- Email: [email protected]
Some comments on deteriorating inventory theory with partial backlogging
Int. J. Production Economics 140 (2012) 554–556
Contents lists available at SciVerse ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Some comments on deteriorating inventory theory with partial backlogging
a r t i c l e i n f o
a b s t r a c t
Keywords: Deteriorating products Pseudoconcave Unimodal
We provide an easier way to prove the main theorems in Dye (2007) and Dye et al. (2007) using a result from nonlinear programming. We also extend the validation of the conclusion in Dye (2007) and Dye et al. (2007) to more cases of the demand function and correct some mistakes on this topic appeared in the literature. & 2012 Elsevier B.V. All rights reserved.
In papers published on Omega (Dye, 2007) and European Journal of Operational Research (Dye et al., 2007), the authors investigate a joint pricing and ordering problem for deteriorating products with time-dependent backlogging rate. The former paper employs the hyperbolic backlogging rate, while the latter uses the exponential backlogging rate. Both papers first prove that given a selling price, there exists a unique optimal replenishment schedule (Theorem 1 in Dye, 2007; Dye et al., 2007). They then show given that the marginal revenue with respect to price is decreasing, the total profit is a concave function of price when the replenishment schedule is given. However, in the numerical example provided in both papers, demand rate dðpÞ ¼ 16 107 p3:21 does not satisfy the condition that the marginal revenue with respect to price is decreasing since pd(p) is convex. We first take a look at the proof of Theorem 1 in Dye (2007). We refer readers to Dye (2007) for the definition of the notation used below. The total profit per unit time is given as follows:
Pðp,t1 ,t2 Þ ¼
1 fðpcÞdðpÞðt 1 þ t 2 ÞKcdðpÞ t 1 þt 2 Z t1 Z t1 Z ½egðtÞ 1 dthdðpÞ egðtÞ 0
0
½s þ dðpc þ pÞdðpÞ
d
2
@ Pðt 1 ,t 2 9pÞ 2zðt 1 ,t 2 9pÞ @zðt 1 ,t 2 9pÞ 1 ¼ @t 1 @t 2 @t 2 ðt 1 þ t 2 Þ2 ðt 1 þt 2 Þ3 @zðt 1 ,t 2 9pÞ 1 @t 1 ðt 1 þ t 2 Þ2 Substituting the first-order conditions into the equation above, we have @2 Pðt 1 ,t 2 9pÞ=@t 1 @t 2 9ðtn ,tn Þ ¼ 0. 1
t1
2
The proof of Theorem 1 in Dye (2007) is very complicated. Here we show that there exists a finite global maximizer for Pðt 1 ,t2 9pÞ using a different approach. Lemma 1. For any given p, zðt 1 ,t 2 9pÞ is a concave function of ðt1 ,t 2 Þ. Proof. @2 zðt 1 ,t 2 9pÞ @t 21
¼ cdðpÞyðt 1 Þegðt1 Þ hdðpÞyðt 1 Þegðt1 Þ
Z
t1
egðtÞ dthdðpÞ o 0
0
egðuÞ du dt
t
½dt 2 ln ð1 þ dt 2 Þg,
Rz where gðzÞ ¼ 0 yðuÞ du and yðtÞ is the instantaneous decay rate. In general, given a twice differentiable function f ðt 1 ,t 2 Þ, if ðt n1 ,t n2 Þ is a stationary point, @2 f ðtn1 ,t n2 Þ=@t 1 @t 2 ¼ 0 does not hold. For example, f ðt1 ,t 2 Þ ¼ t 21 þ 2t 2 t1 t 2 . Therefore, the claim @2 Pðt 1 ,t 2 9pÞ n n ¼ 0, @t 1 @t 2 ðt ,t Þ 1
It is easy to see that @2 zðt1 ,t 2 9pÞ=@t 1 @t 2 ¼ 0 in this case. Thus, 2
2
in the proof of part (b) of Theorem 1 is not straightforward. Define zðp,t1 ,t 2 Þ ¼ ðt1 þ t 2 ÞPðp,t 1 ,t 2 Þ. For any given p, the first-order conditions of the original problem are @Pðt 1 ,t 2 9pÞ zðt1 ,t 2 9pÞ 1 @zðt 1 ,t 2 9pÞ ¼ þ ¼0 @t 1 @t 1 t 1 þt 2 ðt 1 þ t 2 Þ2 @Pðt 1 ,t 2 9pÞ zðt1 ,t 2 9pÞ 1 @zðt 1 ,t 2 9pÞ ¼ þ ¼0 @t 2 @t 2 t 1 þt 2 ðt 1 þ t 2 Þ2 0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2012.06.036
@2 zðt 1 ,t 2 9pÞ @t 22
¼
½s þ dðpc þ pÞdðpÞ ðt 1 þt 2 Þ2
o0
Since @2 zðt 1 ,t 2 9pÞ=@t 1 @t 2 ¼ 0, the Hessian of zðt1 ,t 2 9pÞ is negative definite, which completes the proof. & Lemma 2. Let f and c be functions defined on a set G Rn and c a 0 on G. Let G be open, let x A G, and let f and c be differentiable at x, then f=c is pseudoconcave at x if f is concave at x, c 40, and c is linear on Rn (Mangasarian, 1994, p. 148). Corollary 1. For any given p, Pðt 1 ,t 2 9pÞ is a pseudoconcave function of ðt1 ,t 2 Þ on ð0, þ1Þ ð0, þ1Þ. Theorem 1. If the instantaneous decay rate yðtÞ is an increasing function of time t, then there exists a finite global maximizer for Pðt 1 ,t2 9pÞ. Proof. Due to the positive backorder and lost sale costs, the optimal t2 cannot be þ 1. If yðtÞ increases with t, the instantaneous decay rate goes to infinity as t1 goes to infinity, which
Letter to the Editor / Int. J. Production Economics 140 (2012) 554–556
means that the optimal t1 cannot be þ1. However, if the backorder and lost sale costs are large enough, it is not optimal to have backlogged and lost demand, i.e., t n2 ¼ 0. Based on Corollary 1, a finite global maximizer is guaranteed and it is easy to find this global maximizer numerically. & Dye et al. (2007) investigate the same problem under the assumption that the backlogging rate is exponential and prove that Theorem 1 above holds as well. Similarly, their proof is also very complicated. The authors argue that the optimal length of the shortage period is infinity under certain condition. Based on common sense, if backlog and lost sale costs are positive, it will never be optimal to stay on shortage forever. Abad (2008) tries to generalize Theorem 1 above for any decreasing backlogging rate B(t), where t is the amount of time that the customer waits before receiving the product. Unfortunately, the main result is wrong. In the proof of Proposition 2 in page 184, the author claims that if p Z v þ c1 c3 4 v þ c1 c3 c2 c then pvc1 þ c3 c2 c Z 0 by mistake. For general decreasing backlogging rate B(t), the inventory level can be written as 8 R < dðpÞegðtÞ tt1 egðuÞ du R t2 IðtÞ ¼ : dðpÞ t þ t t BðxÞ dx 1 2
if 0 rt o t1 ,
e
hdðpÞ 0
pdðpÞt 2 þ pdðpÞ
0
Z
t1
gðuÞ
e
du dtsdðpÞ
t
Z
If t 2 4ðpc þ pÞ=s, ½ðpc þ pst2 ÞBðt 2 Þpðt 1 þ t 2 Þ o 0. Now it is easy to see that for any t1 and t 2 4 ðpc þ pÞ=s, Gðt 1 ,t 2 9pÞ r maxfGðt 1 ,ðpc þ pÞ=s9pÞ,0g. & We now turn our attention to the existence of the unique selling price when the replenishment schedule is given, which is shown by Dye (2007) and Dye et al. (2007) when pd(p) is concave and d(p) is convex. However, the example provided, dðpÞ ¼ 16 107 p3:21 , does not satisfy previous condition. Abad (2008) proves pseudoconcavity when 1=dðpÞ is convex. However, the assumption of the convexity of 1=dðpÞ is not weaker than the assumption of the concavity of pd(p). For example, when dðpÞ ¼ p0:5 , pd(p) is concave, but 1=dðpÞ is not convex. We now show the existence of the unique selling price for both types of demand functions when the replenishment schedule is given.
0
Z t2 BðxÞ dx K Fðt1 ,t 2 9pÞ ¼ p dðpÞt 1 þdðpÞ 0 Z t2 Z t1 gðuÞ e du þ dðpÞ BðxÞ dx c dðpÞ gðtÞ
½ðpc þ pst2 ÞBðt 2 Þpðt 1 þt 2 ÞFðt 1 ,t 2 9pÞ @G ¼ : @t2 ðt 1 þ t 2 Þ2
00
f ðxÞ ¼ 0 ) f ðxÞ r0
The number of units sold in one cycle is dðpÞt 1 þ dðpÞ 0 BðxÞ dx. Rt Rt The order quantity is dðpÞ 01 egðuÞ du þ dðpÞ 02 BðxÞ dx. The inventory holding cost is as same as the holding cost in Dye (2007). Rt Backlog and lost sale costs in one cycle are sdðpÞ 02 xBðxÞ dx and R t2 p½dðpÞt2 dðpÞ 0 BðxÞ dx, respectively. Given p, the total profit in one cycle is
0 t1
Now we show that the optimal t2 for Gðt 1 ,t 2 9pÞ satisfies the condition t2 r ðpc þ pÞ=s. Since
Lemma 3. If f(x) is twice differentiable and if the condition
if t 1 rt o t1 þ t 2 : R t2
Z
holds, then f(x) is unimodal (Boyd and Vandenberghe, 2006, p. 101). Theorem 3. If 1=dðpÞ is a convex function of p, or pd(p) is concave and d(p) is convex, then there exists a unique optimal selling price for Gðp9t 1 ,t2 Þ ¼ Fðp9t 1 ,t2 Þ=ðt1 þ t2 Þ, where Fðp9t1 ,t2 Þ has the same expression as Fðt 1 ,t 2 9pÞ in Eq. (1). Proof. When the replenishment schedule is given, the total profit per unit time can be written as
Gðp9t 1 ,t2 Þ ¼ pdðpÞUðt1 ,t2 ÞdðpÞVðt1 ,t 2 Þ
Z
t2
t2
BðxÞ dx:
ð1Þ
Given p, the total profit per unit time is
Uðt 1 ,t 2 Þ ¼
t1 þ
R t2
0 BðxÞ dx 4 0, t 1 þt 2
Z t1 Z t2 1 egðuÞ du þ BðxÞ dx t1 þ t2 0 0 Z t1 Z t1 gðtÞ gðuÞ e e du dt þh 0 t Z t2 Z t2 xBðxÞ dx þ p t 2 BðxÞ dx 4 0 þs
Vðt1 ,t 2 Þ ¼
Fðt 1 ,t 2 9pÞ : t1 þ t2
0
Theorem 2. If the instantaneous decay rate yðtÞ is an increasing function of time t, then there exists a finite global maximizer for Gðt1 ,t2 9pÞ. Proof. The proof of Theorem 1 is based on Corollary 1, which does not hold for Gðt 1 ,t 2 9pÞ. However, we know that
0
If pd(p) is concave and d(p) is convex, then Gðp9t 1 ,t 2 Þ is obviously concave. If 1=dðpÞ is convex, the first-order condition can be written as 0
2
@ F ¼ cdðpÞegðt1 Þ yðt 1 ÞhdðpÞegðt1 Þ yðt 1 Þ @t 21 @F ¼ ðpc þ pst 2 ÞBðt 2 Þp, @t 2
Z
t1
K , t1 þ t2
where
xBðxÞ dx 0
0
Gðt1 ,t2 9pÞ ¼
555
0
pd ðpÞUðt 1 ,t 2 Þ þdðpÞUðt 1 ,t 2 Þd ðpÞVðt 1 ,t 2 Þ ¼ 0: egðtÞ dthdðpÞ o0,
0
If p satisfies the first-order condition, then 2
d Gðp9t 1 ,t 2 Þ
@2 F dB ¼ ðpc þ pst2 Þ sBðt 2 Þ, dt2 @t 22
dp
When t 2 r ðpc þ pÞ=s, the Hessian of Fðt 1 ,t 2 9pÞ is negative definite, which means that Gðt 1 ,t 2 9pÞ is pseudoconcave on ð0, þ1Þ ð0,ðpc þ pÞ=s.
0
00
00
¼ 2d ðpÞUðt 1 ,t 2 Þ þ pd ðpÞUðt 1 ,t 2 Þd ðpÞVðt 1 ,t 2 Þ 0
¼
2
@ F ¼ 0: @t 1 @t 2
2
00
2½d ðpÞ2 d ðpÞdðpÞ Uðt 1 ,t 2 Þ 0 d ðpÞ 0
00
Since 1=dðpÞ is convex, we have 2½d ðpÞ2 d ðpÞdðpÞ Z 0. Also, d ðpÞ o 0. Therefore, Gðp9t 1 ,t 2 Þ is unimodal according to Lemma 3. Since demand will be zero if p is large enough, the optimal price is finite. This completes the proof. & 0
556
Letter to the Editor / Int. J. Production Economics 140 (2012) 554–556
References Abad, P.L., 2008. Optimal price and order size under partial backordering incorporating shortage, backorder and lost sale costs. International Journal of Production Economics 114 (1), 179–186. Boyd, S., Vandenberghe, L., 2006. Convex Optimization. Cambridge University Press, Cambridge, UK. Dye, C.-Y., 2007. Joint pricing and ordering policy for a deteriorating inventory with partial backlogging. Omega 35 (2), 184–189. Dye, C.-Y., Hsieh, T.-P., Ouyang, L.-Y., 2007. Determining optimal selling price and lot size with a varying rate of deterioration and exponential partial backlogging. European Journal Of Operational Research 181 (2), 668–678. Mangasarian, O.L., 1994. Nonlinear Programming. SIAM, Philadelphia, PA.
Rongbing Huang n School of Administrative Studies, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3 E-mail address: [email protected] Received 22 June 2012; accepted 28 June 2012 Available online 17 July 2012
n
Tel.: þ1 416 7362100x22898; fax: þ1 416 7365963.
Contents lists available at SciVerse ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Some comments on deteriorating inventory theory with partial backlogging
a r t i c l e i n f o
a b s t r a c t
Keywords: Deteriorating products Pseudoconcave Unimodal
We provide an easier way to prove the main theorems in Dye (2007) and Dye et al. (2007) using a result from nonlinear programming. We also extend the validation of the conclusion in Dye (2007) and Dye et al. (2007) to more cases of the demand function and correct some mistakes on this topic appeared in the literature. & 2012 Elsevier B.V. All rights reserved.
In papers published on Omega (Dye, 2007) and European Journal of Operational Research (Dye et al., 2007), the authors investigate a joint pricing and ordering problem for deteriorating products with time-dependent backlogging rate. The former paper employs the hyperbolic backlogging rate, while the latter uses the exponential backlogging rate. Both papers first prove that given a selling price, there exists a unique optimal replenishment schedule (Theorem 1 in Dye, 2007; Dye et al., 2007). They then show given that the marginal revenue with respect to price is decreasing, the total profit is a concave function of price when the replenishment schedule is given. However, in the numerical example provided in both papers, demand rate dðpÞ ¼ 16 107 p3:21 does not satisfy the condition that the marginal revenue with respect to price is decreasing since pd(p) is convex. We first take a look at the proof of Theorem 1 in Dye (2007). We refer readers to Dye (2007) for the definition of the notation used below. The total profit per unit time is given as follows:
Pðp,t1 ,t2 Þ ¼
1 fðpcÞdðpÞðt 1 þ t 2 ÞKcdðpÞ t 1 þt 2 Z t1 Z t1 Z ½egðtÞ 1 dthdðpÞ egðtÞ 0
0
½s þ dðpc þ pÞdðpÞ
d
2
@ Pðt 1 ,t 2 9pÞ 2zðt 1 ,t 2 9pÞ @zðt 1 ,t 2 9pÞ 1 ¼ @t 1 @t 2 @t 2 ðt 1 þ t 2 Þ2 ðt 1 þt 2 Þ3 @zðt 1 ,t 2 9pÞ 1 @t 1 ðt 1 þ t 2 Þ2 Substituting the first-order conditions into the equation above, we have @2 Pðt 1 ,t 2 9pÞ=@t 1 @t 2 9ðtn ,tn Þ ¼ 0. 1
t1
2
The proof of Theorem 1 in Dye (2007) is very complicated. Here we show that there exists a finite global maximizer for Pðt 1 ,t2 9pÞ using a different approach. Lemma 1. For any given p, zðt 1 ,t 2 9pÞ is a concave function of ðt1 ,t 2 Þ. Proof. @2 zðt 1 ,t 2 9pÞ @t 21
¼ cdðpÞyðt 1 Þegðt1 Þ hdðpÞyðt 1 Þegðt1 Þ
Z
t1
egðtÞ dthdðpÞ o 0
0
egðuÞ du dt
t
½dt 2 ln ð1 þ dt 2 Þg,
Rz where gðzÞ ¼ 0 yðuÞ du and yðtÞ is the instantaneous decay rate. In general, given a twice differentiable function f ðt 1 ,t 2 Þ, if ðt n1 ,t n2 Þ is a stationary point, @2 f ðtn1 ,t n2 Þ=@t 1 @t 2 ¼ 0 does not hold. For example, f ðt1 ,t 2 Þ ¼ t 21 þ 2t 2 t1 t 2 . Therefore, the claim @2 Pðt 1 ,t 2 9pÞ n n ¼ 0, @t 1 @t 2 ðt ,t Þ 1
It is easy to see that @2 zðt1 ,t 2 9pÞ=@t 1 @t 2 ¼ 0 in this case. Thus, 2
2
in the proof of part (b) of Theorem 1 is not straightforward. Define zðp,t1 ,t 2 Þ ¼ ðt1 þ t 2 ÞPðp,t 1 ,t 2 Þ. For any given p, the first-order conditions of the original problem are @Pðt 1 ,t 2 9pÞ zðt1 ,t 2 9pÞ 1 @zðt 1 ,t 2 9pÞ ¼ þ ¼0 @t 1 @t 1 t 1 þt 2 ðt 1 þ t 2 Þ2 @Pðt 1 ,t 2 9pÞ zðt1 ,t 2 9pÞ 1 @zðt 1 ,t 2 9pÞ ¼ þ ¼0 @t 2 @t 2 t 1 þt 2 ðt 1 þ t 2 Þ2 0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2012.06.036
@2 zðt 1 ,t 2 9pÞ @t 22
¼
½s þ dðpc þ pÞdðpÞ ðt 1 þt 2 Þ2
o0
Since @2 zðt 1 ,t 2 9pÞ=@t 1 @t 2 ¼ 0, the Hessian of zðt1 ,t 2 9pÞ is negative definite, which completes the proof. & Lemma 2. Let f and c be functions defined on a set G Rn and c a 0 on G. Let G be open, let x A G, and let f and c be differentiable at x, then f=c is pseudoconcave at x if f is concave at x, c 40, and c is linear on Rn (Mangasarian, 1994, p. 148). Corollary 1. For any given p, Pðt 1 ,t 2 9pÞ is a pseudoconcave function of ðt1 ,t 2 Þ on ð0, þ1Þ ð0, þ1Þ. Theorem 1. If the instantaneous decay rate yðtÞ is an increasing function of time t, then there exists a finite global maximizer for Pðt 1 ,t2 9pÞ. Proof. Due to the positive backorder and lost sale costs, the optimal t2 cannot be þ 1. If yðtÞ increases with t, the instantaneous decay rate goes to infinity as t1 goes to infinity, which
Letter to the Editor / Int. J. Production Economics 140 (2012) 554–556
means that the optimal t1 cannot be þ1. However, if the backorder and lost sale costs are large enough, it is not optimal to have backlogged and lost demand, i.e., t n2 ¼ 0. Based on Corollary 1, a finite global maximizer is guaranteed and it is easy to find this global maximizer numerically. & Dye et al. (2007) investigate the same problem under the assumption that the backlogging rate is exponential and prove that Theorem 1 above holds as well. Similarly, their proof is also very complicated. The authors argue that the optimal length of the shortage period is infinity under certain condition. Based on common sense, if backlog and lost sale costs are positive, it will never be optimal to stay on shortage forever. Abad (2008) tries to generalize Theorem 1 above for any decreasing backlogging rate B(t), where t is the amount of time that the customer waits before receiving the product. Unfortunately, the main result is wrong. In the proof of Proposition 2 in page 184, the author claims that if p Z v þ c1 c3 4 v þ c1 c3 c2 c then pvc1 þ c3 c2 c Z 0 by mistake. For general decreasing backlogging rate B(t), the inventory level can be written as 8 R < dðpÞegðtÞ tt1 egðuÞ du R t2 IðtÞ ¼ : dðpÞ t þ t t BðxÞ dx 1 2
if 0 rt o t1 ,
e
hdðpÞ 0
pdðpÞt 2 þ pdðpÞ
0
Z
t1
gðuÞ
e
du dtsdðpÞ
t
Z
If t 2 4ðpc þ pÞ=s, ½ðpc þ pst2 ÞBðt 2 Þpðt 1 þ t 2 Þ o 0. Now it is easy to see that for any t1 and t 2 4 ðpc þ pÞ=s, Gðt 1 ,t 2 9pÞ r maxfGðt 1 ,ðpc þ pÞ=s9pÞ,0g. & We now turn our attention to the existence of the unique selling price when the replenishment schedule is given, which is shown by Dye (2007) and Dye et al. (2007) when pd(p) is concave and d(p) is convex. However, the example provided, dðpÞ ¼ 16 107 p3:21 , does not satisfy previous condition. Abad (2008) proves pseudoconcavity when 1=dðpÞ is convex. However, the assumption of the convexity of 1=dðpÞ is not weaker than the assumption of the concavity of pd(p). For example, when dðpÞ ¼ p0:5 , pd(p) is concave, but 1=dðpÞ is not convex. We now show the existence of the unique selling price for both types of demand functions when the replenishment schedule is given.
0
Z t2 BðxÞ dx K Fðt1 ,t 2 9pÞ ¼ p dðpÞt 1 þdðpÞ 0 Z t2 Z t1 gðuÞ e du þ dðpÞ BðxÞ dx c dðpÞ gðtÞ
½ðpc þ pst2 ÞBðt 2 Þpðt 1 þt 2 ÞFðt 1 ,t 2 9pÞ @G ¼ : @t2 ðt 1 þ t 2 Þ2
00
f ðxÞ ¼ 0 ) f ðxÞ r0
The number of units sold in one cycle is dðpÞt 1 þ dðpÞ 0 BðxÞ dx. Rt Rt The order quantity is dðpÞ 01 egðuÞ du þ dðpÞ 02 BðxÞ dx. The inventory holding cost is as same as the holding cost in Dye (2007). Rt Backlog and lost sale costs in one cycle are sdðpÞ 02 xBðxÞ dx and R t2 p½dðpÞt2 dðpÞ 0 BðxÞ dx, respectively. Given p, the total profit in one cycle is
0 t1
Now we show that the optimal t2 for Gðt 1 ,t 2 9pÞ satisfies the condition t2 r ðpc þ pÞ=s. Since
Lemma 3. If f(x) is twice differentiable and if the condition
if t 1 rt o t1 þ t 2 : R t2
Z
holds, then f(x) is unimodal (Boyd and Vandenberghe, 2006, p. 101). Theorem 3. If 1=dðpÞ is a convex function of p, or pd(p) is concave and d(p) is convex, then there exists a unique optimal selling price for Gðp9t 1 ,t2 Þ ¼ Fðp9t 1 ,t2 Þ=ðt1 þ t2 Þ, where Fðp9t1 ,t2 Þ has the same expression as Fðt 1 ,t 2 9pÞ in Eq. (1). Proof. When the replenishment schedule is given, the total profit per unit time can be written as
Gðp9t 1 ,t2 Þ ¼ pdðpÞUðt1 ,t2 ÞdðpÞVðt1 ,t 2 Þ
Z
t2
t2
BðxÞ dx:
ð1Þ
Given p, the total profit per unit time is
Uðt 1 ,t 2 Þ ¼
t1 þ
R t2
0 BðxÞ dx 4 0, t 1 þt 2
Z t1 Z t2 1 egðuÞ du þ BðxÞ dx t1 þ t2 0 0 Z t1 Z t1 gðtÞ gðuÞ e e du dt þh 0 t Z t2 Z t2 xBðxÞ dx þ p t 2 BðxÞ dx 4 0 þs
Vðt1 ,t 2 Þ ¼
Fðt 1 ,t 2 9pÞ : t1 þ t2
0
Theorem 2. If the instantaneous decay rate yðtÞ is an increasing function of time t, then there exists a finite global maximizer for Gðt1 ,t2 9pÞ. Proof. The proof of Theorem 1 is based on Corollary 1, which does not hold for Gðt 1 ,t 2 9pÞ. However, we know that
0
If pd(p) is concave and d(p) is convex, then Gðp9t 1 ,t 2 Þ is obviously concave. If 1=dðpÞ is convex, the first-order condition can be written as 0
2
@ F ¼ cdðpÞegðt1 Þ yðt 1 ÞhdðpÞegðt1 Þ yðt 1 Þ @t 21 @F ¼ ðpc þ pst 2 ÞBðt 2 Þp, @t 2
Z
t1
K , t1 þ t2
where
xBðxÞ dx 0
0
Gðt1 ,t2 9pÞ ¼
555
0
pd ðpÞUðt 1 ,t 2 Þ þdðpÞUðt 1 ,t 2 Þd ðpÞVðt 1 ,t 2 Þ ¼ 0: egðtÞ dthdðpÞ o0,
0
If p satisfies the first-order condition, then 2
d Gðp9t 1 ,t 2 Þ
@2 F dB ¼ ðpc þ pst2 Þ sBðt 2 Þ, dt2 @t 22
dp
When t 2 r ðpc þ pÞ=s, the Hessian of Fðt 1 ,t 2 9pÞ is negative definite, which means that Gðt 1 ,t 2 9pÞ is pseudoconcave on ð0, þ1Þ ð0,ðpc þ pÞ=s.
0
00
00
¼ 2d ðpÞUðt 1 ,t 2 Þ þ pd ðpÞUðt 1 ,t 2 Þd ðpÞVðt 1 ,t 2 Þ 0
¼
2
@ F ¼ 0: @t 1 @t 2
2
00
2½d ðpÞ2 d ðpÞdðpÞ Uðt 1 ,t 2 Þ 0 d ðpÞ 0
00
Since 1=dðpÞ is convex, we have 2½d ðpÞ2 d ðpÞdðpÞ Z 0. Also, d ðpÞ o 0. Therefore, Gðp9t 1 ,t 2 Þ is unimodal according to Lemma 3. Since demand will be zero if p is large enough, the optimal price is finite. This completes the proof. & 0
556
Letter to the Editor / Int. J. Production Economics 140 (2012) 554–556
References Abad, P.L., 2008. Optimal price and order size under partial backordering incorporating shortage, backorder and lost sale costs. International Journal of Production Economics 114 (1), 179–186. Boyd, S., Vandenberghe, L., 2006. Convex Optimization. Cambridge University Press, Cambridge, UK. Dye, C.-Y., 2007. Joint pricing and ordering policy for a deteriorating inventory with partial backlogging. Omega 35 (2), 184–189. Dye, C.-Y., Hsieh, T.-P., Ouyang, L.-Y., 2007. Determining optimal selling price and lot size with a varying rate of deterioration and exponential partial backlogging. European Journal Of Operational Research 181 (2), 668–678. Mangasarian, O.L., 1994. Nonlinear Programming. SIAM, Philadelphia, PA.
Rongbing Huang n School of Administrative Studies, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3 E-mail address: [email protected] Received 22 June 2012; accepted 28 June 2012 Available online 17 July 2012
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