 Open Access
 Total Downloads : 106
 Authors : Satyajit Sahu, Gobinda Chandra Panda, Ajit Kumar Das
 Paper ID : IJERTV6IS060360
 Volume & Issue : Volume 06, Issue 06 (June 2017)
 DOI : http://dx.doi.org/10.17577/IJERTV6IS060360
 Published (First Online): 23062017
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Fullly Backlogged Deteriorating Inventory Model with Price Dependent Demand using Preservation Technology Investment and Trade Credit Policy
1Satyajit Sahu,
1Reseach Scholar, Fakir Mohan University, Balasore,Odisha,India
3Ajit Kumar Das
2*Gobinda Chandra Panda,
2*Department of Mathemtics, Mahavir Institute of Engg. and Technology,
BBSR, Odisha, India
3Department of Mathematics, Fakir Mohan Autonomous College,
Balasore,Odisha,India
Abstract: Here we developed an EOQ inventory model using price dependent demand taking deterioration as a factor. Also we use preservation technology cost to control deterioration and apply trade credit policy to attraact customers to buy more products.In this work shortages are allowed and fullly backlogged during the specific time period.Our main objective in this paper is to find optimal cycle length and preservation technology strategies while maximized the total profit. We have presented a numerical example to validate the work and analysed the sensitivity of different parameters used in this work using LINGO software and also shown graphically the profit function is concave using MATLAB software.
Keywords: Deterioration, Preservation technology, Trade credit policy, Shortage,Backlogged.

INTRODUCTION
Generally the word inventory defined as stock of goods and it has three different stages i.e raw materials,workin process products and fininished goods that are considered to be the part of a business assets that are ready or will be ready for sale.So, inventory is a most important part in a business, considering the vital role of inventory in a business , business organisation always put emphasis on proper management of inventory to run their business smoothly.So proper inventory management gives profit to a business organisation. Here we have taken some factors related to management of inventory properly which helps the business organisation to take better decisions. Deterioration of inventory is a key factor in almost all business organisation which affects the decision related to inventory management.The word deterioration defined as decay or damage or worst or out dated etc. according to the different products. There are some products deteriorate or decay during their storage period such as fruits , vegetables,eggs, fishes, rice,wheat and seasonal products etc. and some products are outdated due to arrival of new products in the market with new technology such as electronic items, automobiles and radioactive substances etc. Many researchers developed their work taking
deterioration in their model such as Darwiash & Odah(2010) has developed vendor managed inventory for singlevendor multi retailer supply chains and Chang et el (2010) has developed a noninstantaneous deteriorating inventory models with stockdependent demand . In this way researcher like Huang et al(2011) has introduced preservation technology and developed an inventory model. In this connection we may refer several related research work was discussed by Bhunia and Shaikh (2011 a,b) , Lee & Dye(2012) ,Bhunia et al (2013) , Hseih & Dye (2013) , Bhunia et al (2015), Bhunia & Shaikh(2016), Shaikh (2016 a,b) and Bhunia et al (2017) and others.
Demand plays an important role in a business.So researcher gives importance to demand and developed their model taking different types of demand according to the market needs. In earlier inventory models , generally demand rate assumed to be either constant, time dependent and stock dependent etc. How ever it is observed that selling price of a product is also most important factor in customer point of view because selling price of a products always present in the mind of a customer before buying a product. So business organisation always changing the price of their products according to the market demand to attracts the customers. So sometimes demand is dependent upon price of a products. Several researchers developed inventory model taking priced dependent demand like Sana(2011) has developed an inventory model taking price sensitive demand with perishable items. Similarly researcher like Maihami et al (2012) ,Avinadav et al (2013) and Shaikh et al (2017) and others have discussed on price dependent demand in their inventory models.
Nowadays, trade credit policy plays an important role in a business scenario. Trade credit is the credit extended by the supplier to the customers for the puchase of goods and services. Trade credit helps the retailer to purchase the supplies of goods by the supplier without immediate payment. Trade credit is commonly used by the business organisation as a source of shortterm financing. There are many forms of trade credit in many forms , different
business organisation use various specialized forms of trade to attracts the customers to bye more products from their organisation. Many researchers discussed trade credit in their research work. Researcher like Min et al (2010) has discussed an inventory model with stock dependent demand and two level trade credit. In this connection we may refer several related research work discussed by Liang & Zhou(2011) , Mahata (2012),Teng et al (2013) , Shaikh (2017 a, b) and others.
In this work , we developed an EOQ inventory model using price dependent demand taking deterioration as a factor.Also we use preservation technology cost to control
deterioration and apply trade credit policy to attraact customers to buy more products.In this work shortages are allowed and fullly backlogged during the specific time period.Our main objective in this paper is to find optimal cycle length and preservation technology strategies while maximized the total profit. The We have presented a numerical example to validate the work and analysed the sensitivity of different parameters used in this work. Lastly we have given some concluding remarks and future research .

NOTATIONS AND ASSUMPTIONS
In order to develop the inventory models we have been used the following notations and assumptions: Notation:
Notations Units Description
c $/order Purchasing cost per order.
h $/unit Holding cost per unit
b $/unit Shortage cost per unit
A $/unit Replenishment cost per order.
$/unit Backlogging parameter
p $/unit Selling price per unit
Constant Deterioration rate
M Month Period of permissible delay in payments offered by the supplier.

Constant

Constant
$/unit Preservation technology cost
Ie $/unit Rate of interest earned by the retailer.
Ic $/unit Rate of interest payable by the supplier
m1 $/unit Mark up rate
R Units Maximum shortage quantity per cycle.
S Units Initial inventory level.
Q Units Order size per cycle
t1 Month Time point at which the inventory level reaches zero
T Month The total length of the inventory cycle.
1
Z1 (t ,T , ) $/month The total profit per unit time for the interval 0 M t
1
1
2
1
Z (t ,T , )
$/month
The total profit per unit time for the interval t M T
Decision variable
t1
Month
Time at which the stock reaches zero
T
Month
The total length of the inventory cycle.
$/unit
Preservation tecnology cost
Assumptions:
The model is developed for a single deteriorating item for linearly price a dependent demand pattern
D( p) a bp
i.e.,
demand function depends on price ,where a 0 and b 0 .
The deterioration rate (0 1) is constant and depends on the stock amount.
There is a no replacement or repair for deteriorated products during the period under consideration. Replenishment rate is infinite and Leadtime is negligible or zero.
The total planning horizon of the inventory system is infinite.
The relationship of deterioration rate and the preservation technology investment parameter satisfies the following
m
0 ,
2m
2
0. Therefore, this research work considers that
m e
a1
; where,
m
is the
1
deterioration rate when there is investing preservation technology, is the deterioration rate without preservation technology investment , and a is the sensitive parameter of investment to the deterioration rate.


MATHEMATICAL FORMULATION
During the time period 0, t1 , the inventory level decreases due to both demand and deerioration and drop to zero.Thus
the inventory level can be represented in the form of the following differential equation.
dI1 t m I t a bp
dt 1
0 t t1
(1)
With the boundary condition I1 t1 0 .Solving equation (1) , we have
m
1
I t a bp e m t1 t 1
0 t t1
(2)
The inventory level reaches zero at t t1 , then shortage occured during the time period t1 ,T and the unsatisfied demand is completely backlogged. The level of inventory during the time period t1 ,T can be represented in the form of following differential equation:
dI2 t a bp
dt t1 t T
(3)
With the boundary conditions I2 T R . Solving equation (3), we have
I2 t a bpT t R
t1 t T
(4)
Now using the continuity property at which is backogged per cycle
R a bpT t1
t t1
, we have
I1 t1 I2 t1
which gives the maximum amout of shortages
(5)
The maaximum invetory level is I1 t S
at t t1 is
S e 1 1
a bp m t
m
Hence the total ordering quantity per cycles is given as follows
Q S R
a bp e m t1 1 a bpT t
(6)
(7)
m 1
Now the different cost assoccieated in this model is Sales revenue
t1
SR p Ddt PR
0
PDt1 PR
Purchasing cost
PC cQ
c a bp e m t1 1 a bpT t
m 1
Holding cost
t1
HC h I1 t dt
0
e m t1 1
h m m t1
Backlogging cost
T
BC b I2 t dt
t1
T 2 t 2
b R T t a bp
1
Tt
1
2
1 2
Preservation technology cost
PTC T
Trade credit is described in two different interval. Case 1: 0 M t1
Case 2: t1 M t2
0 M t1
t1 t
IE1 pIe Ddudt pIe Rt1
0 0
pI Dt 2
e 1 pI Rt
2 e 1
t1
IC1 cIc I1 t dt
M
a bp
e m t1 M 1
cIc m
m
t1 M
Hence the total profit function is for two case is
Z1 t ,T , X
1
Where
T
X SR PC OC HC BC IE1 IC1
a bp m t1
X PDt PR c e 1 a bpT t
1 m 1
e m t1 1
A h m m t1
T 2 t 2
b R T t a bp Tt 1 T
1
2 1 2
pI Dt 2

e 1 pI Rt

cI
a bp
e m t1 M 1
t
M
2 e 1
t1 M t2
c m
m 1
In this case the interest earned is
t1 t
IE2 pIe Ddudt M t1 pIe D pIe RM
0 0
There is no interest charged for this case . Now the total profit function is
Z 2 t ,T , X
1
Where
T
X SR PC OC HC BC IE2
a bp m t1
X PDt PR c e 1 a bpT t
1
e m t1
m 1
1
A h m m t1
T 2 t 2
b R T t a bp Tt 1 T
1
pI Dt 2
2 1 2

e 1 pI RM PI D M t
2 e e
1


NUMERICAL EXAMPLE
To illustrate and validate of our proposed inventory model, we have considered two numerical examples with the following values of different parameters as given below:
Example 1:
A $200 / odrer, p $60 / unit, h $2 / unit / year, b $6 / unit,
Ic $0.12 / $ / year, Ie $0.06 / $ / year, M 90 / 365 / year, .5, a1 0.09, .5, a 220,
c 40, b .4 .From the above numerical example, we have obtained case one gives better optimal solution which are described below:
1 1
Z1* t ,T , $3223.413 ,t* 0.2617378,T * 0.2648429 and * 7.703062.
Figure1 The above figure represent the concavity of the profit function.

SENSITIVITY ANALYSIS
The above described numerical, we have performed sensitivity analysis for example1 to study the effect of under or over estimation of the inventory system parameters on the optimal values of the initial time period, cycle length, preservation cost, initial stock level , maximum shortage rate along with the maximum profit of the system. The percentage changes in the above mentioned
optimal values are taken as measures of sensitivity. The analysis is carried out by changing (increasing and decreasing) the parameters by 20% to +20%. The results are obtained by changing one parameter at a time and keeping the other parameters at their original values. The results of these analyses are given in Tables 1.
Table 1: Sensitivity analysis with respect to different parameters
…………….
Parameter
% Change In Parameters
% Change in
% Change in
Z1
R
S
t
1
T
c
20
4389.374
0
50.61842
0.2464362
0.2464362
23.31297
10
4100.322
0.1487365
60.83170
0.3103647
0.3108706
7.702175
10
2769.370
0.2468636E03
110.3731
0.5631272
0.5631280
7.701725
20
1972.910
8.274615
70.38822
0.3591233
0.3872683
7.701733
A
20
3333.390
0.7834220
48.40948
0.2469867
0.2496514
7.701943
10
…………….
…………….
…………….
…………….
…………….
10
3060.025
0.2170516E01
48.34907
0.2466788
0.2467526
7.701699
20
…………….
…………………
…………….
…………….
…………….
…………….
p
20
………..
………..
………..
………..
……………..
………..
10
2193.700
0.1235149
63.84792
0.3218141
0.3222291
7.701639
10
…………….
………………..
………………
………………
………………
……………..
20
…………….
………………..
………………
…………….
…………….
…………….
h
20
3642.240
0.000000
167.3874
0.8540155
0.8540155
7.701739
10
4898.210
205.8461
48.32878
0.2465753
0.9467322
7.701673
10
3154.611
0.8343884
48.39348
0.2464680
0.2493061
8.026030
20
…………….
……………….
…………….
…………….
…………….
…………….
b
20
3938.709
26.81266
52.08820
0.2657551
0.3569546
7.702283
10
…………….
………………
…………….
…………….
…………….
…………….
10
3119.092
0.000000
0.2464483
0.2464483
0.2464483
8.165308
20
…………….
…………….
…………….
…………….
…………….
…………….
a1
20
4895.013
206.0336
48.32881
0.2465753
0.9473700
9.627235
10
…………….
…………….
…………….
…………….
…………….
…………….
10
2916.870
0.000000
49.65414
0.2465221
0.2465221
12.86717
20
4830.743
132.5408
48.32877
0.2465753
0.6973945
6.418031
20
3628.266
1.946967
178.7438
0.9119577
0.9185801
10.18103
10
3607.586
0.1082727E01
163.2144
0.8327254
0.8327622
8.872389
10
3619.749
0.7871395
164.2208
0.8378601
0.8405375
6.642698
20
2885.661
0.000000
49.84394
0.2464831
0.2464831
11.73518
20
…………….
…………….
…………….
…………….
…………….
…………….
10
…………….
…………….
…………….
…………….
…………….
…………….
10
3143.858
0.7204667E01
48.37213
0.2467964
0.2470192
7.701728
20
3168.504
0.8781295E04
50.01328
0.2551697
0.2551700
7.701675
a
20
2748.050
0.1613506E01
145.0886
0.9545303
0.9545892
7.701645
10
2997.844
0.6726495E03
68.75979
0.3951692
0.3951713
7.702207
10
3470.987
0.000000
54.42541
0.2465443
0.2465443
10.22527
20
4044.856
0.6072942E04
60.32534
0.2513555
0.2513556
7.701704
b
20
5329.602
97.23111
49.65601
0.2472908
0.5163012
7.701703
10
10
3566.794
0.8474837E02
153.8597
0.7947300
0.7947543
7.701638
20
3579.261
7.410067
230.2746
1.204365
1.225896
7.701637
M
20
…………….
…………….
…………….
…………….
…………….
…………….
10
…………….
…………….
…………….
…………….
…………….
…………….
10
3217.480
0.000000
53.16452
0.2712473
0.2712473
7.701748
20
3296.985
0.000000
59.15894
0.3018312
0.3018312
7.701692

CONCLUSION
The purpose of this study , is to present a deteriorating inventory model with price dependent demand with shortage and fully backlogged. Here we have introduced preservation technology investment to control the deterioration rate for highly deteriorated products. Also we have apply trade credit policy in the perspective of retailers. We also provide a useful solution procedure to find the optimal cycle length and preservation technology investment strategies while maximizing the total profit per unit time. Here we have solved numerical examples which explained the importance of preservation technology investment and trade creit policy. Finally we have shown graphically the profit function is concave by using matlab software and analysed sensitivity of different parameters of the model by Lingo software which helps the business organisation for better managerial decisions.

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BIOGRAPHY
Satyajit Sahu is a faculty member in dept of mathematics in EAST, BBSR, India. He has obtained M.Sc degree in mathematics from G.M College, Sambalpur , Odisha, India. He has published 3 research papers in different journals. His research interest in inventory control theory.
Gobinda Chandra Panda is working as an Asst Prof in mathematics in M.I.E.T ,BBSR, Odisha India. He has obtained his M.Phil and PhD in Mathematics from Sambalpur University, odisha , India. He has published 15 research papers in different national and international journals. His research interests include inventory control theory.
Ajit Kumar Das is a faculty member in dept of mathematics in F.M Autonomous College,, Balasore, Odisha, India. He has obtained his Ph.D degree in mathematics from Utkal University, Vani Vihar, BBSR, Odisha, India. He has published research papers in different journals.