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An inventory model under inflation for stock dependent consumption rate items
Engineering Costs and Production Economics, 19
319
( 1990) 319-383
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
AN INVENTORY MODEL UNDER INFLATION FOR STOCK DEPENDENT CONSUMPTION RATE ITEMS Prem Vrat and G. Padmanabhan Department of Mechanical Engineering, Indian Institute of Technology, New Delhi (India)
ABSTRACT
Under inflation the assumption of constant unit price is not valid. The tendency in injlationary environment is to buy more in order to reduce the total system cost, which may be true in certain situations but it is not true when consumption rate of items is dependent on initial stock level since buying more quantity under injlationary environment leads to more consumption resulting in higher total system cost. The
model developed in this paper helps to determine optimum ordering quantity for stock dependent consumption rate items under injlationary environment with injinite replenishment rate without permitting shortages. The results are illustrated with a numerical example.
INTRODUCTION
decisions. Brahmbhatt [ 2 ] developed an EOQ model under variable rate of inflation and mark-up prices. Recently Gupta and Vrat [ 3 ] have developed a multi-item inventory model with a resource constraint system under variable rate of inflation. For certain commodities, such as consumer goods, food grains, stationery items, consumables etc., the consumption rate may depend on the size of the replenishment and may increase as the ordering quantity increases. It may be a behavioural trait born out of complacency on the part of the consumer to consume at a rate more than the normal rate of consumption owing to the higher value of order size and hence, the initial stock level. This phenomenon has been termed as “stock dependent consumption rate” by Gupta [ 41, Gupta and Vrat [ 3,5,6]. They have studied the effect of this phenomenon on various ordering policy decisions.
The assumptions that unit price, ordering and carrying costs are constant over the planning period while designing an inventory policy may not hold under inflationary environment. Buzacott [ 1 ] developed deterministic inventory models under inflation for two cases, first when the price is subject to the same inflation rate as costs and in the second case when the price is dependent on ordering policy assuming constant rate of inflation which means that cost C(t) at time t becomes C( t+&) at time ( t + lit) through inflation, where c(t+&)=C(t)+KC(t)Gt
dC(t)
=s- p=KC(t) as~t-o
dt
which has solution C(t) = CoeKf, where C, is cost at time zero. Later, a few papers have been published considering aspects of inflation in inventory 0167-188X/90/$03.50
0 1990 Elsevier Science Publishers B.V.
380
The general tendency in inflationary environment is to buy higher quantities to affect savings in material cost which ultimately results in reduced total system cost. But this may not be true in the case of items having stock dependent consumption rate. The model developed in this paper will take care of both inflation and stock dependent consumption rate to determine optimum ordering quantity under conditions cited above. The expression obtained for the optimum ordering quantity is solved using the Newton-Raphson method. Finally, a numerical example is included to study the impact of inflation and stock dependent consumption rate on the ordering quantity. Sensitivity analysis of the model is also reported.
= length of planning horizon = ordering quantity Q K =constant rate of inflation (Rs./ Rs./unit time) =unit price of an item at time t C(t) = CoeKT, where C, is unit price at time zero = ordering cost per order at time t A(t) = AoeKz,where A,, is ordering cost at time zero =inventory carrying charge exi pressed as a fraction of unit cost/ unit/unit time H( t,t + w ) = the inventory carrying cost for an item bought at time t and held in stock until time ( t + w )
DEVELOPMENT
Mathematical
OF THE MODEL
L
=iC(t)w.
Assumptions
Total system cost during the planning period
The following assumptions are made: ( 1 )The unit price is subject to the same inflation rate as other inventory related costs, thus implying that the ordering quantity can be found by minimizing the total system cost over a planning period. (2) Rate of inflation is constant. (3) Rate of replenishment is infinite. (4) Shortages are not permitted and lead time is zero. (5 ) Stock dependent consumption rate D is assumed to follow the function D = (x+ PQ’, where (x, p, r are positive constants and Q is ordering quantity. (6 )The inventory carrying cost for an item bought at time t and held in stock until time ( t+ w ) is assumed to be proportional to investment on the item at time t and the period w for which it is held in stock. Notation
The notation follows:
modelling
L is the sum of ordering costs, carrying costs and material costs. Assume that L = m T, where m is an integer and T is a constant interval of
time between replenishments. during the period (0,L) A,+A(T)+A(2T)+
***SA((m-I)T)=A,-----
in this paper is as
(eKL- 1) ( eK*- 1)’
Carrying cost per cycle 7 T Dh(t,t+w) dw= DiC(t)wdw. 5 5cl 0 Carrying costs during the period (0,L) ?-
2T
DiCowdw+ s 0
s
dw+.*.
DiC(T)w
T mT
DiC((m-l)T)wdw
+ s
(rn-,,I-
nT,nT+
adopted
Ordering costs
W) dw
=iDiT2Co(eKL-l)/(eKT-1).
381 Material costs during the period (0,L)
SPECIAL CASES
QC,+QC(T)+QC(ZT)+---
Case 1
+QC((m-1)7J
=CoDT(eKL-l)/(eKT-1)
Hence, the total system cost over the planning horizon L is given by
When K= 0, Eqn. (4) for the optimum value of Q becomes Q2Coi-2Aocx+2Aoj?QT(r-1)+2j?rCoQT+’=0.
TC(L,T)
= [A, + fDiT%‘o +DTColm.
eKL- 1 (1)
Substituting the quadratic approximation of (l), eKT- 1 =KT+iK2T2andDT=QinEqn. we get
(5)
Eqn. (5) is the same as given by Gupta and Vrat [ 5 ] when the costs are constant over the planning period. Case 2
When p=O, i.e., D=a, Eqn. (4) for the optimum ordering quantity can be written as In the present model consumption rate D is assumed to be stock dependent following relationship: D= o + PQ’ substituting this in Eqn. (2 ) gives:
(6)
Case 3
When K= 0 and p= 0, Eqn. (4) for the optimum ordering quantity reduces to the same as the classical EOQ formula.
TC(LQ) C,,iQ* A”+Qco+2(cx+PQT)
Q*C,i-2aAo-K(2QAo+Q2Co)=0.
1
2(eKL-l)(cx+/?QT)* 2(a+PQ’)KQ+K*Q” (3)
The total cost in Eqn. ( 3 ) is convex and hence, to determine optimum ordering quantity, it can be differentiated with respect to Q and equated to zero:
NUMERICAL
EXAMPLE
The values follows:
of various
variables
a=200.00
/I= 1.00
i=O.20
units
L= 1 year
C,=Rs.
1.00
A,=Rs.
are as
5.00
For 7= 0.7 and K= 0.10 the optimum ordering quantity is determined by the NewtonRaphson method using Eqn. (4) as Q2Coi+4(~+PQ')(Ao+Qco,+QCoi 2(a+PQ')Q
-
~(Ao+QCO)((~+PQ’(~-~)+KQ)
Q
+c
0
Q= 64.17 units.
The optimum total system cost is determined using Eqn. ( 3 ).
=o
(4)
This equation can be solved for the optimum value of ordering quantity Q using numerical methods such as the Newton-Raphson method.
TC=Rs.
250.67.
To study the effect of inflation parameter K on Q and TC for various values of r, values considered for inflation rate Kare: 0.0,0.025,0.05, 0.075, 0.10, 0.125, 0.15. The optimum value
382 TABLE
1
0
0.3
0.5
0.7
1.0
TC
100.00 220.00
94.97 224.15
81.08 230.00
57.17 247.67
30.15 271.33
tion K, the parameter r is varied from 0 to 1.O. The results for Q and TC are exhibited in Table I. Sensitivity analysis of the model with respect to ordering cost, unit price, inventory carying cost fraction, inflation rate and stock dependent consumption rate parameters p and r on ordering quantity and total system cost has been reported in Table 2.
0.025
Q TC
107.62 221.33
101.59 225.62
85.25 231.75
59.20 243.90
30.33 274.30
ANALYSIS
0.05
Q TC
117.15 222.59
109.73 227.02
90.09 233.47
60.74 246.14
30.50 277.32
0.075
Q TC
129.53 223.72
120.07 228.32
95.84 235.15
62.40 248.40
30.68 280.38
0.10
Q TC
146.51 224.70
133.82 229.50
102.79 236.77
64.17 250.67
30.89 283.49
0.125
Q TC
171.84 225.43
153.40 230.50
111.47 238.33
66.10 252.96
3 1.05 286.64
0.15
Q TC
215.56 225.79
184.59 231.21
122.72 239.79
68.19 255.25
31.24 289.85
Effect of K and mum total system Inflation rate K 0
7 on
optimum
Stock dependent
Q
ordering
quantity
and
opti-
cost consumption
parameter
T
TABLE 2 Sensitivity
of Q and TC with respect to .4,,, C,, i, K, /3, T
Variation parameter
change in parameter
(%)
-10
-5
0
5
10
A0
Q TC
103.75 226.05
106.78 226.54
109.73 227.02
112.61 227.48
115.44 227.94
(;
Q TC
116.05 205.23
112.77 216.13
109.73 227.02
106.92 237.89
104.31 248.75
i
Q TC
117.19 225.87
113.27 226.45
109.73 227.02
106.51 227.56
103.56 228.10
K
Q TC
107.96 226.75
108.83 226.88
109.73 227.02
110.65 227.15
111.60 227.29
P
Q TC
110.43 226.58
110.08 226.80
109.73 227.02
109.39 227.24
109.04 227.46
r
Q 111.38 ‘TC 226.43
110.59 226.72
109.73 227.02
108.80 227.34
107.78 221.68
Allparameters
Q TC
110.59 204.36
109.73 227.02
108.80 250.90
107.78 276.18
11 1.38 182.95
of Q and TC are determined and exhibited in Table 1. Similarly, to study the behaviour of r on ordering quantity for a particular rate of infla-
OF RESULTS
The results obtained for the illustrative example provide certain important insights about the problem studied. Some of the inferences derived are as follows: ( 1) As the inflation rate increases, ordering quantity increases and total system cost also increases. (2)For a particular rate of inflation, as the value of the stock dependent consumption parameter r increases, the ordering quantity decreases and the total system cost increases. (3)The effect of stock dependent consumption rate is more pronounced on ordering quantity for items having higher inflation rate. (4)The effect of inflation rate is less pronounced on ordering quantity for items with higher stock dependent consumption rate. From the sensitivity analysis it is seen that Q is more sensitive to &, C,, i than K, j?, r, whereas total system cost is more sensitive to C, than &, i, K, /I, z. Further, it is seen that overestimation is costlier than underestimation of parameters. CONCLUSIONS
As a first step in studying the interaction of stock dependent consumption rate and inflation, the model is developed with infinite replenishment rate and without shortages. This model helps to avoid excess consumption of material in the case of stock dependent con-
383
sumption which results in reduced total system cost. It is noticed from the results that the stock dependent consumption phenomenon cannot be ignored in the case of a highly inflationary environment. The model can be further enriched by considering different inflation rates for different inventory related costs. A more general model could be developed for items having probabilistic stock dependent consumption rate along with inflation and perishability. REFERENCES 1 Buzacott, J.A., 1975. Economic order quantities with inflation. Oper. Res. Q., 26: 553-558.
2 Brahmbhatt, A.C., 1982. Economic order quantity under variable rate of inflation and mark-up prices. Productivity, 23: 127-130. 3 Gupta, R. and Vrat, P., 1986. Inventory model with multiitems under constraint systems for stock dependent consumption rate. Proc. of XIX Annual Convention of Oper. Res. Sot. India, 2: 579-609. Also in Oper. Res., 24 ( 1): 41-42. 4 Gupta, R., 1987. Modelling and analysis of inventory systems under inflation and stock dependent consumption. Ph.D Thesis, Indian Inst. Technol., New Delhi, India. 5 Gupta, R. and Vrat, P., 1986. Inventory models for stock dependent consumption rate. Oper. Res., 23: 19-24. 6 Gupta, R. and Vrat, P., 1986. Inventory model with quantity discounts under inflation. Indian J. Manage. Syst., 2 (1): 19-24.
319
( 1990) 319-383
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
AN INVENTORY MODEL UNDER INFLATION FOR STOCK DEPENDENT CONSUMPTION RATE ITEMS Prem Vrat and G. Padmanabhan Department of Mechanical Engineering, Indian Institute of Technology, New Delhi (India)
ABSTRACT
Under inflation the assumption of constant unit price is not valid. The tendency in injlationary environment is to buy more in order to reduce the total system cost, which may be true in certain situations but it is not true when consumption rate of items is dependent on initial stock level since buying more quantity under injlationary environment leads to more consumption resulting in higher total system cost. The
model developed in this paper helps to determine optimum ordering quantity for stock dependent consumption rate items under injlationary environment with injinite replenishment rate without permitting shortages. The results are illustrated with a numerical example.
INTRODUCTION
decisions. Brahmbhatt [ 2 ] developed an EOQ model under variable rate of inflation and mark-up prices. Recently Gupta and Vrat [ 3 ] have developed a multi-item inventory model with a resource constraint system under variable rate of inflation. For certain commodities, such as consumer goods, food grains, stationery items, consumables etc., the consumption rate may depend on the size of the replenishment and may increase as the ordering quantity increases. It may be a behavioural trait born out of complacency on the part of the consumer to consume at a rate more than the normal rate of consumption owing to the higher value of order size and hence, the initial stock level. This phenomenon has been termed as “stock dependent consumption rate” by Gupta [ 41, Gupta and Vrat [ 3,5,6]. They have studied the effect of this phenomenon on various ordering policy decisions.
The assumptions that unit price, ordering and carrying costs are constant over the planning period while designing an inventory policy may not hold under inflationary environment. Buzacott [ 1 ] developed deterministic inventory models under inflation for two cases, first when the price is subject to the same inflation rate as costs and in the second case when the price is dependent on ordering policy assuming constant rate of inflation which means that cost C(t) at time t becomes C( t+&) at time ( t + lit) through inflation, where c(t+&)=C(t)+KC(t)Gt
dC(t)
=s- p=KC(t) as~t-o
dt
which has solution C(t) = CoeKf, where C, is cost at time zero. Later, a few papers have been published considering aspects of inflation in inventory 0167-188X/90/$03.50
0 1990 Elsevier Science Publishers B.V.
380
The general tendency in inflationary environment is to buy higher quantities to affect savings in material cost which ultimately results in reduced total system cost. But this may not be true in the case of items having stock dependent consumption rate. The model developed in this paper will take care of both inflation and stock dependent consumption rate to determine optimum ordering quantity under conditions cited above. The expression obtained for the optimum ordering quantity is solved using the Newton-Raphson method. Finally, a numerical example is included to study the impact of inflation and stock dependent consumption rate on the ordering quantity. Sensitivity analysis of the model is also reported.
= length of planning horizon = ordering quantity Q K =constant rate of inflation (Rs./ Rs./unit time) =unit price of an item at time t C(t) = CoeKT, where C, is unit price at time zero = ordering cost per order at time t A(t) = AoeKz,where A,, is ordering cost at time zero =inventory carrying charge exi pressed as a fraction of unit cost/ unit/unit time H( t,t + w ) = the inventory carrying cost for an item bought at time t and held in stock until time ( t + w )
DEVELOPMENT
Mathematical
OF THE MODEL
L
=iC(t)w.
Assumptions
Total system cost during the planning period
The following assumptions are made: ( 1 )The unit price is subject to the same inflation rate as other inventory related costs, thus implying that the ordering quantity can be found by minimizing the total system cost over a planning period. (2) Rate of inflation is constant. (3) Rate of replenishment is infinite. (4) Shortages are not permitted and lead time is zero. (5 ) Stock dependent consumption rate D is assumed to follow the function D = (x+ PQ’, where (x, p, r are positive constants and Q is ordering quantity. (6 )The inventory carrying cost for an item bought at time t and held in stock until time ( t+ w ) is assumed to be proportional to investment on the item at time t and the period w for which it is held in stock. Notation
The notation follows:
modelling
L is the sum of ordering costs, carrying costs and material costs. Assume that L = m T, where m is an integer and T is a constant interval of
time between replenishments. during the period (0,L) A,+A(T)+A(2T)+
***SA((m-I)T)=A,-----
in this paper is as
(eKL- 1) ( eK*- 1)’
Carrying cost per cycle 7 T Dh(t,t+w) dw= DiC(t)wdw. 5 5cl 0 Carrying costs during the period (0,L) ?-
2T
DiCowdw+ s 0
s
dw+.*.
DiC(T)w
T mT
DiC((m-l)T)wdw
+ s
(rn-,,I-
nT,nT+
adopted
Ordering costs
W) dw
=iDiT2Co(eKL-l)/(eKT-1).
381 Material costs during the period (0,L)
SPECIAL CASES
QC,+QC(T)+QC(ZT)+---
Case 1
+QC((m-1)7J
=CoDT(eKL-l)/(eKT-1)
Hence, the total system cost over the planning horizon L is given by
When K= 0, Eqn. (4) for the optimum value of Q becomes Q2Coi-2Aocx+2Aoj?QT(r-1)+2j?rCoQT+’=0.
TC(L,T)
= [A, + fDiT%‘o +DTColm.
eKL- 1 (1)
Substituting the quadratic approximation of (l), eKT- 1 =KT+iK2T2andDT=QinEqn. we get
(5)
Eqn. (5) is the same as given by Gupta and Vrat [ 5 ] when the costs are constant over the planning period. Case 2
When p=O, i.e., D=a, Eqn. (4) for the optimum ordering quantity can be written as In the present model consumption rate D is assumed to be stock dependent following relationship: D= o + PQ’ substituting this in Eqn. (2 ) gives:
(6)
Case 3
When K= 0 and p= 0, Eqn. (4) for the optimum ordering quantity reduces to the same as the classical EOQ formula.
TC(LQ) C,,iQ* A”+Qco+2(cx+PQT)
Q*C,i-2aAo-K(2QAo+Q2Co)=0.
1
2(eKL-l)(cx+/?QT)* 2(a+PQ’)KQ+K*Q” (3)
The total cost in Eqn. ( 3 ) is convex and hence, to determine optimum ordering quantity, it can be differentiated with respect to Q and equated to zero:
NUMERICAL
EXAMPLE
The values follows:
of various
variables
a=200.00
/I= 1.00
i=O.20
units
L= 1 year
C,=Rs.
1.00
A,=Rs.
are as
5.00
For 7= 0.7 and K= 0.10 the optimum ordering quantity is determined by the NewtonRaphson method using Eqn. (4) as Q2Coi+4(~+PQ')(Ao+Qco,+QCoi 2(a+PQ')Q
-
~(Ao+QCO)((~+PQ’(~-~)+KQ)
Q
+c
0
Q= 64.17 units.
The optimum total system cost is determined using Eqn. ( 3 ).
=o
(4)
This equation can be solved for the optimum value of ordering quantity Q using numerical methods such as the Newton-Raphson method.
TC=Rs.
250.67.
To study the effect of inflation parameter K on Q and TC for various values of r, values considered for inflation rate Kare: 0.0,0.025,0.05, 0.075, 0.10, 0.125, 0.15. The optimum value
382 TABLE
1
0
0.3
0.5
0.7
1.0
TC
100.00 220.00
94.97 224.15
81.08 230.00
57.17 247.67
30.15 271.33
tion K, the parameter r is varied from 0 to 1.O. The results for Q and TC are exhibited in Table I. Sensitivity analysis of the model with respect to ordering cost, unit price, inventory carying cost fraction, inflation rate and stock dependent consumption rate parameters p and r on ordering quantity and total system cost has been reported in Table 2.
0.025
Q TC
107.62 221.33
101.59 225.62
85.25 231.75
59.20 243.90
30.33 274.30
ANALYSIS
0.05
Q TC
117.15 222.59
109.73 227.02
90.09 233.47
60.74 246.14
30.50 277.32
0.075
Q TC
129.53 223.72
120.07 228.32
95.84 235.15
62.40 248.40
30.68 280.38
0.10
Q TC
146.51 224.70
133.82 229.50
102.79 236.77
64.17 250.67
30.89 283.49
0.125
Q TC
171.84 225.43
153.40 230.50
111.47 238.33
66.10 252.96
3 1.05 286.64
0.15
Q TC
215.56 225.79
184.59 231.21
122.72 239.79
68.19 255.25
31.24 289.85
Effect of K and mum total system Inflation rate K 0
7 on
optimum
Stock dependent
Q
ordering
quantity
and
opti-
cost consumption
parameter
T
TABLE 2 Sensitivity
of Q and TC with respect to .4,,, C,, i, K, /3, T
Variation parameter
change in parameter
(%)
-10
-5
0
5
10
A0
Q TC
103.75 226.05
106.78 226.54
109.73 227.02
112.61 227.48
115.44 227.94
(;
Q TC
116.05 205.23
112.77 216.13
109.73 227.02
106.92 237.89
104.31 248.75
i
Q TC
117.19 225.87
113.27 226.45
109.73 227.02
106.51 227.56
103.56 228.10
K
Q TC
107.96 226.75
108.83 226.88
109.73 227.02
110.65 227.15
111.60 227.29
P
Q TC
110.43 226.58
110.08 226.80
109.73 227.02
109.39 227.24
109.04 227.46
r
Q 111.38 ‘TC 226.43
110.59 226.72
109.73 227.02
108.80 227.34
107.78 221.68
Allparameters
Q TC
110.59 204.36
109.73 227.02
108.80 250.90
107.78 276.18
11 1.38 182.95
of Q and TC are determined and exhibited in Table 1. Similarly, to study the behaviour of r on ordering quantity for a particular rate of infla-
OF RESULTS
The results obtained for the illustrative example provide certain important insights about the problem studied. Some of the inferences derived are as follows: ( 1) As the inflation rate increases, ordering quantity increases and total system cost also increases. (2)For a particular rate of inflation, as the value of the stock dependent consumption parameter r increases, the ordering quantity decreases and the total system cost increases. (3)The effect of stock dependent consumption rate is more pronounced on ordering quantity for items having higher inflation rate. (4)The effect of inflation rate is less pronounced on ordering quantity for items with higher stock dependent consumption rate. From the sensitivity analysis it is seen that Q is more sensitive to &, C,, i than K, j?, r, whereas total system cost is more sensitive to C, than &, i, K, /I, z. Further, it is seen that overestimation is costlier than underestimation of parameters. CONCLUSIONS
As a first step in studying the interaction of stock dependent consumption rate and inflation, the model is developed with infinite replenishment rate and without shortages. This model helps to avoid excess consumption of material in the case of stock dependent con-
383
sumption which results in reduced total system cost. It is noticed from the results that the stock dependent consumption phenomenon cannot be ignored in the case of a highly inflationary environment. The model can be further enriched by considering different inflation rates for different inventory related costs. A more general model could be developed for items having probabilistic stock dependent consumption rate along with inflation and perishability. REFERENCES 1 Buzacott, J.A., 1975. Economic order quantities with inflation. Oper. Res. Q., 26: 553-558.
2 Brahmbhatt, A.C., 1982. Economic order quantity under variable rate of inflation and mark-up prices. Productivity, 23: 127-130. 3 Gupta, R. and Vrat, P., 1986. Inventory model with multiitems under constraint systems for stock dependent consumption rate. Proc. of XIX Annual Convention of Oper. Res. Sot. India, 2: 579-609. Also in Oper. Res., 24 ( 1): 41-42. 4 Gupta, R., 1987. Modelling and analysis of inventory systems under inflation and stock dependent consumption. Ph.D Thesis, Indian Inst. Technol., New Delhi, India. 5 Gupta, R. and Vrat, P., 1986. Inventory models for stock dependent consumption rate. Oper. Res., 23: 19-24. 6 Gupta, R. and Vrat, P., 1986. Inventory model with quantity discounts under inflation. Indian J. Manage. Syst., 2 (1): 19-24.