- Email: [email protected]
An EOQ model for retailers partial permissible delay in payment linked to order quantity with shortages
Accepted Manuscript An EOQ model for retailers partial permissible delay in payment linked to order quantity with shortages Vandana, B.K. Sharma PII: DOI: Reference:
S0378-4754(15)00258-X http://dx.doi.org/10.1016/j.matcom.2015.11.008 MATCOM 4268
To appear in:
Mathematics and Computers in Simulation
Received date: 23 November 2014 Revised date: 25 September 2015 Accepted date: 25 November 2015 Please cite this article as: Vandana, B.K. Sharma, An EOQ model for retailers partial permissible delay in payment linked to order quantity with shortages, Math. Comput. Simulation (2015), http://dx.doi.org/10.1016/j.matcom.2015.11.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Manuscript Click here to view linked References
An EOQ model for retailers partial permissible delay in payment linked to order quantity with shortages Vandana∗ and B. K. Sharma School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, (C.G.), 492010, India.
Abstract Living in the business world, maximizing owner’s happiness or getting paid by the sales of goods or services on an open account at some reasonable profit is the main purpose of any business. The credit functions play a vital, role within the organization. In this paper, we develop an inventory model for retailers partial permissible delay-in-payment linked to order quantity with shortage, which is partial backlogged. Here, we consider two different cases, i.e. in first, the trade-credit period (M ) is greater than or equal to the time interval tw , that w units are depleted to zero due to demand; and later, the trade-credit period (M ) is less than to the time interval tw . The principle objective of the introduced model is to minimize the total inventory cost by finding an optimal replenishment policy. Required theorems are provided to verify the optimal solutions. Various numerical examples and managerial implications are discussed to check and substantiated the intransigent results.
2010 Mathematics Subject Classification 90B05. Keywords Inventory model; EOQ model; Partial backlogging; Partial trade-credit.
1
Introduction
Inventory control [12] is a science-based art of controlling the amount of stocks held in various forms, within the business to meet economically the demands placed upon that business. To minimize the total relevant cost the first and basic inventory model was developed by F.W. Harris [7] in 1915, known as an economic order quantity model. In 1935, H. P. Dutton [5] said that, “The principle of determining the economic lot size, i.e. (EOQ), for setting the maximum in stocks and limits in stock control have been known for many years, although these formulas are even yet, in everyday use in a comparatively small number of concerns.” In Harris [7] model, they did not mention the shortage condition. However, in practice sometimes customers demand cannot be fulfilled by the suppliers from the current stocks, this situation is known as a stock-out or shortage condition. In shortages period, two situations has occurred as first, where customers are willing to wait until the arrival of the next order, is complete backlogging and second, when some customers are not willing to 1
∗
Corresponding author Email: [email protected]
1
wait for the next replenishment, they leave the system, has lost sales case. In most of the inventory models, researchers allow shortages, which are completely backlogged. In 1973, Montgomery et al. [11] developed a model for non-perishable items with partial back-order. Now many researchers work on these aspects and modify the partial backlogging phenomena (for details see [18, 16, 17, 19, 4] and references therein). In 1999, Chang and Dye [1], developed an EOQ model with partial backlogged shortage. They assumed that, in case of trendy/fashionable goods, the backlogging are depending on the length of the waiting time. Therefore, the backlogging rates are variable and are dependent on the waiting time for the next replenishment. Hence, the proportion of customers, who would like to accept backlogging at a time (t), decreases with increasing waiting time (T − t) waiting for the next replenishment. After that, many researchers are working to consider this assumption. The main purpose of any business is to maximize owner’s happiness, in other words “getting paid” because business is essentially afraid with the sale of goods or services on an open account at a profit, the credit function plays a vital, role within the organization. When the supplier delivered goods to their customers, they often do not require to be paid immediately. Instead, suppliers offer credit terms that allow the buyers to delay the payment. This is known as trade-credit. This is very beneficial to the customers because, they do not have to pay the supplier immediately, after receiving the product, but instead, can delay their payment until, the end of the allowed period. The related first model was given by Goyal [6] in 1985. After that, many researchers works on those aspects, see ([14, 15, 9, 10, 20, 21] and references therein). In 2003, Chang et al. [2] developed a model for deteriorating items under supplier credits linked to order quantity. In 2007, Huang [8] developed a model in which he breaks the assumption of delay-in-payments, i.e. the supplier will offer to delay-in-payment to his/her retailers, if he/she orders more than the predetermined quantities say w. In Huang [8] model, he considered that, to increase their own sales or profit manufacturers/vendors or supplier offers to the retailers a permissible delay-in-payment even if, they ordered less than the predetermined quantity w. But, for enjoying the permissible delay-in-payments the retailers must pay a portion of the total purchasing cost, say (1 − α)cQ immediately. After that, he/she would pay the remaining balance at the end of the trade-credit period. Next, Ouyang et al. [13] extended Huangs [8] model with considering partially permissible delay in payments linked to order quantity. After that, many researchers works on those aspects, see ([22, 23] and references therein). In 2014, Chen et al. [3], have developed a model to overcome the shortcomings of Huangs [8] and Ouyang et al. [13] models. In this paper, we extend the model of Chen et al. [3] with allowable shortages (which is partial backlogged) and analyze the partial permissible delay-in-payment in an EOQ model, which is linked to order quantity. Also, we assume that, the supplier offers to the retailers a permissible delay-in-payment, even if they order less than the predetermined quantity w. For this situation, to enjoy a fully permissible delay-in-payments the retailers must pay a portion of the total purchasing cost, say (1 − α)cQ immediately. The main purpose of this article, is to investigate the optimal replenishment policy to minimize the total relevant cost.
2
Table 1: Summary of symbols used and their meanings Symbol A c h s o p α β δ Imax Ie Ip M tw t1 t∗1 T Q T C(t1 ) T C∗
Meaning ordering cost per unit purchasing cost per unit holding cost per unit per unit time excluding the capital cost shortage cost for backlogged items per unit per year cost of lost sales per unit selling price per unit the fraction of the delay payments permitted by suppliers per order, 0 ≤ α ≤ 1 1 denote the shortage fraction, i.e. β(t) = 1+δt backlogging parameter, which is positive and constant maximum inventory level interest earned per dollar interest charged per dollar trade-credit period w time interval that w units are depleted to zero due to demand D, where tw = D length of time in which there are no inventory shortage (t1 > td ) optimal length of time in which there are no inventory shortage duration of the replenishment cycle (T > t1 ) order quantity total minimum relevant cost for the inventory system optimal total minimum relevant cost per unit time
The rest of the paper is given as follows: In Section 2, we discuss some notations and assumptions of our model. The mathematical formulation to minimize the total annual inventory cost is established in Section 3. To characterize the optimal solution we derive some theorems and lemmas in Section 4. Several numerical examples for testing and verifying the models are provided in Section 5. Finally, the managerial implications and conclusion of the proposed model is presented in Section 6 and Section 7 respectively.
2
Notation and Assumptions
Notation In this paper, we are following the Chen et al. [3] (2014) notations, given in Table 1.
Assumptions 1. Demand rate is known and constant. 2. Replenishment rate is instantaneous. 3. Shortages are allowed. Unsatisfied demand is partially backlogged, and the fraction of shortages back-ordered is a differentiable and decreasing function of time, denoted as δ(t), where t is the waiting time up to the next replenishment with 0 ≤ δ(t) ≤ 1. 1 Let, β(t) denotes this fraction then, β(t) = 1+δt . 3
4. There is no deterioration. 5. The time horizon is infinite. 6. Only one type of item is considered. 7. The supplier offers to the retailers a permissible delay-in-payment even if, they order less than the predetermined quantity w. For this situation, the retailers must pay a portion of the total purchasing cost, say (1 − α)cQ immediately, where α the fraction of the delay payments permitted by suppliers per order, 0 ≤ α ≤ 1.
3
Mathematical Formulation
During the time period [0, t1 ] the differential equations delineate the inventory status as
Figure 1: Graphical representation of the inventory system
dI1 = −D, 0 ≤ t ≤ t1 (3.1) dt with the boundary condition I1 (t1 ) = 0. Till the time period [t1 , T ] the shortages are occurred and which is partial backlogged. Thus, the differential equation formulate the inventory status as dI2 −D0 = , t1 ≤ t ≤ T dt 1 + δ(T − t)
(3.2)
with the boundary condition I2 (t1 ) = 0. Now, we solve the equations(3.1) and (3.2) with a given differential equation, we get 4
I1 (t) = D(t1 − t) I2 (t) =
D δ
1+δ(T −t) log( 1+δ(T −t1 ) )
(3.3) (3.4)
After that, we calculate the value of maximum inventory level, i.e. Imax . Since, I1 (0) = Imax , then we obtain Imax = Dt1
(3.5)
After that, we put t = T in Eq.(3.4), we can obtain the maximum amount of backlogged demand per cycle as S = −I2 (T ) =
D δ (log(1
+ δ(T − t1 )))
(3.6)
Now, we calculate the order quantity of per cycle, as Q = Imax + S
(3.7)
D δ (log(1
= Dt1 +
+ δ(T − t1 )))
(3.8)
Now, we can obtain the value of the time interval that, w units are depleted to zero due w to demand given as: tw = D . After that, retailers start to accumulate the total cost with calculating the several components • Ordering cost (OC) = Firstly, the retailers will calculate the ordering cost, i.e. the incremental costs of processing an order of goods from a supplier is OC = A T • Holding cost (HC) = holding cost is a major component of supply chain management. The retailers will calculate the holding or carrying cost of items.
HC = =
h T
Z
t1
0
hDt21 2T
I1 (t)dt
(3.9) (3.10)
• Shortage cost (SC) = shortage cost or stock-out cost is the total of all the costs, which associated with shortage units.
SC = =
Z
s T(
T
t1
−I2 (t)dt)
sD (δ(T δ2
− t1 ) − log(1 + δ(T − t1 )))
• Opportunity cost due to lost sale (LS) Z T LS = To D(1 − =
(3.11)
1 1+δ(T −t) )dt
t1 oD (δ(T −t1 )−log(1+δ(T −t1 ))) T δ
5
(3.12)
(3.13)
• Interest earn and payable = Based on the values of M and tw , retailers has two possible cases and we discussed each in details Case (1) tw ≤ M - In this case, there are three possible sub cases are arises, given as below: • subcase(1) tw ≤ M ≤ t1 - In this case, the retailers receive the fully permissible delay, then the interest payable per cycle is the
Ip1 = = =
Z
t1
cIp T (
I1 (t)dt) M D(t21 −M 2 ) cIp + T (− 2 2 cIp D(t1 −M ) 2T
Dt1 (t1 − M ))
(3.14)
and the interest earn is given as Z
pIe T (
Ie1 =
M
(3.15)
Dtdt)
0 pIe DM 2 2T
=
(3.16)
• subcase(2) tw ≤ t1 ≤ M - In this case tw ≤ t1 , thus the retailer receives the fully permissible delay-in-payment, but there are no interest charge, because t1 ≤ M . Thus, the interest charge is Ip2 = 0
(3.17)
and the interest earn is as
Ie2 = =
Z
pIe T (
t1
0
Dtdt + (M − t1 )
t pIe Dt1 (M − 21 ) T
Z
t1
Ddt)
(3.18)
0
(3.19)
• subcase(3) t1 < tw ≤ M - In this case t1 < tw thus, the retailers are receives the partial permissible delay. For the immediate payment the interest payable is Ip3 =
(1−α)Dt21 cIp 2T
(3.20)
and the interest earn is given as below: Ie3 =
6
αpDt1 Ie (M T
−
t1 2)
(3.21)
Therefore, the total minimum relevant cost T C(t1 ) per unit is given as T C(t1 )
=
Thus, we get T C1 (t1 , T )
T C2 (t1 , T )
=
=
T C1 (t1 , T ) T C2 (t1 , T ) T C3 (t1 , T ) A T
+
hDt1 2 2T
+
+
Ip cD(t1 −M ) 2T
A T
+
hDt1 2 2T
2
+
= = =
=
A T
+
+
hDt1 2 2T
+
Ip (1−α)cDt1 2t
OC + HC + SC + LS + Ip2 − Ie2 , tw < t1 ≤ M
OC + HC + SC + LS + Ip3 − Ie3 , t1 < tw ≤ M
sD 1 δ 2 (log( (1+δ(T −t1 )) ) Ie pDM 2T
−
2
+ δ(T − t1 )) +
oD δT (δ(T
+ δT − δt1 ) +
oD δT (δ(T
sD 1 δ 2 (log( (1+δT −δt1 ) )
+ δT − δt1 ) +
oD δT (δ(T
2
t1 2)
−
− t1 ) − log(1 + δ(T − t1 ))) (3.22)
sD 1 δ 2 (log( (1+δT −δt1 ) )
1 − Ie pDt (M − T
T C3 (t1 , T )
OC + HC + SC + LS + Ip1 − Ie1 , M ≤ t1
Ie αpDt1 (M T
−
− t1 ) − log(1 + δ(T − t1 ))) (3.23)
− t1 ) − log(1 + δ(T − t1 )))
t1 2)
(3.24)
Case (2) M < tw : In this case, there are three possible subcases are arises, such as • subcase(1) M < tw ≤ t1 - In this case, the retailers receive the fully permissible delay-in-payment, then the interest charge is
Ip4 = = =
Z
cIp T (
t1
I1 (t)dt) M D(t21 −M 2 ) cIp + T (− 2 2 cIp D(t1 −M ) 2T
Dt1 (t1 − M ))
(3.25)
and the interest earn is as
Ie4 = =
Z
pIe T (
M
0 pIe DM 2 2T
Dtdt) (3.26)
• subcase(2) M < t1 ≤ tw - Since t1 ≤ tw , then the retailers will receives partial permissible delay-in-payment. For the immediate payment the retailers will pay (1−α)Dt21 cIp the interest charge, which is same as Ip3 , i.e. , again the annual 2T interest charged for the delayed payment is 7
Ip5 =
Ip (1−α)cDt21 2T
+
=
Ip (1−α)cDt21 2T Ip (1−α)cDt21 2T
+
=
+
Z
Ip c T (
t1
I1 (t)dt) M D(t21 −M 2 ) Ip c + T (− 2 2 cIp D(t1 −M ) 2T
Dt1 (t1 − M ))
(3.27)
and the interest earn is Ie5 =
Ie pDM 2 2T
(3.28)
• subcase(3) t1 < M ≤ tw - In this case, the interest charge and the interest earn both are the same as Ip3 and Ie3 . Therefore, the total minimum relevant cost T C(t1 ) per unit time is
T C(t1 )
T C4 (t1 , T )
T C5 (t1 , T )
=
=
=
A T
T C4 (t1 , T ) T C5 (t1 , T ) T C6 (t1 , T )
+
hDt1 2 2T
+
2
+
cIp D(t1 −M ) 2T
A T
+
hDt1 2 2T
+
I (1−α)cDt21 + p 2T
T C6 (t1 , T )
=
A T
+
4
sD δ2 T
+
hDt1 2 2T
+
Ip (1−α)cDt1 2T
= =
OC + HC + SC + LS + Ip5 − Ie5 , M < t1 ≤ tw
=
OC + HC + SC + LS + Ip3 − Ie3 , t1 < M ≤ tw
(log( (1+δ(T1 −t1 )) ) + δ(T − t1 )) + Ie pDM 2T
−
OC + HC + SC + LS + Ip4 − Ie4 , M < tw ≤ t1
2
cIp D(t1 −M )2 2T
−
−
Ie αpDt1 (M T
−
+ δT − δt1 ) +
Ie αpDM 2T
sD 1 T δ 2 (log( (1+δT −δt1 ) )
2
− t1 ) − log(1 + δ(T − t1 ))) (3.29)
sD 1 T δ 2 (log( (1+δT −δt1 ) )
+
oD δT (δ(T
oD δT (δ(T
2
+ δT − δt1 ) +
− t1 ) − log(1 + δ(T − t1 ))) (3.30)
oD δT (δ(T
t1 2)
− t1 ) − log(1 + δ(T − t1 ))) (3.31)
Theoretical results
Now, we discuss the theoretical aspects of our proposed model. Case(1) When tw ≤ M - In this case, there are three subcases. subcase(1) when M ≤ t1 The necessary conditions for the total annual cost of T C1 (t1 , T ) in (3.22) to be minimum are dTdtC1 1 = 0 and dTdTC1 = 0. Thus, we get 8
η1 (t1 ) =
= 0 and υ1 (T ) =
dT C1 dt1
η1 =
υ1
=
−D T2
A+
+Ip c(−
ht21 2
D T
+
dT C1 dT
ht1 +
= 0. Thus, we get
s+oδ 1 δ ( (1+δ(T −t1 ))
(s+oδ)T δ
( (1+δ(T1 −t1 )) − 1) − Ie pM 2 + t1 (t1 − M )) − 2
(t21 −M 2 ) 2
− 1) + Ip c(t1 − M ) s+oδ δ 2 (−δT
(4.1)
+ δt1 + log(1 + δ(T − t1 ))) (4.2)
= U and then we solve (4.1) for T , i.e. For the sake of convenience, we assume that s+oδ δ ht1 +Ip c(t1 −M ) T = t1 + δ(U −ht1 −Ip c(t1 −M )) = χ1 , put this value in (4.2), we have υ1 (t1 )
=
A+
ht21 2
+ U χ1 ( (1+δ(χ11 −t1 )) − 1) − Uδ (−δχ1 + δt1 + log(1 + δ(χ1 − t1 ))) (t21 −M 2 ) Ie pM 2 +Ip c(− + t1 (t1 − M )) − 2 (4.3) 2 −D χ1 2
Again, we put t1 = M in (4.3) ∆1
=
−D χ1 2
2 1 A + hM 2 + U χ1 ( (1+δ(χ1 −M )) − 1) −
− Ie pM 2
Lemma 4.1.
U δ (−δχ1
+ δM + log(1 + δ(χ1 − M )))
2
(4.4)
1. If ∆1 < 0, then the total annual cost has its minimum value at t1 > M .
2. If ∆1 ≥ 0, then the total annual cost has its minimum value in (t1 , T ) = (M, T1 ), i.e. t1 = M . Proof. Refer to Appendix 1. The necessary conditions for the total annual cost of T C2 (t1 , T ) in (3.23) to be minimum are dTdtC1 2 = 0 and dTdTC2 = 0. Thus, we have η2 (t1 ) = dTdtC1 2 = 0 and υ2 (T ) = dTdTC2 = 0. Thus, we get η2 =
υ2
=
−D T2
A+
ht21 2
D T
+
ht1 +
(s+oδ)T δ
t1 −Ie pt1 (M − 2 )
s+oδ 1 δ ( (1+δ(T −t1 ))
( (1+δ(T1 −t1 )) − 1) −
− 1) − Ie p(M − t1 )
s+oδ δ 2 (−δT
(4.5)
+ δt1 + log(1 + δ(T − t1 ))) (4.6)
9
For the sake of convenience, we assume that, s+oδ δ = U after that, we solve (4.5) for T , i.e. ht1 −Ie p(M −t1 ) T = t1 + δ(U −ht1 +Ie p(M −t1 )) = χ2 , put this value in (4.6), we have
ht21 2
+ U χ2 ( (1+δ(χ12 −t1 )) − 1) − t1 −Ie pt1 (M − 2 )
υ2 =
−D χ2 2
A+
U δ (−δχ2
+ δt1 + log(1 + δ(χ2 − t1 ))) (4.7)
Now, we put t1 = tw in (4.7), we get ∆2
=
A+
ht2w 2
+ U χ2 ( (1+δ(χ12 −tw )) − 1) − −Ie ptw (M − t2w ) −D χ2 2
U δ (−δχ2
+ δtw + log(1 + δ(χ2 − tw )))
(4.8)
Lemma 4.2. 1. If ∆2 ≤ 0 ≤ ∆1 , then the total annual cost has its minimum value at tw < t1 ≤ M . 2. If ∆2 > 0, then the total annual cost has its minimum value at (t1 , T ) = (tw , T2 ), i.e. t1 = tw . 3. If ∆1 < 0, then the total annual cost has its minimum value at (t1 , T ) = (M, T2 ), i.e. t1 = M . Proof. Refer to Appendix 1. The necessary conditions for the total annual cost of T C3 (t1 , T ) in (3.24) to be minimum are dTdtC1 3 = 0 and dTdTC3 = 0. Thus, we get η3 (t1 ) = dTdtC1 3 = 0 and υ3 (T ) = dTdTC3 = 0. Thus, we get η3 =
υ3
=
D T
−D T2
+
ht1 +
A+
s+oδ 1 δ ( (1+δ(T −t1 ))
ht21 2
Ip (1−α)ct21 2
+
(s+oδ)T δ
( (1+δ(T1 −t1 )) − 1) − − Ie αpt1 (M − t21 )
For our convenience, we assume that, t1 +
− 1) + Ip (1 − α)ct1 − Ie αp(M −
s+oδ δ
I αpt t ht1 +Ip (1−α)ct1 −Ie αp(M − 21 )+ e 2 1 I αpt t δ(U −ht1 −Ip (1−α)ct1 +Ie αp(M − 21 )− e 2 1 )
υ3
=
−D χ3 2
+
A+
+
Ie αpt1 2
(4.9)
+ δt1 + log(1 + δ(T − t1 ))) (4.10)
= U and then we solve (4.9) for T , i.e. T =
= χ3 , put this value in (4.10), we have
ht21 2
Ip (1−α)ct21 2
s+oδ δ 2 (−δT
t1 2)
+ U χ3 ( (1+δ(χ13 −t1 )) − 1) − t1 − Ie αpt1 (M − 2 )
10
U δ (−δχ3
+ δt1 + log(1 + δ(χ3 − t1 ))) (4.11)
Now, we put t1 = tw in (4.11), we get υ3
=
−D χ3 2
+
Lemma 4.3.
A+
ht2w 2
Ip (1−α)ctw 2 2
+ U χ3 ( (1+δ(χ13 −tw )) − 1) − − Ie αptw (M − t2w )
U δ (−δχ3
+ δtw + log(1 + δ(χ3 − tw ))) (4.12)
1. If ∆3 ≥ 0, then the total annual cost has its minimum value in t1 < M .
2. If ∆3 < 0, then the value of T ∈ (0, tw ), which minimize T C3 (t1 , T ) does not exist. Proof. Refer to the Appendix 2. Theorem 4.4. For tw ≤ M , we have T C(t1 , T ) T C(t1 , T ) = T C1 (t∗1 , T1 ) T C(t1 , T ) = min (T C1 (t∗1 , T1 ), T C3 (t1 , T3 )) T C(t1 , T ) = min(T C2 (t1 , T2 ), T C3 (t1 , T3 )) T C(t1 , T ) = min(T C2 (tw , T2 ), T C3 (t1 , T3 )) T C(t1 , T ) = (T C2 (t1 , T2 ))
Conditions ∆1 ≤ 0 & ∆3 < 0 ∆1 ≤ 0, ∆3 ≥ 0 ∆1 > 0, ∆2 < 0 & ∆3 ≥ 0 ∆2 ≥ 0 ∆1 > 0, ∆3 < 0
T T1 T1 /T3 T2 /T3 T2 /T3 T2
Case(2) when M ≤ tw - Now, in this case there are three subcases are arises. subcase (1) when M ≤ tw ≤ t1 - In this case, the total cost is the same as the total cost of subcase 1, then the necessary condition for the total cost of T C1 (t1 , T ), for this sub case is same as in (4.1) and (4.2). To show that uniqueness, we put t1 = tw in (4.2). Thus, we have ∆4 (t1 )
=
A+
htw 2 2
+Ip c(−
Lemma 4.5.
+ U χ4 ( (1+δ(χ14 −tw )) − 1) −
(t2w −M 2 ) 2
+ tw (tw − M )) −
U δ (−δχ4
Ie pM 2 2
+ δtw + log(1 + δ(χ4 − tw ))) (4.13)
1. If ∆4 ≤ 0, then the total annual cost has its minimum value at t1 > M .
2. If ∆4 ≥ 0, then the total annual cost has its minimum value at (t1 , T ) = (tw , T ), i.e. t1 = tw . Proof. Refer to Appendix 1. subcase (2) when M ≤ t1 ≤ tw The necessary conditions for the total annual cost of T C5 (t1 , T ) in (3.30) to be minimum are dTdtC1 5 = 0 and dTdTC5 = 0, which given as η4 (t1 ) = dTdtC1 5 = 0 and υ4 (T ) = dTdTC5 = 0, thus, we get 11
υ4
=
η4
=
−D T2
A+
D T
+Ip c(−
ht1 +
ht21 2
+
(t21 −M 2 ) 2
s+oδ 1 δ ( (1+δ(T −t1 ))
− 1) + Ip c((1 − α)t1 + (t1 − M ))
(s+oδ)T δ
( (1+δ(T1 −t1 )) − 1) − 2 + t1 (t1 − M )) − Ie αpM 2
s+oδ δ 2 (−δT
(4.14)
+ δt1 + log(1 + δ(T − t1 ))) (4.15)
For our convenience we assume that, s+oδ = U and then we solve (4.14) for T , i.e. T = δ ht1 +Ip c((1−α)t1 +(t1 −M )) t1 + δ(U −ht1 +Ip c((1−α)t1 +(t1 −M ))) = χ4 , then putting this value in (4.15), we have υ4
=
−D χ4 2
+
A+
ht21 2
Ip (1−α)ct21 2
+ U χ4 ( (1+δ(χ14 −t1 )) − 1) −
+ Ip c(−
(t21 −M 2 ) 2
U δ (−δχ4
+ t1 (t1 − M )) −
+ δt1 + log(1 + δ(χ4 − t1 )))
Ie αpM 2 2
(4.16)
Now, we put t1 = M and t1 = tw in (4.16), we get ∆5
=
−D χ4 2
+
∆6
=
A+
hM 2 2
Ip (1−α)cM 2 2
−D χ4 2
+
A+
ht2w 2
Ip (1−α)ct2w 2
+ U χ4 ( (1+δ(χ14 −M )) − 1) − 2 − Ie αpM 2
U δ (−δχ4
+ U χ4 ( (1+δ(χ14 −tw )) − 1) −
U δ (−δχ4
+ Ip c(−
(t2w −M 2 ) 2
+ δM + log(1 + δ(χ4 − M ))) (4.17)
+ tw (tw − M )) −
+ δtw + log(1 + δ(χ4 − tw )))
Ie αpM 2 2
(4.18)
Lemma 4.6. 1. If ∆5 ≤ 0 ≤ ∆6 , then the total annual cost T C5 (t1 , T ) has the unique minimum value at (t1 , T ) = (t1 , T ), where t1 ∈ [M, tw ). 2. If ∆5 > 0, then the total annual cost T C5 (t1 , T ) has a minimum value at (t1 , T ) = (M, T ), i.e. t1 = M . 3. If ∆6 < 0, then the value of T ∈ [M, tw ), which minimize the total cost does not exists.
Proof. Refer the Appendix 1 and 2.
subcase (3) when t1 ≤ M ≤ tw - Since, for this subcase the necessary condition for the minimization of T C3 is the same as (4.9) and (4.11). Now, we put t1 = M in (4.11), we get ∆7
=
−D χ3 2
+
A+
hM 2 2
Ip (1−α)cM 2 2
+
−
(s+oδ)χ3 ( (1+δ(χ13 −M )) δ
Ie αpM 2 2
− 1) −
s+oδ δ 2 (−δχ3
+ δM + log(1 + δ(χ3 − M ))) (4.19)
12
Now, it is clear that, the value of T C3 (M, T ) = T C5 (M, T ). Thus, we can obtain the following lemma Lemma 4.7.
1. If ∆5 ≥ 0, then the total cost T C3 (t1 , T ) has the unique value at t1 < M .
2. If ∆5 < 0, then the total cost T C3 (t1 , T ) has a minimum value at (t1 , T ) = (M, T3 ), i.e. t1 = M . Proof. Refer the Appendix 1. Theorem 4.8. For tw ≤ M , we have Conditions ∆6 < 0 ∆4 < 0, ∆5 < 0 & ∆6 ≥ 0 ∆4 < 0 & ∆5 ≥ 0 ∆4 > 0 & ∆5 < 0 ∆4 ≥ 0 & ∆5 ≥ 0
5
T C(t1 , T ) T C(t1 , T ) = min(T C1 (t1 , T1 ), T C3 (M, T3 )) T C(t1 , T ) = min(T C1 (t1 , T1 ), T C4 (t1 , T4 )) T C(t1 , T ) = min(T C1 (t1 , T1 ), T C3 (t1 , T3 )) T C(t1 , T ) = min(T C1 (tw , T1 ), T C4 (t1 , T4 )) T C(t1 , T ) = min(T C1 (tw , T1 ), T C3 (t1 , T3 ))
T T1 /T3 T1 /T4 T1 /T3 T1 /T4 T1 /T3
Numerical example
Example 5.1. To illustrate the solution procedure of the proposed model we provide the following numerical examples. For this, we take the values of parameters are as follows The constant demand rate (D) The holding cost (h) Retailer’s purchasing cost is (c) Shortage cost for backlogged items are (s) Suppliers selling price (p) Cost of lost sales (o) The interest earned by retailers is (Ie ) The interest pay by retailers is (Ip ) trade-credit period given by suppliers to retailers in settling the account (M ) Fraction of shortage back-orders is (δ)
= = = = = = = =
100/units. $5/unit/year. $70/unit/year. $30/units/year. $80/units/year. $25/units/year. $0.04/year. $0.04/year.
= =
0.20/year. $0.56/year.
Now, we discuss the effect of changes in the parameters A(120, 80, 60, 40)/order, α(0.2, 0.5) w 125 and w = (150, 250). It is clear that, if w = 150, then the value of tw = D = 1000 = 0.1250, which is less than to M and hence, we go to the theorem (4.4) of our section 5 and obtain the optimal result of our model. Again, if we consider w = 250, then the value w 250 of tw = D = 1000 = 0.25, this is greater than the value of M. So, we go to the theorem (4.8) of our theoretical results and obtain the optimal results. Thus, the optimal results of aforementioned parameters are given as below in Table(2).
6
Managerial Implication
Now, based on our optimal results given in Table 2 and Table 3, we discuss the managerial implications of the proposed model as 13
1. In the trendy/fashionable goods, the backlogging rate is depends on the length of the waiting time. Therefore, the backlogging rate depends on the times. 2. If, the order quantity of retailers or buyers is greater than the predefined quantity say (w), i.e. (Q ≥ w), then buyers will enjoy the fully permissible delay in the time M (time given by the suppliers to the retailers) and if, the order quantity of retailers or buyers is less than, the predefined quantities say (w), i.e. (Q ≤ w), then buyers/retailers will immediately pay some amount and enjoy the fully permissible delay period M , until the remaining balance. 3. In a backlogging, there are two situations arise, when customers are willing to wait and second, they not willing to wait for next replenishment. But, in case of trendy/fashionable goods, the backlogging was dependent on the length of the waiting times. Therefore, Chang and Day [1] considered the backlogging rate is variable and is dependent on the waiting time for the next replenishment. 4. From Table 2 and Table 3, it is clear that, when we increase the predetermined quantity, i.e. the value of parameter w, then we found the changes as: when w = 125, then tw = 0.1250, which is less than the delay period M (because M = 0.20). Thus, retailer will receives the full delay time M . When w = 250, then tw = 0.25 and in that case tw ≥ M , thus the retailer receives partial trade-credit. 5. From Table 2, it is clear that, when we increase retailers ordering cost, i.e. parameter A, then the effects on both the cases is given as below: • For w = 125, the value of the total cost is increases, but the optimum order quantity (Q∗ ) and the optimum cycle times (T ∗ ) are unchanged, i.e. when (A = 40, 60, 80 and 120), then the value of total costs is increases. • For w = 250, the value of total costs is increases, but the optimum order quantity (Q∗ ) and the optimum cycle times (T ∗ ) are unchanged, i.e. when (A = 40, 60, 80 and 120), then the value of total costs be increases. 6. From Table 3, it is clear that, when we increase the purchasing cost of retailers, i.e. parameter c, then the effects on both cases is given as below: • For w = 125, the total cost, the optimum order quantity Q∗ and the optimum cycle times T ∗ are unchanged, i.e. when (c = 35, 70, 105, and 140), then the total cost of retailers and optimal order quantity of retailers are unchanged. • For w = 250, the total cost, the optimum order quantity (Q∗ ) and the optimum cycle times (T ∗ ) are increases.
7
Conclusion
In this article, we extend the model of Chen et al. [3], by considering the shortage condition, which is partial backlogged and examine our model, for partial trade-credit in an EOQ model linked to order quantity. Furthermore, we assume that the demand rate is constant. Since, demand arrives continuously at a constant and known rate of per units, per year. Arrival of demand at a continuous rate implies that, the optimal order quantity may be non-integer. Same way, the assumptions that demand arrive at a constant and known rate is hardly 14
Table 2: Optimal solutions of different parametric values of cases, when tw ≤ M & M ≤ tw Case
w
A
∆i , i = 1, 2, 3
t∗ 1
T∗
∗ T C(t∗ 1, T )
Q∗
120
∆2 ≥ 0
T ∗ = T2 = 0.1338
∗ T C2 (t∗ 1 , T ) = 790.43
133.78
∆2 ≥ 0
t∗ 1 = tw = 0.1250 t∗ 1 = tw = 0.1250
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∆2 ≥ 0
T ∗ = T2 = 0.1338
∆2 ≥ 0
t∗ 1 = tw = 0.1250 t∗ 1 = tw = 0.1250
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
A
∆i , i = 3, 4, 5, 6
t∗ 1
T∗
∗ T C(t∗ 1, T )
Q∗
120
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
∗ T C1 (t∗ 1 , T ) = 844.40
281.91
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25 t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
α
80 0.2 tw ≤ M
60
150
40 120 80 0.5
60 40
Case
w
α
80 0.2 M ≤ tw
60
250
40 120 80 0.5
60 40
T ∗ = T2 = 0.1338
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∆4 ≥ 0, ∆5 ≥ 0 ∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25
T ∗ = T2 = 0.1338
T ∗ = T1 = 0.2822 T ∗ = T1 = 0.2822
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
∗ T C2 (t∗ 1 , T ) = 491.48 ∗ T C2 (t∗ 1 , T ) = 342.00 ∗ T C2 (t∗ 1 , T ) = 192.53 ∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 491.48 ∗ T C2 (t∗ 1 , T ) = 342.00 ∗ T C2 (t∗ 1 , T ) = 192.53
∗ T C1 (t∗ 1 , T ) = 702.65 ∗ T C1 (t∗ 1 , T ) = 631.78 ∗ T C1 (t∗ 1 , T ) = 560.91 ∗ T C1 (t∗ 1 , T ) = 844.40 ∗ T C1 (t∗ 1 , T ) = 702.65 ∗ T C1 (t∗ 1 , T ) = 631.78 ∗ T C1 (t∗ 1 , T ) = 560.91
133.78 133.78 133.78 133.78 133.78 133.78 133.78
281.91 281.91 281.91 281.91 281.91 281.91 281.91
Table 3: Optimal solutions of different parametric values of cases when tw ≤ M & M ≤ tw Case
w
α
c
∆i , i = 4, 5, 6
t∗ 1
T∗
∗ T C(t∗ 1, T )
Q∗
35
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∗ T C2 (t∗ 1 , T ) = 790.43
133.78
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∆2 ≥ 0
t∗ 1 = tw = 0.1250 t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∆2 ≥ 0
T ∗ = T2 = 0.1338
∆2 ≥ 0
t∗ 1 = tw = 0.1250 t∗ 1 = tw = 0.1250
A
∆i , i = 3, 4, 5, 6
t∗ 1
T∗
∗ T C(t∗ 1, T )
Q∗
35
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25 t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2805
∗ T C1 (t∗ 1 , T ) = 835.0639
280.2425
70 0.2 tw ≤ M
150
105 140 35 70
0.5
105 140
Case
w
α
70 0.2 M ≤ tw
250
105 140 35 70
0.2 250
105 140
∆2 ≥ 0 ∆2 ≥ 0
∆4 ≥ 0, ∆5 ≥ 0 ∆4 ≥ 0, ∆5 ≥ 0 ∆4 ≥ 0, ∆5 ≥ 0
∆4 ≥ 0, ∆5 ≥ 0 ∆4 ≥ 0, ∆5 ≥ 0 ∆4 ≥ 0, ∆5 ≥ 0 ∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338 T ∗ = T2 = 0.1338 T ∗ = T2 = 0.1338
T ∗ = T1 = 0.2822
t∗ 1 = tw = 0.25 t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2838
t∗ 1 = tw = 0.25 t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
t∗ 1 = tw = 0.25
t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2854 T ∗ = T1 = 0.2805 T ∗ = T1 = 0.2838 T ∗ = T1 = 0.2854
∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 790.43
∗ T C1 (t∗ 1 , T ) = 844.4009 ∗ T C1 (t∗ 1 , T ) = 853.8441 ∗ T C1 (t∗ 1 , T ) = 863.5616
∗ T C1 (t∗ 1 , T ) = 835.0639 ∗ T C1 (t∗ 1 , T ) = 844.4009 ∗ T C1 (t∗ 1 , T ) = 853.8441 ∗ T C1 (t∗ 1 , T ) = 863.5616
133.78 133.78 133.78 133.78 133.78 133.78 133.78
281.9131 283.4841 285.0537 280.2425 281.9131 283.4841 285.0537
ever satisfied in practice. In addition, we assume that the planning horizon is infinite. The infinite planning horizon means that, we are choosing the time path of the control variables, for eternity at the initial date of the planning horizon. At first glance, the infinite horizon assumption may seem to be an arbitrary and extreme one, but in fact, it is often less extreme than it first appears. For example, it is often just an arbitrary and extreme to assume that a firm would stop planning at some finite date in the future. However, the model produces good results, when the demand is relatively stable over time and infinite planning horizon [12] are considered. For the optimal solution, we established some lemmas and theorems. Several numerical examples, are provided to check and established the intransigent results. Sensitivity analysis of major parameters was also discussed. For future work, the proposed model can be extended by in several ways like, by considering a variable demand, to allow for deterioration, inflation, quantity discount, finite rate of replenishment, time dependent or variable deterioration’s rates, etc.
15
Acknowledgement The authors wish to thanks the editor and unknown referees, who have patiently gone through the article and whose suggestions have considerably improved, its presentation and readability. The research works of the first author is supported by the Pt. Ravishankar Shukla University, Raipur (C.G.) under University Fellowship Program No. / 412-01/ Finance-Scholarship / 2014.
Appendix 1 First, we discuss the proof of part (a) of Lemma (4.1). For this, we first set t1 = Φ in (4.3), we get
X1 (Φ) =
A+
hΦ2 2
+ U χ1 ( (1+δ(χ11 −Φ)) − 1) − Uδ (−δχ1 + δΦ + log(1 + δ(χ1 − Φ))) (Φ2 −M 2 ) Ie pM 2 + Φ(Φ − M )) − 2 (7.1) +Ip c(− 2 −D χ1 2
For the condition of minimum relevant cost, we differentiate the X1 (Φ) with respect to 1 (Φ) Φ ∈ [M, ∞), we get dXdΦ > 0. Thus, X1 (Φ) is strictly increasing function of Φ ∈ [M, ∞). By our assumption, X1 (M ) = ∆1 and limΦ→∞ X1 (Φ) = +∞. Therefore, by the intermediate value theorem, there exist a unique t∗1 , such that X1 (t∗1 ) = 0. 2 d2 T C1 (t1 ,T ) d2 T C1 (t1 ,T ) d2 T C1 (t1 ,T ) Now, we can find − . For this, we take the second dt1 dT dt1 2 dT 2 1
derivatives of T C1 (t1 , T ) with respect to t1 and T . We get
d2 T C1 (t1 ,T ) dt1 2
d2 T C1 (t1 ,T ) dT1 2
=
D T3
2A D
(1 −
d2 T C1 (t1 ,T ) dt1 dT
Thus, we have
=
=
+ ht21 +
D T
ht1 +
h+
s (1+δT −δt1 )2
+
oδ (1+δT −δt1 )2
2(s+oδ)(δ(T −t1 )−log(1+δ(T −t1 ))) δ2
1 (1+δT −δt1 ) )
− TD2
+
(s+o)T 2 (1+δT −δt1 )2
(s+oδ) (−1 δ
d2 T C1 (t1 ,T ) d2 T C1 (t1 ,T ) dt1 2 dT1 2
−
+
+
(t2 −M 2 ) 2Ip c(− 1 2
1 (1+δT −δt1 ) )
d2 T C1 (t1 ,T ) dt1 dT
unique minimum solution of T C1 (t1 , T ).
16
−
2
+
+ Ip c
(7.2)
2(s+oδ)T δ
+ t1 (t1 − M )) − Ie pM(7.3)
s+oδ (1+δT −δt1 )2
2
+ Ip c(t1 − M ) (7.4)
> 0, it is clear that t1 ∈ [M, ∞) is the
Secondly, we see the proof of part (b) of Lemma (4.1), If ∆1 > 0, then we have X1 (Φ) > 0 for 1 (t1 ,T )) all Φ ∈ [M, ∞). Thus, d(T CdT > 0, for all t1 ∈ [M, ∞). Thus, T C1 is strictly increasing function of (t1 , T ). Therefore, T C1 (t1 , T ) has a minimum value at the point (t∗1 , T1 ∗ ) = M . This complete the proof of Lemma4.1. Similarly, we can prove the Lemma (4.2), Lemma (4.5) and Lemma (4.6) and first part of Lemma 4.7.
Appendix 2 Here, we discuss the proof of first part of Lemma (4.3). For this, we first set t1 = Ψ in (4.19), and let
G(Ψ) =
−D χ3 2
+
A+
hΨ2 2
Ip (1−α)cΨ2 2
+
(s+oδ)χ3 ( (1+δ(χ13 −Ψ)) δ
− Ie αpΨ(M −
Ψ 2)
− 1) −
s+oδ δ 2 (−δχ3
+ δΨ + log(1 + δ(χ3 − Ψ))) (7.5)
To prove the uniqueness, we differentiate (7.5) with respect to Ψ, where Ψ ∈ (0, tw ). > 0. Now, it is clear that, G(Ψ) is a strictly increasing function of So, we have dG(Ψ) dΨ Ψ ∈ (0, tw ). Again, we take the limit as, limΨ→0 G(Ψ) < 0 and limΨ→t− G(Ψ) = υ3 . w If υ3 > 0, then by the intermediate value theorem there exists a unique t1 , say t1 ∈ (0, tw ). The sufficient condition that these values is minimize the total cost T C3 are 2 d2 T C3 (t1 ,T ) d2 T C3 (t1 ,T ) d2 T C3 (t1 ,T ) d2 T C3 (t1 ,T ) d2 T C3 (t1 ,T ) > 0 and > 0 and − > 0. Now, we 2 2 2 dt dT dt1 dt1 dT dT 2 1 1
1
differentiate T C3 with respect to t1 and T . We get, d2 T C3 (t1 ,T ) dt1 2
d2 T C3 (t1 ,T ) dt1 dT
=
=
D T3
D T
2A D
(1 −
d2 T C3 (t1 ,T ) dT1 2
h+
s (1+δT −δt1 )2
+ ht21 +
=
+ Ip (1 − α)c + Ie αp
2(s+oδ)(δ(T −t1 )−log(1+δ(T −t1 ))) δ2
1 (1+δT −δt1 ) )
− TD2
+
oδ (1+δT −δt1 )2
+
(s+o)T 2 (1+δT −δt1 )2
ht1 +
(s+oδ) (−1 δ
+ Ip (1 −
+
1 (1+δT −δt1 ) )
+Ip (1 − α)ct1 + Ie αp(M −
t1 2)
+
2(s+oδ)T δ
−
α)ct21
− 2Ie αpt1 (M −
+
(7.6)
t1 2)
(7.7)
s+oδ (1+δT −δt1 )2
Ie αpt1 2
(7.8)
Hence, we obtain d2 T C3 (t1 ,T ) d2 T C3 (t1 ,T ) dt1 2 dT1 2
−
17
d2 T C3 (t1 ,T ) dt1 dT
2
>0
(7.9)
Thus, this proves that t1 is exists and unique. G(Ψ) = υ3 < 0, For the proof of the second part of Lemma (4.3), we consider that, if limΨ→t− w dT C3 then G(Ψ) < 0 for all t1 ∈ (0, tw ). Thus, we have dT < 0, where t1 ∈ (0, tw ). This implies that, T C3 (t1 , T ) is a strictly decreasing function of T . hence, the value of T does not exist during these conditions. This complete the proof of Lemma 4.3. Similarly, we can prove the second part of Lemma 4.7.
References [1] H.J. Chang, C. Y. Dye, An EOQ model for deteriorating items with time varying demand and partial backlogging, J. Oper. Res. Soc. 50(11)(1999)1176 - 1182. [2] C.T. Chang, L.Y. Ouyang, J.T. Teng, An EOQ model for deteriorating items under supplier credits linked to order quantity, Appl. Math. Model. 27(12)(2003)983-996. [3] S.C. Chen, L.E. Crdenas-Barrn, J.T. Teng, Retailers economic order quantity when the supplier offers conditionally permissible delay-in-payments link to order quantity, Int. J. Prod. Econ. 155(2014)284-291. [4] S.K. De, A. Goswami, S.S. Sana, An interpolating by pass to Pareto optimality in intuitionistic fuzzy technique for a EOQ model with time sensitive backlogging, Appl. Math. Comput. 230(1)(2014)664-674. [5] H.P. Dutton, Production control, Fact. Manag. Maint. 93(1935)545-560. [6] S.K. Goyal, Economic order quantity under condition of permissible delay in payments, J. Oper. Res. Soc. 36(4)(1985)335-338. [7] F. Harris, How many parts to make at once, Factory, The Mag. Manag. 10(1915)135136. [8] Y.F. Huang, Economic order quantity under conditionally permissible delay in payments, Eur. J. Oper. Res. 176(2)(2007)911-924. [9] A.M. Jamal, B.R. Sarker, S. Wang, An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, J. Oper. Res. Soc. 48(8)(1997)826 - 833. [10] H.C. Liao, C.H. Tsai, C.T. Su, An inventory model with deteriorating items under inflation when a delay in payment is permissible, Int. J. Prod. Econ. 63(2)(2000)207 214. [11] D.C. Montgomery, M.S. Bazaraa, A.K. Keswani, Inventory models with a mixture of backorders and lost sales, Nav. Res. Logist. Q. 20(2)(1973)255 - 263. [12] J.A. Muckstadt, A. Sapra, Principles of Inventory Management When You Are Down to Four, Order More, Springer New York Dordrecht Heidelberg London, 2010.
18
[13] L.Y. Ouyang, J.T. Teng, S.K. Goyal, C.T. Yang, An economic order quantity model for deteriorating items with partially permissible delay in payments linked to order quantity, Eur. J. Oper. Res. 194(2)(2009)418 - 431. [14] B. Pal, S.S. Sana, K.S. Chaudhuri, Two-echelon competitive integrated supply chain model with price and credit period dependent demand, Int. J. Sys. Sci. 45(2014)1-13. [15] B. Pal, S.S. Sana, K.S. Chaudhuri, Three stage trade-credit policy in a three-layer supply chaina production-inventory model, Int. J. Sys. Sci. 45(9)(2014)1844-1868. [16] K.S. Park, Inventory model with partial backorders, Int. J. Sys. Sci. 13(12)(1982)13131317. [17] D. Rosenberg, A new analysis of a lot-size model with partial backlogging, Nav. Res. Logist. Q. 26(2)(1979)349 - 353. [18] M.D. Roy, S.S. Sana, K.S. Chaudhuri, An economic production lot size model for defective items with stochastic demand, backlogging and rework, IMA J. Manag. Math. 25(2)(2014)159-183. [19] S.S. Sana, S.K. Goyal, (Q, r, L) model for stochastic demand with lead-time dependent partial backlogging, Ann. Oper. Res. 220(2014)1-10. [20] J.T. Teng, On the economic order quantity under conditions of permissible delay in payments, J. Oper. Res. Soc. 53(8)(2002)915-918. [21] J.T. Teng, C.T. Chang, S.K. Goyal, Optimal pricing and ordering policy under permissible delay in payments, Int. J. Prod. Econ. 97(2)(2005)121-129. [22] J.T. Teng, H.L. Yang, M.S. Chern, An inventory model for increasing demand under two levels of trade-credit linked to order quantity, Appl. Math. Model. 37(1415)(2013)7624-7632. [23] C.T. Yang, Q. Pan, L.Y. Ouyang, J.T. Teng, Retailers optimal order and credit policies when a supplier offers either a cash discount or a delay payment linked to order quantity, Eur. J. Ind. Eng. 7(3)(2013)370-392. ***
19
S0378-4754(15)00258-X http://dx.doi.org/10.1016/j.matcom.2015.11.008 MATCOM 4268
To appear in:
Mathematics and Computers in Simulation
Received date: 23 November 2014 Revised date: 25 September 2015 Accepted date: 25 November 2015 Please cite this article as: Vandana, B.K. Sharma, An EOQ model for retailers partial permissible delay in payment linked to order quantity with shortages, Math. Comput. Simulation (2015), http://dx.doi.org/10.1016/j.matcom.2015.11.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Manuscript Click here to view linked References
An EOQ model for retailers partial permissible delay in payment linked to order quantity with shortages Vandana∗ and B. K. Sharma School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, (C.G.), 492010, India.
Abstract Living in the business world, maximizing owner’s happiness or getting paid by the sales of goods or services on an open account at some reasonable profit is the main purpose of any business. The credit functions play a vital, role within the organization. In this paper, we develop an inventory model for retailers partial permissible delay-in-payment linked to order quantity with shortage, which is partial backlogged. Here, we consider two different cases, i.e. in first, the trade-credit period (M ) is greater than or equal to the time interval tw , that w units are depleted to zero due to demand; and later, the trade-credit period (M ) is less than to the time interval tw . The principle objective of the introduced model is to minimize the total inventory cost by finding an optimal replenishment policy. Required theorems are provided to verify the optimal solutions. Various numerical examples and managerial implications are discussed to check and substantiated the intransigent results.
2010 Mathematics Subject Classification 90B05. Keywords Inventory model; EOQ model; Partial backlogging; Partial trade-credit.
1
Introduction
Inventory control [12] is a science-based art of controlling the amount of stocks held in various forms, within the business to meet economically the demands placed upon that business. To minimize the total relevant cost the first and basic inventory model was developed by F.W. Harris [7] in 1915, known as an economic order quantity model. In 1935, H. P. Dutton [5] said that, “The principle of determining the economic lot size, i.e. (EOQ), for setting the maximum in stocks and limits in stock control have been known for many years, although these formulas are even yet, in everyday use in a comparatively small number of concerns.” In Harris [7] model, they did not mention the shortage condition. However, in practice sometimes customers demand cannot be fulfilled by the suppliers from the current stocks, this situation is known as a stock-out or shortage condition. In shortages period, two situations has occurred as first, where customers are willing to wait until the arrival of the next order, is complete backlogging and second, when some customers are not willing to 1
∗
Corresponding author Email: [email protected]
1
wait for the next replenishment, they leave the system, has lost sales case. In most of the inventory models, researchers allow shortages, which are completely backlogged. In 1973, Montgomery et al. [11] developed a model for non-perishable items with partial back-order. Now many researchers work on these aspects and modify the partial backlogging phenomena (for details see [18, 16, 17, 19, 4] and references therein). In 1999, Chang and Dye [1], developed an EOQ model with partial backlogged shortage. They assumed that, in case of trendy/fashionable goods, the backlogging are depending on the length of the waiting time. Therefore, the backlogging rates are variable and are dependent on the waiting time for the next replenishment. Hence, the proportion of customers, who would like to accept backlogging at a time (t), decreases with increasing waiting time (T − t) waiting for the next replenishment. After that, many researchers are working to consider this assumption. The main purpose of any business is to maximize owner’s happiness, in other words “getting paid” because business is essentially afraid with the sale of goods or services on an open account at a profit, the credit function plays a vital, role within the organization. When the supplier delivered goods to their customers, they often do not require to be paid immediately. Instead, suppliers offer credit terms that allow the buyers to delay the payment. This is known as trade-credit. This is very beneficial to the customers because, they do not have to pay the supplier immediately, after receiving the product, but instead, can delay their payment until, the end of the allowed period. The related first model was given by Goyal [6] in 1985. After that, many researchers works on those aspects, see ([14, 15, 9, 10, 20, 21] and references therein). In 2003, Chang et al. [2] developed a model for deteriorating items under supplier credits linked to order quantity. In 2007, Huang [8] developed a model in which he breaks the assumption of delay-in-payments, i.e. the supplier will offer to delay-in-payment to his/her retailers, if he/she orders more than the predetermined quantities say w. In Huang [8] model, he considered that, to increase their own sales or profit manufacturers/vendors or supplier offers to the retailers a permissible delay-in-payment even if, they ordered less than the predetermined quantity w. But, for enjoying the permissible delay-in-payments the retailers must pay a portion of the total purchasing cost, say (1 − α)cQ immediately. After that, he/she would pay the remaining balance at the end of the trade-credit period. Next, Ouyang et al. [13] extended Huangs [8] model with considering partially permissible delay in payments linked to order quantity. After that, many researchers works on those aspects, see ([22, 23] and references therein). In 2014, Chen et al. [3], have developed a model to overcome the shortcomings of Huangs [8] and Ouyang et al. [13] models. In this paper, we extend the model of Chen et al. [3] with allowable shortages (which is partial backlogged) and analyze the partial permissible delay-in-payment in an EOQ model, which is linked to order quantity. Also, we assume that, the supplier offers to the retailers a permissible delay-in-payment, even if they order less than the predetermined quantity w. For this situation, to enjoy a fully permissible delay-in-payments the retailers must pay a portion of the total purchasing cost, say (1 − α)cQ immediately. The main purpose of this article, is to investigate the optimal replenishment policy to minimize the total relevant cost.
2
Table 1: Summary of symbols used and their meanings Symbol A c h s o p α β δ Imax Ie Ip M tw t1 t∗1 T Q T C(t1 ) T C∗
Meaning ordering cost per unit purchasing cost per unit holding cost per unit per unit time excluding the capital cost shortage cost for backlogged items per unit per year cost of lost sales per unit selling price per unit the fraction of the delay payments permitted by suppliers per order, 0 ≤ α ≤ 1 1 denote the shortage fraction, i.e. β(t) = 1+δt backlogging parameter, which is positive and constant maximum inventory level interest earned per dollar interest charged per dollar trade-credit period w time interval that w units are depleted to zero due to demand D, where tw = D length of time in which there are no inventory shortage (t1 > td ) optimal length of time in which there are no inventory shortage duration of the replenishment cycle (T > t1 ) order quantity total minimum relevant cost for the inventory system optimal total minimum relevant cost per unit time
The rest of the paper is given as follows: In Section 2, we discuss some notations and assumptions of our model. The mathematical formulation to minimize the total annual inventory cost is established in Section 3. To characterize the optimal solution we derive some theorems and lemmas in Section 4. Several numerical examples for testing and verifying the models are provided in Section 5. Finally, the managerial implications and conclusion of the proposed model is presented in Section 6 and Section 7 respectively.
2
Notation and Assumptions
Notation In this paper, we are following the Chen et al. [3] (2014) notations, given in Table 1.
Assumptions 1. Demand rate is known and constant. 2. Replenishment rate is instantaneous. 3. Shortages are allowed. Unsatisfied demand is partially backlogged, and the fraction of shortages back-ordered is a differentiable and decreasing function of time, denoted as δ(t), where t is the waiting time up to the next replenishment with 0 ≤ δ(t) ≤ 1. 1 Let, β(t) denotes this fraction then, β(t) = 1+δt . 3
4. There is no deterioration. 5. The time horizon is infinite. 6. Only one type of item is considered. 7. The supplier offers to the retailers a permissible delay-in-payment even if, they order less than the predetermined quantity w. For this situation, the retailers must pay a portion of the total purchasing cost, say (1 − α)cQ immediately, where α the fraction of the delay payments permitted by suppliers per order, 0 ≤ α ≤ 1.
3
Mathematical Formulation
During the time period [0, t1 ] the differential equations delineate the inventory status as
Figure 1: Graphical representation of the inventory system
dI1 = −D, 0 ≤ t ≤ t1 (3.1) dt with the boundary condition I1 (t1 ) = 0. Till the time period [t1 , T ] the shortages are occurred and which is partial backlogged. Thus, the differential equation formulate the inventory status as dI2 −D0 = , t1 ≤ t ≤ T dt 1 + δ(T − t)
(3.2)
with the boundary condition I2 (t1 ) = 0. Now, we solve the equations(3.1) and (3.2) with a given differential equation, we get 4
I1 (t) = D(t1 − t) I2 (t) =
D δ
1+δ(T −t) log( 1+δ(T −t1 ) )
(3.3) (3.4)
After that, we calculate the value of maximum inventory level, i.e. Imax . Since, I1 (0) = Imax , then we obtain Imax = Dt1
(3.5)
After that, we put t = T in Eq.(3.4), we can obtain the maximum amount of backlogged demand per cycle as S = −I2 (T ) =
D δ (log(1
+ δ(T − t1 )))
(3.6)
Now, we calculate the order quantity of per cycle, as Q = Imax + S
(3.7)
D δ (log(1
= Dt1 +
+ δ(T − t1 )))
(3.8)
Now, we can obtain the value of the time interval that, w units are depleted to zero due w to demand given as: tw = D . After that, retailers start to accumulate the total cost with calculating the several components • Ordering cost (OC) = Firstly, the retailers will calculate the ordering cost, i.e. the incremental costs of processing an order of goods from a supplier is OC = A T • Holding cost (HC) = holding cost is a major component of supply chain management. The retailers will calculate the holding or carrying cost of items.
HC = =
h T
Z
t1
0
hDt21 2T
I1 (t)dt
(3.9) (3.10)
• Shortage cost (SC) = shortage cost or stock-out cost is the total of all the costs, which associated with shortage units.
SC = =
Z
s T(
T
t1
−I2 (t)dt)
sD (δ(T δ2
− t1 ) − log(1 + δ(T − t1 )))
• Opportunity cost due to lost sale (LS) Z T LS = To D(1 − =
(3.11)
1 1+δ(T −t) )dt
t1 oD (δ(T −t1 )−log(1+δ(T −t1 ))) T δ
5
(3.12)
(3.13)
• Interest earn and payable = Based on the values of M and tw , retailers has two possible cases and we discussed each in details Case (1) tw ≤ M - In this case, there are three possible sub cases are arises, given as below: • subcase(1) tw ≤ M ≤ t1 - In this case, the retailers receive the fully permissible delay, then the interest payable per cycle is the
Ip1 = = =
Z
t1
cIp T (
I1 (t)dt) M D(t21 −M 2 ) cIp + T (− 2 2 cIp D(t1 −M ) 2T
Dt1 (t1 − M ))
(3.14)
and the interest earn is given as Z
pIe T (
Ie1 =
M
(3.15)
Dtdt)
0 pIe DM 2 2T
=
(3.16)
• subcase(2) tw ≤ t1 ≤ M - In this case tw ≤ t1 , thus the retailer receives the fully permissible delay-in-payment, but there are no interest charge, because t1 ≤ M . Thus, the interest charge is Ip2 = 0
(3.17)
and the interest earn is as
Ie2 = =
Z
pIe T (
t1
0
Dtdt + (M − t1 )
t pIe Dt1 (M − 21 ) T
Z
t1
Ddt)
(3.18)
0
(3.19)
• subcase(3) t1 < tw ≤ M - In this case t1 < tw thus, the retailers are receives the partial permissible delay. For the immediate payment the interest payable is Ip3 =
(1−α)Dt21 cIp 2T
(3.20)
and the interest earn is given as below: Ie3 =
6
αpDt1 Ie (M T
−
t1 2)
(3.21)
Therefore, the total minimum relevant cost T C(t1 ) per unit is given as T C(t1 )
=
Thus, we get T C1 (t1 , T )
T C2 (t1 , T )
=
=
T C1 (t1 , T ) T C2 (t1 , T ) T C3 (t1 , T ) A T
+
hDt1 2 2T
+
+
Ip cD(t1 −M ) 2T
A T
+
hDt1 2 2T
2
+
= = =
=
A T
+
+
hDt1 2 2T
+
Ip (1−α)cDt1 2t
OC + HC + SC + LS + Ip2 − Ie2 , tw < t1 ≤ M
OC + HC + SC + LS + Ip3 − Ie3 , t1 < tw ≤ M
sD 1 δ 2 (log( (1+δ(T −t1 )) ) Ie pDM 2T
−
2
+ δ(T − t1 )) +
oD δT (δ(T
+ δT − δt1 ) +
oD δT (δ(T
sD 1 δ 2 (log( (1+δT −δt1 ) )
+ δT − δt1 ) +
oD δT (δ(T
2
t1 2)
−
− t1 ) − log(1 + δ(T − t1 ))) (3.22)
sD 1 δ 2 (log( (1+δT −δt1 ) )
1 − Ie pDt (M − T
T C3 (t1 , T )
OC + HC + SC + LS + Ip1 − Ie1 , M ≤ t1
Ie αpDt1 (M T
−
− t1 ) − log(1 + δ(T − t1 ))) (3.23)
− t1 ) − log(1 + δ(T − t1 )))
t1 2)
(3.24)
Case (2) M < tw : In this case, there are three possible subcases are arises, such as • subcase(1) M < tw ≤ t1 - In this case, the retailers receive the fully permissible delay-in-payment, then the interest charge is
Ip4 = = =
Z
cIp T (
t1
I1 (t)dt) M D(t21 −M 2 ) cIp + T (− 2 2 cIp D(t1 −M ) 2T
Dt1 (t1 − M ))
(3.25)
and the interest earn is as
Ie4 = =
Z
pIe T (
M
0 pIe DM 2 2T
Dtdt) (3.26)
• subcase(2) M < t1 ≤ tw - Since t1 ≤ tw , then the retailers will receives partial permissible delay-in-payment. For the immediate payment the retailers will pay (1−α)Dt21 cIp the interest charge, which is same as Ip3 , i.e. , again the annual 2T interest charged for the delayed payment is 7
Ip5 =
Ip (1−α)cDt21 2T
+
=
Ip (1−α)cDt21 2T Ip (1−α)cDt21 2T
+
=
+
Z
Ip c T (
t1
I1 (t)dt) M D(t21 −M 2 ) Ip c + T (− 2 2 cIp D(t1 −M ) 2T
Dt1 (t1 − M ))
(3.27)
and the interest earn is Ie5 =
Ie pDM 2 2T
(3.28)
• subcase(3) t1 < M ≤ tw - In this case, the interest charge and the interest earn both are the same as Ip3 and Ie3 . Therefore, the total minimum relevant cost T C(t1 ) per unit time is
T C(t1 )
T C4 (t1 , T )
T C5 (t1 , T )
=
=
=
A T
T C4 (t1 , T ) T C5 (t1 , T ) T C6 (t1 , T )
+
hDt1 2 2T
+
2
+
cIp D(t1 −M ) 2T
A T
+
hDt1 2 2T
+
I (1−α)cDt21 + p 2T
T C6 (t1 , T )
=
A T
+
4
sD δ2 T
+
hDt1 2 2T
+
Ip (1−α)cDt1 2T
= =
OC + HC + SC + LS + Ip5 − Ie5 , M < t1 ≤ tw
=
OC + HC + SC + LS + Ip3 − Ie3 , t1 < M ≤ tw
(log( (1+δ(T1 −t1 )) ) + δ(T − t1 )) + Ie pDM 2T
−
OC + HC + SC + LS + Ip4 − Ie4 , M < tw ≤ t1
2
cIp D(t1 −M )2 2T
−
−
Ie αpDt1 (M T
−
+ δT − δt1 ) +
Ie αpDM 2T
sD 1 T δ 2 (log( (1+δT −δt1 ) )
2
− t1 ) − log(1 + δ(T − t1 ))) (3.29)
sD 1 T δ 2 (log( (1+δT −δt1 ) )
+
oD δT (δ(T
oD δT (δ(T
2
+ δT − δt1 ) +
− t1 ) − log(1 + δ(T − t1 ))) (3.30)
oD δT (δ(T
t1 2)
− t1 ) − log(1 + δ(T − t1 ))) (3.31)
Theoretical results
Now, we discuss the theoretical aspects of our proposed model. Case(1) When tw ≤ M - In this case, there are three subcases. subcase(1) when M ≤ t1 The necessary conditions for the total annual cost of T C1 (t1 , T ) in (3.22) to be minimum are dTdtC1 1 = 0 and dTdTC1 = 0. Thus, we get 8
η1 (t1 ) =
= 0 and υ1 (T ) =
dT C1 dt1
η1 =
υ1
=
−D T2
A+
+Ip c(−
ht21 2
D T
+
dT C1 dT
ht1 +
= 0. Thus, we get
s+oδ 1 δ ( (1+δ(T −t1 ))
(s+oδ)T δ
( (1+δ(T1 −t1 )) − 1) − Ie pM 2 + t1 (t1 − M )) − 2
(t21 −M 2 ) 2
− 1) + Ip c(t1 − M ) s+oδ δ 2 (−δT
(4.1)
+ δt1 + log(1 + δ(T − t1 ))) (4.2)
= U and then we solve (4.1) for T , i.e. For the sake of convenience, we assume that s+oδ δ ht1 +Ip c(t1 −M ) T = t1 + δ(U −ht1 −Ip c(t1 −M )) = χ1 , put this value in (4.2), we have υ1 (t1 )
=
A+
ht21 2
+ U χ1 ( (1+δ(χ11 −t1 )) − 1) − Uδ (−δχ1 + δt1 + log(1 + δ(χ1 − t1 ))) (t21 −M 2 ) Ie pM 2 +Ip c(− + t1 (t1 − M )) − 2 (4.3) 2 −D χ1 2
Again, we put t1 = M in (4.3) ∆1
=
−D χ1 2
2 1 A + hM 2 + U χ1 ( (1+δ(χ1 −M )) − 1) −
− Ie pM 2
Lemma 4.1.
U δ (−δχ1
+ δM + log(1 + δ(χ1 − M )))
2
(4.4)
1. If ∆1 < 0, then the total annual cost has its minimum value at t1 > M .
2. If ∆1 ≥ 0, then the total annual cost has its minimum value in (t1 , T ) = (M, T1 ), i.e. t1 = M . Proof. Refer to Appendix 1. The necessary conditions for the total annual cost of T C2 (t1 , T ) in (3.23) to be minimum are dTdtC1 2 = 0 and dTdTC2 = 0. Thus, we have η2 (t1 ) = dTdtC1 2 = 0 and υ2 (T ) = dTdTC2 = 0. Thus, we get η2 =
υ2
=
−D T2
A+
ht21 2
D T
+
ht1 +
(s+oδ)T δ
t1 −Ie pt1 (M − 2 )
s+oδ 1 δ ( (1+δ(T −t1 ))
( (1+δ(T1 −t1 )) − 1) −
− 1) − Ie p(M − t1 )
s+oδ δ 2 (−δT
(4.5)
+ δt1 + log(1 + δ(T − t1 ))) (4.6)
9
For the sake of convenience, we assume that, s+oδ δ = U after that, we solve (4.5) for T , i.e. ht1 −Ie p(M −t1 ) T = t1 + δ(U −ht1 +Ie p(M −t1 )) = χ2 , put this value in (4.6), we have
ht21 2
+ U χ2 ( (1+δ(χ12 −t1 )) − 1) − t1 −Ie pt1 (M − 2 )
υ2 =
−D χ2 2
A+
U δ (−δχ2
+ δt1 + log(1 + δ(χ2 − t1 ))) (4.7)
Now, we put t1 = tw in (4.7), we get ∆2
=
A+
ht2w 2
+ U χ2 ( (1+δ(χ12 −tw )) − 1) − −Ie ptw (M − t2w ) −D χ2 2
U δ (−δχ2
+ δtw + log(1 + δ(χ2 − tw )))
(4.8)
Lemma 4.2. 1. If ∆2 ≤ 0 ≤ ∆1 , then the total annual cost has its minimum value at tw < t1 ≤ M . 2. If ∆2 > 0, then the total annual cost has its minimum value at (t1 , T ) = (tw , T2 ), i.e. t1 = tw . 3. If ∆1 < 0, then the total annual cost has its minimum value at (t1 , T ) = (M, T2 ), i.e. t1 = M . Proof. Refer to Appendix 1. The necessary conditions for the total annual cost of T C3 (t1 , T ) in (3.24) to be minimum are dTdtC1 3 = 0 and dTdTC3 = 0. Thus, we get η3 (t1 ) = dTdtC1 3 = 0 and υ3 (T ) = dTdTC3 = 0. Thus, we get η3 =
υ3
=
D T
−D T2
+
ht1 +
A+
s+oδ 1 δ ( (1+δ(T −t1 ))
ht21 2
Ip (1−α)ct21 2
+
(s+oδ)T δ
( (1+δ(T1 −t1 )) − 1) − − Ie αpt1 (M − t21 )
For our convenience, we assume that, t1 +
− 1) + Ip (1 − α)ct1 − Ie αp(M −
s+oδ δ
I αpt t ht1 +Ip (1−α)ct1 −Ie αp(M − 21 )+ e 2 1 I αpt t δ(U −ht1 −Ip (1−α)ct1 +Ie αp(M − 21 )− e 2 1 )
υ3
=
−D χ3 2
+
A+
+
Ie αpt1 2
(4.9)
+ δt1 + log(1 + δ(T − t1 ))) (4.10)
= U and then we solve (4.9) for T , i.e. T =
= χ3 , put this value in (4.10), we have
ht21 2
Ip (1−α)ct21 2
s+oδ δ 2 (−δT
t1 2)
+ U χ3 ( (1+δ(χ13 −t1 )) − 1) − t1 − Ie αpt1 (M − 2 )
10
U δ (−δχ3
+ δt1 + log(1 + δ(χ3 − t1 ))) (4.11)
Now, we put t1 = tw in (4.11), we get υ3
=
−D χ3 2
+
Lemma 4.3.
A+
ht2w 2
Ip (1−α)ctw 2 2
+ U χ3 ( (1+δ(χ13 −tw )) − 1) − − Ie αptw (M − t2w )
U δ (−δχ3
+ δtw + log(1 + δ(χ3 − tw ))) (4.12)
1. If ∆3 ≥ 0, then the total annual cost has its minimum value in t1 < M .
2. If ∆3 < 0, then the value of T ∈ (0, tw ), which minimize T C3 (t1 , T ) does not exist. Proof. Refer to the Appendix 2. Theorem 4.4. For tw ≤ M , we have T C(t1 , T ) T C(t1 , T ) = T C1 (t∗1 , T1 ) T C(t1 , T ) = min (T C1 (t∗1 , T1 ), T C3 (t1 , T3 )) T C(t1 , T ) = min(T C2 (t1 , T2 ), T C3 (t1 , T3 )) T C(t1 , T ) = min(T C2 (tw , T2 ), T C3 (t1 , T3 )) T C(t1 , T ) = (T C2 (t1 , T2 ))
Conditions ∆1 ≤ 0 & ∆3 < 0 ∆1 ≤ 0, ∆3 ≥ 0 ∆1 > 0, ∆2 < 0 & ∆3 ≥ 0 ∆2 ≥ 0 ∆1 > 0, ∆3 < 0
T T1 T1 /T3 T2 /T3 T2 /T3 T2
Case(2) when M ≤ tw - Now, in this case there are three subcases are arises. subcase (1) when M ≤ tw ≤ t1 - In this case, the total cost is the same as the total cost of subcase 1, then the necessary condition for the total cost of T C1 (t1 , T ), for this sub case is same as in (4.1) and (4.2). To show that uniqueness, we put t1 = tw in (4.2). Thus, we have ∆4 (t1 )
=
A+
htw 2 2
+Ip c(−
Lemma 4.5.
+ U χ4 ( (1+δ(χ14 −tw )) − 1) −
(t2w −M 2 ) 2
+ tw (tw − M )) −
U δ (−δχ4
Ie pM 2 2
+ δtw + log(1 + δ(χ4 − tw ))) (4.13)
1. If ∆4 ≤ 0, then the total annual cost has its minimum value at t1 > M .
2. If ∆4 ≥ 0, then the total annual cost has its minimum value at (t1 , T ) = (tw , T ), i.e. t1 = tw . Proof. Refer to Appendix 1. subcase (2) when M ≤ t1 ≤ tw The necessary conditions for the total annual cost of T C5 (t1 , T ) in (3.30) to be minimum are dTdtC1 5 = 0 and dTdTC5 = 0, which given as η4 (t1 ) = dTdtC1 5 = 0 and υ4 (T ) = dTdTC5 = 0, thus, we get 11
υ4
=
η4
=
−D T2
A+
D T
+Ip c(−
ht1 +
ht21 2
+
(t21 −M 2 ) 2
s+oδ 1 δ ( (1+δ(T −t1 ))
− 1) + Ip c((1 − α)t1 + (t1 − M ))
(s+oδ)T δ
( (1+δ(T1 −t1 )) − 1) − 2 + t1 (t1 − M )) − Ie αpM 2
s+oδ δ 2 (−δT
(4.14)
+ δt1 + log(1 + δ(T − t1 ))) (4.15)
For our convenience we assume that, s+oδ = U and then we solve (4.14) for T , i.e. T = δ ht1 +Ip c((1−α)t1 +(t1 −M )) t1 + δ(U −ht1 +Ip c((1−α)t1 +(t1 −M ))) = χ4 , then putting this value in (4.15), we have υ4
=
−D χ4 2
+
A+
ht21 2
Ip (1−α)ct21 2
+ U χ4 ( (1+δ(χ14 −t1 )) − 1) −
+ Ip c(−
(t21 −M 2 ) 2
U δ (−δχ4
+ t1 (t1 − M )) −
+ δt1 + log(1 + δ(χ4 − t1 )))
Ie αpM 2 2
(4.16)
Now, we put t1 = M and t1 = tw in (4.16), we get ∆5
=
−D χ4 2
+
∆6
=
A+
hM 2 2
Ip (1−α)cM 2 2
−D χ4 2
+
A+
ht2w 2
Ip (1−α)ct2w 2
+ U χ4 ( (1+δ(χ14 −M )) − 1) − 2 − Ie αpM 2
U δ (−δχ4
+ U χ4 ( (1+δ(χ14 −tw )) − 1) −
U δ (−δχ4
+ Ip c(−
(t2w −M 2 ) 2
+ δM + log(1 + δ(χ4 − M ))) (4.17)
+ tw (tw − M )) −
+ δtw + log(1 + δ(χ4 − tw )))
Ie αpM 2 2
(4.18)
Lemma 4.6. 1. If ∆5 ≤ 0 ≤ ∆6 , then the total annual cost T C5 (t1 , T ) has the unique minimum value at (t1 , T ) = (t1 , T ), where t1 ∈ [M, tw ). 2. If ∆5 > 0, then the total annual cost T C5 (t1 , T ) has a minimum value at (t1 , T ) = (M, T ), i.e. t1 = M . 3. If ∆6 < 0, then the value of T ∈ [M, tw ), which minimize the total cost does not exists.
Proof. Refer the Appendix 1 and 2.
subcase (3) when t1 ≤ M ≤ tw - Since, for this subcase the necessary condition for the minimization of T C3 is the same as (4.9) and (4.11). Now, we put t1 = M in (4.11), we get ∆7
=
−D χ3 2
+
A+
hM 2 2
Ip (1−α)cM 2 2
+
−
(s+oδ)χ3 ( (1+δ(χ13 −M )) δ
Ie αpM 2 2
− 1) −
s+oδ δ 2 (−δχ3
+ δM + log(1 + δ(χ3 − M ))) (4.19)
12
Now, it is clear that, the value of T C3 (M, T ) = T C5 (M, T ). Thus, we can obtain the following lemma Lemma 4.7.
1. If ∆5 ≥ 0, then the total cost T C3 (t1 , T ) has the unique value at t1 < M .
2. If ∆5 < 0, then the total cost T C3 (t1 , T ) has a minimum value at (t1 , T ) = (M, T3 ), i.e. t1 = M . Proof. Refer the Appendix 1. Theorem 4.8. For tw ≤ M , we have Conditions ∆6 < 0 ∆4 < 0, ∆5 < 0 & ∆6 ≥ 0 ∆4 < 0 & ∆5 ≥ 0 ∆4 > 0 & ∆5 < 0 ∆4 ≥ 0 & ∆5 ≥ 0
5
T C(t1 , T ) T C(t1 , T ) = min(T C1 (t1 , T1 ), T C3 (M, T3 )) T C(t1 , T ) = min(T C1 (t1 , T1 ), T C4 (t1 , T4 )) T C(t1 , T ) = min(T C1 (t1 , T1 ), T C3 (t1 , T3 )) T C(t1 , T ) = min(T C1 (tw , T1 ), T C4 (t1 , T4 )) T C(t1 , T ) = min(T C1 (tw , T1 ), T C3 (t1 , T3 ))
T T1 /T3 T1 /T4 T1 /T3 T1 /T4 T1 /T3
Numerical example
Example 5.1. To illustrate the solution procedure of the proposed model we provide the following numerical examples. For this, we take the values of parameters are as follows The constant demand rate (D) The holding cost (h) Retailer’s purchasing cost is (c) Shortage cost for backlogged items are (s) Suppliers selling price (p) Cost of lost sales (o) The interest earned by retailers is (Ie ) The interest pay by retailers is (Ip ) trade-credit period given by suppliers to retailers in settling the account (M ) Fraction of shortage back-orders is (δ)
= = = = = = = =
100/units. $5/unit/year. $70/unit/year. $30/units/year. $80/units/year. $25/units/year. $0.04/year. $0.04/year.
= =
0.20/year. $0.56/year.
Now, we discuss the effect of changes in the parameters A(120, 80, 60, 40)/order, α(0.2, 0.5) w 125 and w = (150, 250). It is clear that, if w = 150, then the value of tw = D = 1000 = 0.1250, which is less than to M and hence, we go to the theorem (4.4) of our section 5 and obtain the optimal result of our model. Again, if we consider w = 250, then the value w 250 of tw = D = 1000 = 0.25, this is greater than the value of M. So, we go to the theorem (4.8) of our theoretical results and obtain the optimal results. Thus, the optimal results of aforementioned parameters are given as below in Table(2).
6
Managerial Implication
Now, based on our optimal results given in Table 2 and Table 3, we discuss the managerial implications of the proposed model as 13
1. In the trendy/fashionable goods, the backlogging rate is depends on the length of the waiting time. Therefore, the backlogging rate depends on the times. 2. If, the order quantity of retailers or buyers is greater than the predefined quantity say (w), i.e. (Q ≥ w), then buyers will enjoy the fully permissible delay in the time M (time given by the suppliers to the retailers) and if, the order quantity of retailers or buyers is less than, the predefined quantities say (w), i.e. (Q ≤ w), then buyers/retailers will immediately pay some amount and enjoy the fully permissible delay period M , until the remaining balance. 3. In a backlogging, there are two situations arise, when customers are willing to wait and second, they not willing to wait for next replenishment. But, in case of trendy/fashionable goods, the backlogging was dependent on the length of the waiting times. Therefore, Chang and Day [1] considered the backlogging rate is variable and is dependent on the waiting time for the next replenishment. 4. From Table 2 and Table 3, it is clear that, when we increase the predetermined quantity, i.e. the value of parameter w, then we found the changes as: when w = 125, then tw = 0.1250, which is less than the delay period M (because M = 0.20). Thus, retailer will receives the full delay time M . When w = 250, then tw = 0.25 and in that case tw ≥ M , thus the retailer receives partial trade-credit. 5. From Table 2, it is clear that, when we increase retailers ordering cost, i.e. parameter A, then the effects on both the cases is given as below: • For w = 125, the value of the total cost is increases, but the optimum order quantity (Q∗ ) and the optimum cycle times (T ∗ ) are unchanged, i.e. when (A = 40, 60, 80 and 120), then the value of total costs is increases. • For w = 250, the value of total costs is increases, but the optimum order quantity (Q∗ ) and the optimum cycle times (T ∗ ) are unchanged, i.e. when (A = 40, 60, 80 and 120), then the value of total costs be increases. 6. From Table 3, it is clear that, when we increase the purchasing cost of retailers, i.e. parameter c, then the effects on both cases is given as below: • For w = 125, the total cost, the optimum order quantity Q∗ and the optimum cycle times T ∗ are unchanged, i.e. when (c = 35, 70, 105, and 140), then the total cost of retailers and optimal order quantity of retailers are unchanged. • For w = 250, the total cost, the optimum order quantity (Q∗ ) and the optimum cycle times (T ∗ ) are increases.
7
Conclusion
In this article, we extend the model of Chen et al. [3], by considering the shortage condition, which is partial backlogged and examine our model, for partial trade-credit in an EOQ model linked to order quantity. Furthermore, we assume that the demand rate is constant. Since, demand arrives continuously at a constant and known rate of per units, per year. Arrival of demand at a continuous rate implies that, the optimal order quantity may be non-integer. Same way, the assumptions that demand arrive at a constant and known rate is hardly 14
Table 2: Optimal solutions of different parametric values of cases, when tw ≤ M & M ≤ tw Case
w
A
∆i , i = 1, 2, 3
t∗ 1
T∗
∗ T C(t∗ 1, T )
Q∗
120
∆2 ≥ 0
T ∗ = T2 = 0.1338
∗ T C2 (t∗ 1 , T ) = 790.43
133.78
∆2 ≥ 0
t∗ 1 = tw = 0.1250 t∗ 1 = tw = 0.1250
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∆2 ≥ 0
T ∗ = T2 = 0.1338
∆2 ≥ 0
t∗ 1 = tw = 0.1250 t∗ 1 = tw = 0.1250
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
A
∆i , i = 3, 4, 5, 6
t∗ 1
T∗
∗ T C(t∗ 1, T )
Q∗
120
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
∗ T C1 (t∗ 1 , T ) = 844.40
281.91
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25 t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
α
80 0.2 tw ≤ M
60
150
40 120 80 0.5
60 40
Case
w
α
80 0.2 M ≤ tw
60
250
40 120 80 0.5
60 40
T ∗ = T2 = 0.1338
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∆4 ≥ 0, ∆5 ≥ 0 ∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25
T ∗ = T2 = 0.1338
T ∗ = T1 = 0.2822 T ∗ = T1 = 0.2822
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
∗ T C2 (t∗ 1 , T ) = 491.48 ∗ T C2 (t∗ 1 , T ) = 342.00 ∗ T C2 (t∗ 1 , T ) = 192.53 ∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 491.48 ∗ T C2 (t∗ 1 , T ) = 342.00 ∗ T C2 (t∗ 1 , T ) = 192.53
∗ T C1 (t∗ 1 , T ) = 702.65 ∗ T C1 (t∗ 1 , T ) = 631.78 ∗ T C1 (t∗ 1 , T ) = 560.91 ∗ T C1 (t∗ 1 , T ) = 844.40 ∗ T C1 (t∗ 1 , T ) = 702.65 ∗ T C1 (t∗ 1 , T ) = 631.78 ∗ T C1 (t∗ 1 , T ) = 560.91
133.78 133.78 133.78 133.78 133.78 133.78 133.78
281.91 281.91 281.91 281.91 281.91 281.91 281.91
Table 3: Optimal solutions of different parametric values of cases when tw ≤ M & M ≤ tw Case
w
α
c
∆i , i = 4, 5, 6
t∗ 1
T∗
∗ T C(t∗ 1, T )
Q∗
35
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∗ T C2 (t∗ 1 , T ) = 790.43
133.78
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∆2 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∆2 ≥ 0
t∗ 1 = tw = 0.1250 t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338
∆2 ≥ 0
T ∗ = T2 = 0.1338
∆2 ≥ 0
t∗ 1 = tw = 0.1250 t∗ 1 = tw = 0.1250
A
∆i , i = 3, 4, 5, 6
t∗ 1
T∗
∗ T C(t∗ 1, T )
Q∗
35
∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.25 t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2805
∗ T C1 (t∗ 1 , T ) = 835.0639
280.2425
70 0.2 tw ≤ M
150
105 140 35 70
0.5
105 140
Case
w
α
70 0.2 M ≤ tw
250
105 140 35 70
0.2 250
105 140
∆2 ≥ 0 ∆2 ≥ 0
∆4 ≥ 0, ∆5 ≥ 0 ∆4 ≥ 0, ∆5 ≥ 0 ∆4 ≥ 0, ∆5 ≥ 0
∆4 ≥ 0, ∆5 ≥ 0 ∆4 ≥ 0, ∆5 ≥ 0 ∆4 ≥ 0, ∆5 ≥ 0 ∆4 ≥ 0, ∆5 ≥ 0
t∗ 1 = tw = 0.1250
T ∗ = T2 = 0.1338 T ∗ = T2 = 0.1338 T ∗ = T2 = 0.1338
T ∗ = T1 = 0.2822
t∗ 1 = tw = 0.25 t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2838
t∗ 1 = tw = 0.25 t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2822
t∗ 1 = tw = 0.25
t∗ 1 = tw = 0.25
T ∗ = T1 = 0.2854 T ∗ = T1 = 0.2805 T ∗ = T1 = 0.2838 T ∗ = T1 = 0.2854
∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 790.43 ∗ T C2 (t∗ 1 , T ) = 790.43
∗ T C1 (t∗ 1 , T ) = 844.4009 ∗ T C1 (t∗ 1 , T ) = 853.8441 ∗ T C1 (t∗ 1 , T ) = 863.5616
∗ T C1 (t∗ 1 , T ) = 835.0639 ∗ T C1 (t∗ 1 , T ) = 844.4009 ∗ T C1 (t∗ 1 , T ) = 853.8441 ∗ T C1 (t∗ 1 , T ) = 863.5616
133.78 133.78 133.78 133.78 133.78 133.78 133.78
281.9131 283.4841 285.0537 280.2425 281.9131 283.4841 285.0537
ever satisfied in practice. In addition, we assume that the planning horizon is infinite. The infinite planning horizon means that, we are choosing the time path of the control variables, for eternity at the initial date of the planning horizon. At first glance, the infinite horizon assumption may seem to be an arbitrary and extreme one, but in fact, it is often less extreme than it first appears. For example, it is often just an arbitrary and extreme to assume that a firm would stop planning at some finite date in the future. However, the model produces good results, when the demand is relatively stable over time and infinite planning horizon [12] are considered. For the optimal solution, we established some lemmas and theorems. Several numerical examples, are provided to check and established the intransigent results. Sensitivity analysis of major parameters was also discussed. For future work, the proposed model can be extended by in several ways like, by considering a variable demand, to allow for deterioration, inflation, quantity discount, finite rate of replenishment, time dependent or variable deterioration’s rates, etc.
15
Acknowledgement The authors wish to thanks the editor and unknown referees, who have patiently gone through the article and whose suggestions have considerably improved, its presentation and readability. The research works of the first author is supported by the Pt. Ravishankar Shukla University, Raipur (C.G.) under University Fellowship Program No. / 412-01/ Finance-Scholarship / 2014.
Appendix 1 First, we discuss the proof of part (a) of Lemma (4.1). For this, we first set t1 = Φ in (4.3), we get
X1 (Φ) =
A+
hΦ2 2
+ U χ1 ( (1+δ(χ11 −Φ)) − 1) − Uδ (−δχ1 + δΦ + log(1 + δ(χ1 − Φ))) (Φ2 −M 2 ) Ie pM 2 + Φ(Φ − M )) − 2 (7.1) +Ip c(− 2 −D χ1 2
For the condition of minimum relevant cost, we differentiate the X1 (Φ) with respect to 1 (Φ) Φ ∈ [M, ∞), we get dXdΦ > 0. Thus, X1 (Φ) is strictly increasing function of Φ ∈ [M, ∞). By our assumption, X1 (M ) = ∆1 and limΦ→∞ X1 (Φ) = +∞. Therefore, by the intermediate value theorem, there exist a unique t∗1 , such that X1 (t∗1 ) = 0. 2 d2 T C1 (t1 ,T ) d2 T C1 (t1 ,T ) d2 T C1 (t1 ,T ) Now, we can find − . For this, we take the second dt1 dT dt1 2 dT 2 1
derivatives of T C1 (t1 , T ) with respect to t1 and T . We get
d2 T C1 (t1 ,T ) dt1 2
d2 T C1 (t1 ,T ) dT1 2
=
D T3
2A D
(1 −
d2 T C1 (t1 ,T ) dt1 dT
Thus, we have
=
=
+ ht21 +
D T
ht1 +
h+
s (1+δT −δt1 )2
+
oδ (1+δT −δt1 )2
2(s+oδ)(δ(T −t1 )−log(1+δ(T −t1 ))) δ2
1 (1+δT −δt1 ) )
− TD2
+
(s+o)T 2 (1+δT −δt1 )2
(s+oδ) (−1 δ
d2 T C1 (t1 ,T ) d2 T C1 (t1 ,T ) dt1 2 dT1 2
−
+
+
(t2 −M 2 ) 2Ip c(− 1 2
1 (1+δT −δt1 ) )
d2 T C1 (t1 ,T ) dt1 dT
unique minimum solution of T C1 (t1 , T ).
16
−
2
+
+ Ip c
(7.2)
2(s+oδ)T δ
+ t1 (t1 − M )) − Ie pM(7.3)
s+oδ (1+δT −δt1 )2
2
+ Ip c(t1 − M ) (7.4)
> 0, it is clear that t1 ∈ [M, ∞) is the
Secondly, we see the proof of part (b) of Lemma (4.1), If ∆1 > 0, then we have X1 (Φ) > 0 for 1 (t1 ,T )) all Φ ∈ [M, ∞). Thus, d(T CdT > 0, for all t1 ∈ [M, ∞). Thus, T C1 is strictly increasing function of (t1 , T ). Therefore, T C1 (t1 , T ) has a minimum value at the point (t∗1 , T1 ∗ ) = M . This complete the proof of Lemma4.1. Similarly, we can prove the Lemma (4.2), Lemma (4.5) and Lemma (4.6) and first part of Lemma 4.7.
Appendix 2 Here, we discuss the proof of first part of Lemma (4.3). For this, we first set t1 = Ψ in (4.19), and let
G(Ψ) =
−D χ3 2
+
A+
hΨ2 2
Ip (1−α)cΨ2 2
+
(s+oδ)χ3 ( (1+δ(χ13 −Ψ)) δ
− Ie αpΨ(M −
Ψ 2)
− 1) −
s+oδ δ 2 (−δχ3
+ δΨ + log(1 + δ(χ3 − Ψ))) (7.5)
To prove the uniqueness, we differentiate (7.5) with respect to Ψ, where Ψ ∈ (0, tw ). > 0. Now, it is clear that, G(Ψ) is a strictly increasing function of So, we have dG(Ψ) dΨ Ψ ∈ (0, tw ). Again, we take the limit as, limΨ→0 G(Ψ) < 0 and limΨ→t− G(Ψ) = υ3 . w If υ3 > 0, then by the intermediate value theorem there exists a unique t1 , say t1 ∈ (0, tw ). The sufficient condition that these values is minimize the total cost T C3 are 2 d2 T C3 (t1 ,T ) d2 T C3 (t1 ,T ) d2 T C3 (t1 ,T ) d2 T C3 (t1 ,T ) d2 T C3 (t1 ,T ) > 0 and > 0 and − > 0. Now, we 2 2 2 dt dT dt1 dt1 dT dT 2 1 1
1
differentiate T C3 with respect to t1 and T . We get, d2 T C3 (t1 ,T ) dt1 2
d2 T C3 (t1 ,T ) dt1 dT
=
=
D T3
D T
2A D
(1 −
d2 T C3 (t1 ,T ) dT1 2
h+
s (1+δT −δt1 )2
+ ht21 +
=
+ Ip (1 − α)c + Ie αp
2(s+oδ)(δ(T −t1 )−log(1+δ(T −t1 ))) δ2
1 (1+δT −δt1 ) )
− TD2
+
oδ (1+δT −δt1 )2
+
(s+o)T 2 (1+δT −δt1 )2
ht1 +
(s+oδ) (−1 δ
+ Ip (1 −
+
1 (1+δT −δt1 ) )
+Ip (1 − α)ct1 + Ie αp(M −
t1 2)
+
2(s+oδ)T δ
−
α)ct21
− 2Ie αpt1 (M −
+
(7.6)
t1 2)
(7.7)
s+oδ (1+δT −δt1 )2
Ie αpt1 2
(7.8)
Hence, we obtain d2 T C3 (t1 ,T ) d2 T C3 (t1 ,T ) dt1 2 dT1 2
−
17
d2 T C3 (t1 ,T ) dt1 dT
2
>0
(7.9)
Thus, this proves that t1 is exists and unique. G(Ψ) = υ3 < 0, For the proof of the second part of Lemma (4.3), we consider that, if limΨ→t− w dT C3 then G(Ψ) < 0 for all t1 ∈ (0, tw ). Thus, we have dT < 0, where t1 ∈ (0, tw ). This implies that, T C3 (t1 , T ) is a strictly decreasing function of T . hence, the value of T does not exist during these conditions. This complete the proof of Lemma 4.3. Similarly, we can prove the second part of Lemma 4.7.
References [1] H.J. Chang, C. Y. Dye, An EOQ model for deteriorating items with time varying demand and partial backlogging, J. Oper. Res. Soc. 50(11)(1999)1176 - 1182. [2] C.T. Chang, L.Y. Ouyang, J.T. Teng, An EOQ model for deteriorating items under supplier credits linked to order quantity, Appl. Math. Model. 27(12)(2003)983-996. [3] S.C. Chen, L.E. Crdenas-Barrn, J.T. Teng, Retailers economic order quantity when the supplier offers conditionally permissible delay-in-payments link to order quantity, Int. J. Prod. Econ. 155(2014)284-291. [4] S.K. De, A. Goswami, S.S. Sana, An interpolating by pass to Pareto optimality in intuitionistic fuzzy technique for a EOQ model with time sensitive backlogging, Appl. Math. Comput. 230(1)(2014)664-674. [5] H.P. Dutton, Production control, Fact. Manag. Maint. 93(1935)545-560. [6] S.K. Goyal, Economic order quantity under condition of permissible delay in payments, J. Oper. Res. Soc. 36(4)(1985)335-338. [7] F. Harris, How many parts to make at once, Factory, The Mag. Manag. 10(1915)135136. [8] Y.F. Huang, Economic order quantity under conditionally permissible delay in payments, Eur. J. Oper. Res. 176(2)(2007)911-924. [9] A.M. Jamal, B.R. Sarker, S. Wang, An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, J. Oper. Res. Soc. 48(8)(1997)826 - 833. [10] H.C. Liao, C.H. Tsai, C.T. Su, An inventory model with deteriorating items under inflation when a delay in payment is permissible, Int. J. Prod. Econ. 63(2)(2000)207 214. [11] D.C. Montgomery, M.S. Bazaraa, A.K. Keswani, Inventory models with a mixture of backorders and lost sales, Nav. Res. Logist. Q. 20(2)(1973)255 - 263. [12] J.A. Muckstadt, A. Sapra, Principles of Inventory Management When You Are Down to Four, Order More, Springer New York Dordrecht Heidelberg London, 2010.
18
[13] L.Y. Ouyang, J.T. Teng, S.K. Goyal, C.T. Yang, An economic order quantity model for deteriorating items with partially permissible delay in payments linked to order quantity, Eur. J. Oper. Res. 194(2)(2009)418 - 431. [14] B. Pal, S.S. Sana, K.S. Chaudhuri, Two-echelon competitive integrated supply chain model with price and credit period dependent demand, Int. J. Sys. Sci. 45(2014)1-13. [15] B. Pal, S.S. Sana, K.S. Chaudhuri, Three stage trade-credit policy in a three-layer supply chaina production-inventory model, Int. J. Sys. Sci. 45(9)(2014)1844-1868. [16] K.S. Park, Inventory model with partial backorders, Int. J. Sys. Sci. 13(12)(1982)13131317. [17] D. Rosenberg, A new analysis of a lot-size model with partial backlogging, Nav. Res. Logist. Q. 26(2)(1979)349 - 353. [18] M.D. Roy, S.S. Sana, K.S. Chaudhuri, An economic production lot size model for defective items with stochastic demand, backlogging and rework, IMA J. Manag. Math. 25(2)(2014)159-183. [19] S.S. Sana, S.K. Goyal, (Q, r, L) model for stochastic demand with lead-time dependent partial backlogging, Ann. Oper. Res. 220(2014)1-10. [20] J.T. Teng, On the economic order quantity under conditions of permissible delay in payments, J. Oper. Res. Soc. 53(8)(2002)915-918. [21] J.T. Teng, C.T. Chang, S.K. Goyal, Optimal pricing and ordering policy under permissible delay in payments, Int. J. Prod. Econ. 97(2)(2005)121-129. [22] J.T. Teng, H.L. Yang, M.S. Chern, An inventory model for increasing demand under two levels of trade-credit linked to order quantity, Appl. Math. Model. 37(1415)(2013)7624-7632. [23] C.T. Yang, Q. Pan, L.Y. Ouyang, J.T. Teng, Retailers optimal order and credit policies when a supplier offers either a cash discount or a delay payment linked to order quantity, Eur. J. Ind. Eng. 7(3)(2013)370-392. ***
19