IJCST Vol. 2, Issue 3, September 2011
ISSN : 2229-4333(Print) | ISSN : 0976-8491(Online)
Optimal Inventory Managament for Exponentially
Increasing Demand with Finite Rate of Production
and Deterioration
1
1
Avikar, 2Ram Kumar Tuli, 3Abhilash sharma
Dept. of Computer Science and Engg. G.N.D.U. Regional Campus, Gurdaspur, Pb., India
2
Dept. of Mathematics, S.S.M. College, Dinanagar, Gsp, India
3
S.M.D.R.S.D College, Pathankot, Gsp, India
Abstract
In this paper, a production inventory model for deteriorating
items with Exponentially increasing demand over a fixed time
horizon is presented. Production rate is considered as constant.
The deterioration rate is also constant. Approximation procedure
is used to solve the model.
Keywords
Optiomal Inventory Management, EOQ Model
I. Introduction
Order level inventory models for items deteriorating with time
have engaged attention of researchers in recent years. An EOQ
model for items with a variable rate of deterioration, an infinite
rate of production and no shortage was introduced by Covert and
Philip (1973). In 1975 Misra, formulated an inventory model
with a variable rate of deterioration without shortages. Dev and
Choudhary (1987) developed an EOQ model for items with finite
rate of production and variable rate of deterioration. Shah and
Jaiswal (1977) developed an order level inventory model for the
items with constant rate of deterioration but they made a wrong
assumption regarding the inventory holding cost. Actually the
aurthors have considered the inventory level of the system exactly
linear in prescribed interval which is incorrect. Aggarwal(1978)
developed an order level inventory model of rectifying the
error in the analysis of Shah and Jaiswal (1977) in calculating
the average inventory holding cost in the model. Misra (1975)
developed deterministic model with finite rate of replenishment
without shortages. He considered both constant and variable rate
of deterioration. Choudhary and Choudhary (1981) derived an
order level inventory model for constantly deteriorating items
with finite rate of replenishment. There are some mistakes in
their calculations of the value of time t1 at which the production
stops and attains highest level S and therefore the average
inventory holding cost is also incorrect. Kumar and Sharma
(2001) presented inventory model for deteriorating items taking
uniform deteriorating rate and production rate is assumed as linear
combination of on hand inventory and demand at any instant.
Demand is an exponential function of time, shortage are allowed
and excess demand is backlogged as well. They obtain expression
production scheduling period, maximum inventory level, unfilled
order backlog and the total average cost. They used approximate,
solution procedure to get the required result. Yang and Wee (2003)
developed an integrated multi-lot-size production inventory model
for deteriorating item with constant production and demand rates.
The present paper deals with an EOQ model for items with
constant rate of deterioration and finite rate of production taking
shortages into account. The form taken into consideration seem
to be reasonably realistic. Only the deterministic version of model
is discussed.
w w w. i j c s t. c o m
II. Assumptions and Notations
A deterministic production model for deteriorating product for
finite time horizon is developed under following assumption and
notation.
Assumptions
1. The lead time is zero.
2. A single item is considered over the prescribed period T units
of time which is subject to a constant deterioration rate.
3. Demand rate is known and increases exponentially.
4. Deterioration of the items is considered only after they have
been received in to the inventory.
5. No replenishment or repair of deteriorated items is made
during a given cycle.
6. Shortage are allowed and backlogged.
7. Production rate is finite.
Notations
1. q(t) is the inventory level at any time t, t ≥ 0.
2.
is the constant deterioration rate, 0 ≤ ≤ 1.
3. P is the production rate.
4. T is the length of one-cycle.
5. C1 is the constant carrying cost per unit per unit time.
6. C2 is the constant shortage cost per unit per unit time.
7. C3 is the constant set up cost for each new cycle.
8. C4 is the constant cost of each deteriorated items.
9. T ( = t1 + t2 + t3 + t4 ) is the cycle time.
10. S1 is the maximum inventory level.
11. S2 is the unfilled order backlog.
12. Demand rate
, 0 ≤ α < 1, a > 0.
13. C is the total average cost for a production cycle.
III. Mathematical Model and Analysis for The System
Let us consider that the initially stock is zero. Procurement through
production or purchase starts just after t = 0 and continue up to t =
t1 when the stock attains a level S1. Procurement is now stopped.
The inventory declines up to t = t1 until it again reaches the zero
level. At this time shortages start developing and at time t = t3
it reaches to S2, maximum shortage level. At this time fresh
production start to clear the backlog by the time t = t4. We wish
to determine the optimum values of t1, t2, t3, t4, S1, and S2 that
minimize C, the total average cost over the time horizon [0,T].
The differential equations governing the system are given by
International Journal of Computer Science and Technology 357
IJCST Vol. 2, Issue 3, September 2011
ISSN : 2229-4333(Print) | ISSN : 0976-8491(Online)
Maximum shortage level can be obtained from equation (9) at t
= t3, q(t) = S2. Hence
Boundary conditions are
q(t) = 0 at t = 0, t1 + t2 and T (5)
q(t1) = S1, q(t1 + t2 +t3 ) = S2
(6)
Solutions of equations (1) to (4) after adjusting the constants of
integration are obtained as follows:
Equation (1) is linear differential equation in q(t). Its solution
is given
by
Again, we know from equation (7) that at t = t1, q(t) = S1
Also, from equation (10) at t = 0, q(t) = S2,
From equations (11) and (13), we get
Where is the constant of integration.
Now using boundary condition q(t) = 0, at t = 0, we get the value
of
i.e
This is equivalent to
Hence, solution of (1) is
From equations (12) and (14), we get
Also, solution of (2) is given by
Where
is the constant of integration.
Now using boundary condition q(t) = 0, at t = t2, we get the value
of
or
Now, Deterioration cost for the period (0,T) is given by
Hence, we get the solution of (2) is
Inventory carrying cost over the period (0,T) is given by
Solution of equation (3) after using the boundary condition q(t)
= 0 at t = 0, is given by
Shortage cost is given by
Now the solution of equation (4) after using the boundary
condition
q(t) = 0 at t = t4, is given by
Using the boundary conditions (5) and (6), we can get
expression
For maximum stock level from equation (8) at t = 0, q(t) = S1.
Hence
358 International Journal of Computer Science and Technology
Now total average cost of the inventory system is given by (C)
C = Set up cost + Deterioration cost + Carrying cost + Shortage
cost
w w w. i j c s t. c o m
IJCST Vol. 2, Issue 3, September 2011
ISSN : 2229-4333(Print) | ISSN : 0976-8491(Online)
and
Now differentiating (22) with respect to t1 and t4, we get
After integrating w.r.t ‘t’, we get
and
IV. The Approximate Solution Procedure
In most of the cases and α are very small. Hence the Maclaurin’s
series for approximation can be used in equation (21) and this
equation reduces to
Or
Or
Equation (22) contains four variables t1, t2, t3 and t4. Moreover
they are not independent.
For maximum C, we must have
Provided that these values of ti, i = 1,4 satisfy the conditions
w w w. i j c s t. c o m
(24)
Where R(t1) and R(t4) can be obtained by Maclaurin’s series
approximation from equation (13) and (14) respectively. From
these non-linear equation (23) and (24), the optimum values of
t1* and t4* can be found out by computational numerical method.
The optimum values t2*, t3*, S1, S2 and minimum average cost C
can be obtained from the above equation (23).
References
[1] Yang,P.C, Wee, H.M. Computers and Operations Research, 30, 2003, pp:671-682.
[2] Dev,M, Chaudhury, K.S. Journal of Operation Research
Society,38, 1987, pp:459-463.
[3] Misra,R.B. International Journal of Production Research,
15, 1975, pp495-505.
[4] Roy Choudhuri, M, Chaudhauri,K.S. Opsearch, 20, 1983,
pp:99-106.
[5] Henery,R.J. Journal of Operation Research Society. 30(7),
1975, 1979, pp: 611-617.
[6] Goyal,S.K, Morin,D., Nebebe,F. Journal of operation
Research Society. 43(12), 1992, pp:1173-1178.
[7] Silver,E.A , Pyke,D.F, Peterson,R. (1998). John Wiley and
Sons Inc, New York
[8] Goyal,S.K, Giri,B.C. Eur.Journal of Operation Research.134,
2001, pp:1-16.
[9] Mak, K.L. Computer and Industrial Engineering 6, 1982,
pp:309-17.
[10] Aggarwal,S.P. Opsearch ,15, 1978, pp:184-187.
[11] Shah,N.H. International Journal Of Production Economics
32, 1993, pp:77-82.
[12] Shah, Y.K, Jaiswal, M.L Opsearch 14,174-184.
[13] Shah, Y.K. AIIE Transactions 9, 108-112.
[14] Sachan, R.S. Journal of Operational Research society, 35(11)
1013-1019.
[15] Bahari-Kashani, H. Journal Of Operational Research society
40,1989, pp:75-81.
[16] Dave, U, Patel, L.K. Journal of Operational Research
society 32, 1981, pp:137- 143.
[17] Henery, R.J. Journal of Operational Research society 30(7),
1989, pp: 611-617.
[18] Covert, R.P, Philip, G.C. AIIE Transactions, 5, 1993, pp:323326.
[19] Dave, U ( 1989a ) Naval Research Logistics, 36, 507-514.
[20] Dave, U ( 1989b ) Journal of Operational Research society,
International Journal of Computer Science and Technology 359
IJCST Vol. 2, Issue 3, September 2011
ISSN : 2229-4333(Print) | ISSN : 0976-8491(Online)
40(9) , 827-830.
[21] Goyal, S.K, Giri, B.C. Naval Research Logistics, 47, 2000,
pp: 602-06.
[22] Hariga,M.A, Goyal,S.K. Journal Of Operational Research
society, 46(4), 1995, pp: 521-527.
[23] Ghare, P.M, Schradev,S.F. Journal Of Industrial Engineering
14, 1963, pp: 238-243.
[24] Hollier, R.H, Mak, K.L. International Journal of Production
Research, 21, 1983, pp: 813-826.
[25] Teng, J.T, Ouyang, L.Y, Chang, C.T. Eur Journal of Operation
Research, 93, 2005, pp: 476-489
360 International Journal of Computer Science and Technology
w w w. i j c s t. c o m