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A
BSTRACTIn this paper ,We developed an inventory model for deteriorating items with the effect of inflation over a fixed planning horizon. Deterioration rate is taken as time dependent. Two cases have been discussed; first is when shortages are not allowed and second is shortages are allowed with complete backlogging. A numerical assessment of the theoretical model has been done to illustrate the theory. The solution obtained has also been checked for sensitivity with the result that the model is found to be quite suitable and stable.
Key Words : Purchasing cost, Holding cost, Ordering quantity, Replenishment cost
1.
Introduction:
One of the important concerns of the management is to decide when and how much to order or to manufacture so that the total cost associated with the inventory system should be minimum. This is somewhat more important, when the inventory undergo decay or deterioration. Most of the researchers in inventory system were directed towards non-deteriorating products. However there are certain substances, whose utility do not remain same with the passage of time. Deterioration of these items plays an important role and items cannot be stored for a long time. Deterioration of an item may be defined as decay, evaporation, obsolescence, loss of utility or marginal value of an item that results in the decreasing usefulness of an inventory from the original condition.
The analysis of deteriorating inventory began with Ghare and Schrader (1963), who established the classical no-shortage inventory model with a constant rate of decay. Misra (1975-b) presented a production lot size model for an inventory system with deteriorating items with variable rate of deterioration while rate of production was finite. An order level inventory model for a system with constant rate of deterioration was presented by Shah and Jaiswal (1977). Roychowdhury and Chaudhuri (1983) formulated an order level inventory model for deteriorating items with finite rate of replenishment. Hollier and Mak (1983) developed inventory replenishment policies for deteriorating item with demand rate decreases negative exponentially and constant rate of deterioration. Dave (1986) presented an order level inventory model for deteriorating items. An EOQ model for deteriorating items with a linear trend in demand was formulated by Goswami and Chaudhuri (1991). An inventory model with exponential demand and constant rate of deterioration was proposed by Kishan and Mishra (1995). An order level inventory model for deteriorating items was proposed by Gupta and Agarwal (2000). Aggarwal and Jain (2001) presented an inventory model for exponentially increasing demand rate with time. The items were deteriorating at a constant rate and shortages were allowed. An order-level inventory problem for a deteriorating item with time dependent demand was presented by Khanra and Chaudhuri (2003). The inventory was assumed to deteriorate at a constant rate and shortages was not allowed. An inventory model for a deteriorating item over a finite planning horizon was presented by Sana et al. (2004). Deterioration rate was taken as constant fraction of the on-hand inventory. An order level inventory system for deteriorating items has been discussed by Manna and Chaudhuri (2006). Order level inventory systems with ramp type demand rate for deteriorating items were discussed by Panda et al. (2007). Shah, N.H. and Mishra, P. (2010) developed an order level inventory model for deteriorating items with stock dependent demand.
2.
Assumptions and Notations:
1. The replenishment rate is finite and lead time is zero.
2. A single item is considered over a prescribed period of H units of time. 3. The demand rate, α units per year, is known and constant.
4. ‘m’ denotes the number of replenishment periods during the time horizon H.
5. When inventory system allows shortages, m+1replenishment are made during the entire time horizon H. The last replenishment is made at time t=H just to replenish any shortages generated in the last cycle.
6. The rate of deterioration is dependent on time.
7. Two models are analyzed; Model I in which backlogging is not permitted and model II in which complete backlogging is permitted with a finite shortages cost C2 per unit per unit time.
Inventory Model for Deteriorating Items
with the Effect of inflation
Dr.Kapil Kumar Bansal
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8. C, the unit cost, C1, the inventory holding cost per unit per unit time and A, the ordering cost per order.
9. R, representing the discount rate net of inflation.
10. Tj is the total time that is elapsed up to and including the jth replenishment cycle (j=1,2,…..m), where Tm=H and
T0=0.
11. tj is the time at which the inventory level in the jth replenishment cycle (j=1,2,…..m).
3.
Model Formulation and Solution
We have discussed two models:
3.1. Model I: No Shortages Permitted
The total time horizon ‘H’ has been divided into ‘m’ equal parts of length T so that T=H/m. Hence, the reorder times over the planning horizon H are Tj=jT (j=0,1,2……..m-1). To start with, consider the inventory level I(t) during the first replenishment cycle. The inventory level is depleted by the effects of demand and deterioration. So, the variation of I(t) w.r.t. ‘t’ is governed by the following differential equation:
dI(t)
bt I(t)
dt
0 t T
…(1)
With the boundary condition
I T
0
, So, the solution of equation (1) is given by
I t
2
2 2
b
3 3T
t
T
t
T
t
2
6
2 2 t1 t b
2
,0 t T …(2)
Since there are ‘m’ replenishments in the entire time horizon H, the present values of the total replenishment are given by:
ince there are ‘m’ replenishments in the entire time horizon H, the present values of the total
R C m 1 RTj 1 j 0
A
e
RC
RH RH m
1 e
A
1 e
…(3)The present values of total purchasing costs are
P
C
m
RTj 1
j 1
C I(0)e
2 3 m T2 j 1
j 1
T T
C T b 1 0 e
2 6
RH 2 3 2 RH/ m 1 e T TC T b
2 6 1 e
…(4)
The present values of the holding costs during the first replenishment cycle are:
1
H
T Rt 1 0
C I t e dt
2
2 Rt
3 1
b
T e 1
C T T
2 6 R R
2
32 2 RT RT
2 2
b T
T Te e 1
1 T
2 6 R R R
Hence, the present values of the total holding costs during the entire time horizon H are given as
H
C m RTj 1 1 j 1
H e
2 3 2 Rt m 1 j 1 2 32 2 RT RT
RTj 1 2 2
b T
T e 1
C T
2 6 R R
b T
T Te e 1
1 T e
2 6 R R R
HC
2 3
2 RH/ m
1 2 3
b
H
H
H
e
1
C
m
2m
6m
R
R
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2
32 2 RT RH/ m RH
2 3 2 2 RH /m
b H
H H He e 1 1 e
1
m 2m 6m Rm R R 1 e
….(5)
Consequently, the present value of the total variable cost of the system during the entire time period H is given by:
TC m CRCPCH
2 3 2 3
2 2
1 2 3
T T H H H
A C T b C b
2 6 m 2m 6m
RH/m 2 3
2
1 2 3
e 1 H H H
C 1 b
R R 2m 2m 6m
RH RH/m RH/ m
2 2 RH/ m
1 e
He e 1
Rm R R 1 e
…(6)
Optimal solution procedure
If we treat the variable m as a continuous variable, and the second order derivative d2TC(m)/dm2 is positive. Consequently, TC(m) is the smallest positive integer m such that TC(m+1)≥TC(m). Using the optimal solution
procedure described above, we can find that the optimal order quantity is:
I t
2
2 2
b
3 3T
t
T
t
T
t
2
6
2
t
21
t
b
2
Numerical Example:
To illustrate all the results obtained in the present study, following numerical examples has been solved by the proposed method.
α=500 units, θ=150, A=240, C1=1.50 per unit per year, C=4 per unit, R=0.15, b=0.04, H=10 yr.
By using the solution procedure that we developed, the optimal values replenishment number, order quantity and total variable cost are m*=20, Q*=257.345 and TC*(m)=17654.64
Sensitivity Analysis:
The change in the values of parameters can take place due to uncertainties in any decision making situation. In order to examine the implications of these changes, the sensitivity analysis will be of great help in decision making.
Parameters
Variation of the different parameters
Percentage -50 -25 25 50
‘α’ Q 0.863 0.924 1.123 1.210
TC 0.679 0.821 1.172 1.342
‘θ’ Q 1.024 1.008 0.946 0.879
TC 0.912 0.973 1.004 1.018
A Q 0.906 0.922 1.126 1.130
TC 0.821 0.863 1.128 1.147
C Q 1.136 1.084 0.937 0.892
TC 0.679 0.891 1.129 1.154
C1 Q 1.044 1.032 0.986 0.954
TC 0.759 0.875 1.012 1.023
R Q 1.052 1.047 0.980 0.978
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3.2. Model II: Shortages permitted with complete backlogging
Suppose that the planning horizon H is divided into m equal parts of length T=H/m. Hence, the reorder times over the planning horizon H are Tj=jT (j=0,1,2…….m). We further assume that the period for which there is no-shortages in each interval [jT, (j+1)T] is a fraction of the scheduling period T and is equal to KT (0<K<1). Shortages occur at time tj=(K+j-1)T, (j=1,2…..m).
Let us consider the level of inve
ntory at time t, I(t), during
the first replenishment cycle, i.e. 0≤t≤T. This inventory is depleted due to demand and deterioration.
Therefore, the variation of I(t) w.r.t. time is governed by the following differential equation:
1 dI(t)
bt I t ,0 t t
dt …(10)
Using the condition
I(t)
0
, the solution of the equation is:I(t)
2
2 2 3 3
1 1 1
b
t t t t t t
2 6
2
1 T
1 t b ,0 t t
2
…(11)
T=0
t1=KH/m
T1=H/m
t1= (K+1)H/m
Time
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As the level of shortages, S(t) during the first replenishment cycle may be represented by the following differential equation, since demand (backlogging) rate is constant.
dS t
dt
1
t t T …(12)
With the boundary conditions
S t
1
0
, the solution of the equation is:
S t
tt , t1
1 t T …(13)Since there are m+1 replenishments in the entire time horizon H, the present values of the total replenishment costs are given by R C m RTj 1 j 0
A e
R
C
RH/m RH RH/ m e e A e 1 …(14)Let
I
1 be the initial inventory level and letS
1 be the maximum shortage quantity during the first replenishment cycle. Using equation (11) and (13), we get:1
I
2
2 3 1
1 1
t 1
t b t
2 6
2
2 22 1 b
KH K H KH
1
2 m 6 m m
…(15) 1S
Tt1
H KH m m 1
S
1 K
Hm
…(16)
Because shortages during the first replenishment cycle should be backordered during the next replenishment cycle and shortages during the last cycle is replenished at time Tm=H. Therefore, the present values of total purchasing cost
during the entire time horizon H are:
P
C
m m
RTj 1 RTj
1 1
j 1 j 1
C I e S e
RH 2 2 m 2 RH/m j 1 1 eKH 1 K H KH
C 1 b
2 m 6 m m 1 e
RH RH/m 1 e H 1 Km e 1
…(17)
The present values of holding costs during the first replenishment cycle are:
1
HC 1 t1
Rt0
C I t e dt
2
Rt2 1
3 1
1 1 1
b
t
e
1
C
t
t
2
6
R
R
2
3 Rt t2 2 1 1
1
1 1
1 2 2
b
t
t
t e
e
1
1
t
2
6
R
R
R
Hence the present values of the total holding costs during the entire time horizon H are given as: Now, H C m RTj 1 1 j 1
HC e
2
32 RH/m
1 2 3
b
H
H
H
e
1
C
m
2m
6m
R
R
2
32 2 RT RH/m
2 3 2 2
b
H
H
H
He
e
1
1
m
2m
6m
Rm
R
R
RH RH / m1 e 1 e …. (18)
Present values of the total shortage costs during the first replenishment cycle are:
1
SC T1
Rt2 t1
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R (K 1)H/m
RH/m 22
C H
R K 1 e 1 e
R m
Hence, the present values of the total shortages costs during the entire time horizon H are:
CS
m
RTj 1 1 j 1
SC e
RH R (1 K )H/ m RH/m
2
2 RH/m
1 e
C HR
K 1 e 1 e
R m e 1
…(19)
Consequently, the present value of the total variable cost of the system during the entire time horizon H is:
TC m, K CRCPCHCS
2 2 2
2
KH 1 K H KH H
AD C 1 b E C 1 K F
2 m 6 m m m
2 2
2 RKH/m
1 2
HK 1 H K KH
C 1 b 1 e
2m 6 m mR
2 2 2 3 3
2 RKH/m
2 3
HK H K H K H
1 b e
m 2m 6m mR
RH/m 2 2
e 1
E
R R
R (1 K) H/m 2
2
C HR
K 1 e 1 F
R m
…(20)
Where
D
RH/m RH
RH/m
e
e
e
1
E
RH
RH/m
1 e
1 e
F
RH
RH/m
1 e
e
1
Optimal solution procedure:
The present value of total variable cost function TC(m,K) is a function of two variables K and m where K is a continuous variable and m is a discrete variable. For a given value of m, the necessary condition for TC(m,K) to be minimized is dTC(m, K)
0
dK
and also shows that
2 2 d TC(m, K)
0
dK
Numerical Example:
To illustrate all the results obtained in the present study, following numerical examples has been solved by the proposed method.
α=500 units, θ=0.05, A=240, C1=1.50 per unit per year, C=4 per unit, R=0.15, C2=3.5 per unit per year, H=10 yr
By using the solution procedure that we developed, the optimal values replenishment number, order quantity and total variable cost are m*=20, Q*=175.673 and TC*(m)=16832.728
Sensitivity Analysis:
In order to examine the implications of these changes, the sensitivity analysis will be of great help in decision making.
Parameters Variation of the different parameters
Percentage -50 -25 25 50
‘α’ Q 0.857 0.956 1.115 1.210
TC 0.739 0.821 1.119 1.216
‘θ’ Q 1.016 1.010 0.979 0.862
TC 0.935 0.989 1.005 1.011
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TC 0.841 0.889 1.115 1.132
C Q 1.234 1.013 0.977 0.901
TC 0.799 0.961 1.124 1.143
C1 Q 1.124 1.025 0.956 0.811
TC 0.704 0.859 1.016 1.053
R Q 1.045 1.028 0.964 0.950
TC 1.221 1.193 0.888 0.713
C2 Q 0.965 0.978 1.114 1.217
TC 0.842 0.980 1.028 1.045
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In this chapter, an inventory system has been developed with time deteriorating items over a finite planning horizon. To make our study more suitable to present-day market, we have done our research in an inflationary environment. The study of inflation, gives a viability that makes it more pragmatic and acceptable. The setup that has been chosen boasts of uniqueness in terms of the conditions under which the model has been developed. Even till now, most of the researchers have been either completely ignoring the decay factor or are considering a constant rate of deterioration in the inventory model which is not practical. We have considered an inventory with deterioration rate increasing linearly with time.
The problem has been formulated analytically and has been used to arrive at the optimal solution. Numerical assessment and sensitivity analysis are implemented to illustrate the theoretical model. Hence, from the economical point of view, the proposed model will be useful to the business situations in the present context as it gives better inventory control system.
The model presents ample scope for further extension and development. This study may be extended to multi-items. Another possible extension of this study may consider the assumption of the stochastic demand and deterioration rate.
References:
[1.] Aggarwal, S.P. and Jain, V. (2001): Optimal inventory management for exponentially increasing demand with deterioration. International Journal of Management and Systems (I.J.M.S.), 17(1), 1-10.
[2.] Dave, U. (1986): An order level inventory model for deteriorating items with variable instantaneous demand and discrete opportunities for replenishment. Opsearch 23, 244-249.
[3.] Gupta, P.N. and Aggarwal, R.N. (2000): An order level inventory model with time dependent deterioration. Opsearch, 37(4), 351-359.
[4.] Ghare, P.M. and Schrader, G.P. (1963): A model for exponentially decaying inventory. Journal of Industrial Engineering (J.I.E.), 14, 228-243.
[5.] Goswami, A. and Chaudhuri, K.S. (1991): An EOQ model for deteriorating items with a linear trend in demand. J.O.R.S., 42(12), 1105-1110.
[6.] Hollier, R.H. and Mak, K.L. (1983): Inventory replenishment policies for deteriorating items in a declining market. I.J.P.E., 21, 813-826
[7.] Khanra, S. and Chaudhuri, K.S. (2003): A note on an order-level inventory model for a deteriorating item with time-dependent quadratic demand. C.O.R., 30, 1901-1916.
[8.] Kishan, H. and Mishra, P.N. (1995): An inventory model with exponential demand and constant deterioration with shortages. Indian Journal of Mathematics, 37(3), 275-279.
[9.] Misra, R.B. (1975-b): Optimum production lot size model for a system with deteriorating inventory. I.J.P.E., 13, 495-505.
[10.] Manna, S.K. and Chaudhuri, K.S. (2006): An EOQ model with ramp type demand rate, time dependent deterioration rate, unit production cost and shortages. E.J.O.R., 171, 557-566.
[11.] Panda, S., Saha, S. and Basu, M. (2007): An EOQ model with generalized ramp-type demand and Weibull distribution deterioration. Asia Pacific Journal of Operational Research, 24(1), 1-17.
[12.] Roychowdhury, M. and Chaudhuri, K.S. (1983): An order level inventory model for deteriorating items with finite rate of replenishment. Opsearch, 20, 99-106.
[13.] Shah, Y.K. and Jaiswal, M.C. (1977): An order level inventory model for a system with constant rate of deterioration. Opsearch, 14(3), 174-184.
[14.] Sana, S., Goyal, S.K. and Chaudhuri, K.S. (2004): A production-inventory model for a deteriorating item with trended demand and shortages. E.J.O.R., 157, 357-371.
