Formulation of a Combined Transportation and Inventory Optimization Model with Multiple Time Periods

G.Krishnakumari. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 3, (Part - 1) March 2016, pp.80-87
www.ijera.com 80 | P a g e
Formulation of a Combined Transportation and Inventory
Optimization Model with Multiple Time Periods
G.Krishnakumari,
Professor, Vidya Jyothi Institute of Technology,Hyderabad-59,
ABSTRACT
Most distribution network design models existing in the literature have focused on minimizing the costs of
inventory and transportation. During the analysis of supply chain of currency management problem it is
observed that the transportation of currency from various sources to various destinations and the required
inventory to be maintained to meet the emerging demands requires formulation of a combined problem. This
framework aims to support the coordination of inventory and transportation activities to properly manage the
inventory profiles and currency flows between source locations and distribution centers. This paper considers a
multi-period inventory and transportation model for a single commodity. The key contribution of this paper is, a
mathematical programming formulation of transportation cum inventory problem is proposed and an algorithm
for this new formulation as a multi period decision process is intended. A numerical example of currency
transportation cum inventory is presented to illustrate the proposed algorithm.
Keywords: Transportation, and Inventory, Supply chain, Algorithm, multi- period.
I. INTRODUCTION
Here we briefly present transportation problem
and inventory problem. We also present the
importance of these two problems in the supply
chain management and the motivation for the
combined transportation and inventory
optimization model along with the organization of
the paper.
In a transportation problem (TP) there are
multiple sources of supply units and multiple
destination centers which demand certain quantity
of single or multiple commodities from the supply
centers. The transportation problem has the
objective of minimizing the overall transportation
cost by determining how much quantity from the
respective sources to be sent to the various
destinations keeping in view the available units at
the source centers and the requirement at the
destinations. The TP formulation was first given by
F.L.Hitchcock (1941).. The source centers (Si) may
be production centers, plants, warehouses etc,
whereas the destination centers (Dj) may be
distributors, wholesale units, retail outlets etc.
In an inventory problem there may be one or
multiple sources or production centers where single
or multiple commodities in the form of raw
materials or finished products are maintained for
either production process or delivery to the various
destinations. If the inventory of the required raw
materials is not available in time the production
process will be disturbed. On the other hand
inadequate stock of finished products gives raise to
reputation loss, orders loss etc.
The objective of the inventory problem is to
determine the optimal quantity of the required
items to be stocked which will minimize the overall
inventory costs. The optimal quantity is called
EOQ (Economic Order Quantity) and the optimal
cost consists of purchasing cost, set up cost,
holding cost and shortage cost.
A supply chain SC is a path from a production
or supply source of a product or service to an end
customer through a set of facilitating intermediate
entities. These entities in a given market segment
may be manufacturers, suppliers, transporters,
wholesalers, retailers and agents. The objective of
supply chain is to deliver a final product or service
to a customer efficiently in time as per the
specifications of the customer.
This has motivated us to look into the
combined formulation of transportation and
inventory in the supply chain. From the literature
survey it is understood that much work is not done
with multiple periods which minimizes the
transportation and inventory costs together for
single commodity with shortages. In fact, to the
best of our knowledge there is no study at all
which is incorporating the inventory costs in to the
transportation model and extending it to the
multiple periods. Our present work will concentrate
on single commodity with multi periods.
This paper will combine the two approaches
of transportation and inventory problem to develop
a joint transportation cum inventory problem to
determine the preferred compromised solution for
multi period inventory cum transportation problem
(MPICTP) for a single product, single stage with
RESEARCH ARTICLE OPEN ACCESS

multiple suppliers, multiple destinations with
deterministic demand for multiple periods. This
approach leverages the advantages of considering
both at a time to produce a powerful method to
solve MPICTP. After observing the relevant
models and important aspects presented in the
previous literature a modified model is proposed. In
the previous models, it is observed that the joint TP
& IP model is formulated without taking either
shortage costs or multiple time periods. Multiple
time periods though considered, have not been
taken continuously from one time period to
another. With this proposed model period by period
analysis is possible.
In section 2 Literature review of developments
of transportation, inventory problems and the
supply chain management of various supply chain
networks are given. Problem description and an
algorithm for the solution of the proposed
formulation is given under section 3.Section 4
contains a numerical illustration by using the
proposed Mathematical formulation.. The
observations of the above proposed method and the
numerical illustration are given under 5.The
concluding remarks and the scope for further work
are explained in 6. References are being given in
the succeeding section 7.
II. RELATED WORK
The TP formulation was first given by F.L.
Hitchcock (1941) and first inventory problem
formulation(EOQ) was given by F.W. Harris in
1913.
In the literature development of the
formulation of both inventory and transportation
problems have taken independently. However
several researchers have made attempts to combine
both transportation and inventory for joint
formulation. The first attempt in this direction was
made by Brian Q. Rieksts, et al developed Optimal
inventory policies with two modes of freight
transportation European [1] (2008). Zhendong
Pan,Jiafu Tang etc (2009) [2] Lihui Liu and
Chunming Ye (2009) [3], Jae-Hun Kang Yeong-
Dae Kim,[4] (2009), , Erhan Kutanoglu,Divi
Lohiya (2008) presented an optimization based
model to gain insights into the integrated inventory
& transportation problem for a single- echelon,
multi facility service parts logistics system with
time based service level constraints.[5] Nikhil. A
.Pujari, Trever. S Hale, Fazul Huq (2008)
presented a continuous approximations procedure
for determining inventory distribution schemas
within supply chains. The model shows how
inventory policy decisions directly impact expected
transportation costs and provides a new method for
setting stock levels that jointly minimizes inventory
and transportation costs.[6]. Xiuli Wang,T.C.E
Cheng [7] (2009) developed the heuristic
algorithms considering both inventory and
transportation costs simultaneously. Kadir Eritrogal
(2007) suggested a Lagrangean decomposition
based solution procedure for the problem of multi
item single source ordering problem with
transportation cost.[8] Simone Zanoni,Lucio
Zavanella(2007) considered the problem of
shipping products from a single vendor to a buyer
with the objective of minimizing the sum of the
inventory and transportation costs when the
products are perishable. They have given a mixed
integer linear programming formulation for it. [9].
K. Ertogral, M. Darwish, and M. Ben-Daya (2007)
incorporated the transportation cost explicitly into
the model and develop optimal solution procedures
for solving the integrated models of joint vendor-
buyer problem. [10]
A. Noorul Haq and G. Kannan.(2006) have
proposed a two-echelon distribution-inventory
supply chain model for the bread industry using
genetic algorithm. Their paper presents an approach
of optimizing the inventory level for a two-echelon
supply chain by considering the distribution costs
and various production related costs in meeting the
customer demand.[11]. Linda van Norden et al
dealt with a Multi-product lot-sizing with a
transportation capacity reservation contract [12]
(2005) , Cheng Liang Cen,Wen cheng Lee(2004)
have given a multi product, multi stage & multi
period scheduling model to deal with multiple
incommensurable goals for a multi echelon supply
chain network with uncertain market demands &
product prices.[13]. Qiu Hong Zhao Shou-Yang
Wangb, K.-K. Laic, Guo-Ping Xiaa. (2004) have
presented a modified EOQ model including
transportation cost.[14]. Linda K Nozick, Mark A
Turnquist (2001) presented a modeling approach
that provides an integrated view of inventory,
transportation, serviced quality and the location of
distribution centres.[15]
Wendy W Qu etc(1997) have proposed an
integrated inventory transportation system with
modified periodic policy for multiple products.[16]
B.Q Rieksts et al [17] (2010) developed Two-stage
inventory models with a bi-modal transportation
cost, Nonihal Singh Dhakry et al [18] (2013) did
research in Minimization of Inventory &
Transportation Cost of an Industry – A supply
chain optimization. They developed a new method
by combining the transportation & inventory costs
and minimizing them. Vivek K Agarwal et al [19]
2015 discussed an Inventory Model with Discrete
Transportation cost and price break, Barry R Cobb
[20] ( 2016) approached a method to an Inventory
control for returnable transport itmes in a closed
loop supply chain.

Looking at the previous works we have
observed that the attempt to have a joint
formulation of inventory and transportation
originated with the extension of inventory problem.
III. PROPOSED TRANSPORTATION
CUM INVENTORY MODEL
We present the mathematical programming
formulation of a multi-period transportation cum
inventory for a single item but for multi periods.
3.1 Assumptions
The various assumptions of the proposed model :
i). Transportation cost of each item to transport it
from source locations to the destinations may vary
across all the periods.
ii). Inventory-carrying cost of each item per period
at all source locations and destinations may vary
across multiple periods.
iii). Unit cost of the commodity at any source
location is fixed, throughout the period of study at
that plant.
iv). Transportation cost per product from the
manufacturer to the source location is included in
the unit commodity cost.
v). Daily demand of the demand centre is
deterministic and the supply capacity of the
selected supplier is limited. It is also assumed that
the time taken to transport the item between the
source location and the destination is homogenous
and is not taken into consideration.
vi) Every item is considered as non perishable item
for a certain number of periods.
vii) The centralized body which manages all the
supply centers is concerned for over all
optimization of inventory and transportation costs
of these supply centers.
viii) Inventory and the transportation costs at the
supply centers are only considered.
3.2 Notations:
M : Number of source locations of supply
N : Number of destination locations of
demand
Si : ith
source location point where i=1(1) m
Dj : jth
destination point where j= 1(1) n
TTC : Total transportation cost from all sources to
destinations.
TIC : Total inventory cost at the source locations
S : Total number of time periods.
T k
: kth
time period, where k=1(1) s.
s i
k
: Total quantity of shortage of the commodity
at ith
source Si during time period Tk
.
cs
k
i : Unit shortage cost at source Si during time
period Tk
.
h
k
i : Total quantity of a commodity held at ith
source Si from the beginning of time period Tk
ch
k
i : Unit holding cost at source Si during time
period Tk
k
i
o : Ordering cost per order at Si during k -th
time period Si.
k
i
N : Number of orders placed at ith
source Si
during time period Tk
k
i
a : Quantity of commodity available at Si at
the beginning of the time period T k
.
k
j
b : Quantity of commodity required at Dj at
the beginning of the time period T k
c ij : Cost of transporting one unit of commodity
from Si to Dj .
x ij : Quantity of commodity transported from Si
to Dj .
p
k
i : Unit cost of the commodity available at Si
during the time period Tk
k
iT a : Total quantity of commodity along with the
carried forward number of units, if
any available at Si during Tk
Tbj
k
: Total quantity of commodity demanded
along with the unfulfilled demand in
the previous time period at Dj during Tk
.
k
j
u b : Total quantity of unfulfilled demand at Dj
during Tk
k
i
N C : Total number of units of commodity
available at Si during Tk
3.3 Mathematical Programming Formulation of
Transportation cum Inventory Model
The objective is to
Min Z = W1*TIC + W2 *TTC =
1
1 1
* [ * * * ]
m s
k k k k k k
i i i i i i
i k
w h ch s cs p N C
 
  
+ 
1 1
*
s m
k k
i i
k i
o N
 
  +
2w * 
1 1 1
s m n
k k
ij ij
k i j
x c
  
   (1)
Where TIC = Total Holding Cost +Total Shortage
Cost + Total Commodity Cost +Total Ordering
Cost
TTC= Total Transportation Cost.
Subject to the Constraints
T
1

k
i
k
i
k
i
haa for i=1(1) m,
k=1(1) s Where 0
0ih 
(2)

T
1

k
j
k
j
k
j
ubbb for j=1(1) n,
k=1(1) s where 0
0ju b 
(3)
T
1
n
k k k
i ij i
j
a x h

  for i=1(1) m, k=1(1)
s (4)
T
1
m
k k k
j ij j
i
b x ub

  for j=1(1) n, k=1(1) s
(5)
k
i
n
j
k
ij
Tax 
1
for k=1(1) s
(6)
k
j
m
i
k
ij
Tbx 
1
for j=1(1) n, k=1(1) s
(7)



m
i
k
i
n
j
k
j
sub
11
for k=1(1) s
(8)
h
k
i * s i
k
=0 for i=1(1) m, k=1(1) s
(9)
xij N for i=1(1) m,
j=1(n) 1
(10)
w1+w2=1 for 0 ≤w1 ≤ 1, 0 ≤ w2 ≤ 1
(11)
On simplification, the k-th time period problem
where k=1(1)s formulated results into:
Min Z=
1 1 1
( * * * * )
m m n
k k k k k k k k k k
i i i i i i i i ij ij
i i j
h ch s cs p N C o N c x
  
     
(12)
Subject to the Constraints
T
1

k
i
k
i
k
i
haa for i=1(1) m,
k=1(1) s Where 0
0ih 
(13)
T
1

k
j
k
j
k
j
ubbb for j=1(1) n,
k=1(1) s Where
0
0jub 
(14)
T
1
n
k k k
i ij i
j
a x h

  for i=1(1) m, j=1(n)
1, k=1(1) (s-1 (15)
T
1
m
k k k
j ij j
i
b x ub

  for i=1(1) m, j=1(1)
n, k=1(1) (s-1) (16)



m
i
k
i
n
j
k
j
sub
11
for k=1(1) s
(17)
* 0
k k
i ih s  for i=1(1) m,
k=1(1) s
(18)
xij N for i=1(1) m,
j=1(n) 1
(19)
w1+w2=1 for 0 ≤w1 ≤ 1, 0 ≤ w2
≤ 1
(20)
Equation 1 gives the objective which is the total
cost of the supply chain network.
Constraint (2)-(3) represents the modified supply
and demand values from the previous time period
respectively. Eq (4)-(5) represents the units held,
shortage respectively
Eq (6) describes the shortage at ith
supply centre is
equal to the unfulfilled demand at jth
destination.
Eq (7) describes either holding or shortage of units
will occur at the source i.
The mathematical formulation for the proposed
transportation cum inventory problem is first
defined for the sum of k time periods and specially
the k-th time period problem is defined as above.
The objective of this problem is to minimize the
transportation and inventory like holding, shortage,
ordering and unit commodity costs. The
corresponding algorithm for the solution of the
above proposed transportation cum inventory
problem is given below.
3.4 Algorithm for the solution of the proposed
Transportation cum Inventory Problem:
Step1: Initialize time period k=1.
Step-2:Construct the transportation matrix by
entering the supply amount i
a at ith
origin for
i=1(1)m, demand bj at jth
destination for 1(1) n and
the unit costs cij in the (i ,j)th
cell for
i=1(1)m,j=1(1)n cells for the kth
time period k =1.
Step 3:-Find the initial basic feasible solution. We
used the Vogel’s approximation method after
balancing the given transportation matrix.
Step 4:- Check for optimality of the obtained basic
feasible solution and compute optimal solution by
MODI’s method.
Step 5:- Compute total transportation
cost=Minimum transportation cost.

Identify the number of units carried forward or
number of units of shortage and find the shortage
cost or carrying cost.
Step-6: Compute total inventory cost=production
cost + carrying cost + shortage cost.
Total cost = Total transportation cost + Total
Inventory cost.
Step-7: Increment time period k=k+1 and go to
step-2. Repeat the process until k=s.
Find the Cumulative total cost for the time periods
k=1 (1)s
Stop if k>s
IV. VERIFICATION OF THE
FORMULATION & ALGORITHM
WITH A NUMERICAL EXAMPLE
(STEP BY STEP EVALUATION)
A numerical example is given to illustrate
the above algorithm. A general transportation
problem is considered with 3 source locations and 4
destinations for 7 time periods. Capacities and
demands are given for the respective source
locations and destination points. The unit
transportation cost is also given to transport 1 unit
from any source to any destination. The number of
commodities carried forward or number of units of
shortage is identified and it will carry forward to
the next time period. Initial basic feasible solution
and the Optimum solutions are calculated by the
Vogel’s Approximation Method & MODI’ s
method respectively. The procedure will be
continued for 7 time periods. Detailed illustration is
given for k=1 and the same procedure is continued
for the remaining time periods.
Time period k=1
Step -1: k=1
Step-2: Suppose the give transportation problem is
given as follows.
The above considered transportation problem is not
a balanced TP problem.(43≠45).To make it as a
balanced add a 4th
source which is dummy.
Modified problem is given below.
Step-3: Initial basic feasible solution is identified
by VAM.
x12=12,x13=2,x14=8,x23=15,x31=5,x34=1,x41=2
Step-4: The optimal solution is found out by using
MODI method which is given below
x12=12,x13=1,x14=9,x23=15,x31=6,x41=1,x43=1
1 unit shortage in first destination& 1 unit shortage
in destination 3 is identified .This will be carried to
day-2.
Step-5: Total Transportation cost=
3*12+5*1+4*9+2*15+5*6+0*1+0*1=137/-
Step-6: Inventory cost =43*10/-+2*2/-+ 1*1=
430+4+1 =435/-
Total cost=Total transportation cost+ Total
inventory cost. =137/- + 435/- = 572/-
Step-7: Increment the time period by 1. ie k =1+1
=2<7 proceed to the next day.
Formulation:-
The objective function is
Min Z
=
3
1i 

3 4
1 1 1 1 1 1 1 1 1 1
1 1
{ * * * * }i i i i i i i i ij ij
i j
h ch s cs p N C o N c x
 
     
 Min Z =
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 3 3 3 3 3 3 3 3 11 11 12 12 13 13 14 14
1 1 1 1 1 1 1 1 1 1 1
21 21 22 22 23 23 24 24 31 31 32
* * * * * * * *
* * * * * * * *
* * * * * *
h ch s cs p NC o N h ch s cs p NC o
N h ch s cs p NC o N c x c x c x c x
c x c x c x c x c x c x
      
       
    
1 1 1 1 1
32 33 33 34 34
* *c x c x 
=
1 1 1 1 1 1 1 1 1 1 1 1
11 12 13 14 21 22 23 24 31 32 33 34
3* 5 * 4 5 9 2 7 5 7 8 6x x x x x x x x x x x x          
1
11
1 1 1 1 1 1 1 1 1 1 1
12 13 14 21 22 23 24 31 32 33 34
0 1* 3 10 * 22 0 0 10 *15 0 1* 3 10 * 6 6 * 6
3 5 4 5 9 2 7 5 7 8 6
x
x x x x x x x x x x x
          
         
After solving the problem by Vogel’s
approximation method and MODI’s method the
optimal solution is
1 1 1 1 1
1 2 1 3 1 4 2 3 3 1
1 2 , 1, 9 , 1 5, 6x x x x x     .Subs
titute these values in the above equation, then
Min Z = 435/-+3*12/-+5*1/-+4*9/-+2*15/-+5*6/-
=572/-
The no. of units of commodity available at the 3
source locations are as follows.
a 1=22,a 2=15, a3 =6
The no. of units of commodity required at the 4
destinations is as follows.
b 1=7,b2=12,b3=17,b4=9.
The given problem is an unbalanced problem. The
total availability=43
I
II
III
1 2 3 4
22
15
6
6 3 5 4
5 9 2 7
5 7 8 6
7 12 17 9
I
II
III
IV
1 2 3 4
22
15
6
2
6 3 5 4
5 9 2 7
5 7 8 6
0 0 0 0
7 12 17 9

The total requirement is =45.
The total availability is less than that of the total
requirement. So 2 units of shortage is identified.
One dummy source location is added. We have
0
0i
h  and
0
0j
u b 
4
1 1 1
3
1
i j i
j
a x h

  i=1(1)3
3
1 1 1
1
j ij j
i
b x u b

  j=1(1)4
In this problem availability is less than that of the
requirement. So no units will be carried forward to
the next time period. But 2 units of shortage is
identified. So these 2 units of unfulfilled demand
will be carried forward to the next time period.
3
1 1 1
1 1 1
1
i
i
b x ub

   7-(0+0+6) =1
3
1 1 1
2 2 2
1
i
i
b x ub

   12-(12+0+0) =0
3
1 1 1
3 3 3
1
i
i
b x ub

   17-(1+15+0)=1
3
1 1 1
4 4 4
1
i
i
b x ub

   9-(9+0+0)=0
1 unit of shortage in D1 & 1 unit of shortage in D3
is identified. These units will be carried forward to
the next time period to the corresponding
destinations.
Time Period k=2:
The current time period inputs are given as
After including the previous time periods carried
forward inputs to the current inputs the problem is
After solving by Vogel’s approximation & MODI’s
methods the optimal allotments are x14=6,
x22=6,x24=2,x31=6,x33=4
No shortage or held units are identified because
the problem is a balanced transportation
problem.
Total transportation cost =50/-
Total inventory cost= 24*8/-+1*1/- = 193/-
Total cost =Total transportation cost + Total
Inventory cost = 50/- + 193/- =243/-
Cumulative total cost = Total cost of time period
1+ Total cost of time period 2
=572/-+243/- = 815/-
Increment the time period by 1 so k=2+1=3<7
Go to next time period.
Similarly the 3rd
, 4th, 5th
and 6th
time periods
inventory and transportation costs are calculated.
Also the no. of units of shortage which is carried
forward from the previous time period 6 is 2 units
and it it in destination D2.So to the existing current
inputs of D2 , 2 other units are added in the time
period-7.
Time Period-7:
The current inputs for the fifth time period are as
follows:
After balancing the problem by adding the dummy
column and including the carried forward 2 units to
D2 the resultant input is as follows.
1 2 3 4
6
8
10
1 3 4 2
4 2 5 2
3 6 1 5
5 6 3 8
I II
III
1 2 3 4
1 3 4 2 6
4 2 5 2 8
3 6 1 5 10
6 6 4 8
1 2 3 4
I
II
III
1 3 4 2 2
2
3
4 2 5 2
3 6 1 5
1 2 1 1
1 2 3 4
I
II
III
1 3 4 2 2
2
3
4 2 5 2
3 6 1 5
1 4 1 1
1 1 0
1 1 1
1 1 0
2 2 2
1 1 0
3 3 3
1 1 0
1 1 1
1 1 0
2 2 2
1 1 0
3 3 3
1 1 0
4 4 4
2 2
1 5
6
7
1 2
1 7
9
T a a h
T a a h
T a a h
T b b u b
T b b u b
T b b u b
T b b u b
  
  
  
  
  
  
  

The optimum solution by MODI’s method which is
been obtained after finding the IBFS by Vogel’s
approximation method is given
below.x12=1,x14=1,x22=2,x32=1,x33=1,x34=1
Total cost =Total transportation cost +total
inventory cost
= Rs (137/-+50/-+14/-+21/-+21/-+24/-
+19/-) + (435/-+193/-+129/-+95/-
+84/- +82/-+71/-) =Rs 1375/-
K=7+1=8th
day > s
Stop, since k > s
The problem is a balanced transportation problem.
No. of units required is equal to the no. of units
demanded.
The consolidated results of the above problem are
given below.
The original input table for 3 sources and 4
destinations:
The Transportation & inventory input cost
matrix:
For every time period shortage, holding, ordering
& commodity costs are given. According to the
requirement either shortage or holding cost is
considered in every time period.
Optimal solutions for the 7 time periods which
includes the minimized transportation and
inventory costs.
V. Concluding remarks
1 Here a mathematical programming formulation
combining transportation and inventory problem is
given. 2. This formulation considers giving priority
to either transportation cost or inventory cost
however in reality the transportation cost
overshoots inventory cost. 3. From the proposed
formulation a decision maker can get the
information related to Inventory costs (shortage
costs & holding costs) and Transportation costs
separately in each time period. 4. The number of
commodities that are carried forward and shortage
of the number of items in each stage can also be
known by the decision maker. 5. This is a multi
period formulation for a single objective and can be
extended to multi objectives and for multi products.
In the above numerical example, all the types of
transportation problems like balanced, unbalanced,
degeneracy in the initial basic feasible solution and
degeneracy in the optimum solution are observed.
Also for some time periods, carried forward
commodities and for some other time periods,
shortage of commodities is observed. This
proposed formulation will work though the cost per
unit Cij values is different for different time
periods. This problem is solved again by a familiar
linear programming technique (penalty method)
and the results are compared. It is concluded that
the optimal solution is same for both the above said
methods.
REFERENCES
[1]. Brian Q. Rieksts, Jose A. Ventura, , Optimal
inventory policies with two modes of freight
transportation European Journal of
Operational Research ,186 , (2008) ,576–
585.
[2]. Zhendong Pan, Jiafu Tang & Richard Y.K.
Fung. Synchronization of inventory and
transportation under flexible vehicle
constraint: A heuristics approach using
sliding windows and hierarchical tree
structure. European Journal of Operational
Research, 192, (2009) 824–836.
[3]. Lihui Liu and Chunming YeModel and
Algorithm for Inventory-Transportation
Integrated Optimization Problem -In view
of many-one distribution network . IEEE,
978-1-4244-3662-0/09. .(2009).
[4]. Jae-Hun Kang, Yeong-Dae Kim.
Coordination of inventory and transportation
managements in a two-level supply chain.
Int. J. Production Economics doi:10.1016/
j.ijpe. (2009
[5]. Erhan Kutanoglu & Divi Lohiya..Integrated
inventory and transportation mode
selection: A service parts logistics system.

Transportation Research , Part E 44, (2008)
,665–683
[6]. Nikhil A. Pujari , Trevor S. Hale & Faizul
Huq . A continuous approximation
procedure for determining inventory
distribution schemas within supply chains
European Journal of Operational Research,
186 , (2008), 405–422.
[7]. Xiuli Wang, T. C. E. Cheng Logistics
scheduling to minimize inventory and
transport costs. Int. J. Production
Economics, 121, (2009). 266–273.
[8]. Kadir Ertogral, Multi-item single source
ordering problem with transportation cost: A
Lagrangian decomposition approach.
European Journal of Operational Research.
(2007)
[9]. Simone Zanoni, Lucio Zavanella..Single-
vendor single-buyer with integrated
transport-inventory system: Models and
heuristics in the case of perishable goods,
Computers & Industrial Engineering, 52,
(2007) ,107–123.
[10]. K. Ertogral, M. Darwish, M. Ben-Daya
,.Production and shipment lot sizing in a
vendor–buyer supply chain with
transportation cost. European Journal of
Operational Research. 176, .(2007) ,1592–
1606.
[11]. A. Noorul Haq and G. Kannan. Two-echelon
distribution-inventory supply chain model
for the bread industry using genetic
algorithm. Int. J. Logistics Systems and
Management, Vol. 2, No. 2. 2006.
[12]. Linda van Norden , Steef van de Velde
..Multi-product lot-sizing with a
transportation capacity reservation contract,
European Journal of Operational Research,
165, (2005) ,127–138.
[13]. Cheng-Liang Chen, Wen-Cheng Lee . Multi-
objective optimization of multi-echelon
supply chain networks with uncertain
product demands and prices. Computers and
Chemical Engineering,728 , (2004). ,1131–
1144.
[14]. Qiu-Hong Zhaoa, Shou-Yang Wangb, K.-K.
Laic, Guo-Ping Xiaa. Model and algorithm
of an inventory problem with the
consideration of transportation cost.
Computers & Industrial Engineering, 46,
(2004), 389–397.
[15]. . Linda K. Nozick & Mark A. Turnquist.
Inventory, transportation, service quality and
the location of distribution centers .European
Journal of Operational Research ,129 ,
(2001). ,362-371.
[16]. Wendy W. Qu , James H. Bookbinder &
Paul Iyogun An integrated inventory -
transportation system with modified periodic
policy for multiple products European
Journal of Operational Research, 115,
.(1999). , 254-269.
[17]. B.Q Rieksts , Two-stage inventory models
with a bi-modal transportation cost,
Computers & Operations Research 37(1):20-
31 · January 2010.
[18]. Nonihal Singh Dhakry , Minimization of
Inventory & Transportation Cost of an
Industry – A supply chain optimization. Int
Journal of Engineering Research and
Applications Vol 3, Issue 5, Sep- Oct 2013,
pp 96- 101.
[19]. Vivek K Agarwal, An Inventory Model with
Discrete Transportation cost and price break,
Journal of chemical, biological and physical
sciences, peer review E-3 Journal of
Sciences.vol 5 no 2, 1912-1920, ISSN 2249-
1929, April 2015
[20]. Barry R Cobb, Inventory control for
returnable transport itmes in a closed loop
supply chain, Transportation Research Part
E: Logistics and Transportation
Review.volume 86 Feb 2016, pp 53-68

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