LESSON 22: MATERIAL REQUIREMENTS
PLANNING: LOT SIZING
Outline
• Lot Sizing
• Lot Sizing Methods
– Lot-for-Lot (L4L)
– EOQ
– Silver-Meal Heuristic
– Least Unit Cost (LUC)
– Part Period Balancing
1
Lot-Sizing
• In Lesson 21
– We employ lot for lot ordering policy and order
production as much as it is needed.
– Exception are only the cases in which there are
constraints on the order quantity.
– For example, in one case we assume that at least 50
units must be ordered. In another case we assume that
the order quantity must be a multiple of 50.
• The motivation behind using lot for lot policy is minimizing
inventory. If we order as much as it is needed, there will
be no ending inventory at all!
2
Lot-Sizing
• However, lot for lot policy requires that an order be placed
each period. So, the number of orders and ordering cost
are maximum.
• So, if the ordering cost is significant, one may naturally try
to combine some lots into one in order to reduce the
ordering cost. But then, inventory holding cost increases.
• Therefore, a question is what is the optimal size of the lot?
How many periods will be covered by the first order, the
second order, and so on until all the periods in the
planning horizon are covered. This is the question of lot
sizing. The next slide contains the statement of the lot
sizing problem.
3
Lot-Sizing
• The lot sizing problem is as follows: Given net
requirements of an item over the next T periods, T >0, find
order quantities that minimize the total holding and
ordering costs over T periods.
• Note that this is a case of deterministic demand. However,
the methods learnt in Lessons 11-15 are not appropriate
because
– the demand is not necessarily the same over all
periods and
– the inventory holding cost is only charged on ending
inventory of each period
4
Lot-Sizing
• Although we consider a deterministic model, keep in mind
that in reality the demand is uncertain and subject to
change.
• It has been observed that an optimal solution to the
deterministic model may actually yield higher cost
because of the changes in the demand. Some heuristic
methods give lower cost in the long run.
• If the demand and/or costs change, the optimal solution
may change significantly causing some managerial
problems. The heuristic methods may not require such
changes in the production plan.
• The heuristic methods require fewer computation steps
and are easier to understand.
• In this lesson we shall discuss some heuristic methods.
The optimization method is discussed in the text,
Appendix 7-A, pp 406-410 (not included in the course). 5
Lot-Sizing
• Some heuristic methods:
– Lot-for-Lot (L4L):
• Order as much as it is needed.
• L4Lminimizes inventory holding cost, but maximizes
ordering cost.
– EOQ:
• Every time it is required to place an order, lot size
equals EOQ.
• EOQ method may choose an order size that covers
partial demand of a period. For example, suppose
that EOQ is 15 units. If the demand is 12 units in
period 1 and 10 units in period 2, then a lot size of
15 units covers all of period 1 and only (15-12)=3
units of period 2. So, one does not save the
ordering cost of period 2, but carries some 3 units in
6
Lot-Sizing
• Some heuristic methods:
the inventory when that 3 units are required in
period 2. This is not a good idea because if an order
size of 12 units is chosen, one saves on the holding
cost without increasing the ordering cost!
• So, what’s the mistake? Generally, if the order
quantity covers a period partially, one can save on
the holding cost without increasing the ordering
cost. The next three methods, Silver-Meal heuristic,
least unit cost and part period balancing avoid order
quantities that cover a period partially. These
methods always choose an order quantity that
covers some K periods, K >0.
• Be careful when you compute EOQ. Express both
holding cost and demand over the same period. If
the holding cost is annual, use annual demand. If
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the holding cost is weekly, use weekly demand.
Lot-Sizing
• Some heuristic methods:
– Silver-Meal Heuristic
• As it is discussed in the previous slide, Silver-Meal
heuristic chooses a lot size that equals the demand
of some K periods in future, where K>0.
• If K =1, the lot size equals the demand of the next
period.
• If K =2, the lot size equals the demand of the next 2
periods.
• If K =3, the lot size equals the demand of the next 3
periods, and so on.
• The average holding and ordering cost per period is
computed for each K=1, 2, 3, etc. starting from K=1
and increasing K by 1 until the average cost per
period starts increasing. The best K is the last one
8
up to which the average cost per period decreases.
Lot-Sizing
• Some heuristic methods:
– Least Unit Cost (LUC)
• As it is discussed before, least unit cost heuristic
chooses a lot size that equals the demand of some
K periods in future, where K>0.
• The average holding and ordering cost per unit is
computed for each K=1, 2, 3, etc. starting from K=1
and increasing K by 1 until the average cost per unit
starts increasing. The best K is the last one up to
which the average cost per unit decreases.
• Observe how similar is Silver-Meal heuristic and
least unit cost heuristic. The only difference is that
Silver-Meal heuristic chooses K on the basis of
average cost per period and least unit cost on
average cost per unit.
9
Lot-Sizing
• Some heuristic methods:
– Part Period Balancing
• As it is discussed before, part period balancing
heuristic chooses a lot size that equals the demand
of some K periods in future, where K>0.
• Holding and ordering costs are computed for each
K=1, 2, 3, etc. starting from K=1 and increasing K
by 1 until the holding cost exceeds the ordering
cost. The best K is the one that minimizes the
(absolute) difference between the holding and
ordering costs.
• Note the similarity of this method with the SilverMeal heuristic and least unit cost heuristic. Part
period balancing heuristic chooses K on the basis of
the (absolute) difference between the holding and
ordering costs.
10
Lot-Sizing
• Some important notes
– Inventory costs are computed on the ending inventory.
– L4L minimizes carrying cost
– Silver-Meal Heuristic, LUC and Part Period Balancing
are similar
– Silver-Meal Heuristic and LUC perform best if the costs
change over time
– Part Period Balancing perform best if the costs do not
change over time
– The problem extended to all items is difficult to solve
11
Lot-Sizing
Example 2: The MRP gross requirements for Item A are
shown here for the next 10 weeks. Lead time for A is three
weeks and setup cost is $10. There is a carrying cost of
$0.01 per unit per week. Beginning inventory is 90 units.
Week
1
2
3
4
5
Gross requirements Week
30
6
50
7
10
8
20
9
70
10
Determine the lot sizes.
Gross requirements
80
20
60
200
50
12
Lot-Sizing: Lot-for-Lot
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0
Inventory
Net
0 0 0 20
Requirements
Time-phased Net
Requirements
Planned order
Release
Planned
Deliveries
Ending
60 10 0
Inventory
13
Use the above table to compute ending inventory of various periods.
Lot-Sizing: Lot-for-Lot
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0
Inventory
Net
0 0 0 20
Requirements
Time-phased Net
20
Requirements
Planned order
Release
Planned
Deliveries
Ending
60 10 0
Inventory
14
Week 4 net requirement = 20 > 0. So, an order is required.
Lot-Sizing: Lot-for-Lot
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0
Inventory
Net
0 0 0 20
Requirements
Time-phased Net
20
Requirements
Planned order
20
Release
Planned
20
Deliveries
Ending
60 10 0
Inventory
15
A delivery of 20 units is planned for the 4th period..
Lot-Sizing: Lot-for-Lot
Exercise
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0 0
Inventory
Net
0 0 0 20 70
Requirements
Time-phased Net
20
Requirements
Planned order
20
Release
Planned
20
Deliveries
Ending
60 10 0 0
Inventory
16
The net requirement of the 5th period is 70 periods.
Lot-Sizing: EOQ
• First, compute EOQ
– Annual demand is not given. Annual demand is
estimated from the known demand of 10 weeks.
Estimated annual demand, 
Total demand over 10 weeks

 52 weeks/yea r
10
30  50  10  20  70  80  20  60  200  50

 52
10
590

 52
10
 3,068 units/year
– Compute annual holding cost per unit
h  $0.01/unit/week  $0.52/unit /year
17
Lot-Sizing: EOQ
• First, compute EOQ
  3,068 units/year
K  $10 /order
h  $0.52 /unit/year
2 K
2 10  3,068
EOQ 

 343.51  344 units
h
0.52
• Therefore, whenever it will be necessary to place an
order, the order size will be 344 units. This will now be
shown in more detail.
18
Lot-Sizing: EOQ
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0
Inventory
Net
0 0 0 20
Requirements
Time-phased Net
Requirements
Planned order
Release
Planned
Deliveries
Ending
60 10 0
Inventory
19
Use the above table to compute ending inventory of various periods.
Lot-Sizing: EOQ
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0
Inventory
Net
0 0 0 20
Requirements
Time-phased Net 20
Requirements
Planned order
Release
Planned
Deliveries
Ending
60 10 0
Inventory
20
Week 4 net requirement = 20 > 0. So, an order is required.
Lot-Sizing: EOQ
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0
Inventory
Net
0 0 0 20
Requirements
Time-phased Net 20
Requirements
Planned order
344
Release
Planned
344
Deliveries
Ending
60 10 0
Inventory
21
Order size = EOQ = 344, whenever it is required to place an order.
Lot-Sizing: EOQ
Exercise
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0 324
Inventory
Net
0 0 0 20
Requirements
Time-phased Net 20
Requirements
Planned order
344
Release
Planned
344
Deliveries
Ending
60 10 0 324
Inventory
22
Week 5 b. inv=344-20=324>70= gross req. So, no order is required.
Lot-Sizing: Silver-Meal-Heuristic
j
rj
Order for weeks
1 week, week 4
2 weeks, weeks 4 to 5
3 weeks, weeks 4 to 6
4 weeks, weeks 4 to 7
5 weeks, weeks 4 to 8
6 weeks, weeks 4 to 9
7 weeks, weeks 4 to 10
1
2
3
4
5
6
7
20 70 80 20
60 200 50
Units in the inventory at the end of Week
Q
4
5
6
7
8
9 10
Per
H. Ord. Period
Cost Cost Cost
The order is placed for K periods, for some K>0. Use the above table
to find K.
23
Lot-Sizing: Silver-Meal-Heuristic
j
rj
Order for weeks
1 week, week 4
2 weeks, weeks 4 to 5
3 weeks, weeks 4 to 6
4 weeks, weeks 4 to 7
5 weeks, weeks 4 to 8
6 weeks, weeks 4 to 9
7 weeks, weeks 4 to 10
1
2
3
4
5
6
7
20 70 80 20
60 200 50
Units in the inventory at the end of Week
Q
4
5
6
7
8
9 10
20
Per
H. Ord. Period
Cost Cost Cost
0.00 10 10.0
If K=1, order is placed for 1 week and the order size = 20. Then, the
ending inventory = inventory holding cost =0. The order cost = $10.
24
Average cost per period = (0+10)/1=$10.
Lot-Sizing: Silver-Meal-Heuristic
Exercise
j
rj
Order for weeks
1 week, week 4
2 weeks, weeks 4 to 5
3 weeks, weeks 4 to 6
4 weeks, weeks 4 to 7
5 weeks, weeks 4 to 8
6 weeks, weeks 4 to 9
7 weeks, weeks 4 to 10
1
2
3
4
5
6
7
20 70 80 20
60 200 50
Units in the inventory at the end of Week
Q
4
5
6
7
8
9 10
20
90 70
Per
H. Ord. Period
Cost Cost Cost
0.00 10 10.0
0.70 10 5.35
If K=2, order is placed for 2 weeks and the order size = 20+70=90.
Then, inventory at the end of week 4 = 90-20=70 and holding cost
25
=70 0.01. = 0.70. Average cost per period = (0.70+10)/2=$5.35.
Lot-Sizing: Silver-Meal-Heuristic
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0
Inventory
Net
0 0 0 20
Requirements
Time-phased Net
Requirements
Planned order
Release
Planned
Deliveries
Ending
60 10 0
Inventory
26
Use the above table to compute ending inventory of various periods.
Lot-Sizing: Silver-Meal-Heuristic
Exercise
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0
Inventory
Net
0 0 0 20
Requirements
Time-phased Net 20
Requirements
Planned order
Release
Planned
Deliveries
Ending
60 10 0
Inventory
27
Week 4 net requirement = 20 > 0. So, an order is required.
Lot-Sizing: Least Unit Cost
j
rj
Order for weeks
1 week, week 4
2 weeks, weeks 4 to 5
3 weeks, weeks 4 to 6
4 weeks, weeks 4 to 7
5 weeks, weeks 4 to 8
6 weeks, weeks 4 to 9
7 weeks, weeks 4 to 10
1
2
3
4
5
6
7
20 70 80 20
60 200 50
Units in the inventory at the end of Week
Q
4
5
6
7
8
9 10
H. Ord. Unit
Cost Cost Cost
The order is placed for K periods, for some K>0. Use the above table
to find K.
28
Lot-Sizing: Least Unit Cost
j
rj
Order for weeks
1 week, week 4
2 weeks, weeks 4 to 5
3 weeks, weeks 4 to 6
4 weeks, weeks 4 to 7
5 weeks, weeks 4 to 8
6 weeks, weeks 4 to 9
7 weeks, weeks 4 to 10
1
2
3
4
5
6
7
20 70 80 20
60 200 50
Units in the inventory at the end of Week
Q
4
5
6
7
8
9 10
20
H. Ord. Unit
Cost Cost Cost
0.00 10 .500
If K=1, order is placed for 1 week and the order size = 20. Then, the
ending inventory = inventory holding cost =0. The order cost = $10.
29
Average cost per unit = (0+10)/20=$0.50
Lot-Sizing: Least Unit Cost
Exercise
j
rj
Order for weeks
1 week, week 4
2 weeks, weeks 4 to 5
3 weeks, weeks 4 to 6
4 weeks, weeks 4 to 7
5 weeks, weeks 4 to 8
6 weeks, weeks 4 to 9
7 weeks, weeks 4 to 10
1
2
3
4
5
6
7
20 70 80 20
60 200 50
Units in the inventory at the end of Week
Q
4
5
6
7
8
9 10
20
90 70
H. Ord. Unit
Cost Cost Cost
0.00 10 .500
0.70 10 .119
If K=2, order is placed for 2 weeks and the order size = 20+70=90.
Then, inventory at the end of week 4 = 90-20=70 and holding cost
30
=70 0.01. = 0.70. Average cost per unit = (0.70+10)/90=$0.119.
Lot-Sizing: Least Unit Cost
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0
Inventory
Net
0 0 0 20
Requirements
Time-phased Net
Requirements
Planned order
Release
Planned
Deliveries
Ending
60 10 0
Inventory
31
Use the above table to compute ending inventory of various periods.
Lot-Sizing: Least Unit Cost
Exercise
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0
Inventory
Net
0 0 0 20
Requirements
Time-phased Net 20
Requirements
Planned order
Release
Planned
Deliveries
Ending
60 10 0
Inventory
32
Week 4 net requirement = 20 > 0. So, an order is required.
Lot-Sizing: Part Period Balancing
j
rj
Order for weeks
1 week, week 4
2 weeks, weeks 4 to 5
3 weeks, weeks 4 to 6
4 weeks, weeks 4 to 7
5 weeks, weeks 4 to 8
6 weeks, weeks 4 to 9
7 weeks, weeks 4 to 10
1
2
3
4
5
6
7
20 70
80 20 60 200 50
Units in the inventory at the end of Week
Q
4
5
6
7
8
9 10
H. Ord.
Cost Cost
The order is placed for K periods, for some K>0. Use the above table
to find K.
33
Diff
Lot-Sizing: Part Period Balancing
j
rj
Order for weeks
1 week, week 4
2 weeks, weeks 4 to 5
3 weeks, weeks 4 to 6
4 weeks, weeks 4 to 7
5 weeks, weeks 4 to 8
6 weeks, weeks 4 to 9
7 weeks, weeks 4 to 10
1
2
3
4
5
6
7
20 70
80 20 60 200 50
Units in the inventory at the end of Week
Q
4
5
6
7
8
9 10
20
90
170
190
250
450
1 week, week 9
200
2 weeks, weeks 9 to 10 250
70
150 80
170 100 20
230 160 80 60
430 360 280 260 200
NOT COMPUTED
50
H. Ord.
Cost Cost
0.00
0.70
2.30
2.90
5.30
15.30
10
10
10
10
10
10
Diff
10.0
9.30
7.70
7.10
4.70
5.30
0.00 10 10.0
0.50 10 9.50
The above computation is similar to that of the Silver-Meal heuristic.
The primary difference is that the (absolute) difference between
34
holding and ordering cost is shown in the last column.
Lot-Sizing: Part Period Balancing
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0
Inventory
Net
0 0 0 20
Requirements
Time-phased Net
Requirements
Planned order
Release
Planned
Deliveries
Ending
60 10 0
Inventory
35
Use the above table to compute ending inventory of various periods.
Lot-Sizing: Part Period Balancing
Period
1 2 3 4 5 6 7 8 9 10
Gross
30 50 10 20 70 80 20 60 200 50
Requirements
Beginning
90 60 10 0 230 160 80 60 0 50
Inventory
200
Net
0 0 0 20
Requirements
Time-phased Net 20
200
Requirements
Planned order
250
250
Release
Planned
250
250
Deliveries
Ending
60 10 0 230 160 80 60 0 50 0
Inventory
The computation is similar to that of the Silver-Meal heuristic. 36
Cost Comparison
• Lot-for-Lot
– See the last slide entitled “lot-sizing: lot-for-lot”
– Number of orders: 7
– Ordering cost = 7  $10/order = $70
– Holding cost = (60+10)  $0.01/unit/week = $0.70
– Total cost = 70+0.70 =$70.70
• EOQ
– See the last slide entitled “lot-sizing: EOQ”
– Number of orders: 2
– Ordering cost = 2  $10/order = $20
– Holding cost = (60 +10 +324 +254 +174 +154 +94
+237 +187)  $0.01/unit/week = $14.94
– Total cost = 20+14.94 =$34.94
37
Cost Comparison
• Silver-Meal Heuristic
– See the last slide entitled “lot-sizing: Silver-Meal
heuristic”
– Number of orders: 2
– Ordering cost = 2  $10/order = $20
– Holding cost = (60 +10 +230 +160 +80 +60 +50) 
$0.01/unit/week = $6.50
– Total cost = 20+6.50 =$26.50
• Least Unit Cost
– See the last slide entitled “lot-sizing: least unit cost”
– Number of orders: 2
– Ordering cost = 2  $10/order = $20
– Holding cost = (60 +10 +430 +360 +280 +260 +200) 
$0.01/unit/week = $16.00
38
– Total cost = 20+16.00 =$36.00
Cost Comparison
• Part-Period Balancing
– See the last slide entitled “lot-sizing: part-period
balancing”
– Number of orders: 2
– Ordering cost = 2  $10/order = $20
– Holding cost = (60 +10 +230 +160 +80 +60 +50) 
$0.01/unit/week = $6.50
– Total cost = 20+6.50 =$26.50
• Conclusion: In this particular case, Silver-Meal heuristic
and part period balancing yield the least total holding and
ordering cost of $26.50 over the planning period of 10
weeks.
39
READING AND EXERCISES
Lesson 22
Reading:
Section 7.2-7.3 pp. 366-375 (4th Ed.), pp. 358-366
(5th Ed.)
Exercise:
17 and 25 pp. 371-373, 375 (4th Ed.), pp. 363, 366
40