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A simple heuristic method for determining economic order interval for linear demand
Engineering Costs and Production Economics, 11 (1987) 53-57 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
53
Short Communication
A SIMPLE HEURISTIC METHOD FOR DETERMINING ECONOMIC ORDER INTERVAL FOR LINEAR DEMAND S.K. Goyal Department of Quantitative Methods,
Concordia University,
1455 de Maisonneuve
Blvd. West, Montreal, Quebec H3G lM8
(Canada)
ABSTRACT
In this paper a simple heuristic method is suggested for determining the economic ordering policy for an item having linear trend in
demand. The performance of the method is compared with other known methods.
INTRODUCTION
over the time horizons are more than four. In this paper a method is given which uses the heuristic method of Goyal et al. [ 91 and Ritchic’s [ lo] observation, that the stock holding cost after several orders is equal to the fixed ordering cost.
The classical square root formula for determining the economic order quantity was established almost seventy years ago but it was Donaldson [ 1 ] who established the following optimality condition for linear trend in demand: “the quantity ordered at a replenishment point must be the product of the previous replenishment interval and the current rate of demand”. Donaldson used this optimality condition to determine the replenishment policy for a given number of replenishments. Buchanan [ 21 and Henry [ 31 have also suggested alternative solution methods for determining the replenishment intervals for linearly increasing demand. Silver [ 41, Ritchie [ 51 and Goyal [ 61 have suggested heuristic methods for determining the economic replenishment intervals. Phelps [ 71 and Mitra et al. [ 81 have suggested approaches for determining the constant economic replenishment intervals; however, their approach provides much higher cost solutions when the number of replenishments
0167-188X/87/$03.50
0 1987 Elsevier Science Publishers B.V.
ASSUMPTIONS
AND NOTATIONS
In deriving the total cost model we are making the following assumptions: (1) Rate of demand increases linearly with time. (2) Shortages are not allowed. (3) Time horizon is finite. (4) Minimization of total cost in the objective. The following notations are being used for developing the total cost model: r stock holding cost per item per unit of time S fixed cost of a replenishment a rate of demand at time zero increase in demand per unit of time b a+bt rate of demand at time t time horizon H n total number of replenishments
W total cost For the ith replenishment: rate of demand at the time of making ai the replenishment timing of the replenishment ri replenishment interval replenishment quantity the ith w( ti) the total cost during replenishment Note that al =a, T, =O. The total cost, the sum of the fixed cost of replenishment and the stock holding cost, during the time interval covered by the ith replenishment is given b$
Li
W (ti) = S + r+ (ai +$ bti)
(1)
See Silver [ 41 for the above model. It may be pointed out that: i-l
ai=a+bTi=a+bC
tj j=O
(2)
TABLE 1 Data for the sample examples Example 1 9 2 900t 1 9
S r
Demand pattern H M
Example 2 9 0.1 6+t 11 180
gests transforming the origin to t = -x. From this new origin, the optimal ordering policy can be determined. The approach works well if one of the orders is placed exactly at t = x; i.e. t = 0 for the original problem. In case no order is placed exactly at time x then Ritchie suggests alternative approaches. According to the optimality condition of Donaldson [ 1 ] the order quantity in the last order, is given by:
Qn = (a+bW
K-C-d
(4)
If n orders are placed during the time horizon then the total cost is given by:
where T, - T,_ 1= (u - 1) th order interval. However, the demand during the last interval equals,
W=nS+ii$ltf(ai+$bti)
Q,, =a,,& +y
(3)
Unfortunately the computational effort required for determining the optimal ordering policy by using the method developed by Donaldson [ 1] is by no means simple. Due to this reason a number of heuristic methods have been proposed by Silver [ 41, Ritchie [ 51, Goyal [6],Mitra [8],Goyaletal. [9]. Ritchie [ lo] observed that for an optimal ordering policy, the cost of stock holding after a few iterations is almost equal to the fixed ordering cost and suggested a simple approach for determining almost optimal ordering policy. Instead of bounding the demand interval O-H, he suggests extending the horizon, so that it no longer affects the ordering times. This simplifies the computations for determining the optimal ordering policy. His approach works well when the rate of demand, a, at time zero is 0. However when a = bx, then he sug-
Qn = (a+bT,) (Tn+l-L) +
b (Tn+l -TJ2 2
(5)
Note a,=a+bT,, and t,= T,,+,- T,,) On equating eqns. (4) and (5) and after simplification the timing of the (n - 1) th order is obtained: T,,_, = 2T,,-T,,,,
-
b(Tn+, -Tn)’ 2(a+bT,)
(6)
Unfortunately, T,_ 1 can not be determined without knowing T,, which is given by -t ). Hence, at first an initial estimate z?nz t A’; and a, = a $’‘are obtained as follows: t(l)
n
=
2s r(a+bH)
J
(7)
55 TABLE 2 Result of computations
for determining
order intervals
Order i
T,, ,
r,
tt
Modified times when n = 7 1 t:=-t 1.003 t
n
1.ooo 0.898 0.790 0.673 0.540 0.400 0.230
0.898 0.790 0.673 0.544 0.400 0.230 -0.003
0.102 0.108 0.117 0.129 0.144 0.170 0.233
0.10 0.11 0.12 0.13 0.14 0.17 0.23
n-1
n-2 n-3 n-4 n-5 n-6
Modified times with (n - 1) orders result in a higher total cost.
TABLE 3 Comparison
of results for example 1 by different methods
Order i
Time of placing the ith order, T,, by the method of Donaldson Ill
Total cost
Silver 141
Goyal 161
Ritchie [51
Goyal et al. [91
Suggested method
0
0
0
0
0
0
0.23 0.40 0.54 0.67 0.79 0.90
0.21 0.38 0.53 0.66 0.78 0.89
0.25 0.42 0.56 0.68 0.79 0.90
0.21 0.39 0.54 0.67 0.79 0.90
0.21 0.38 0.52 0.65 0.77 0.89
0.23 0.40 0.54 0.67 0.79 0.90
125.29
125.44
125.48
125.46
125.51
125.29
0.12
0.15
0.14
Percent increase over minimum
0
0.18
0
TABLE 4 Comparison
of results for example 2 by different methods
Order i
Time of placing the ith order, T,, by the method of Donaldson [II
1 2 3 Total cost Percent increase over minimum
Silver [41
Goyal [61
Ritchie [51
0 4.2 7.8
0 4.0 7.5
0 4.3 7.9
0 4.3 7.8
0 4.2 7.8
0 4.2 7.8
51.085
51.185
51.099
51.091
51.085
51.085
0.196
0.027
0.012
0
Goyal et al. [91
0
Suggested method
0
56
ap = a+b(H-tp)
(8)
In general the (k+ 1) th estimates for t, and a, are evaluated as follows: t(k)
n
=
J
r aU-1) n (
2s + 26 +I) 3 n
(9) >
and ULk’ = a+b(H-p)
(10)
(where k>2). With the help of eqns. (7), (8), (9) and (10) the value of t, can be determined. Normally convergence is very rapid. For practical purposes it is unnecessary to go beyond ti2) for estimating tn. In general a prior knowledge of Ti+, and Ti will enable us to determine Ti_, from the following: Ti_1 = 2Ti_Ti+l
- b(Ti+l-ri)’ 2(a+bTi)
ill)
Very often the order intervals obtained by this method or other methods exceed the time horizon if y1orders are placed and fall short of H if ( n - 1) orders are placed. Hence, the order intervals are madified by multiplying ti values by H/C Yz~ ti for ( IZ- 1) orders and by H/C y=1 ti for n orders. The total cost for (y1- 1) orders and n orders using the modified order intervals are obtained. The number of orders resulting in a lower cost gives us the economic policy.
. ... can be evaluated from eqn. (11). Step (3) Obtain the modified times for n orders and (n- 1) orders. Select the policy which provides the lower cost. Table 1 provides the data of two examples. We will now determine the order intervals for the first example based on the method given in this paper. tp
=
J
2s r(a+bH)
= 0.1
a:” = a+b(H-ti”) tA2) =
J
= 810
2s ~(a$‘) + 3 bt:“)
- 0.10171. -
Hence, t, = 0.102 (up to three decimal places). The result of computations for determining order intervals is given in Table 2. Comparisons of the results obtained by various methods for both examples are shown in Tables 3 and 4. CONCLUSIONS
The suggested methods provides a simple approach for determining economic order intervals and generally achieves lower cost policy as compared to other heuristic methods. ACKNOWLEDGEMENT
This research was supported by grant No. A5004 from the National Sciences and Engineering Research Council of Canada REFERENCES
ALGORITHM FOR DETERMINING ORDERING POLICY
Step(1)
Set k= 1 and obtain tf ) from eqn. (7). Substitute the value of t;’ ) in eqn. (8) and evaluate ai’) and then determine t, = ti2) from eqn. (9). Step (2) For i=n set Ti+l= H and Ti=H- t,, determine Ti_ 1from eqn. (11). Similarly Ti_2, Ti_3,
Donaldson, W.A., 1977. Inventory replenishment policy for a linear trend in demand and analytical solution. Oper. Res. Q., 28: 663-670. Buchanan, J.T., 1980. Alternative solution methods for the inventory replenishment problem under increasing demand. J. Oper. Res. Sot., 31: 615-620. Henry, R.J., 1979. Inventory replenishment policy for increasing demands. J. Oper. Res. Sot., 30: 611-617. Silver, E.A., 1979. A simple inventory replenishment decision rule for a linear trend in demand. J. Oper. Res. sot., 30: 71-75. Ritchie, E., 1980. Practical inventory replenishment policies for a linear trend in demand followed by a period of steady demand. J. Oper. Res. Sot., 31: 605-613.
6
7
8
Goyal, SK., 1982. A production inventory model for linear trend in demand. In: Proceedings - Third International Symposium on Inventories, Budapest, Hungary, pp. 705-710. Phelps, R.I., 1980. Optimal inventory rule for a linear trend in demand with a constant replenishment period. J. Oper. Res. Sot., 3 1: 439-442. Mitra, A., Cox, J.F. and Jesse, R.R., 1984. A note on determining order quantities with a linear trend in demand. J. Oper. Res. Sot., 35: 141-144.
9
10
Goyal, S.K., Kusy, M. and Soni, R., 1985. Economic replenishment interval for an item having positive linear trend in demand. J. Eng. Costs Prod. Econ., lO( 3): 253-255. Ritchie, E., 1984. The E.O.Q. for linear increasing demand: A simple optional solution. J. Oper. Res. Sot., 35: 949-952. (Received March 6, 1986; accepted April 15, 7986)
53
Short Communication
A SIMPLE HEURISTIC METHOD FOR DETERMINING ECONOMIC ORDER INTERVAL FOR LINEAR DEMAND S.K. Goyal Department of Quantitative Methods,
Concordia University,
1455 de Maisonneuve
Blvd. West, Montreal, Quebec H3G lM8
(Canada)
ABSTRACT
In this paper a simple heuristic method is suggested for determining the economic ordering policy for an item having linear trend in
demand. The performance of the method is compared with other known methods.
INTRODUCTION
over the time horizons are more than four. In this paper a method is given which uses the heuristic method of Goyal et al. [ 91 and Ritchic’s [ lo] observation, that the stock holding cost after several orders is equal to the fixed ordering cost.
The classical square root formula for determining the economic order quantity was established almost seventy years ago but it was Donaldson [ 1 ] who established the following optimality condition for linear trend in demand: “the quantity ordered at a replenishment point must be the product of the previous replenishment interval and the current rate of demand”. Donaldson used this optimality condition to determine the replenishment policy for a given number of replenishments. Buchanan [ 21 and Henry [ 31 have also suggested alternative solution methods for determining the replenishment intervals for linearly increasing demand. Silver [ 41, Ritchie [ 51 and Goyal [ 61 have suggested heuristic methods for determining the economic replenishment intervals. Phelps [ 71 and Mitra et al. [ 81 have suggested approaches for determining the constant economic replenishment intervals; however, their approach provides much higher cost solutions when the number of replenishments
0167-188X/87/$03.50
0 1987 Elsevier Science Publishers B.V.
ASSUMPTIONS
AND NOTATIONS
In deriving the total cost model we are making the following assumptions: (1) Rate of demand increases linearly with time. (2) Shortages are not allowed. (3) Time horizon is finite. (4) Minimization of total cost in the objective. The following notations are being used for developing the total cost model: r stock holding cost per item per unit of time S fixed cost of a replenishment a rate of demand at time zero increase in demand per unit of time b a+bt rate of demand at time t time horizon H n total number of replenishments
W total cost For the ith replenishment: rate of demand at the time of making ai the replenishment timing of the replenishment ri replenishment interval replenishment quantity the ith w( ti) the total cost during replenishment Note that al =a, T, =O. The total cost, the sum of the fixed cost of replenishment and the stock holding cost, during the time interval covered by the ith replenishment is given b$
Li
W (ti) = S + r+ (ai +$ bti)
(1)
See Silver [ 41 for the above model. It may be pointed out that: i-l
ai=a+bTi=a+bC
tj j=O
(2)
TABLE 1 Data for the sample examples Example 1 9 2 900t 1 9
S r
Demand pattern H M
Example 2 9 0.1 6+t 11 180
gests transforming the origin to t = -x. From this new origin, the optimal ordering policy can be determined. The approach works well if one of the orders is placed exactly at t = x; i.e. t = 0 for the original problem. In case no order is placed exactly at time x then Ritchie suggests alternative approaches. According to the optimality condition of Donaldson [ 1 ] the order quantity in the last order, is given by:
Qn = (a+bW
K-C-d
(4)
If n orders are placed during the time horizon then the total cost is given by:
where T, - T,_ 1= (u - 1) th order interval. However, the demand during the last interval equals,
W=nS+ii$ltf(ai+$bti)
Q,, =a,,& +y
(3)
Unfortunately the computational effort required for determining the optimal ordering policy by using the method developed by Donaldson [ 1] is by no means simple. Due to this reason a number of heuristic methods have been proposed by Silver [ 41, Ritchie [ 51, Goyal [6],Mitra [8],Goyaletal. [9]. Ritchie [ lo] observed that for an optimal ordering policy, the cost of stock holding after a few iterations is almost equal to the fixed ordering cost and suggested a simple approach for determining almost optimal ordering policy. Instead of bounding the demand interval O-H, he suggests extending the horizon, so that it no longer affects the ordering times. This simplifies the computations for determining the optimal ordering policy. His approach works well when the rate of demand, a, at time zero is 0. However when a = bx, then he sug-
Qn = (a+bT,) (Tn+l-L) +
b (Tn+l -TJ2 2
(5)
Note a,=a+bT,, and t,= T,,+,- T,,) On equating eqns. (4) and (5) and after simplification the timing of the (n - 1) th order is obtained: T,,_, = 2T,,-T,,,,
-
b(Tn+, -Tn)’ 2(a+bT,)
(6)
Unfortunately, T,_ 1 can not be determined without knowing T,, which is given by -t ). Hence, at first an initial estimate z?nz t A’; and a, = a $’‘are obtained as follows: t(l)
n
=
2s r(a+bH)
J
(7)
55 TABLE 2 Result of computations
for determining
order intervals
Order i
T,, ,
r,
tt
Modified times when n = 7 1 t:=-t 1.003 t
n
1.ooo 0.898 0.790 0.673 0.540 0.400 0.230
0.898 0.790 0.673 0.544 0.400 0.230 -0.003
0.102 0.108 0.117 0.129 0.144 0.170 0.233
0.10 0.11 0.12 0.13 0.14 0.17 0.23
n-1
n-2 n-3 n-4 n-5 n-6
Modified times with (n - 1) orders result in a higher total cost.
TABLE 3 Comparison
of results for example 1 by different methods
Order i
Time of placing the ith order, T,, by the method of Donaldson Ill
Total cost
Silver 141
Goyal 161
Ritchie [51
Goyal et al. [91
Suggested method
0
0
0
0
0
0
0.23 0.40 0.54 0.67 0.79 0.90
0.21 0.38 0.53 0.66 0.78 0.89
0.25 0.42 0.56 0.68 0.79 0.90
0.21 0.39 0.54 0.67 0.79 0.90
0.21 0.38 0.52 0.65 0.77 0.89
0.23 0.40 0.54 0.67 0.79 0.90
125.29
125.44
125.48
125.46
125.51
125.29
0.12
0.15
0.14
Percent increase over minimum
0
0.18
0
TABLE 4 Comparison
of results for example 2 by different methods
Order i
Time of placing the ith order, T,, by the method of Donaldson [II
1 2 3 Total cost Percent increase over minimum
Silver [41
Goyal [61
Ritchie [51
0 4.2 7.8
0 4.0 7.5
0 4.3 7.9
0 4.3 7.8
0 4.2 7.8
0 4.2 7.8
51.085
51.185
51.099
51.091
51.085
51.085
0.196
0.027
0.012
0
Goyal et al. [91
0
Suggested method
0
56
ap = a+b(H-tp)
(8)
In general the (k+ 1) th estimates for t, and a, are evaluated as follows: t(k)
n
=
J
r aU-1) n (
2s + 26 +I) 3 n
(9) >
and ULk’ = a+b(H-p)
(10)
(where k>2). With the help of eqns. (7), (8), (9) and (10) the value of t, can be determined. Normally convergence is very rapid. For practical purposes it is unnecessary to go beyond ti2) for estimating tn. In general a prior knowledge of Ti+, and Ti will enable us to determine Ti_, from the following: Ti_1 = 2Ti_Ti+l
- b(Ti+l-ri)’ 2(a+bTi)
ill)
Very often the order intervals obtained by this method or other methods exceed the time horizon if y1orders are placed and fall short of H if ( n - 1) orders are placed. Hence, the order intervals are madified by multiplying ti values by H/C Yz~ ti for ( IZ- 1) orders and by H/C y=1 ti for n orders. The total cost for (y1- 1) orders and n orders using the modified order intervals are obtained. The number of orders resulting in a lower cost gives us the economic policy.
. ... can be evaluated from eqn. (11). Step (3) Obtain the modified times for n orders and (n- 1) orders. Select the policy which provides the lower cost. Table 1 provides the data of two examples. We will now determine the order intervals for the first example based on the method given in this paper. tp
=
J
2s r(a+bH)
= 0.1
a:” = a+b(H-ti”) tA2) =
J
= 810
2s ~(a$‘) + 3 bt:“)
- 0.10171. -
Hence, t, = 0.102 (up to three decimal places). The result of computations for determining order intervals is given in Table 2. Comparisons of the results obtained by various methods for both examples are shown in Tables 3 and 4. CONCLUSIONS
The suggested methods provides a simple approach for determining economic order intervals and generally achieves lower cost policy as compared to other heuristic methods. ACKNOWLEDGEMENT
This research was supported by grant No. A5004 from the National Sciences and Engineering Research Council of Canada REFERENCES
ALGORITHM FOR DETERMINING ORDERING POLICY
Step(1)
Set k= 1 and obtain tf ) from eqn. (7). Substitute the value of t;’ ) in eqn. (8) and evaluate ai’) and then determine t, = ti2) from eqn. (9). Step (2) For i=n set Ti+l= H and Ti=H- t,, determine Ti_ 1from eqn. (11). Similarly Ti_2, Ti_3,
Donaldson, W.A., 1977. Inventory replenishment policy for a linear trend in demand and analytical solution. Oper. Res. Q., 28: 663-670. Buchanan, J.T., 1980. Alternative solution methods for the inventory replenishment problem under increasing demand. J. Oper. Res. Sot., 31: 615-620. Henry, R.J., 1979. Inventory replenishment policy for increasing demands. J. Oper. Res. Sot., 30: 611-617. Silver, E.A., 1979. A simple inventory replenishment decision rule for a linear trend in demand. J. Oper. Res. sot., 30: 71-75. Ritchie, E., 1980. Practical inventory replenishment policies for a linear trend in demand followed by a period of steady demand. J. Oper. Res. Sot., 31: 605-613.
6
7
8
Goyal, SK., 1982. A production inventory model for linear trend in demand. In: Proceedings - Third International Symposium on Inventories, Budapest, Hungary, pp. 705-710. Phelps, R.I., 1980. Optimal inventory rule for a linear trend in demand with a constant replenishment period. J. Oper. Res. Sot., 3 1: 439-442. Mitra, A., Cox, J.F. and Jesse, R.R., 1984. A note on determining order quantities with a linear trend in demand. J. Oper. Res. Sot., 35: 141-144.
9
10
Goyal, S.K., Kusy, M. and Soni, R., 1985. Economic replenishment interval for an item having positive linear trend in demand. J. Eng. Costs Prod. Econ., lO( 3): 253-255. Ritchie, E., 1984. The E.O.Q. for linear increasing demand: A simple optional solution. J. Oper. Res. Sot., 35: 949-952. (Received March 6, 1986; accepted April 15, 7986)