International Journal of Modern Mathematical Sciences, 2017, 15(2): 237-247
International Journal of Modern Mathematical Sciences
Journal homepage: www.ModernScientificPress.com/Journals/ijmms.aspx
ISSN: 2166-286X
Florida, USA
Article
Optimal Ordering and Transfer Policy for an Inventory with
Non-Increasing Time –Dependent Demand
R.P.Tripathi1,* and Manjit Kaur 2
1
Department of Mathematics, Graphic Era University, Dehradun (UK), India
2
Department of Mathematics, Banasthali Vidhiyapeeth, Rajasthan India
*Author to whom correspondence should be addressed; E-Mail: [email protected]
Article history: Received 27 December 2016; Revised 15 May 2017; Accepted 30 May 2017;
Published 1 June 2017.
Abstract: This paper develops optimal ordering and transfer policy for an inventory with
non-increasing dependent demand from the warehouse to the display area. Mathematical
model is developed for finding optimal order quantity, cycle time and total profit. The main
aim is to find maximum the average profit per unit time provided by the retailer. Moreover,
a numerical example is provided to illustrate the proposed model. Next, sensitivity analysis
with respect to different parameters is established to demonstrate the model developed.
Mathematica 5.1 software is used to find numerical result.
Keywords: Non-decreasing demand; inventory; transfer; deterioration; warehouse
Mathematics Subject Classification 2010: 90B05
1. Introduction
In the traditional inventory model, demand rate is considered to be either constant or timedependent. In real life demand rate is not always constant, particularly in case of seasonal products, the
demand rate is time-dependent. Several research papers have been published considering timedependent demand. Silver & Meal [1] presented an EOQ (Economic Order Quantity) for the case of a
varying demand. Linearly time-dependent demand was established by Donaldon [2]. Khanna et al. [3]
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Int. J. Modern Math. Sci.2017, 15(2): 237-247
238
developed an EOQ (Economic Order Quantity) model for deteriorating item having time dependent
demand when delay is payment is permissible. Teng et al. [4] presented EOQ model under trade credit
financing with increasing demand. Khanra et al. [5] established an EOQ model for a deteriorating
items having time dependent demand when delay in payment is permissible. Jalan and Chaudhuri [6]
established EOQ model for exponentially demand pattern. Large numbers of researchers like Mitra et
al. [7], Silver [8], Tripathi & Pandey [9], Singh et al. [10], Khanna and Chaudhuri. [11], Ghosh and
Chaudhuri [12] presented their research work considering time- dependent demand.
Commonly, at most all items deteriorate over time. Therefore, the effect of deterioration cannot
be ignored in the study of inventory problems. Ghare & Schrader [13] developed an EOQ model for
exponential deterioration rate. Chung et al. [14] established a new economic production quantity.
(EPQ) inventory model for deteriorating items under two levels of trade credits, in the supplier offers
to the retailer. A permissible delay period and simultaneously the retailer in terms provide a maximal
trade credit period to its customer in a supply chain system comprised of three stages. The valuable
work is this direction came from researchers like, Giri et al. [15], Jalan et al. [16], Gowsami &
Chaudhuri [17], Chung & Ting [18], Lin et al.[19], Wee [20],Yhmadi et al. [21] , Modarres &
Taimury [22] , Rabbani & Manavizadeh [23],Tripathi & Uniyal [24], in this direction etc.
During the last few decades, the study of transfer policy, the integration of production and
inventory model as well as the development of inventory policy have been considered by large several
researchers. For instance Goyal &Chang [25] dealt with an ordering transfer inventory model to
determine the retailer' optimal order quantity and the number of transfer per order from the warehouse
to the display area. Goyal [26] initially developed a single supplier-single retailer integrated inventory
model. Banerjee [27] presented a joint economic lot-size model and assumed that the supplier followed
a lot for shipment policy with respect to a retailer. Goyal [28], extended model [27] and discussed a
model that customers number for equal-sized shipments, but the production of model had to be
finished before the shipments could start. Yang & Wee [29] presented an integrated multi-lot-size
production inventory model for deteriorating items. Yao et al. [30] established a model that explains
how important supply chain parameters affect the cost saving to be realized from collaborative
initiatives such as vender-managed inventory.
The objective of this paper is to find the ordering and transfer schedule which maximize the
profit per unit time. In this case the amount of display space is limited. Then the cost of inventory
inside the shop may be greater than the back room. In this paper, we develop an inventory model in
which, the demand rate is linearly time-dependent.
The rest of the paper organized as follows. In section 2, we provide assumption and notation
for the proposed model. In section 3 a mathematical model is developed to obtain maximum profit.
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Int. J. Modern Math. Sci.2017, 15(2): 237-247
239
Optimal solutions are discussed in section 4. In section 5, numerical example is given. The sensitivity
analysis of the optimal solution with respect to parameters of the system is carried out in section 6.
Finally, conclusions are discussed in the last section.
2. Assumption and Notations
2.1. Assumptions
(i).The lead time between the retailer and the supplier is negligible
(ii).The time to transfer items from the warehouse to the display area is negligible
(iii).The shortage are not allowed
(iv).The credit-transfer policy is adopted.
(v).Demand rate is linearly time dependent i.e. D = D(t) = a-bt, a > 0, 0 ≤ b ≤ 1
2.2. Notations
h1 &h
: unit carrying cost per item in the warehouse &display area respectively.
n
: integer number of transfers from the warehouse to the display area.
p& c
: unit selling price& purchase cost of the product / unit respectively.
s
: fixed cost per transfer from the warehouse to the display area.
S
: cost of placing order.

: deterioration rate.
t1&T
: replenishment cycle time in the display area& warehouse respectively.
q
: quantity per transfer from the warehouse to the display area.
Q
: order quantity placed on the supplier.
I(t)
: inventory level at time t in the display area.
TP
: total profit per cycle time
R
: the inventory level at t = t1
AP
: average total profit per cycle time
AP*
: optimal average total profit per cycle time
D = D (t): the demand rate at time t. we assume that the demand rate D = D(t) is a function of the
stock on the display I (t). Demand rate is D = D (t) = a-bt, a > 0, 0 ≤ b ≤ 1
3. Mathematical Formulation
According to assumption the following two cases may arise to form the mathematical model:
(i)The total cost per unit cycle in the warehouse
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Int. J. Modern Math. Sci.2017, 15(2): 237-247
240
The retailer stocks Q items per order in the warehouse provided by the supplier. The inventory
level in the warehouse falls to zero after the quantity q per transfer is transferred from the warehouse to
the display area. We obtain Q = nq. The total cost during [0, T] in the warehouse containing:
(a) The cost of placing order = S and
 n(n  1) 
(b) The cost of stock holding is h1 
q  t1 .
 2

(ii) The total cost per unit cycle in the display area
Initially at time t = 0, the inventory level I(t) reaches the maximum due to the commodities are
transferred from the warehouse in the display area. The inventory level decreases gradually to R at the
end of the cycle. The differential equation involving the inventory level at time t can be written as
follows:
dI (t )
  I (t )  (a  bt ), a  0, 0  t  1, 0  t  t1
dt
(1)
The solution of the above differential equation along with the boundary condition I( t1 ) = R
b
a b 
I (t )  Re (t1 t )    2  e (t1 t )  1  t1 e (t1 t )  t

  
(2)
The total cost during [0, t1 ] consist of cost of placing orders = S and the cost of stock holding is

bt 2 
1
1
b b  1
b
h  I (t )dt  h (e t1  1)  R   a    t1    a   t1  1 
      
2 

0
t1
(3)
The Sales revenue per cycle is
t1
t1
bt 

( p  c)  DI (t )dt  ( p  c)  (a  bt )dt  ( p  c)t1  a  1 
2 

0
0
(4)
Using equation (2) and I (0) = q + R, we get
 t1
q  (e

1
b   bt e
 1)  R   a     1

  

 t1
(5)
 t1
The second approximation is used for exponential term that is e
( t1 )2
.
 1   t1 
2
b
 bt12 
1
b 
 t 
q   t1 1  1   R   a     t1   bt1 

2 

 
2 


(6)
 n(n  1) 
Then, the cost of stock holding in the warehouse is h1 
q  t1
 2

t
 n(n  1)  t1

1
b   bt1e 1 
 n(n  1) 
h1 
q  t1  h1 
(e  1)  R   a   
t
      1
 2


 2
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(7)
Int. J. Modern Math. Sci.2017, 15(2): 237-247
241
From the above result, the total profit (TP) during [0, T] is given by
TP ( t1 ) = Revenue – total cost in the warehouse-total cost in the display area
bt  
a b

 n(n  1)  t1

 n( p  c)t1  a  1    S  h1 
(e  1)  R   2
2  
 
 2


t
 bt1e 1  
t 

  1 


bt 2 
1
1
b b  1
b
 ns  nh (e t1  1)  R   a    t1    a   t1  1 
      
2 

  S  ns 
nhat1

2  t1
1
h bn(n  1)t e
 1
2
(8)
nhbt12 nhbt1e t1

1
b    h n(n  1)t1 nh 

 (e t1  1)  R   a      1
 
2
2

    
2
 

bt 

 n( p  c)t1  a  1 
2 


(9)
The average profit per unit time is
AP ( t1 ) =


TP
T
, where T= n t1
S s ha hbt1 hbe t1

1
b    h (n  1) h 
 

 2  (e t1  1)  R   a     1
 
nt1 t1 
2

     2
 t1 

h1b(n  1)t1e t1
bt 

 ( p  c)  a  1 
2
2 

(10)
4. Determination of Optimal Solution
Taking the first and second partial derivative of equation (10) with respect to t1, we obtain
of AP (n, R, t1 ) with respect to t1 .
dAP (t1 )
h (n  1)
S
s hb hbe t1 
1
b 
 2  2 

  R   a    e t1 1
dt1
2


 
2
nt1
t1

 t1

1
b   he t1
1 
1
b  h
b( p  c) h1b(n  1)e

 R   a  
     R   a    2 
2

    t1
t1  

    t1
2

(1  t1 )
d 2 AP (t1 )
h (n  1) he t1
2S 2s

1
b  

  3  3  hbe t1   R   a     2 e t1 1

2

  
2
 t1
dt1
nt1
t1



1
  
t1 


he t1
 t12
t

2  2h 
 h1b(n  1)e 1




(2  t1 )  0.



t1   t13 
2


(11)
(12)
2
Since d AP 2(t1 ) < 0, it means that total profit is concave function of t1 .The optimal solution is
dt1
obtained by following dAP (t1 ) =0.
dt1
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Int. J. Modern Math. Sci.2017, 15(2): 237-247
The second approximation is used for exponential term
e t1  1   t1 
242
in equation (11) i.e.
( t1 )2 , providedӨt <1.
1
2

1
b 
2S  2n s  hbnt12  2hbn t13  hbn 2t14   R   a    h1n(n  1) 2t12  h1n(n  1) 3t13

 

 0.5h1 n(n  1) 4 t14  2nh( t1  1)  2nh t1 ( t1  1)  nh 2 t12 ( t1  1)  2nh   h1n(n  1)bt12
(1   t1 )  h1 n(n  1)b t13 (1   t1 ) 
h1n(n  1)b 2 t14 (1   t1 )
 ( p  c)bn t12  0 .
2
The second order approximation is used of exponential term i.e. e t1  1   t1 
(13)
( t1 )2
, to find
2
closed form solution provided Өt1<1.The closed form solution is not valid for Өt1>1 .
2
2 2


AP ( t1 )   S  s  ha  hbt1  hb2  hbt1   R  1  a  b    h1 (n  1) t1  h1 (n  1) t1  h  h t1 
nt1 t1 
2 
2
    
2
4
2 


h1b(n  1)t1 h1b(n  1)t12 h1b (n  1)t13
bt 



 ( p  c)  a  1 
2
2
4
2 

(14)
Note: The second order approximation of exponential terms are valid if Өt1<1 etc, only.
5. Numerical Examples and Sensitivity Analysis
Let a =100units/unit time, b= 0.2 units/ unit time, n=2, h = $10/unit/unit time, h1 =$0.3/unit/unit
time, S=20, s = 20, p=$3per unit, c = $1/ unit,  = 0.05. Let T = n t1 and Q = nq and obtained the
numerical result as shown in tables 1 to 4.
To study the effect of changes is the system of key parameters t1 , T, h, h1 , q,  , on the optimal
average total profit AP*. Sensitivity analysis has been performed by changing the parameters ,
n=2,3,4,5,6,7,8,9 , h=10,11,12,1314,15,16 , h1 = 0.3,0.4,0.5, 0.6,0.7,0.8,0.9  =0.05,0.10, 0.15, 0.20,
0.25, 0.30, 0.35 ,S = 20, 40, 60, 80, 100, 130, 170, 200, 250, 300 , s = 20, 25, 30, 55,70, 80, 90,100,
and changing one parameter at a time, keeping the remaining parameter at their original values. The
results are shown in the following table.
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Int. J. Modern Math. Sci.2017, 15(2): 237-247
243
Table 1.Variation of number of transfers from the warehouse n
n
t1
2
3
4
5
6
7
8
9
0.228826
0.212746
0.203183
0.196426
0.191161
0.186800
0.183040
0.179707
T = n t1
0.457652
0.638238
0.812732
0.98213
1.146966
1.307600
1.464320
1.617363
q
23.4685
21.8109
20.8258
20.1299
19.5879
19.1391
18.7521
18.4092
Q=nq
46.937
65.4327
83.3032
100.6495
117.5274
133.9737
150.0168
165.6828
AP*
22268.9
22274.7
22277.0
22277.8
22277.9
22277.6
22277.1
22276.3
Table 2. Variation of unit carrying cost in display area h
h
10
11
12
13
14
15
16
t1
0.228826
0.218523
0.209499
0.201510
0.194372
0.187943
0.182113
T=n t1
0.457652
0.437046
0.418998
0.40302
0.388744
0.375886
0.364226
q
23.4685
22.4063
21.4764
20.6535
19.9185
19.2567
18.6568
Q=nq
AP*
46.937
44.8126
42.9528
41.307
39.837
38.5134
37.3136
22268.9
24262.7
26256.8
28251.1
30245.7
32240.4
34235.3
Table 3. Variation of parameter unit carrying cost per item in the warehouse h1
h1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
t1
0.228826
0.227729
0.226648
0.225583
0.224532
0.223496
0.222474
T=n t1
0.457652
0.455458
0.453296
0.451166
0.449064
0.446992
0.444948
q
23.4685
23.3554
23.2439
23.1341
23.0257
22.9189
22.8136
Q=nq
46.937
46.7108
46.4878
46.2682
46.0514
45.8378
45.6272
AP*
22268.9
22268.3
22267.6
22267.0
22266.4
22265.8
22265.2
Table 4. Variation of deterioration rate 

0.05
0.10
0.15
0.20
0.25
0.30
0.35
t1
0.228826
0.217968
0.208451
0.200021
0.192487
0.185702
0.179550
T=n t1
0.457652
0.435936
0.416902
0.400042
0.384974
0.3714040
0.359100
q
23.4685
25.555
26.9532
28.1509
29.2485
30.2818
31.2675
Q=nq
46.937
51.11
53.9064
56.3018
58.497
60.5636
62.535
AP*
22268.9
11262.4
7589.42
5750.02
4644.14
3905.12
3375.77
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Int. J. Modern Math. Sci.2017, 15(2): 237-247
244
Table 5. Variation of the cost of placing order S
S
20
40
60
80
100
130
170
200
250
300
t1
0.228826
0.264004
0.294949
0.322886
0.348545
0.383758
0.426097
0.455205
0.499855
0.540715
T=n t1
0.457652
0.528008
0.589898
0.645772
0.69709
0.767516
0.852194
0.91041
0.99971
1.08143
q
23.4685
27.0991
30.2978
33.1896
35.849
39.5038
43.9063
46.9381
51.5907
55.8685
Q=nq
46.937
54.1982
60.5956
66.3792
71.698
79.0076
87.8126
93.8762
103.1814
111.737
AP*
22268.9
22248.5
22230.0
22214.2
22199.2
22178.6
22153.7
22136.5
22110.1
22085.8
Table 6. Variation of the fixed cost per transfer from the warehouse to the display area s
s
t1
20
25
30
40
55
70
80
90
100
0.228826
0.247053
0.264004
0.294949
0.335965
0.372400
0.394781
0.415928
0.436022
T=n t1
0.457652
0.494106
0.528008
0.589898
0.671930
0.74480
0.789562
0.831856
0.872044
q
23.4685
24.3489
27.0991
30.2978
34.5448
38.3243
40.6492
42.8481
44.9396
Q=nq
46.937
48.6978
54.1982
60.5956
69.0896
76.6486
81.2984
85.6962
89.8792
AP*
22268.9
22258.3
22248.5
22230.5
22206.6
22185.2
22172.1
22159.7
22147.8
All the above observation mentioned in table 1-6 can be summed up as follows:
 From Table 1, it can easily see that increase of n results, increase in order quantity Q and
average total profit AP. That is, change in n causes positive change in both Q and AP.
 From Table 2, we see that increase of h, results decrease in order quantity Q and average total
profit AP. That is, change in h causes negative change in both Q and AP.
 From Table 3, we see that increase of h1 result slight decrease in order quantity Q and
 Average total profit AP. That is change in h1 leads slight negative change in both Q and AP.
 From Table 4, we can easily seen that increase of deterioration rate  cause increase in order
quantity Q and decrease in average total profit AP. That is change in  will leads positive
change in Q and negative change in AP.
 From Table 5, we see that increase of S, results increase of order quantity Q and decrease in
average total profit AP. That is, change in S will leads positive change in Q and negative
change in AP.
Copyright © 2017 by Modern Scientific Press Company, Florida, USA
Int. J. Modern Math. Sci.2017, 15(2): 237-247
245
 From Table 6, we see that, increase of s result increase in order quantity Q, and decrease in
average total profit AP. That is, change in s, leads positive in Q and negative change in AP.
6. Conclusion
In this paper, we have developed optimal ordering and transfer policy for an inventory with
linearly time dependent demand for deteriorating items. The mathematical formulation has been
developed for (i). The total cost per unit cycle in the warehouse.(ii). The total cost per unit cycle in the
display area. The optimal shows that the average total profit is concave function of t1 . From sensitivity
analysis the following observation is obtained.
 The change in n result positive change in Q and AP.
 The change in h result negative change in Q and AP.
 The change in h1 result slight negative change in Q and AP.
 The change in deterioration rate  result positive change in Q and negative change in AP.
 The change in S results positive change in Q and negative change in AP.
The paper may be extended for several ways. For instance, we may extend for nondeteriorating items. We may also modify the paper by extending the two parameter Weibull
distribution deterioration. Finally, we could generalize the model for allowing shortage, inflation etc.
References
[1] E.A. Silver, H.C. Meal. A simple modification of EOQ for the case of a varying demand rate. Prod.
and Invent. Manag., 10(1969): 52-65.
[2] W.A. Donaldson. Inventory replenishment policy for a linear trend in demand-an analytical
solution. Operat. Res. Quar., 28(1977):663-670.
[3] S. Khanna, S.K. Ghosh, K.S. Chaudhuri. An EOQ model for a deteriorating with time-dependent
quadratic demand under permissible delay is payment. Appl. Math. and Comp.,218(2011): 1-9.
[4] J. T. Teng, J. Min, Q. Pan. Economic order quantity model with trade credit financing for nondecreasing demand. Omega, 40(2012): 328-335.
[5] S. Khanra, S.K. Ghosh, K.S. Chaudhuri. An EOQ model for deteriorating items with Time dependent quadratic demand under permissible delay in payment. Appl. Math. and Comp.,
218(2011): 1-9.
[6] A.K. Jalan, K.S. Chaudhuri .An EOQ model for deteriorating items in a declining market with SFI
policy. Korean J. of Comp. and App. Math., 6(1999):437- 449.
Copyright © 2017 by Modern Scientific Press Company, Florida, USA
Int. J. Modern Math. Sci.2017, 15(2): 237-247
246
[7] A. Mitra, J.F. Fox, R.R. Jessejr. A note on determining order quantities with a linear trend in
demand. J. of the Oper. Res. Soc., 35(1984):141-144.
[8] E.A. Silver .A simple inventory replenishment decision rate for a linear trend in demand. J. of the
Oper. Res. So., 30(1979): 71-75.
[9] R.P. Tripathi, H.S. Pandey. An EOQ model for deteriorating items with Weibull time-dependent
demand rate under trade credit. Int. J. of Inf. and Man. Sci., 24(2013): 320-347.
[10] D. Singh, R.P. Tripathi, T. Mishra. Inventory model with non –instantaneous receipt and the
Exponential demand rate under Trade credits. Inte. J. of Mod. Math. Sci, 8(2013): 185-194.
[11] S. Kharna, S.K. Ghosh, K.S. Chaudhuri.A note on an order level inventory model for a
deteriorating item with time-dependent quadratic demand. Comp. and Oper. Res.,30(2003): 19011916.
[12] S.K. Ghosh, K.S. Chaudhuri. An EOQ model with a quadratic demand, time-proportional
deterioration and shortages in all cycles. Int. J. of Sys. Sci., 37(2006): 663-672.
[13] P.M. Ghare, G.F. Schrader. A model for exponential decaying inventories. J. of Ind. Eng.,
14(1963):238-243.
[14] K.J. Chung, L.E.C. Barron, P.S. Ting. An inventory model with non-instantaneous receipt and
exponentially deteriorating items for an integrated three layer supply chain system under two
levels of trade credit. Int. J. of Prod. Eco., 155(2014): 310-317.
[15] B.C. Giri, A. Goswami, K.S. Chaudhuri. An inventory model for deteriorating item with timevaring demand rate. Eur. J. of Oper. Res., 47(1996): 1398-1405.
[16] A.K. Jalan, R.R. Giri, K.S. Chaudhuri. EOQ model for items with weibull distribution
deterioration, shortage and trended demand. Int. J. of Sys. Sci., 27(1996):851-855.
[17] A. Goswami, K.S. Chaudhuri. An EOQ model for deteriorating items with a linear trend in
demand. J. of the Oper. Res. Soc., 42 (1991): 1105-1110.
[18] K.J. Chung, P.S. Ting. A heuristic for replenishment of deteriorating items with a linear trend in
demand. Journal of the operational Research Society, 44 (1993): 1235- 1241.
[19] C. Lin, B. Tan, W.C. Lee. An EOQ model for deteriorating items with time-varying demand and
shortages. Int. J. of Sys. Sci., 31(2000): 391-400.
[20] H.M. Wee . A deterministic lot-size inventory model for deteriorating items with shortages and a
declining market. Comp. & Oper. Res, 22 (1995): 345-356.
[21]. A.A. Yhmadi, M. Honarvar. A two stage stochastic programming model of the price decision
problem in the dual channel closed-loop supply chain. Int. J. of Eng., 5(2015): 738-745.
[22] M. Modarres, E. Taimury. Optimal solution in a Constrained Distribution. Sys. Int. J. of Eng.,
15(2002): 179-190.
Copyright © 2017 by Modern Scientific Press Company, Florida, USA
Int. J. Modern Math. Sci.2017, 15(2): 237-247
247
[23] M. Rabbani, N. Manavizadeh, S. Balati. A Stochastic model for indirect condition Monitoring
using Proportional covariate model, Int. J. of Eng., .21(2008): 45-56.
[24] R.P. Tripathi, A.K. Uniyal. EOQ model with cash flow oriented and quantity. Int. J. of Eng.,
27(2014): 1107-1112.
[25] S.K. Goyal, C.T. Chang. Optimal ordering and transfer policy for an inventory with stockdependent demand. Eur. J. of Oper. Res., 196(2009):177-185.
[26] S.K. Goyal. An integrated inventory model for a single supplier-single customer problem. Int. J.
of Pro. Res., 5(1977): 107-111.
[27] A. Banerjee. A joint economic-lot-size model for purchaser and vender. Dec. Sci. 17(1986): 292311.
[28] S.K. Goyal. A Joint economic-lot-size model for purchaser and vendor a comment. Decision
Science, 19(1988): 236-241.
[29] P.C. Yang, H.M. Wee. An integrated multi-lot-size production inventory model for deteriorating
items. Comp. and Oper. Res., 30(2003): 671-682.
[30] Y. Yao P.T. Evers, M.E. Dresner. Supply Chain integration in vender-managed Inventory. Dec.
Supp. Sys., 43(2007): 663-674.
Copyright © 2017 by Modern Scientific Press Company, Florida, USA