Hierarchy of Production Decisions
• Forecasting: First, a firm must forecast demand for
aggregate sales over the planning horizon.
• Aggregate planning: The forecasts provide inputs for
determining aggregate production and workforce levels
over the planning horizon.
• Master production schedule (MPS): Recall, that the
aggregate production plan does not consider any “real”
product but a “fictitious” aggregate product. The MPS
translates the aggregate plan output in terms of specific
production goals by product and time period.
1
Hierarchy of Production Decisions
Forecast of Demand
Aggregate Planning
Master Production Schedule
Inventory Control
Operations Scheduling
Vehicle Routing
2
Hierarchy of Production Decisions
Suppose that a firm produces three types of
chairs:
1.ladder-back chair,
2.kitchen chair
3. desk chair.
The aggregate production considers a fictitious
aggregate unit of chair and finds that the firm
should produce 550 units of chairs in April. The
MPS then translates this output in terms of three
product types and four work-weeks in April.
3
Hierarchy of Production Decisions
• The MPS suggests that the firm produce
200 units of desk chairs in Week 1, 150
units of ladder-back chair in Week 2, and
200 units of kitchen chairs in Week 3.
• Material Requirements Planning (MRP):
A product is manufactured from some
components or subassemblies. For example
a chair may require two back legs, two front
legs, 4 leg supports, etc.
4
Hierarchy of Production Decisions
Master Production Schedule
April
1
Ladder-back chair
2
3
Aggregate
production plan
for chair family
4
5
150
6
7
8
150
200
Kitchen chair
Desk chair
May
200
120
120
200
550
200
790
5
Hierarchy of Production Decisions
While forecasting, aggregate plan and MPS consider the
volume of finished products, MRP plans for the
components, and subassemblies. A firm may obtain the
components by in-house production or purchasing. MRP
prepares a plan of in-house production or purchasing
requirements of components and subassemblies.
• Scheduling: Scheduling allocates resources over time in
order to produce the products. The resources include
workers, machines and tools.
• Vehicle Routing: After the products are produced, the firm
may deliver the products to some other manufacturers, or
warehouses. The vehicle routing allocates vehicles and
prepares a route for each vehicle.
6
Hierarchy of Production Decisions
Materials Requirement Planning
Back
legs
Back slats
Seat cushion
Leg supports
Seat-frame
boards
Front
legs
7
Material Requirements Planning
• The demands for the finished goods are obtained
from forecasting. These demands are called
independent demand.
• The demands for the components or subassemblies
depend on those for the finished goods. These
demands are called dependent demand.
• Material Requirements Planning (MRP) is used for
dependent demand and for both assembly and
manufacturing
• If the finished product is composed of many
components, MRP can be used to optimize the
inventory costs.
8
Importance of an MRP System
• Next two slides explain the importance of an MRP
system. The first one shows inventory levels when an
MRP system is not used. The next one shows the same
when an MRP system is used.
• The chart at the top shows inventory levels of the
finished goods and the chart on the bottom shows the
same of the components.
• If the production is stopped (like it is at the beginning of
the chart), the finished goods inventory level decreases
because of sales. However, the component inventory
level remains unchanged. When the production
resumes, the finished goods inventory level increases,
but the component inventory level decreases.
9
Importance of an MRP System
Inventory
without
an MRP
System
10
Importance of an MRP System
Inventory
with an
MRP
System
11
Importance of an MRP System
• Without an MRP system:
– Component is ordered at time A, when the inventory
level of the component hits reorder point, R
– So, the component is received at time B.
– However, the component is actually needed at time C,
not B. So, the inventory holding cost incurred
between time B and C is a wastage.
• With an MRP system:
– We shall see in this lesson that given the production
schedule of the finished goods and some other
information (see the next slide), it is possible to
predict the exact time, C when the component will be
required. Order is placed carefully so that it is
received at time C.
12
MRP Input and Output
• MRP Inputs:
– Master Production Schedule (MPS): The MPS of
the finished product provides information on the net
requirement of the finished product over time.
– Bill of Materials: For each component, the bill of
materials provides information on the number of units
required, source of the component (purchase/
manufacture), etc. There are two forms of the bill of
materials:
• Product Structure Tree: The finished product is
shown at the top, at level 0. The components
assembled to produce the finished product is
shown at level 1 or below. The sub-components
used to produce the components at level 1 is
13
MRP Input and Output
Master
Production
Schedule
Forecasts
Bill of
Materials
file
MRP
computer
program
Inventory
file
To Production
Reports
Orders
To Purchasing
14
MRP Input and Output
shown at level 2 or below, and so on.
The number in the parentheses shows the
requirement of the item. For example, “G(4)”
implies that 4 units of G is required to produce 1
unit of B.
The levels are important. The net requirements of
the components are computed from the low levels
to high. First, the net requirements of the
components at level 1 is computed, then level 2,
and so on.
15
MRP Input and Output
• Bill of Materials: For each item, the name,
number, source, and lead time of every
component required is shown on the bill of
materials in a tabular form.
– Inventory file: For each item, the number of units on
hand is obtained from the inventory file.
• MRP Output:
– Every required item is either produced or purchased.
So, the report is sent to production or purchasing.
16
Bill of Materials: Product Structure Tree
Level 0
Level 1
Level 2
Level 3
17
Bill of Materials
Item
B
C
D
E
BILL OF MATERIALS
Product Description: Ladder-back chair
Item: A
Component
Quantity
Source
Required
Description
Ladder-back
1
Manufacturing
Front legs
2
Purchase
Leg supports
4
Purchase
Seat
1
Manufacturing
18
Bill of Materials
Item
H
I
BILL OF MATERIALS
Product Description: Seat
Item: E
Component
Quantity
Required
Description
Seat frame
1
Seat cushion
1
Source
Manufacturing
Purchase
19
On Hand Inventory and Lead time
Component
Units in
Inventory
Lead
time
(weeks)
Seat
Subassembly
25
2
Seat frame
50
3
Seat frame
boards
75
1
20
MRP Calculation
• Suppose that 150 units of ladder-back chair is
required.
• The net requirement is computed from top to bottom.
Since 150 units of ladder-back chair is required, and
since 1 unit of seat subassembly is required for each
unit of ladder-back chair, the gross requirement of seatsubassembly is 150*1 =150 units. Since there are 25
units of seat-subassembly in the inventory, the net
requirement of the seat-subassembly is 150-25 = 125
units.
• Since 1 unit of seat frames is required for each unit of
seat subassembly, the gross requirement of the seat
frames is 125*1 = 125 units. Since there are 50 units of
seat frames in the inventory, the net requirement of the
21
seat frames is 125-50 = 75 units.
MRP Calculation: Some Definitions
• Scheduled Receipts:
– Items ordered prior to the current planning period
and/or
– Items returned from the customer
• Lot-for-lot (L4L)
– Order quantity equals the net requirement
– Sometimes, lot-for-lot policy cannot be used. There
may be restrictions on minimum order quantity or
order quantity may be required to multiples of 50, 100
etc.
22
Example 1:MRP Calculation
Level 0
Level 1
Level 2
23
Example 1:MRP Calculation
 Items A, C, D, and E have on-hand inventories of 20, 10,
20, and 10 units, respectively.
 Item B has a scheduled receipt of 10 units in period 1,
and C has a scheduled receipt of 50 units in Period 1.
 Lot-for-lot (L4L) is used for Items A and B. Item C
requires a minimum lot size of 50 units. D and E are
required to be purchased in multiples of 100 and 50,
respectively.
 Lead times are 1 period for Items A, B, and C, and 2
periods for Items D and E. The gross requirements for A
are 30 in Period 2, 30 in Period 5, and 40 in Period 8. Find
the planned order releases for all items.
24
MRP Calculation
Period
1
Item Gross
Requirements
A Scheduled
receipts
LT= On hand from
prior period
Net
requirements
Q= Time-phased Net
Requirements
Planned order
releases
Planned order
delivery
2
3
4
5
6
7
8
9
10
25
MRP Calculation
Period
1 2 3
Item Gross
30
Requirements
A Scheduled
receipts
LT= On hand from
20
prior period
1
Net
WK requirements
Q= Time-phased Net
Requirements
L4L Planned order
releases
Planned order
delivery
All the information above are given.
4
5
30
6
7
8
9
10
40
26
MRP Calculation
Period
1 2 3 4 5 6
Item Gross
30
30
Requirements
A Scheduled
receipts
LT= On hand from
20 20
prior period
1
Net
WK requirements
-Q= Time-phased Net
Requirements
L4L Planned order
releases
Planned order
delivery
20 units are just transferred from Period 1 to 2.
7
8
9
10
40
27
MRP Calculation
Period
1 2 3 4 5 6 7 8 9 10
Item Gross
30
30
40
Requirements
A Scheduled
receipts
LT= On hand from
20 20
prior period
1
Net
WK requirements
-- 10
Q= Time-phased Net
10
Requirements
L4L Planned order
10
releases
Planned order
10
delivery
28
The net requirement of 30-20=10 units must be ordered in week 1.
MRP Calculation
Period
1 2 3 4 5 6
Item Gross
30
30
Requirements
A Scheduled
receipts
LT= On hand from
20 20 0 0 0
prior period
1
Net
WK requirements
-- 10
Q= Time-phased Net
10
Requirements
L4L Planned order
10
releases
Planned order
10
delivery
On hand in week 3 is (20+10)-30=0 unit.
7
8
9
10
40
29
MRP Calculation
Period
1 2 3 4 5 6 7 8 9 10
Item Gross
30
30
40
Requirements
A Scheduled
receipts
LT= On hand from
20 20 0 0 0
prior period
1
Net
30
WK requirements
-- 10
Q= Time-phased Net
30
10
Requirements
L4L Planned order
30
10
releases
Planned order
10
30
delivery
30
The net requirement of 30-0=30 units must be ordered in week 4.
MRP Calculation
Period
1 2 3 4 5 6 7 8 9 10
Item Gross
30
30
40
Requirements
A Scheduled
receipts
LT= On hand from
20 20 0 0 0
0 0 0
prior period
1
Net
30
40
WK requirements
-- 10
Q= Time-phased Net
30
40
10
Requirements
L4L Planned order
30
40
10
releases
Planned order
10
30
40
delivery
31
The net requirement of 40-0=30 units must be ordered in week 7.
MRP Calculation
Period
1 2 3 4 5 6 7 8 9 10
Item Gross
30
30
40
Requirements
A Scheduled
receipts
LT= On hand from
20 20 0 0 0
0 0 0 0 0
prior period
1
Net
30
40
WK requirements
-- 10
Q= Time-phased Net
30
40
10
Requirements
L4L Planned order
30
40
10
releases
Planned order
10
30
40
delivery
32
The net requirement of 40-0=30 units must be ordered in week 7.
MRP Calculation
Period
1
Item Gross
Requirements
B Scheduled
receipts
LT= On hand from
prior period
Net
requirements
Q= Time-phased Net
Requirements
Planned order
releases
Planned order
delivery
2
3
4
5
Exercise
6
7
8
9
10
33
MRP Calculation
Period
1
Item Gross
Requirements
C
Scheduled
receipts
LT= On hand from
prior period
Net
requirements
Q= Time-phased Net
Requirements
Planned order
releases
Planned order
delivery
2
3
4
5
Exercise
6
7
8
9
10
34
MRP Calculation
Period
1
Item Gross
Requirements
D
Scheduled
receipts
LT= On hand from
prior period
Net
requirements
Q= Time-phased Net
Requirements
Planned order
releases
Planned order
delivery
2
3
4
5
Exercise
6
7
8
9
10
35
MRP Calculation
Period
1
Item Gross
Requirements
E
Scheduled
receipts
LT= On hand from
prior period
Net
requirements
Q= Time-phased Net
Requirements
Planned order
releases
Planned order
delivery
2
3
4
5
Exercise
6
7
8
9
10
36
LOT SIZING
• Lot Sizing Methods
– Lot-for-Lot (L4L)
– EOQ
– Silver-Meal Heuristic
– Least Unit Cost (LUC)
– Part Period Balancing
37
Lot-for-Lot
– Order production or purchasing as much as it
is needed.
– Exception are only the cases in which there are
constraints on the order quantity.
– For example, it may be required to order at
least 50 units or multiples 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!
38
Lot-for-Lot
• 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 concern of lot sizing methods.
39
Lot-Sizing Problem
• 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 and the inventory holding cost is
only charged on ending inventory of each
period.
40
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
41
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, SilverMeal 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 the holding cost is weekly, use
weekly demand.
42
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 up to which the average cost per period
decreases.
43
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.
44
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.
45
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
46
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
47
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
48
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
49
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
50
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
51
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
52
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.
53
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
54
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
55
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
56
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
57
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.
58
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.
59
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
0.01 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
60
=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
61
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
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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.
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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
0.01 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.
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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
0.01 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
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=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
66
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
67
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.
68
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
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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
70
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. 71
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
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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
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– 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.
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