optimal retailer's ordering policy with deteriorating items under twolevel trade credit
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This article was downloaded by: [New York University]On: 10 October 2014, At: 00:33Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK
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Optimal retailer's ordering policy with deterioratingitems under twolevel trade creditYung-Fu Huang a , Ming-Hon Hwang a , Yu-Cheng Tu a & Hao-Wei Yang aa Department of Marketing and Logistics Management , Chaoyang University ofTechnology , Taichung , Taiwan, R.O.C.Published online: 18 Jun 2013.
To cite this article: Yung-Fu Huang , Ming-Hon Hwang , Yu-Cheng Tu & Hao-Wei Yang (2010) Optimal retailer's orderingpolicy with deteriorating items under twolevel trade credit, Journal of Information and Optimization Sciences, 31:4,941-954, DOI: 10.1080/02522667.2010.10700004
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Optimal retailer’s ordering policy with deteriorating items under two-level trade credit2
Yung-Fu Huang ∗
Ming-Hon Hwang4
Yu-Cheng TuHao-Wei Yang6
Department of Marketing and Logistics ManagementChaoyang University of Technology8
TaichungTaiwan, R.O.C.10
AbstractIn practice, in the supply chain there are two levels of trade credit: one is offered to the12
retailer from its supplier; the other is offered to the customer from its retailer. Moreover, thecommodity under consideration usually deteriorates over time. The present study therefore14
proposes a retailer’s EOQ (economic order quantity) model by considering deteriorating itemsunder two-level trade credit. The proposed model takes into account both levels of trade16
credit. In the first place, the authors model the retailer’s inventory decision as a costminimization problem. Secondly, the authors prove the convexity of the inventory function18
in terms of relevant annual costs. The authors then move on to construct an easy-to-usetheorem to efficiently determine the optimal replenishment cycle, hence, the optimal order20
quantity. Finally, the authors provide several numerical examples to illustrate the theoremand to conduct a sensitivity analysis. Based on the proposed model, the authors conclude that22
a longer replenishment cycle (a larger order quantity) is directly related to a higher orderingcost, and, is inversely related to a higher deteriorating rate.24
Keywords and phrases : Economic order quantity, inventory, optimization, sensitivity analysis,
deteriorating item.26
1. Introduction
In the classical economic order quantity (EOQ) model, it was tacitly as-28
sumed that the purchaser must pay for the items purchased in the momentthe items are received. However, in practice, the supplier frequently offers30
∗Corresponding author’s address: No. 168, Jifong E. Rd., Wufong Township, TaichungCounty 41349, Taiwan, R.O.C. E-mail: [email protected]——————————–Journal of Information & Optimization SciencesVol. 31 (2010), No. 4, pp. 941–954c© Taru Publications
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942 Y. F. HUANG ET AL.
the trade credit, i.e., permissible delay in payments, to its purchasers.Goyal [7] established a single-item inventory model under permissible2
delay in payments. Chung [4] developed an efficient decision procedureto determine the economic order quantity under condition of permissible4
delay in payments. Teng [17] assumed that the selling price was not equalto the purchasing price to modify Goyal’s model [7]. Chung and Huang [5]6
investigated this issue within EPQ framework and developed an efficientsolving procedure to determine the optimal replenishment cycle for the8
retailer. Huang and Chung [10] investigated the inventory policy undercash discount and trade credit. Chung and Liao [6] adopted alternative10
payment rules to develop the inventory model and obtain different results.Recently, Huang [9] adopted the payment rule discussed in Chung and12
Liao [6], and, assumed finite replenishment rate, to investigate the buyer’sinventory problem.14
All above published papers modeling trade credit implicitly assumedthat the supplier would offer the retailer a delay payment period but16
the retailer would not offer a delay payment period to its customers.In other words, all previously published papers assume there is only18
one level of trade credit. We know that this assumption is debatablein the practical business situations. Recently, Huang [8] modified this20
assumption to assume that the retailer would adopt a similar trade creditpolicy to stimulate its customers’ demand in developing the retailer’s22
replenishment model. This viewpoint that there are two levels of tradecredit corresponds, more closely to the supply chain models in practice.24
But Huang [8] ignored the deteriorating effect of items in developinghis model. However, in real-life situations, there is an inventory loss26
by deterioration in categories of commodity such as volatile liquids,blood banks, medicines, electronic components, and fashion goods. Many28
articles have studied inventory models for deteriorating items. Someof the prominent papers are discussed below. Aggarwal and Jaggi [1]30
considered the inventory model with an exponential deterioration rateunder the condition of permissible delay in payments. Hwang and Shinn32
[11] modeled an inventory system for retailer’s pricing and lot sizingpolicy for exponentially deteriorating products under the condition of34
permissible delay in payment. Liao et al. [14] and Sarker et al. [15]investigated this topic with inflation. Jamal et al. [12], and Chang and36
Dye [2] examined this issue with allowable shortage. Jamal et al. [13], and
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OPTIMAL RETAILER’S ORDERING POLICY 943
Sarker et al. [16] addressed the optimal payment time under permissibledelay in payment with deterioration. Chang et al. [3] discussed this2
topic with deteriorating items under time-value of money and finite timehorizon. Therefore, this paper is purported to extend Huang’s model [8]4
by considering deteriorating items. Specifically, the authors model theretailer’s inventory decision as a cost minimization problem to determine6
the retailer’s optimal replenishment cycle and order quantity. The rest ofthis paper is organized in the following manner. In Section 2, the authors8
describe the notation and assumptions used throughout this paper. InSection 3, the mathematical models are derived. In Section 4, the authors10
prove the convexity of the inventory model in terms of relevant annualcost functions. In Section 5, the authors search for an approximately12
closed-form solution to the optimal replenishment cycle problem by useof Taylor’s series, and obtain an ease-to-use theorem to determine the14
optimal replenishment cycle. In Section 6, several numerical examples aregiven to illustrate the theorem so established in the present paper and to16
conduct a sensitivity analysis. Finally, in Section 7, the authors present theconclusions.18
2. Notation and assumptions
The following notation and assumptions are used throughout this20
paper.
Notation:22
D = demand rate per year;A = ordering cost per order;c = unit purchasing price per item;h = unit stock holding cost per item per year excluding interest
charges;Ie = interest earned per dollar per year;Ik = interest charged per dollar in stocks per year by the supplier;Θ = the deterioration rate of the inventory;M = the retailer’s trade credit period offered by the supplier in years;N = the customer’s trade credit period offered by the retailer in years;Q = the order quantity;I(t) = the level of inventory at time t, 0 ≤ t ≤ T;T = the replenishment cycle time in years;
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944 Y. F. HUANG ET AL.
TVC(T) = the annual total relevant cost, which is a function of T;T∗ = the optimal replenishment cycle time which minimizes
TVC(T) given T > 0;Q∗ = the optimal order quantity = DT∗.
Assumptions:2
(A) Demand rate, D , is known and remains constant.(B) Shortages are not allowed.4
(C) Time horizon is infinite.(D) Replenishment is instantaneous.6
(E) There is no repair or replacement of the deteriorated inventory duringa given cycle.8
(F) The constant fraction θ of on hand inventory gets deteriorated pertime unit.10
(G) Ik ≥ Ie , M ≥ N .(H) When T > M , the account is settled at T = M and the retailer starts12
paying for the interest charges due on the items in stock with rate Ik .When T ≤ M , the account is settled at T = M and the retailer does14
not need to pay any interest charges.(I) The retailer can accumulate revenue and earn interest after its cus-16
tomer pays for the amount of purchasing cost to the retailer until theend of the trade credit period offered by the supplier. That is, the18
retailer can accumulate revenue and earn interest during the periodfrom N (after the expiry of trade credit offered to the customers by the20
retailer) to M (before the expiry of trade credit offered to the retailerby the supplier) with rate Ie under the condition of trade credit.22
3. Mathematical formulation
The level of inventory I(t) is depleted by the effects of demand24
and deterioration, then the differential equation which describes theinstantaneous states of I(t) over (0, T) is given as:26
dI(t)dt
+θI(t) = −D, where 0 ≤ t ≤ T
with the boundary condition I(T) = 0 . The solution of above equation is28
given by
I(t) =Dθ[eθ(T−1) − 1], where 0 ≤ t ≤ T .30
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OPTIMAL RETAILER’S ORDERING POLICY 945
Noting that I(0) = Q , the quantity ordered in each replenishment cycle is
Q =Dθ(eθT − 1) .2
Furthermore, the annual total relevant cost function consists of the or-dering cost, inventory holding cost, cost of deteriorated units and the4
opportunity cost of the capital. From now on, the individual cost isevaluated before they are grouped together.6
(i) Annual ordering cost = AT .
(ii) Annual inventory holding cost (excluding the opportunity cost of8
the capital) = hT∫ T
0 I(t)dt = hDθ2T (e
θT −θT − 1) .
(iii) Annual cost of deteriorated units = c(Q−DT)T = cD
θT (eθT −θT − 1) .10
(iv) From assumptions (H) and (I), there are three cases in terms ofannual opportunity cost of the capital.12
Case I: T ≥ M
the annual opportunity cost of capital14
=cIk
∫ TM I(t)dt − cIe
∫ MN Dtdt
T
=cIkDθ2T
[eθ(T−M) −θ(T − M)− 1]− cIeD(M2 − N2)
2T16
Case II: N ≤ T < M
the annual opportunity cost of capital18
= − cIe[∫ T
N Dtdt + DT(M − T)]T
= − cIeD2T
[2MT − N2 − T2]20
Case III: T < N
the annual opportunity cost of capital22
= − cIe∫ M
N DTdtT
= cIeD(M − N)
According to the above conditions, we have24
TVC(T) =
TVC1(T) if M ≤ T (1a)
TVC2(T) if N ≤ T < M (1b)
TVC3(T) if 0 < T < N (1c)
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946 Y. F. HUANG ET AL.
where
TVC1(T) =AT+
Dθ2T
(cθ+ h)(eθT −θT − 1)2
+cIkDθ2T
[eθ(T−M) −θ(T − M)− 1]
− cIeD(M2 − N2)
2Tif T > 0, (2)4
TVC2(T) =AT+
Dθ2T
(cθ+ h)(eθT −θT − 1)
− cIeD2T
(2MT − N2 − T2) if T > 0 (3)6
and
TVC3(T) =AT+
Dθ2T
(cθ+ h)(eθT −θT − 1)− cIeD(M − N)8
if T > 0. (4)
Because TVC1(M) = TVC2(M) at T = M and TVC2(N) = TVC3(N) at10
T = N , TVC(T) is continuous and well defined.
4. The convexity12
In this section, the authors demonstrate that three inventory functionsdescribed in Section 3 are convex on their appropriate domains.14
Theorem 1.
(i) TVC1(T) is convex on [M, ∞) .16
(ii) TVC2(T) is convex on (0, ∞) .
(iii) TVC3(T) is convex on (0, ∞) .18
(iv) TVC(T) is convex on (0, ∞) .
Before proving Theorem 1 , we need the following lemma.20
Lemma 1.
eθ(T−M) − 1 −θTeθ(T−M) +θ2T2
2eθ(T−M) +θM − θ2(M2 − N2)
2> 022
if T ≥ M .
Proof. Let g(T) = eθ(T−M) − 1 − θTeθ(T−M) + θ2T2
2 eθ(T−M) + θM −24
θ2(M2−N2)2 , then we have g′(T) = θ3T2
2 eθ(T−M) . So g(T) is increasing
on (M, ∞) and g(T) > g(M) = θ2 N2
2 > 0 if T > M . Consequently,26
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OPTIMAL RETAILER’S ORDERING POLICY 947
eθ(T−M) − 1 −θTeθ(T−M) + θ2T2
2 eθ(T−M) +θM − θ2(M2−N2)2 > 0 if T ≥ M .
This completes the proof of Lemma 1. ¤2
Proof of Theorem 1. (i): Form equation (2), the derivatives yield
TVC′1(T) = − A
T2 +D
θ2T2 (cθ+ h)(θTeθT − eθT + 1)4
+cIkDθ2T2 [θTeθ(T−M) − eθ(T−M) + 1 −θM]
+cIeD(M2 − N2)
2T2 (5)6
and
TVC′′1 (T) =
2AT3 +
2D(cθ+ h)θ2T3
[eθT
(1 −θT +
θ2T2
2
)− 1
]8
+2cIkDθ2T3
[eθ(T−M) − 1 −θTeθ(T−M)
+θ2T2
2eθ(T−M) +θM
]− cIeD(M2 − N2)
T310
≥ 2AT3 +
2D(cθ+ h)θ2T3 [(eθT · e−θT)− 1]
+2cIkDθ2T3
[eθ(T−M) − 1 −θTeθ(T−M)
12
+θ2T2
2eθ(T−M) +θM − θ2(M2 − N2)
2
]
=2AT3 +
2cIkDθ2T3
[eθ(T−M) − 1 −θTeθ(T−M)
14
+θ2T2
2eθ(T−M) +θM − θ2(M2 − N2)
2
]. (6)
Lemma 1 imply that d2TVC1(T)dT2 > 0 if T ≥ M . Therefore, TVC1(T) is16
convex on [M, ∞) .
(ii) and (iii): From equations (3) and (4), the derivatives yield18
TVC′2(T) = − A
T2 +D(cθ+ h)
θ2T2 (θTeθT − eθT + 1)
− cIeD2T2 (N2 − T2), (7)20
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948 Y. F. HUANG ET AL.2
TVC′′2 (T) =
2AT3 +
2D(cθ+ h)θ2T3
[eθT
(1 −θT +
12θ2T2
)− 1
]
+cIeDN2
T34
>2AT3 +
2D(cθ+ h)θ2T3 [eθT · e−θT − 1] +
cIeDN2
T3
=2AT3 +
cIeDN2
T3 > 0, (8)6
TVC′3(T) = − A
T2 +D(cθ+ h)
θ2T2 (θTeθT − eθT + 1) (9)
and8
TVC′′3 (T) =
2AT3 +
2D(cθ+ h)θ2T3
[eθT(1 −θT +
θ2T2
2)− 1
]
>2AT3 +
2D(cθ+ h)θ2T3 [eθT · e−θT − 1]10
=2AT3 > 0. (10)
Therefore, TVC2(T) and TVC3(T) is convex on (0, ∞) , respectively.12
(iv) Case I implies that TVC′1(T) is increasing on [M, ∞) . Cases II and III
implies that TVC′2(T) and TVC′
3(T) is increasing on (0, M] . Since14
TVC′1(M) = TVC′
2(M) and TVC′2(N) = TVC′
3(N) , then TVC′(T) isincreasing on T > 0 . Consequently TVC(T) is convex on T > 0 .16
Combing the above arguments, we have completed the proof. ¤
5. Theoretical results18
Since all TVCi(T) (i = 1, 2, 3) are convex on their appropriatedomains. We are in a position to consider the following equations, which20
are the necessary and sufficient conditions for obtaining the optimalsolution.22
First order conditions:
TVC′1(T) = 0, (11)24
TVC′2(T) = 0 (12)
and26
TVC′3(T) = 0. (13)
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OPTIMAL RETAILER’S ORDERING POLICY 949
We can simplify equations (11)-(13) to obtain
−A +Dθ2 (cθ+ h)(θTeθT − eθT + 1)2
+cIkDθ2 [θTeθ(T−M) − eθ(T−M) + 1 −θM]
+cIeD(M2 − N2)
2= 0 , (14)4
− A +D(cθ+ h)
θ2 (θTeθT − eθT + 1)− cIeD2
(N2 − T2) = 0 (15)
and6
− A +D(cθ+ h)
θ2 (θTeθT − eθT + 1) = 0. (16)
From equations (14)-(16), we cannot obtain the explicit closed-form solu-8
tion of the optimal cycle time, T∗i (i = 1, 2, 3) . In reality, the value for
the deterioration rate θ is usually very small. For simplicity, using the10
truncated Taylor’s series expansion for the exponential term, we have
eθT ≈ 1 +θT + (θT)2/2 as θT is small.12
Then we obtain
θTeθT − eθT + 1 =(θT)2
2+ o(θ2) ≈ (θT)2
2(17)14
and
θTeθ(T−M) − eθ(T−M) + 1 −θM =(θT)2
2− (θM)2
2+ o(θ2)16
≈ (θT)2
2− (θM)2
2. (18)
Substituting equations (17) and (18) into equations (14)-(16), we can get18
T∗1 ≈
√2A + cD[M2(Ik − Ie) + N2 Ie]
D[h + c(θ+ Ik)], (19)
T∗2 ≈
√2A + cDN2 Ie
D[h + c(θ+ Ie)](20)20
and
T∗3 ≈
√2A
D(h + cθ). (21)22
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950 Y. F. HUANG ET AL.
To ensure T∗1 ≥ M , we substitute equation (19) into inequality T∗
1 ≥ M ,and obtain that2
if − 2A + DM2[h + c(θ+ Ie)]− cDN2 Ie ≤ 0, then T∗1 ≥ M. (22)
Similarly, we substitute equation (20) into inequality N ≤ T∗2 < M , and4
obtain that
if − 2A + DM2[h + c(θ+ Ie)]− cDN2 Ie > 06
and − 2A + DN2(h + cθ) ≤ 0, then N ≤ T∗2 < M. (23)
Finally, we substitute equation (21) into inequality T∗3 < N , and obtain8
that
if − 2A + DN2(h + cθ) > 0, then T∗3 < N. (24)10
For convenience, we define
∆1 = −2A + DM2[h + c(θ+ Ie)]− cDN2 Ie (25)12
and
∆2 = −2A + DN2(h + cθ). (26)14
Since M ≥ N , we can easily obtain ∆1 ≥ ∆2 . From equations (19)-(26),we know that a higher value of A causes a longer replenishment cycle,16
and vice versa. In contrast, a higher value of h or θ causes a shorterreplenishment cycle, and vice versa. Putting all together, we have the18
following theorem to determine the optimal replenishment cycle time T∗ .
Theorem 2.20
(i) If ∆2 > 0 , then TVC(T∗) = TVC3(T∗3 ) . Hence, T∗ is equal to T∗
3 .
(ii) If ∆1 > 0 and ∆2 ≤ 0 , then TVC(T∗) = TVC2(T∗2 ) . Hence, T∗ is22
equal to T∗2 .
(iii) If ∆1 ≤ 0 , then TVC(T∗) = TVC1(T∗1 ) . Hence, T∗ is equal to T∗
1 .24
Proof. It immediately follows from equations (22)-(26).
6. Numerical examples26
In this section, the authors provide several numerical examples toillustrate the theoretical results contained in Theorem 2 as reported in28
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OPTIMAL RETAILER’S ORDERING POLICY 951
Section 5. At the same time, the authors also carry out a sensitivity analysisconcerning the impact of a change in the parametric value on the optimal2
replenishment cycle and optimal order quantity.
Example 1 (Optimal solution for T∗3 ). Given A = $50 /order, D = 70004
units/year, M = 0.1 year, N = 0.05 year, c = $100 /unit, Ik =
$0.15 /$/year, Ie = $0.12 /$/year, θ = 0.01 and h = $5 /unit/year.6
Then, we can obtain ∆1 = 950 > 0 and ∆2 = 5 > 0 . Using Theorem 2(i),we get T∗ = T∗
3 = 0.048795 year. The corresponding optimal order8
quantity will be Q∗ = DT∗3 = 341.565 units.
Example 2 (Optimal solution for T∗2 ). Given A = $100 /order, D = 200010
units/year, M = 0.1 year, N = 0.05 year, c = $100 /unit, Ik =
$0.15 /$/year, Ie = $0.12 /$/year, θ = 0.01 and h = $5 /unit/year.12
Then, we can obtain ∆1 = 100 > 0 and ∆2 = −170 < 0 . UsingTheorem 2(ii), we get T∗ = T∗
2 = 0.084984 year. The corresponding14
optimal order quantity will be Q∗ = DT∗2 = 169.968 units.
Example 3 (Optimal solution for T∗1 ). Given A = $100 /order, D = 100016
units/year, M = 0.1 year, N = 0.05 year, c = $100 /unit, Ik =
$0.15 /$/year, Ie = $0.12 /$/year, θ = 0.01 and h = $5 /unit/year.18
Then, we can obtain ∆1 = −50 < 0 and ∆2 = −185 < 0 . UsingTheorem 2(iii), we get T∗ = T∗
1 = 0.11127 year. The corresponding20
optimal order quantity will be Q∗ = DT∗1 = 111.27 units.
Example 4 (Sensitivity analysis on A ). Given D = 5000 units/year,22
M = 0.1 year, N = 0.05 year, c = $100 /unit, Ik = $0.15 /$/year,Ie = $0.12 /$/year, 0 = 0.01 and h = $5 /unit/year. If A = $30 , 60 or 9024
per order, then from Theorem 2 we can easily obtain the optimal solutionsas shown in Table 1. It indicates that a higher value of ordering costal26
implies a longer replenishment cycle T∗ and a larger order quantity Q∗ .
Example 5 (Sensitivity analysis on θ ). Given A = $60 /order, D =28
7000 units/year, M = 0.1 year, N = 0.05 year, c = $100 /unit,Ik = $0.15 /$/year, Ie = $0.12 /$/year and h = $5 /unit/year. If30
θ = 0.01 , 0.02 or 0.03, then from Theorem 2 we can easily obtain theoptimal solutions as shown in Table 2. It indicates that a higher value of32
deteriorating rate θ implies a shorter of replenishment cycle T∗ and asmaller order quantity Q∗ .34
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952 Y. F. HUANG ET AL.
Table 1Sensitivity analysis on A
Ordering ∆1 ∆2 Theorem 2 Replenishment Economic order
cost A cycle T∗ quantity Q∗
30 > 0 > 0 (i) T∗3 = 0.044721 223.6068
60 > 0 < 0 (ii) T∗2 = 0.054772 273.8613
90 > 0 < 0 (ii) T∗2 = 0.060553 302.765
2
Table 2Sensitivity analysis on θ
Deterioration ∆1 ∆2 Theorem 2 Replenishment Economic order
rate θ cycleT∗ quantity Q∗
0.01 > 0 < 0 (ii) T∗2 = 0.051177 358.2364
0.02 > 0 > 0 (i) T∗3 = 0.049487 346.4102
0.03 > 0 > 0 (i) T∗3 = 0.046291 324.037
4
7. Conclusions
In reality, we have witnessed two levels of trade credit: one is offered6
to the retailer from its supplier; the other is offered to the customersfrom its retailer. Moreover, it is usually the case that the commodity8
under consideration will deteriorate over time. The authors are thereforemotivated to develop a retailer’s EOQ model for the deteriorating item in10
the presence of two-level trade credit.
In order to obtain the explicit closed-form solutions to the proposed12
model, the authors use Taylor’s series approximation. Through suchprocedure, the authors establish an easy-to-use Theorem 2 from the14
retailer’s perspective. Finally, some numerical examples are provided toillustrate this Theorem, and to obtain the following managerial insights:16
(1) a higher value of ordering cost brights about longer replenishmentcycle and a larger order quantity; (2) a higher value of deterioration rate18
brights about shorter replenishment cycle and a smaller order quantity.
Acknowledgements. This paper is supported by NSC Taiwan, number20
NSC 96-2221-E-324-007-MY3 and CYUT.
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