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A note on the economic replenishment interval for an item with a linear trend in demand
253
Engineering Costs and Production Economics, I0 (1986) 253-255 Elsevier Science Publishers B.V., Amsterdam
-
Printed in The Netherlands
Short Communication
A NOTE ON THE ECONOMIC REPLENISHMENT INTERVAL FOR AN ITEM WITH A LINEAR TREND IN D E M A N D Suresh Goyal, Martin Kusy and Rajan Soni Department of Quantitative Methods, Faculty of Commerce and Administration, Concordia University, Montreal, Quebec (Canada)
The classical square root formula for determining t h e economic order quantity was established almost seventy years ago but it was Donaldson (1977) 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. The computational effort required to determine the optimal policy was by no means simple. Buchanan (1980), and Henry (1979) have also suggested alternative solution methods for determining the replenishment intervals for linearly increasing demand. Silver (1979), Ritchie (1980), and Goyal (1982) have suggested heuristic methods for determining the economic replishment intervals. Phelps (1980) and Mitra et al. (1984) have suggested approaches for determining the constant economic replenishment intervals; however, their approaches provide higher cost solutions. A common feature of the heuristic methods of Silver (1979) and Goyal (1982) is that it requires a solution of a cubic equation in order to obtain an economic time interval. Ritchie's (1980) method requires evaluation of two square roots per replenishment interval. 0167-188X/86/$03.50
© 1986 Elsevier Science Publishers B.V.
In this paper we will present a heuristic method which avoids solving a cubic equation and requires evaluation of only one square root per replenishment interval.
ASSUMPTIONS AND NOTATIONS In deriving the total cost model we are assuming a finite time horizon H during which the demand increases linearly. At time t, the rate of demand equals (a + bt) where a is the rate of demand at time zero and b is the change in demand per unit of time. A fixed cost S is incurred whenever a replenishment is made and it costs r to keep an item in stock for a unit time. Shortages are not permitted, hence, the total cost during a replenishment is given by the sum of fixed replenishment cost and stock holding cost. For the ith replishment we have: ai = rate of demand at the time of making the replishment; Ti = timing of the replenishment; ti = replenishment interval; and Z(ti) the total cost during the ith replenishment. 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 (Silver, 1979) by 2 Z(t i) = S + r ~ (ai + "zbti) 3
(1)
254 SILVER'S HEURISTIC METHOD Silver used the well known Silver-Meal (1973) heuristic method and determined the replenishment interval which minimized the total cost per unit of time, during the replenishment interval. During each replenishment interval, Silver minimizes:
Silver's and Goyal's heuristic methods are (ai + ~ bti) and (ai + ~ bti) respectively. Taking the mean of these two denominators we get (at + bti) which reduces to a delightfully simple expression, ai+ 1, which is the rate of demand at the time of making the (i + 1)th replenishment. Hence, ti=¢a
Z(t i) - S + r ti(a i+ ~bti). 2 ti ti 2
Jia M tJ
(3)
bti)
(6)
(2)
such that ti =
M
i + 1
Determination of ti from (6) requires working backwards. We first determine the replenishment interval for the last order, tr, as follows: Assume ar+ 1 = a + bH, determine
where (4)
M = 2Sir.
tr =
;
ar =ar+ 1 - b t r.
Silver suggests that (3) is solved iteratively when ai > 0 with t~°) = 0. Convergence is normally very rapid.
Then we determine
GOYAL'S HEURISTIC METHOD
In this way we evaluate tr-2, t r - 3 , .... We stop at t r - , + 1 when the following condition is satisfied
Goyal used the equally well known P a r t Period heuristic due to De Matteis (1968) and consequently determined the replenishment interval such that the fixed cost of replenishment equals the stock holding cost during the replenishment. Hence, we should have S =t~2 r(ai + ~ bti)
tr- I
;
a r - 1 = a t - b t r - 1, a n d so on.
( t r + t r _ l +... + tr_ n+ 2 ) < H , ( t r + t r - l + t r - n + 1).
' ,
+... (7)
It invariably happens that the replenishment intervals obtained by different methods exceed the time horizon if n orders are placed and fall short of H if (n - 1) orders are placed. Hence, the replenishment intervals are modified by multiplying the ti values by
or
n-1
1-1/ i ~ l t i for (n - 1) replenishments, and by ti =
,
(5)
The iterative procedure adopted by Silver can be used to determine the replenishment interval. SUGGESTED METHOD It may be observed that the denominator in
n/
n
z i=l
ti for n
replenishments.
The total cost, Y.Z(tD, for (n - 1) orders and n orders using the modified replenishment intervals are compared and the solution with the lowest cost is selected. The following tables provide the data and the solutions of two problems given in Silver (1979).
255 TABLE 1 Results for example 1. S = 9 , r = 0.1,a = 6, b = 1 a n d H = 11 Time of placing the tth order, T~ by the method of Order i
Total cost Percent over the minimum cost
Donaldson
Silver
Goyal
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
51.085
51.185
51.099
51.091
51.085
0.196
0.027
0.012
0
Ritchie
Suggested method
0
TABLE 2 Results for example 2. S = 9, r = 2, a = 0, b = 900 and H = 1 Time of placing the ith order, T i by the method of Order i
Total cost Percent increase over minimum
Donaldson
Sliver
Goyal
Ritchie
Suggested method
0 0.23 0.40 0.54 0.67 0.79 0.90
0 0.21 0.38 0.53 0.66 0.78 0.89
0 0.25 0.42 0.56 0.68 0.79 0.90
0 0.21 0.39 0.54 0.67 0.79 0.90
0 0.21 0.38 0.52 0.65 0.77 0.89
125.29
125.44
125A8
125.46
125.51
0.12
0.15
0.14
0.18
0
ACKNOWLEDGMENT The authors are grateful to the referees for valuable comments and suggestions.
REFERENCES Buchanan, J.T., 1980. Alternative solution methods for the inventory replenishment problem under increasing demand. J. Oper. Res. Soc., 31: 6 1 5 - 6 2 0 . De Matteis, J.J., 1968. An economic lot sizing technique: the part-period algorithm. IBM Syst. J., 7 (1): 3 0 - 3 8 . Donaldson, W.A., 1977. Inventory replenishment policy for a linear trend in demand and analytical solution. Oper. Res. Q., 28: 6 6 3 - 6 7 0 . Goyal, S.K., 1982. A production inventory model for linear trend in demand. In: A. Chikfia fed.) Prec. Third International Symposium on Inventories, Budapest, Hungary, publ. Akademlai Klado, Budapest, pp. 7 0 5 - 7 1 0 .
Hemy, R.J., 1979. Inventory replenishment policy for increasing demands. J. Oper. Res. Soc., 30: 6 1 1 - 6 1 7 . 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. Soc., 35: 141-144. Phelps, R.I., 1980. Optimal inventory rule for a linear trend in demand with a constant replenishment period. J. Oper. Res. Soc., 31: 4 3 9 - 4 4 2 . Ritchie, E., 1980. Practical inventory replenishment policies for a linear trend in demand followed by a period of steady demand. J. Oper. Res. Soc., 31: 6 0 5 - 6 1 3 . Silver, E.A., 1979. A simple inventory replenishment decision rule for a linear trend in demand. J. Oper. Res. Soc., 30: 7 1 - 7 5 . Silver, E.A. and Meal, H.C., 1973. A heuristic for selecting lot size quantities for the case of deterministic time varying demand rate and discrete opportunities for replenishment. Prod. Invent. Manag., 14: 6 4 - 7 4 .
Engineering Costs and Production Economics, I0 (1986) 253-255 Elsevier Science Publishers B.V., Amsterdam
-
Printed in The Netherlands
Short Communication
A NOTE ON THE ECONOMIC REPLENISHMENT INTERVAL FOR AN ITEM WITH A LINEAR TREND IN D E M A N D Suresh Goyal, Martin Kusy and Rajan Soni Department of Quantitative Methods, Faculty of Commerce and Administration, Concordia University, Montreal, Quebec (Canada)
The classical square root formula for determining t h e economic order quantity was established almost seventy years ago but it was Donaldson (1977) 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. The computational effort required to determine the optimal policy was by no means simple. Buchanan (1980), and Henry (1979) have also suggested alternative solution methods for determining the replenishment intervals for linearly increasing demand. Silver (1979), Ritchie (1980), and Goyal (1982) have suggested heuristic methods for determining the economic replishment intervals. Phelps (1980) and Mitra et al. (1984) have suggested approaches for determining the constant economic replenishment intervals; however, their approaches provide higher cost solutions. A common feature of the heuristic methods of Silver (1979) and Goyal (1982) is that it requires a solution of a cubic equation in order to obtain an economic time interval. Ritchie's (1980) method requires evaluation of two square roots per replenishment interval. 0167-188X/86/$03.50
© 1986 Elsevier Science Publishers B.V.
In this paper we will present a heuristic method which avoids solving a cubic equation and requires evaluation of only one square root per replenishment interval.
ASSUMPTIONS AND NOTATIONS In deriving the total cost model we are assuming a finite time horizon H during which the demand increases linearly. At time t, the rate of demand equals (a + bt) where a is the rate of demand at time zero and b is the change in demand per unit of time. A fixed cost S is incurred whenever a replenishment is made and it costs r to keep an item in stock for a unit time. Shortages are not permitted, hence, the total cost during a replenishment is given by the sum of fixed replenishment cost and stock holding cost. For the ith replishment we have: ai = rate of demand at the time of making the replishment; Ti = timing of the replenishment; ti = replenishment interval; and Z(ti) the total cost during the ith replenishment. 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 (Silver, 1979) by 2 Z(t i) = S + r ~ (ai + "zbti) 3
(1)
254 SILVER'S HEURISTIC METHOD Silver used the well known Silver-Meal (1973) heuristic method and determined the replenishment interval which minimized the total cost per unit of time, during the replenishment interval. During each replenishment interval, Silver minimizes:
Silver's and Goyal's heuristic methods are (ai + ~ bti) and (ai + ~ bti) respectively. Taking the mean of these two denominators we get (at + bti) which reduces to a delightfully simple expression, ai+ 1, which is the rate of demand at the time of making the (i + 1)th replenishment. Hence, ti=¢a
Z(t i) - S + r ti(a i+ ~bti). 2 ti ti 2
Jia M tJ
(3)
bti)
(6)
(2)
such that ti =
M
i + 1
Determination of ti from (6) requires working backwards. We first determine the replenishment interval for the last order, tr, as follows: Assume ar+ 1 = a + bH, determine
where (4)
M = 2Sir.
tr =
;
ar =ar+ 1 - b t r.
Silver suggests that (3) is solved iteratively when ai > 0 with t~°) = 0. Convergence is normally very rapid.
Then we determine
GOYAL'S HEURISTIC METHOD
In this way we evaluate tr-2, t r - 3 , .... We stop at t r - , + 1 when the following condition is satisfied
Goyal used the equally well known P a r t Period heuristic due to De Matteis (1968) and consequently determined the replenishment interval such that the fixed cost of replenishment equals the stock holding cost during the replenishment. Hence, we should have S =t~2 r(ai + ~ bti)
tr- I
;
a r - 1 = a t - b t r - 1, a n d so on.
( t r + t r _ l +... + tr_ n+ 2 ) < H , ( t r + t r - l + t r - n + 1).
' ,
+... (7)
It invariably happens that the replenishment intervals obtained by different methods exceed the time horizon if n orders are placed and fall short of H if (n - 1) orders are placed. Hence, the replenishment intervals are modified by multiplying the ti values by
or
n-1
1-1/ i ~ l t i for (n - 1) replenishments, and by ti =
,
(5)
The iterative procedure adopted by Silver can be used to determine the replenishment interval. SUGGESTED METHOD It may be observed that the denominator in
n/
n
z i=l
ti for n
replenishments.
The total cost, Y.Z(tD, for (n - 1) orders and n orders using the modified replenishment intervals are compared and the solution with the lowest cost is selected. The following tables provide the data and the solutions of two problems given in Silver (1979).
255 TABLE 1 Results for example 1. S = 9 , r = 0.1,a = 6, b = 1 a n d H = 11 Time of placing the tth order, T~ by the method of Order i
Total cost Percent over the minimum cost
Donaldson
Silver
Goyal
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
51.085
51.185
51.099
51.091
51.085
0.196
0.027
0.012
0
Ritchie
Suggested method
0
TABLE 2 Results for example 2. S = 9, r = 2, a = 0, b = 900 and H = 1 Time of placing the ith order, T i by the method of Order i
Total cost Percent increase over minimum
Donaldson
Sliver
Goyal
Ritchie
Suggested method
0 0.23 0.40 0.54 0.67 0.79 0.90
0 0.21 0.38 0.53 0.66 0.78 0.89
0 0.25 0.42 0.56 0.68 0.79 0.90
0 0.21 0.39 0.54 0.67 0.79 0.90
0 0.21 0.38 0.52 0.65 0.77 0.89
125.29
125.44
125A8
125.46
125.51
0.12
0.15
0.14
0.18
0
ACKNOWLEDGMENT The authors are grateful to the referees for valuable comments and suggestions.
REFERENCES Buchanan, J.T., 1980. Alternative solution methods for the inventory replenishment problem under increasing demand. J. Oper. Res. Soc., 31: 6 1 5 - 6 2 0 . De Matteis, J.J., 1968. An economic lot sizing technique: the part-period algorithm. IBM Syst. J., 7 (1): 3 0 - 3 8 . Donaldson, W.A., 1977. Inventory replenishment policy for a linear trend in demand and analytical solution. Oper. Res. Q., 28: 6 6 3 - 6 7 0 . Goyal, S.K., 1982. A production inventory model for linear trend in demand. In: A. Chikfia fed.) Prec. Third International Symposium on Inventories, Budapest, Hungary, publ. Akademlai Klado, Budapest, pp. 7 0 5 - 7 1 0 .
Hemy, R.J., 1979. Inventory replenishment policy for increasing demands. J. Oper. Res. Soc., 30: 6 1 1 - 6 1 7 . 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. Soc., 35: 141-144. Phelps, R.I., 1980. Optimal inventory rule for a linear trend in demand with a constant replenishment period. J. Oper. Res. Soc., 31: 4 3 9 - 4 4 2 . Ritchie, E., 1980. Practical inventory replenishment policies for a linear trend in demand followed by a period of steady demand. J. Oper. Res. Soc., 31: 6 0 5 - 6 1 3 . Silver, E.A., 1979. A simple inventory replenishment decision rule for a linear trend in demand. J. Oper. Res. Soc., 30: 7 1 - 7 5 . Silver, E.A. and Meal, H.C., 1973. A heuristic for selecting lot size quantities for the case of deterministic time varying demand rate and discrete opportunities for replenishment. Prod. Invent. Manag., 14: 6 4 - 7 4 .