- Email: [email protected]
Determining economic replenishment intervals for linear trend in demand
international journal of
production economics
ELSEVIER
Int. J. Production
Economics
34 (1994) 115-l 17
Technical
Determining
economic
note
intervals
replenishment demand
for linear trend in
S.K. Goyal Department qf Decision Sciences and MIS, Concordia University, 1455 de Maisonneuve (Received
19 April
1993; accepted
for publication
Blvd. West, Montreal, 3 November
Quebec, Canada H3G IM8
1993)
Abstract In this
note
a simple
demand. The method replenishment interval
method is given for determining economic replenishments of an item having linear trend in requires solution of only one cubic equation for the first replenishment. Each subsequent requires solution of a quadratic equation. An example is given to illustrate the method.
The problem of determining economic replenishment interval for an inventory item over a finite planning horizon has interested a number of researchers over the past 15 years. Unfortunately, the computations required for the analytical optimal algorithm of Donaldson (1977) are complex and due to this reason a number of heuristic methods have been proposed for determining the inventory policy see Silver (1979), Goyal and Gommes (1982), Ritchie (1984) and Tsado (1985). A common feature of the heuristic methods is that they require solution of a cubic equation for each replenishment. Hence, if n replenishments are to be made during the planning horizon then n cubic equations must be solved. In this paper we suggest a procedure requiring solution of a cubic equation and (n - 1) quadratic equations for determining replenishment intervals for an n replenishments policy. In order to develop the procedure, the following notation is used: A = fixed replenishment cost; H = planning horizon;
g
= =
qi+l
=
rate of demand at time zero; rate of incriase in demand for unit of time; o(t) = a + bt rate of demand at time t; I = stock holding cost per item per unit of time; For the ith replenishment (i = 1,2, 3, . . , n); = rate of demand at the time of making the 4 replenishment = replenishment interval; ti = optimal order quantity. 4i Donaldson (1977) developed the following optimality condition for the quantity to be ordered in the (i + 1)th replenishment: ‘the optimal replenishment quantity for the (i + 1)th replenishment (i > 1) is given by the multiplication of the rate of demand at the time of making the (i + 1)th replenishment by the ith replenishment interval’. The optimal order quantity for the (i + 1)th replenishment as per the optimality condition must be equal to
0925-5273/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved. SSDI 0925-5273(93)E0087-C
ti’ai+l.
(1)
116
S.K. Goyal/Int.
J. Production
The demand during the (i + 1)th replenishment duration ti+ 1 is given by 4i+l
=
ai+l.ti+l
+
b, 2
-ti+l.
of
(2)
On substituting the value of qi+ 1 from (1) in (2) we get a quadratic equation and the only feasible solution of Ti+ 1 is given by ti+l
Ui+l
+
=
Step 3. Obtain modified tj”- ‘) and ty) for the (n policy as given below:
replenishment intervals 1) and n replenishments
n-1 t!“-
l) =
Hti
il;lti
(i=
. ,n-
1,2,.
1. Algorithm for determining the economic ordering policy tr, by solving the cubic equation:
1
I - 2A = 0
Step 2. Determine tZ, t3, . . . , t, using (3) such that the following condition is satisfied:
1)
I and 4”’ = Ht,
(i = 1, 2, . . . , n).
i ti 1 i=l
the total relevant cost (TRC) horizon from the following:
(3)
For any value of ti there exists an optimal value as given by (3). Hence, for a particular value Ofti+ of the first replenishment interval, tl, subsequent optimal replenishment intervals can be determined with the help of Eq. (3).
;t: + at:
115-117
Step 4. Determine during the planning i = 1, 2, . . . ) (n - 1).
(
34 (19941
J~f+l + Zb.Ui+lti b
Step 1. Determine
Economics
TRC(k)
= kA
+
i
1 q
+
b(tlk’)3 3
i=l
for k = n - 1, n. Select the policy with (n) d TRC (n - 1).
n replenishments
if TRC
2. An Example For an inventory item the following data was provided by Donaldson (1977): D(t) = 900t, A = 9 per order, I = 2 per item per unit of time, H = 1, a = 0, b = 900 per unit of time. The solution is as follows. Step 1. (;bt: + at:)1 - 2A = 0, 600t: = 9 tl = 0.2466.
Table 1 Results of computations
for steps 2-4
Step(2)
Step (3) $7) = ~ t, ’ 1.072
i
4
a,+ 1
t, +1 from (3)
t!6’ = - ti ’ 0.9607
1 2 3 4 5 6 7
0.2466 0.1805 0.1531 0.1369 0.1259 0.1177 0.1113
22 1.94 384.39 522.18 645.39 758.70 864.63
0.1805 0.1531 0.1369 0.1259 0.1177 0.1113
0.2567 0.1879 0.1594 0.1425 0.1310 0.1225
0.2300 0.1684 0.1428 0.1277 0.1174 0.1098 0.1039
Step (4)
TRC (6) = 127.4907
TIC (7) = 125.2602
S.K. Goyal/Int.
Table 2 TRC by different
J. Production
Economics
34 (1994)
115-117
117
Acknowledgement methods
Method
TRC
The author is grateful to the referees for their very constructive comments and suggestions.
Donaldson (1977) This note Silver (1979) Ritchie (1984)
125.2600 125.2602 125.44 125.46
References
The computations for steps 2-4 are given in Table 1. Note that the economic policy is to order at (0,0.2300,0.3984,0.5412,0.6689,0.7863 and 0.8961). This policy requiring seven replenishments costs 125.2602. The TRC for the economic ordering policy for this example as obtained by different methods is given in Table 2.
Donaldson, W.A. 1977. Inventory replenishment policy for a linear trend in demand - An analytical solution. Oper. Res. Quart., 28: 663-670. Ritchie, E. 1984. The EOQ for linear increasing demand: A simple optimal solution. J. Oper. Res. Sot., 35: 949-952. Silver, E.A. 1979. A simple inventory replenishment rule for a linear trend in demand. J. Oper. Sot., 30: 71-75. Tsado, A. 1985. Evaluation of the performance of lot-sizing techniques on deterministic and stochastic demand. Unpublished Ph.D. Thesis, University of Lancaster, UK. Goyal, S.K. and L.F. Gommes 1982. Determination of replenishment intervals for a linear trend in demand. Working Paper, Department of Quantitative Methods, Concordia University, Canada.
production economics
ELSEVIER
Int. J. Production
Economics
34 (1994) 115-l 17
Technical
Determining
economic
note
intervals
replenishment demand
for linear trend in
S.K. Goyal Department qf Decision Sciences and MIS, Concordia University, 1455 de Maisonneuve (Received
19 April
1993; accepted
for publication
Blvd. West, Montreal, 3 November
Quebec, Canada H3G IM8
1993)
Abstract In this
note
a simple
demand. The method replenishment interval
method is given for determining economic replenishments of an item having linear trend in requires solution of only one cubic equation for the first replenishment. Each subsequent requires solution of a quadratic equation. An example is given to illustrate the method.
The problem of determining economic replenishment interval for an inventory item over a finite planning horizon has interested a number of researchers over the past 15 years. Unfortunately, the computations required for the analytical optimal algorithm of Donaldson (1977) are complex and due to this reason a number of heuristic methods have been proposed for determining the inventory policy see Silver (1979), Goyal and Gommes (1982), Ritchie (1984) and Tsado (1985). A common feature of the heuristic methods is that they require solution of a cubic equation for each replenishment. Hence, if n replenishments are to be made during the planning horizon then n cubic equations must be solved. In this paper we suggest a procedure requiring solution of a cubic equation and (n - 1) quadratic equations for determining replenishment intervals for an n replenishments policy. In order to develop the procedure, the following notation is used: A = fixed replenishment cost; H = planning horizon;
g
= =
qi+l
=
rate of demand at time zero; rate of incriase in demand for unit of time; o(t) = a + bt rate of demand at time t; I = stock holding cost per item per unit of time; For the ith replenishment (i = 1,2, 3, . . , n); = rate of demand at the time of making the 4 replenishment = replenishment interval; ti = optimal order quantity. 4i Donaldson (1977) developed the following optimality condition for the quantity to be ordered in the (i + 1)th replenishment: ‘the optimal replenishment quantity for the (i + 1)th replenishment (i > 1) is given by the multiplication of the rate of demand at the time of making the (i + 1)th replenishment by the ith replenishment interval’. The optimal order quantity for the (i + 1)th replenishment as per the optimality condition must be equal to
0925-5273/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved. SSDI 0925-5273(93)E0087-C
ti’ai+l.
(1)
116
S.K. Goyal/Int.
J. Production
The demand during the (i + 1)th replenishment duration ti+ 1 is given by 4i+l
=
ai+l.ti+l
+
b, 2
-ti+l.
of
(2)
On substituting the value of qi+ 1 from (1) in (2) we get a quadratic equation and the only feasible solution of Ti+ 1 is given by ti+l
Ui+l
+
=
Step 3. Obtain modified tj”- ‘) and ty) for the (n policy as given below:
replenishment intervals 1) and n replenishments
n-1 t!“-
l) =
Hti
il;lti
(i=
. ,n-
1,2,.
1. Algorithm for determining the economic ordering policy tr, by solving the cubic equation:
1
I - 2A = 0
Step 2. Determine tZ, t3, . . . , t, using (3) such that the following condition is satisfied:
1)
I and 4”’ = Ht,
(i = 1, 2, . . . , n).
i ti 1 i=l
the total relevant cost (TRC) horizon from the following:
(3)
For any value of ti there exists an optimal value as given by (3). Hence, for a particular value Ofti+ of the first replenishment interval, tl, subsequent optimal replenishment intervals can be determined with the help of Eq. (3).
;t: + at:
115-117
Step 4. Determine during the planning i = 1, 2, . . . ) (n - 1).
(
34 (19941
J~f+l + Zb.Ui+lti b
Step 1. Determine
Economics
TRC(k)
= kA
+
i
1 q
+
b(tlk’)3 3
i=l
for k = n - 1, n. Select the policy with (n) d TRC (n - 1).
n replenishments
if TRC
2. An Example For an inventory item the following data was provided by Donaldson (1977): D(t) = 900t, A = 9 per order, I = 2 per item per unit of time, H = 1, a = 0, b = 900 per unit of time. The solution is as follows. Step 1. (;bt: + at:)1 - 2A = 0, 600t: = 9 tl = 0.2466.
Table 1 Results of computations
for steps 2-4
Step(2)
Step (3) $7) = ~ t, ’ 1.072
i
4
a,+ 1
t, +1 from (3)
t!6’ = - ti ’ 0.9607
1 2 3 4 5 6 7
0.2466 0.1805 0.1531 0.1369 0.1259 0.1177 0.1113
22 1.94 384.39 522.18 645.39 758.70 864.63
0.1805 0.1531 0.1369 0.1259 0.1177 0.1113
0.2567 0.1879 0.1594 0.1425 0.1310 0.1225
0.2300 0.1684 0.1428 0.1277 0.1174 0.1098 0.1039
Step (4)
TRC (6) = 127.4907
TIC (7) = 125.2602
S.K. Goyal/Int.
Table 2 TRC by different
J. Production
Economics
34 (1994)
115-117
117
Acknowledgement methods
Method
TRC
The author is grateful to the referees for their very constructive comments and suggestions.
Donaldson (1977) This note Silver (1979) Ritchie (1984)
125.2600 125.2602 125.44 125.46
References
The computations for steps 2-4 are given in Table 1. Note that the economic policy is to order at (0,0.2300,0.3984,0.5412,0.6689,0.7863 and 0.8961). This policy requiring seven replenishments costs 125.2602. The TRC for the economic ordering policy for this example as obtained by different methods is given in Table 2.
Donaldson, W.A. 1977. Inventory replenishment policy for a linear trend in demand - An analytical solution. Oper. Res. Quart., 28: 663-670. Ritchie, E. 1984. The EOQ for linear increasing demand: A simple optimal solution. J. Oper. Res. Sot., 35: 949-952. Silver, E.A. 1979. A simple inventory replenishment rule for a linear trend in demand. J. Oper. Sot., 30: 71-75. Tsado, A. 1985. Evaluation of the performance of lot-sizing techniques on deterministic and stochastic demand. Unpublished Ph.D. Thesis, University of Lancaster, UK. Goyal, S.K. and L.F. Gommes 1982. Determination of replenishment intervals for a linear trend in demand. Working Paper, Department of Quantitative Methods, Concordia University, Canada.