0 and @ @T@b ðn; T; bÞ, which implies that optimal value of n decrease, but T increases as b increases. (2) Taking the mixed derivative of Pðn; TÞ with respect to n, T and m yields
@ 2 Pðn;TÞ aKean T T ¼ 4c þ ln 1 @n@m 1þmT 1þm 4T
2ð2 þ 2m TÞT T þ 4ð1 þ mÞln 1 þh 1þmT 1þm and
( " # @ 2 Pðn;TÞ aKean ð1 þ m 2TÞT T ¼ 4c þ ln 1 @T@m 1þm 4T 2 ð1 þ m TÞ2 " #) Tð2 2m þ 3TÞ T 2ln 1 : þ2hð1 þ mÞ 1þm ð1 þ m TÞ2
C.-Y. Dye et al. / European Journal of Operational Research 239 (2014) 868–871
For convenience, we let
Acknowledgements
þ ln 1
T T 1þmT 1þm
2ð2 þ 2m TÞT T þh þ 4ð1 þ mÞ ln 1 1þmT 1þm
V 1 ¼ 4c
The authors would like to thank the editor and anonymous reviewers for their valuable and constructive comments, which have led to a significant improvement in the manuscript. The first author’s research was partially supported by the National Science Council of the Republic of China under Grant NSC-102-2221-E366-001. The second author’s research was partially supported by the National Science Council of the Republic of China under Grant NSC-102-2221-E-231-014.
and
" ð1 þ m 2TÞT
# T V 2 ¼ 4c 1þm ð1 þ m TÞ2 " # Tð2 2m þ 3TÞ T 2 ln 1 þ 2hð1 þ mÞ : 1þm ð1 þ m TÞ2
For all T 2 ð0; mÞ, since
871
þ ln 1
dV 1 dT
References
ð2cþhT Þ ¼ 2T > 0 and limT!0 V 1 ¼ 0, it is ð1þmT Þ2
straightforward to see that V 1 > 0, which also implies that @ 2 Pðn;TÞ > 0. Using similar arguments, @n@m 4T ½cð1þmþTÞþhð1þmÞT dV 2 ¼ > 0 and limT!0 V 2 ¼ dT ð1þmTÞ3 2
we
can
show
that
0; thus, it is obvious to
Pðn;TÞ > 0. Using the definition see that V 2 > 0. Therefore, we have @ @T@m of supermodular function, we can easily obtain that Pðn; TÞ is supermodular in ðn; mÞ and ðT; mÞ, respectively. Therefore, for a given T, the optimal value of n increases as m increases. Likewise, for a given n, the optimal value of T increases as m increases. h
Amir, R., Dierker, H., Grodal, B., Hamilton, J., francois Mertens, J., Roberts, J., et al. (2005). Supermodularity and complementarity in economics: An elementary survey. Southern Economic Journal, 71, 636–660. Cambini, A., & Martein, L. (2009). Generalized convexity and optimization: Theory and applications. Berlin: Springer. Lou, K. R., & Wang, W. C. (2012). Optimal trade credit and order quantity when trade credit impacts on both demand rate and default risk. Journal of the Operational Research Society, 64, 1551–1556. Teng, J. T., & Lou, K. R. (2012). Seller’s optimal credit period and replenishment time in a supply chain with up-stream and down-stream trade credits. Journal of Global Optimization, 53, 417–430. Wang, W. C., Teng, J. T., & Lou, K. R. (2014). Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime. European Journal of Operational Research, 232, 315–321. Wu, J., Ouyang, L. Y., Cárdenas-Barrón, L. E., & Goyal, S. K. (2014). Optimal credit period and lot size for deteriorating items with expiration dates under twolevel trade credit financing. European Journal of Operational Research, 237, 898–908.
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Short Communication
A note on ‘‘Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime’’ Chung-Yuan Dye a, Chih-Te Yang b,⇑, Fang-Cheng Kung a a b
Graduate School of Business and Administration, Shu-Te University, Yen-Chao, Kaohsiung 824, Taiwan Department of Industrial Management, Chien Hsin University of Science and Technology, Taoyuan 320, Taiwan
a r t i c l e
i n f o
Article history: Received 22 March 2014 Accepted 25 June 2014 Available online 3 July 2014 Keywords: Supply chain management Inventory Maximum lifetime Trade credit
a b s t r a c t In 2014, Wang et al. (2014) extended the model of Lou and Wang (2012) to incorporate the credit period dependent demand and default risk for deteriorating items with maximum lifetime. However, the rates of demand, default risk and deterioration in the model of Wang et al. (2014) are assumed to be specific functions of credit period which limits the contributions. In this note, we first generalize the theoretical results of Wang et al. (2014) under some certain conditions. Furthermore, we also present some structural results instead of a numerical analysis on variation of optimal replenishment and trade credit strategies with respect to key parameters. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Trade credit is an increasingly important payment behavior in real business transactions. Since the trade credit management needs to balance the trade-off between increased sales and the risk of granting credit, Teng and Lou (2012) first incorporated the creditlinked demand and default risks to establish an EOQ model for the retailer in a supply chain with up-stream and down-stream trade credits. In an article published in Journal of the Operational Research Society, Lou and Wang (2012) extended the model of Teng and Lou (2012) by introducing the credit-linked demand and default risks. Later, Wang, Teng, and Lou (2014) extended the model of Lou and Wang (2012) for deterioration with a maximum lifetime. Base on the model of Teng and Lou (2012), Wu, Ouyang, Cárdenas-Barrón, and Goyal (2014) subsequently built an EOQ model for deteriorating items with expiration dates under two-level trade credit financing. However, the rates of demand and default risk in the abovementioned studies are assumed to be specific functions of the credit period. In this short note, we first generalize the theoretical results of Lou and Wang (2012) and Wang et al. (2014) under some certain conditions. Next, a sensitivity analysis with respect to key parameters are further performed in rigorous discussions. 2. The model For easy tractability with Wang et al. (2014), we use the same notations and assumptions. Further, in order to generalize the ⇑ Corresponding author. Tel.: +886 3 4581196x6123. E-mail address: [email protected] (C.-T. Yang). http://dx.doi.org/10.1016/j.ejor.2014.06.037 0377-2217/Ó 2014 Elsevier B.V. All rights reserved.
models of Lou and Wang (2012) and Wang et al. (2014), we make assumptions on the rates of demand and default risk. Let n be the credit period offered by the retailer to its customers, the demand is given by a demand function DðnÞ, which is assumed to be nonnegative, continuous and twice differentiable. Because the trade credit allows customers to enjoy the benefits of delayed payments, lengthening the credit period will stimulate sales. The longer the credit period, the higher the demand; hence, the demand strictly increases in the credit period, that is D0 ðnÞ > 0. Meanwhile, since a longer credit period may tie up the retailer’s capital and then increase the risk of default, the rate of default risk giving the credit period n is assumed to be FðnÞ, where Fð0Þ ¼ 0; 0 6 FðnÞ < 1 and F 0 ðnÞ > 0. Hence, the total profit per unit time constructed by Wang et al. (2014) can be reviewed as follows:
o T
Pðn; TÞ ¼ pDðnÞ½1 FðnÞ
hDðnÞ T
Z 0
T
Z
cDðnÞ T
Z
T
edðtÞ dt
0
T
edðuÞdðtÞ dudt;
ð1Þ
t
Rz where dðzÞ ¼ 0 hðuÞdu and hðtÞ is the time-varying deterioration rate at time t with 0 6 hðtÞ 6 1. Then the problem can be formulated as max0
869
C.-Y. Dye et al. / European Journal of Operational Research 239 (2014) 868–871
is the upper bound for the optimal value n . Hence, we can so n from further consideration, and the probexclude the region n > n lem can be then rewritten as max0
Theorem 2. For a given n, a unique T 2 ð0; m exists such that Pðn; TÞ is maximized. Proof. Assuming there is an interior solution, the first order condition is
RT dPðn; TÞ o cDðnÞ 0 edðTÞ edðtÞ dt ¼ 2 dT T2 T n R o RT RT T dðTÞdðtÞ hDðnÞ T 0 e dt 0 t edðuÞdðtÞ dudt ¼ 0: ð3Þ T2
Proof. If DðnÞ and 1 FðnÞ are both log-concave in n, we have 2 2 ½F 0 ðnÞ D00 ðnÞDðnÞ D0 ðnÞ 6 0 and F 00 ðnÞ P 1FðnÞ , respectively. Fur-
Substituting (3) into the second order derivative and simplifying yields
½1FðnÞ thermore, dDðnÞdn > 0 implies that D0 ðnÞ½1 FðnÞ DðnÞ 0 F ðnÞ > 0. Assuming there is an interior solution, the first order condition is
d Pðn; TÞ
dPðn; TÞ ¼ p D0 ðnÞ½1 FðnÞ DðnÞF 0 ðnÞ dn Z T Z T Z T D0 ðnÞ c edðtÞ dt þ h edðuÞdðtÞ dudt ¼ 0: ð2Þ T 0 0 t Substituting these results and (2) into the second order derivative, it follows that 2
d Pðn; TÞ ¼ p½1 FðnÞD00 ðnÞ 2pD0 ðnÞF 0 ðnÞ pDðnÞF 00 ðnÞ dn2 Z T Z T Z T D00 ðnÞ c edðtÞ dt þ h edðuÞdðtÞ dudt T 0 0 t 0 2 F ðnÞ 6 p½1 FðnÞD00 ðnÞ 2pD0 ðnÞF 0 ðnÞ þ pDðnÞ 1 FðnÞ D00 ðnÞ 0 D ðnÞ½1 FðnÞ DðnÞF 0 ðnÞ p 0 D ðnÞ pF 0 ðnÞ ¼ DðnÞF 0 ðnÞ D0 ðnÞ½1 FðnÞ 1 FðnÞ 2 o pF 0 ðnÞ n 00 D ðnÞDðnÞ D0 ðnÞ < 0: þ 0 D ðnÞ
2
dT 2
¼
n o RT DðnÞ cedðTÞ hðTÞ þ h þ hhðTÞ 0 edðTÞdðtÞ dt T
:
ð4Þ
2
ðn;TÞ Since hðTÞ > 0 for all T 2 ð0; mÞ, we then have d P < 0. Because it dT 2 is strictly concave at any interior critical point, Pðn; TÞ is strictly pseudoconcave in T. Furthermore, since a strictly pseudoconcave function can be either unimodal or monotone, we discuss the two situations separately below. By L’Hospital’s rule, we have
R T dðTÞ e edðtÞ dt 0
lim
T
T!0
2
¼ lim T!0
hðTÞedðTÞ 1 ¼ hð0Þ 2 2
and
lim
T
RT 0
edðTÞdðtÞ dt
RT RT
T
T!0
0 2
t
edðuÞdðtÞ dudt
¼ lim
1þhðTÞ
T!0
RT
edðTÞdðtÞ dt 1 ¼ : 2 2
0
Combining these result yields
lim T!0
If
dPðn; TÞ o cDðnÞhð0Þ hDðnÞ ¼ lim 2 ¼ 1: T!0 T dT 2 2
dPðn;TÞ dT
T¼m
ðn;TÞ < 0, since limT!0 dPdT > 0; Pðn; TÞ is unimodal in T.
Therefore, the solution to (2) is the unique interior maximum. Con dPðn;TÞ ðn;TÞ P 0, since limT!0 dPdT > 0; Pðn; TÞ is strictly dT
According to Theorems 3.4.7 in Cambini and Martein (2009), since
versely, if
Pðn; TÞ is strictly concave at any interior critical point, Pðn; TÞ is
monotonic increasing in T, and it is straightforward to see that T ¼ m. Combining the two arguments stated above, we can conclude that there exists a unique global maximum for Pðn; TÞ. This completes the proof of Theorem 2. h
strictly pseudoconcave in n. Since a strictly pseudoconcave function can be either unimodal or monotone, we discuss the two situations separately below. If dPðn;TÞ ðn;TÞ > 0, since dPdn < 0, Pðn; TÞ is unimodal in n. dn n>n
n¼0
Therefore, the solution to (2) is the unique interior maximum. ðn;TÞ ðn;TÞ 6 0, since dPdn < 0, Pðn; TÞ is strictly Conversely, if dPdn n¼0
n>n
monotonic decreasing in n, and it is straightforward to see that n ¼ 0. Combining the two arguments stated above, we can conclude that there exists a unique global maximum for Pðn; TÞ. This completes the proof of Theorem 1. h
Remark 1. If dPðn;TÞ dn
dDðnÞ½1FðnÞ dn
6 0, it is obvious to see from (2) that
< 0 for a given T. Therefore, Pðn; TÞ is maximized at n ¼ 0. ½1FðnÞ The intuition of the condition dDðnÞdn > 0 in Theorem 1 is sim1FðnÞ ple. Once dDðnÞ½dn 6 0, it represents that the marginal revenue after the default risk is less than or equal to zero. It is no benefit for the retailer to offer the customer a credit period. And thus, the retailer should shorten its credit period as much as possible. We next state our second structural result as the following theorem, which proves that for arbitrary deterioration rate, the optimal replenishment strategy is unique.
T¼m
From the analysis carried out so far, it follows that, if dDðnÞ½1FðnÞ dn
6 0, we can obtain from Remark 1 that max0
½1FðnÞ > 0, since guaranteed by Theorem 2. On the other hand, if dDðnÞdn the optimal value of T can be uniquely determined for a given n, (3) implicitly defines T as a function of n. Let P ðn; TðnÞÞ denote the maximum value function of P for a given n, then the problem becomes max06n6n P ðn; TðnÞÞ. By Berge’s Theorem of the Maximum, since the maximum value function P ðn; TðnÞÞ is continuous , the existence of a n in in the closed and bounded interval ½0; n which maximizes P ðn; TðnÞÞ is guaranteed by Weierstrass’ ½0; n > 0 and P ð0; Tð0ÞÞ > 0, theorem. In addition, if dP ðn;TðnÞÞ dn n¼0
; Tðn ÞÞ < 0, there exists at least one n 2 ð0; n Þ such because P ðn
¼ 0. However, since we cannot show the concavity that dP ðn;TðnÞÞ dn property of P ðn; TðnÞÞ, one can solve P ðn; TðnÞÞ many times from distinct starting values, and take the highest total profit obtained as the global maximum to avoid saddle points or local maximum.
870
C.-Y. Dye et al. / European Journal of Operational Research 239 (2014) 868–871
(4) Taking the mixed derivative of Pðn; TÞ with respect to n, T; c and h yields
3. Optimal replenishment time and trade credit strategies Although we know the optimal replenishment time and trade credit strategies exist from the above discussions, we do not have any further information about the system behavior. In this section we study how the optimal replenishment time and trade credit strategies vary with respect to the key parameters by using the property of supermodular function. In the following, we first introduce the definition of supermodular function.
@ 2 Pðn; TÞ DðnÞ ¼ @T@c
hR
T 0
RT
0
2
edðtÞ dt TedðTÞ T2 edðtÞ dt
i
¼
hR i T DðnÞ 0 edðtÞ edðTÞ dt T2
< 0;
D ðnÞ 0 @ Pðn; TÞ ¼ < 0; @n@c nR TR o RT T T dðuÞdðtÞ dudt T 0 edðTÞdðtÞ dt @ 2 Pðn; TÞ DðnÞ 0 t e ¼ <0 @T@h T2
Definition 1. A function F : X ! R is supermodular if
F ðx _ x0 Þ þ F ðx ^ x0 Þ P F ðxÞ þ F ðx0 Þ; 8x; x0 2 A;
and
where
D ðnÞ @ 2 Pðn; TÞ ¼ @n@h
0
x _ x0 ¼ max x1 ; x01 ; max x2 ; x02 ; . . . ; max xN ; x0N and
x ^ a ¼ min x1 ; x01 ; min x2 ; x02 ; . . . ; min xN ; x0N : 0
2
ðxÞ If F is a smooth function, F is supermodular if and only if @@xF@x P 0. i j Furthermore, if the inequality is strict, then F is strictly supermodular. We now state our third structural result as the following proposition.
Proposition 1. Suppose that the conditions in Theorem 1 are met, then: (1) The optimal value of T decreases as n increases. (2) The optimal value of n increases, but the optimal value of decreases as p increases. (3) The optimal value of n decreases, but the optimal value of increases as o increases. (4) For a given T, the optimal value of n decreases as c or increases. Likewise, for a given n, the optimal value of decreases as c or h increases.
0
@ Pðn; TÞ cD ðnÞ ¼ @n@T þ
RT 0
T h T
@ 2 Pðn;TÞ @n@T
:
< 0. According to Lemma 1 in Amir et al.
@ Pðn;TÞ @n@T
< 0 implies that Pðn; TÞ is supermodular in ðn; TÞ. (2005), Hence the optimal value of T is a decreasing function of n. (2) Taking the mixed derivatives of Pðn; TÞ with respect to n; T and p gives @
Pðn;TÞ @T@p
¼ 0 and @
2
Pðn;TÞ @n@p
¼ D0 ðnÞ½1 FðnÞ DðnÞF 0 ðnÞ > 0. 2
Pðn;TÞ < 0, we obtain that Combining these with the fact that @ @n@T Pðn; TÞ is supermodular in ðn; T; pÞ. Therefore, the optimal value of n increases, but the optimal value of T decreases as p increases. (3) Taking the mixed derivative of Pðn; TÞ with respect to n, T
and o yields
@ 2 Pðn;TÞ @T@o @ 2 Pðn;TÞ @n@T
¼
1 T2
> 0 and
@ 2 Pðn;TÞ @n@o
Finally, the rates of default risk and deterioration defined in the model of Wang et al. (2014) are used to study the impacts of default risk and deterioration on the optimal replenishment time and trade credit strategies. We then state the structural results on the variation of optimal replenishment time and trade credit strategies with respect to b and m as the following proposition. 1 Proposition 2. If DðnÞ ¼ ke ; FðnÞ ¼ 1 ebn and hðtÞ ¼ 1þmt with a > b, then
(1) The optimal value of n decreases, but T increases as b increases. (2) For a given T, the optimal value of n increases as m increases. Likewise, for a given n, the optimal value of T increases as m increases.
Proof. d2 ln DðnÞ dn2
¼0
d ln ½1FðnÞ dn2
2
2
respectively. Using the definition of supermodular function, we obtain that Pðn; TÞ is supermodular in ðn; c; hÞ and ðT; c; hÞ, respectively. Therefore, for a given T, the optimal value of n decreases as c or h increases. Likewise, for a given n, the optimal value of T decreases as c or h increases. h
2
Since D0 ðnÞ > 0 and ddðzÞ ¼ hðzÞ > 0 for all z 2 ð0; mÞ, it is straightfor R Tdz dðuÞdðtÞ RT ward to see that t e du < 0 edðTÞdðtÞ du < T edðTÞdðtÞ , which implies that
edðuÞdðtÞ dudt < 0; T
½1FðnÞ (1) If a > b; dDðnÞdn ¼ ða bÞKeðabÞn > 0. Because
2
T2
t
an
edðtÞ edðTÞ dt
T nR R o RT T T 0 hD ðnÞ 0 t edðuÞdðtÞ dudt T 0 edðTÞdðtÞ dt
0
T
Proof. (1) Taking the mixed derivative of Pðn; TÞ with respect to n and T we get 2
RT RT
¼ 0 . Combining these with
the fact that < 0, we then obtain that Pðn; TÞ is supermodular in ðn; T; oÞ. Hence the optimal value of n decreases, but the optimal value of T increases as o increases.
¼ 0, DðnÞ and 1 FðnÞ are both log-concave and in n. Therefore, based on Proposition 1.1, we can conclude that
@ 2 Pðn;TÞ @n@T
< 0. Further, since
@ 2 Pðn;TÞ @n@b
¼ pK ½1 þ ða bÞn
2
Pðn;TÞ ¼ 0, Pðn; TÞ is supermodular in eðabÞn < 0 and @ @T@b ðn; T; bÞ, which implies that optimal value of n decrease, but T increases as b increases. (2) Taking the mixed derivative of Pðn; TÞ with respect to n, T and m yields
@ 2 Pðn;TÞ aKean T T ¼ 4c þ ln 1 @n@m 1þmT 1þm 4T
2ð2 þ 2m TÞT T þ 4ð1 þ mÞln 1 þh 1þmT 1þm and
( " # @ 2 Pðn;TÞ aKean ð1 þ m 2TÞT T ¼ 4c þ ln 1 @T@m 1þm 4T 2 ð1 þ m TÞ2 " #) Tð2 2m þ 3TÞ T 2ln 1 : þ2hð1 þ mÞ 1þm ð1 þ m TÞ2
C.-Y. Dye et al. / European Journal of Operational Research 239 (2014) 868–871
For convenience, we let
Acknowledgements
þ ln 1
T T 1þmT 1þm
2ð2 þ 2m TÞT T þh þ 4ð1 þ mÞ ln 1 1þmT 1þm
V 1 ¼ 4c
The authors would like to thank the editor and anonymous reviewers for their valuable and constructive comments, which have led to a significant improvement in the manuscript. The first author’s research was partially supported by the National Science Council of the Republic of China under Grant NSC-102-2221-E366-001. The second author’s research was partially supported by the National Science Council of the Republic of China under Grant NSC-102-2221-E-231-014.
and
" ð1 þ m 2TÞT
# T V 2 ¼ 4c 1þm ð1 þ m TÞ2 " # Tð2 2m þ 3TÞ T 2 ln 1 þ 2hð1 þ mÞ : 1þm ð1 þ m TÞ2
For all T 2 ð0; mÞ, since
871
þ ln 1
dV 1 dT
References
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can
show
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