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Improving lot sizing decisions based on heuristic methods
287
Engineering Costs and Production Economics, 10 (1986) 287-291 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
IMPROVING LOT SIZING DECISIONS BASED ON HEURISTIC METHODS S.K. Goyal Department o f Quantitative Methods, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H 3 G 11148 (Canada)
ABSTRACT
A systematic approach is presented for improving ordering policies obtained by applying a heuristic method for determining replenishment intervals for an item, the
demand for which varies over time. An example is given to illustrate the use o f the suggested approach.
INTRODUCTION
(1) Replenishments can be made at discrete time intervals, i.e. at the beginning of each period; (2) The demand in a period is supplied at the beginning of the period; (3) Minimization of cost is the objective. Consider a time interval of T periods, and let: Dj = demand in p e r i o d / ; A = fixed cost of a replenishment; h = unit stock holding cost per item per period; and TC= total cost. The total cost will be determined for when (1) only one replenishment is made, and (2) two replenishments are made to supply the demand during the time interval.
Wagner and Whitin [1] provided an algorithm for determining the optimal replenishment policy of an item, the demand for which varies over time. Due to the computational problems encountered in problems having a longer planning horizon, a number of heuristic methods (e.g. [2, 3]) have been suggested for determining replenishment policies. These heuristic methods have been found to provide nearly optimal replenishment policies. Silver and Miltenburg [4] suggested two modifications to the Silver-Meal [ 3 ] heuristic for imProving the solution obtained for a dynamic lot size problem. In the present paper, a systematic approach is developed for improving the replenishment policies of dynamic lot size problems obtained by applying heuristic methods. In developing the method for improving the solutions, the following assumptions are made: 0167-188X/86/$03.50
TOTAL COST WHEN ONLY ONE ORDER IS PLACED T
TC(1) = A + h
~., ( / - - 1 ) D 1 = A + I ( 1 ) i=2
Where: I ( 1 ) = i n v e n t o r y carrying cost; and TC (1) = total cost with only one replenishment.
© 1986 Elsevier Science Publishers B.V.
288 On the other hand, only one replenishment must be made to supply the demand if
TOTAL COST WHEN TWO ORDERS ARE PLACED
T rc(2)
A >ihr ~ Di
= 2A + 1 ( 2 )
Where: I ( 2 ) = minimum inventory carrying cost when two orders are placed; and TC(2) = total cost with two replenishments. Let the first order supply the demand over X periods and the second order supply the demand over (T - X) periods. Note that as a result of placing two orders the reduction in time-weighted inventory is given by I ( 2 ) = I ( 1 ) --
(3)
,i=r~l
hX(Dx+1 + D x + 2
+ . . • DT),
or
Silver and Miltenburg [4] explored the possibility of adding one more replenishment during the interval, T, covered by the current replenishment. They suggested a heuristic approach for determining those replenishment intervals which might benefit from an additional replenishment. They suggested consideration of an additional replenishment if a replenishment interval satisfies the following condition:
T •(2)
=
I(1)--hX E Di"
(4)
/=X+I
In order to minimize the inventory carrying cost as a result of two replenishments over the policy of a single replenishment, the objective must be to maximize T
hx E /=X+l
for 1 ~ < X ~ < T - - 1 . If Max
(T)
T
X E Dj = h r E D],
\
i=X+I
/
(i)
i=r*)
for 1 <~X <~ T - - 1, then the minimum total cost as a result of two replenishments is given by T TC(2) = 2 A + l ( 1 ) - - h r
~ Dj. /=r+l
It is economical to have 2 replenishments if TC(2) < TC(1), or T
2A+IC1)--hr ~
Dj
+I(1),
j=r,i
or T
A
% j=r+ I
(2)
Where: H = number of periods for which future demand data is available; k = 1.25, for H ~< 30; and k = 1.50, for H >~ 30. Once replenishments satsifying the above condition are identified, then the value of r and hr Z~=r+1 Dj for each replenishment is evaluated from eqn. (1). An additional order is placed if condition (2) is satisfied. Once all the orders satisfying condition (4) are considered for a possible additional order, two replenishments at a time are considered. Let the number of replenishments during the entire planning horizon be m and, for the ith replenishment, t~ = time at which the replenishment is made; Ti = replenishment interval; and Qi = replenishment quantity. For the first pair of replenishments, we consider the first and the second replenishments. The first replenishment is made at tl = 0 and supplies the demand for Ta periods. The second replenishment is made at t2 (note that t2 t> T1) and covers the demand for T2 periods. Hence, the time interval covered by these replenishments is:
289 (T 2 + t 2 ) - - t ,
= T2 + t : .
From the given data for demand during this time interval, the value of r = rl can be determined from eqn. (1), and the second replenishment is made at rl if condition (2) is satisfied. On the other hand, if condition (2) is not satisfied, then only one replenishment is made to satisfy the demand. Next consider replenishments 2 and 3. Note that the second replenishment is made at rz, and the third is made at t 3 and supplies the demand till the period T 3 + t 3. Hence, the time interval covered by these replenishments is ( T 3 + t 2 - - r l ) . l ~ o r the demand data given over this period the value of r = r 2 is obtained from eqn. (1). If two replenishments are to be placed during this interval then the third replenishment is made at r~ + r 2 . In the same manner, consider ( m - 1) pairs of replenishments. Note that, as a result of considering the ith pair of replenishment, a modified timing for the ( i + 1)th replenishment is obtained. The modified times are denoted as t* (for the first replenishment tl = t* = 0).
APPROACH FOR IMPROVING LOT SIZING DECISIONS
Step (1): Determine those replenishments for which condition (4) is satisfied. Add a replenishment if condition (2) is satisfied. Determine timing and interval of each replenishment for m replenishments. Step (2): Consider replenishment pairs (1, 2), (2, 3 ) . . . ( m - - I , m). For the ith pair of replenishments the timing of replenishments is given by (t*, ti+l), where t* is the new timing of the ith replenishment obtained as a result of considering the ( i - 1) pair of replenishments, considering the ( i - 1)th and the ith replenishments.
EXAMPLE For an item the following data is given: Cost of placing an order = $280; Unit stock holding cost per period = $1.00;
Period
Demand
2 7 11 15 22 25 26
179 44 10 123 55 19 174 16
27 28 29 30 31 32
300 266 112 84 56 42
1
Total
1,480
The following solution was obtained on applying the Silver-Meal heuristic [ 2].
Replenishment, i
1
2
3
4
5
Timing, t i
1
11
22
25
27
Replenishment quantity, Qi
233
178
19
190
860
Replenishment interval, T i
7
5
1
2
6
Total cost = 2,916
Applying the suggested method improving the replenishment policy:
Step (1) H = 32 periods, so k = 1.50.
L
j:t
j
for
290 Hence, according to Silver and Miltenburg [4], the first and last replenishment intervals exceed 5.21, so they should be considered for an additional order i.
Pairs o f orders 1, 2 T = 15 periods. /' Dj
Replenishment no. 1 T = 7 periods
1
2
7
11
15
179
44
10
123
55
232
1,128
1,780
770
15 ( / - - 1)Dj
--
i'=2 j Dj
1
2
7
179
44
10
0
54
60
T (/. -- 1) D,/
The timing of the second replenishment is unchanged.
j=z
No change in policy is desirable for this replenishment.
Pairs o f orders 2, 3 T = 12 periods. Period
Replenishment No. 5 T = 6 periods
11
15
22
1
5
11
123
55
19
296
209
! Dj
11 Period
27
28
29
30
31
32
1
2
3
4
5
6
300
266
112
84
56
42
--
560
588
546
392
210
j Dj T (l'--l)Oj 1=2
Hence r = 2. The improved policy as a result of adding an order to the last replenishment is as follows: ti
Qi
1
11
22
25
27
29
233
178
19
190
566
294
(/-
1)Dj
--
j=2
The third replenishment should be made at period 15. Order quqntities for the second and third orders change to 123 and 74, respectively. Pairs of orders (3, 4), (4, 5) and (5, 6) were then considered. No change in the timing of the fourth, fifth and sixth replenishments was found to be desirable. The improved replenishment policy is as given below: ti
1
11
15
25
27
29
233
123
74
190
566
294
For this policy, total cost = $ 2,608.
Qi
Step (2)
Total cost = $2,521 Incidentally, the above policy is the optimal policy as obtained by the application of the Wagner-Whitin [1] algorithm to the problem.
Pairs of orders are considered next in order to improve the current replenishment schedule.
291 CONCLUSION
of applying the approach given in this paper. REFERENCES
The paper represents a simple approach for improving the heuristic lot sizes obtained on applying heuristic methods to the problem of determining economic replenishment for the dynamic inventory problem. The possible reduction in the total cost of the replenishment policy obtained by applying the suggested approach largely depends on the quality of the initial solution obtained by applying the heuristic method. If the initial solution has a total cost much larger than the optimal solution, then a significant reduction in cost can be achieved as a result
1 Wagner, H.M. and Whitin, T.M., 1958. Dynamic version of the economic lot size model. Manage. Sci., 5: 89-96. 2 Dixon, P. and Silver, E.A., 1981. A heuristic solution procedure for the multiitem, single level, limited capacity, lot sizing problem. J. Oper. Manage., 2: 23-40. 3 Silver, E.A. and Meal, H.C., 1973. A heuristic for selecting lot size quantities for the case of deterministic timevarying demand rate and discrete opportunities for replenishment. Prod. Inventory Manage., 14: 64-73. 4 Silver, E.A. and Miltenburg, J., 1984. Two modifications of the Silver-Meal lot sizing heuristic. INFOR, 22: 56-69.
{Received January 2, 1985; accepted in revised form December 5, 1985)
Engineering Costs and Production Economics, 10 (1986) 287-291 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
IMPROVING LOT SIZING DECISIONS BASED ON HEURISTIC METHODS S.K. Goyal Department o f Quantitative Methods, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H 3 G 11148 (Canada)
ABSTRACT
A systematic approach is presented for improving ordering policies obtained by applying a heuristic method for determining replenishment intervals for an item, the
demand for which varies over time. An example is given to illustrate the use o f the suggested approach.
INTRODUCTION
(1) Replenishments can be made at discrete time intervals, i.e. at the beginning of each period; (2) The demand in a period is supplied at the beginning of the period; (3) Minimization of cost is the objective. Consider a time interval of T periods, and let: Dj = demand in p e r i o d / ; A = fixed cost of a replenishment; h = unit stock holding cost per item per period; and TC= total cost. The total cost will be determined for when (1) only one replenishment is made, and (2) two replenishments are made to supply the demand during the time interval.
Wagner and Whitin [1] provided an algorithm for determining the optimal replenishment policy of an item, the demand for which varies over time. Due to the computational problems encountered in problems having a longer planning horizon, a number of heuristic methods (e.g. [2, 3]) have been suggested for determining replenishment policies. These heuristic methods have been found to provide nearly optimal replenishment policies. Silver and Miltenburg [4] suggested two modifications to the Silver-Meal [ 3 ] heuristic for imProving the solution obtained for a dynamic lot size problem. In the present paper, a systematic approach is developed for improving the replenishment policies of dynamic lot size problems obtained by applying heuristic methods. In developing the method for improving the solutions, the following assumptions are made: 0167-188X/86/$03.50
TOTAL COST WHEN ONLY ONE ORDER IS PLACED T
TC(1) = A + h
~., ( / - - 1 ) D 1 = A + I ( 1 ) i=2
Where: I ( 1 ) = i n v e n t o r y carrying cost; and TC (1) = total cost with only one replenishment.
© 1986 Elsevier Science Publishers B.V.
288 On the other hand, only one replenishment must be made to supply the demand if
TOTAL COST WHEN TWO ORDERS ARE PLACED
T rc(2)
A >ihr ~ Di
= 2A + 1 ( 2 )
Where: I ( 2 ) = minimum inventory carrying cost when two orders are placed; and TC(2) = total cost with two replenishments. Let the first order supply the demand over X periods and the second order supply the demand over (T - X) periods. Note that as a result of placing two orders the reduction in time-weighted inventory is given by I ( 2 ) = I ( 1 ) --
(3)
,i=r~l
hX(Dx+1 + D x + 2
+ . . • DT),
or
Silver and Miltenburg [4] explored the possibility of adding one more replenishment during the interval, T, covered by the current replenishment. They suggested a heuristic approach for determining those replenishment intervals which might benefit from an additional replenishment. They suggested consideration of an additional replenishment if a replenishment interval satisfies the following condition:
T •(2)
=
I(1)--hX E Di"
(4)
/=X+I
In order to minimize the inventory carrying cost as a result of two replenishments over the policy of a single replenishment, the objective must be to maximize T
hx E /=X+l
for 1 ~ < X ~ < T - - 1 . If Max
(T)
T
X E Dj = h r E D],
\
i=X+I
/
(i)
i=r*)
for 1 <~X <~ T - - 1, then the minimum total cost as a result of two replenishments is given by T TC(2) = 2 A + l ( 1 ) - - h r
~ Dj. /=r+l
It is economical to have 2 replenishments if TC(2) < TC(1), or T
2A+IC1)--hr ~
Dj
+I(1),
j=r,i
or T
A
% j=r+ I
(2)
Where: H = number of periods for which future demand data is available; k = 1.25, for H ~< 30; and k = 1.50, for H >~ 30. Once replenishments satsifying the above condition are identified, then the value of r and hr Z~=r+1 Dj for each replenishment is evaluated from eqn. (1). An additional order is placed if condition (2) is satisfied. Once all the orders satisfying condition (4) are considered for a possible additional order, two replenishments at a time are considered. Let the number of replenishments during the entire planning horizon be m and, for the ith replenishment, t~ = time at which the replenishment is made; Ti = replenishment interval; and Qi = replenishment quantity. For the first pair of replenishments, we consider the first and the second replenishments. The first replenishment is made at tl = 0 and supplies the demand for Ta periods. The second replenishment is made at t2 (note that t2 t> T1) and covers the demand for T2 periods. Hence, the time interval covered by these replenishments is:
289 (T 2 + t 2 ) - - t ,
= T2 + t : .
From the given data for demand during this time interval, the value of r = rl can be determined from eqn. (1), and the second replenishment is made at rl if condition (2) is satisfied. On the other hand, if condition (2) is not satisfied, then only one replenishment is made to satisfy the demand. Next consider replenishments 2 and 3. Note that the second replenishment is made at rz, and the third is made at t 3 and supplies the demand till the period T 3 + t 3. Hence, the time interval covered by these replenishments is ( T 3 + t 2 - - r l ) . l ~ o r the demand data given over this period the value of r = r 2 is obtained from eqn. (1). If two replenishments are to be placed during this interval then the third replenishment is made at r~ + r 2 . In the same manner, consider ( m - 1) pairs of replenishments. Note that, as a result of considering the ith pair of replenishment, a modified timing for the ( i + 1)th replenishment is obtained. The modified times are denoted as t* (for the first replenishment tl = t* = 0).
APPROACH FOR IMPROVING LOT SIZING DECISIONS
Step (1): Determine those replenishments for which condition (4) is satisfied. Add a replenishment if condition (2) is satisfied. Determine timing and interval of each replenishment for m replenishments. Step (2): Consider replenishment pairs (1, 2), (2, 3 ) . . . ( m - - I , m). For the ith pair of replenishments the timing of replenishments is given by (t*, ti+l), where t* is the new timing of the ith replenishment obtained as a result of considering the ( i - 1) pair of replenishments, considering the ( i - 1)th and the ith replenishments.
EXAMPLE For an item the following data is given: Cost of placing an order = $280; Unit stock holding cost per period = $1.00;
Period
Demand
2 7 11 15 22 25 26
179 44 10 123 55 19 174 16
27 28 29 30 31 32
300 266 112 84 56 42
1
Total
1,480
The following solution was obtained on applying the Silver-Meal heuristic [ 2].
Replenishment, i
1
2
3
4
5
Timing, t i
1
11
22
25
27
Replenishment quantity, Qi
233
178
19
190
860
Replenishment interval, T i
7
5
1
2
6
Total cost = 2,916
Applying the suggested method improving the replenishment policy:
Step (1) H = 32 periods, so k = 1.50.
L
j:t
j
for
290 Hence, according to Silver and Miltenburg [4], the first and last replenishment intervals exceed 5.21, so they should be considered for an additional order i.
Pairs o f orders 1, 2 T = 15 periods. /' Dj
Replenishment no. 1 T = 7 periods
1
2
7
11
15
179
44
10
123
55
232
1,128
1,780
770
15 ( / - - 1)Dj
--
i'=2 j Dj
1
2
7
179
44
10
0
54
60
T (/. -- 1) D,/
The timing of the second replenishment is unchanged.
j=z
No change in policy is desirable for this replenishment.
Pairs o f orders 2, 3 T = 12 periods. Period
Replenishment No. 5 T = 6 periods
11
15
22
1
5
11
123
55
19
296
209
! Dj
11 Period
27
28
29
30
31
32
1
2
3
4
5
6
300
266
112
84
56
42
--
560
588
546
392
210
j Dj T (l'--l)Oj 1=2
Hence r = 2. The improved policy as a result of adding an order to the last replenishment is as follows: ti
Qi
1
11
22
25
27
29
233
178
19
190
566
294
(/-
1)Dj
--
j=2
The third replenishment should be made at period 15. Order quqntities for the second and third orders change to 123 and 74, respectively. Pairs of orders (3, 4), (4, 5) and (5, 6) were then considered. No change in the timing of the fourth, fifth and sixth replenishments was found to be desirable. The improved replenishment policy is as given below: ti
1
11
15
25
27
29
233
123
74
190
566
294
For this policy, total cost = $ 2,608.
Qi
Step (2)
Total cost = $2,521 Incidentally, the above policy is the optimal policy as obtained by the application of the Wagner-Whitin [1] algorithm to the problem.
Pairs of orders are considered next in order to improve the current replenishment schedule.
291 CONCLUSION
of applying the approach given in this paper. REFERENCES
The paper represents a simple approach for improving the heuristic lot sizes obtained on applying heuristic methods to the problem of determining economic replenishment for the dynamic inventory problem. The possible reduction in the total cost of the replenishment policy obtained by applying the suggested approach largely depends on the quality of the initial solution obtained by applying the heuristic method. If the initial solution has a total cost much larger than the optimal solution, then a significant reduction in cost can be achieved as a result
1 Wagner, H.M. and Whitin, T.M., 1958. Dynamic version of the economic lot size model. Manage. Sci., 5: 89-96. 2 Dixon, P. and Silver, E.A., 1981. A heuristic solution procedure for the multiitem, single level, limited capacity, lot sizing problem. J. Oper. Manage., 2: 23-40. 3 Silver, E.A. and Meal, H.C., 1973. A heuristic for selecting lot size quantities for the case of deterministic timevarying demand rate and discrete opportunities for replenishment. Prod. Inventory Manage., 14: 64-73. 4 Silver, E.A. and Miltenburg, J., 1984. Two modifications of the Silver-Meal lot sizing heuristic. INFOR, 22: 56-69.
{Received January 2, 1985; accepted in revised form December 5, 1985)