Dynamic Lot Sizing
35E00100 Service Operations and Strategy
#6 Fall 2015
Topics
Demand management
Lot sizing policies
Order management
Key points
Useful material:
Hopp, W. & Spearman, M. (2000), Factory Physics, Chapter 2.1-2.4 and 3.1.6
Nahmias, S. (2002) “Alternative Lot Sizing Schemes” Ch 7.2 in Production and Operations Analysis
Vollmann, T., W. Berry & C. Whybark (1997) “McLaren’s Order Moment” in Manufacturing Planning and Control
Systems
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Demand management is a part of
the MPC system!
MPC
boundary
Resource
planning
Production
planning
Demand
management
Master
production
scheduling
Marketplace
(customers and
other demand sources)
Front end
Engine
Back end
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Vollmann et al. 1997, 313
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Need for capacity management
depends on the market situation
Sales
forecasts
Business plan
FGI and
Backlog
Define
production
plan rates
Orders
Master
schedule
rough cut
Approvals
Master schedule
Vollmann et al. 1997
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Capacity and Demand Control Tools
Ways to manage capacity
 “Stretch” production capacity
 Speed up the process
 Schedule downtime (e.g. maintenance) during periods of low demand
 Squeeze more people in or rent / share extra facilities equipment
 Workforce management
 Employ part-timers, seasonal workers, flexible work force
 Cross-train employees
 Prepare intelligent schedules for both workers and equipment
Strategies for managing demand
 Organize better
 Avoid needless division of work (finance, customer service, transport planning, etc.)
 Design rules and procedures for providing flexibility
 Manage service levels
 Adjust delivery promises continuously
 Utilize different pricing methods
 Communicate capabilities
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Lot Sizing Schemes
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Example 1
Comparison of Lot Sizing Policies
A component used in a manufacturing facility is ordered from an
outside supplier. Because the component is used in a variety of
end products, the demand is high.
Estimated demand (in thousands) over the next 10 weeks is:
Week
Demand
1
22
2
34
3
32
4
12
5
8
6
44
7
54
8
16
9
76
10
30
Cost per component is 0.65.
The interest rate used to compute holding costs is 0.5 % per week.
The fixed ordering cost is estimated to be 200.
 What ordering policy you recommend and why?
 Which method would result in the lowest-cost policy for this
problem?
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Lot for Lot Ordering
Basic principle of LFL
 Production quantity = time-phased net requirements
 No inventory carried from one period to another
Normal assumption in MRP examples
 For convenience and ease of use
Rarely the optimal production rule
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e.g. Hopp and Spearman 2000, 125
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Economic Order Quantity (EOQ)
The principle
 Production quantity = EOQ quantity
 Information required
 Fixed setup/ ordering cost A
 Holding cost h
Q 
*
 Demand rate D
2 AD
h
Shortcomings are due to the assumptions of the modell
 Instantaneous production
 Immediate delivery
 Deterministic and constant demand over time
 Fixed setup cost
 Products can be analyzed individually
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e.g. Hopp and Spearman 2000, 49-56
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Periodic Order Quantity (POQ)
The principle
 Calculate the time between orders (TBO) using EOQ formula
TBO = EOQ / D
T 
*
2A
hD
 TBO (rounded to closest integer) shows for how many periods
products should be produced or ordered.
Fixed order period (FOP) is a similar method.
 Periods with no demand are skipped.
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e.g. Hopp and Spearman 2000, 126-127
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Part Period Balancing (PPB)
The principle
 Definition of a part-period
 [# of parts in a lot] * [# of period they are carried in inventory]
 Combines the procedure of Wagner-Whitin with the mechanics
of the EOQ
 Set the order horizon equal to the number of periods that most closely
matches the total holding cost with the set-up cost over that period
 Steps of the procedure
 Calculate holding costs per different number of periods
 Compare when holding cost is closest to set-up costs
 Stop and repeat
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e.g. Hopp and Spearman 2000, 127-128
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McLaren’s Order Moment (MOM)
The principle
 Evaluates the set-up cost for an integral number of future periods
 Applies part period (=an unit of inventory carried for one period)
accumulation principle directly
 Lot size is determined by matching the number of accumulated part
periods to the number that would be incurred if an order for an EOQ
were placed under conditions of constant demand
 Calculate order moment target (OMT)
 T *1


OMT  D   j  TBO  T T 
 j 1

 Two tests are used
T* = largest integer less than or equal to TBO
K = period currently under consideration
rj = requirement/demand for period j
 Tentatively order covers the requirements of periods (r) for which
T
  j  1 r
j 1
j
 OMT
 Once accumulated parts reach or exceed the OMT, test if one more
period should be included
h ( j  1) r j  A
e.g. Vollman et al. 1997, 445-446
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Least Unit Cost (LUC)
The principle
 Choose order horizon that minimizes the cost per unit of
demand
 Define C(T) as the average holding and set-up cost per unit if
the current order spans the next T periods
 Let (r1,…,rj) be the requirements over the j-period horizon
A
C (1) 
r1
A  hr2
C ( 2) 
r1  r2
 .
 .
A  hr2  2 hr3  ...  ( j  1) hrj
C ( j) 
r1  r2  ...  rj
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e.g. Nahmias 2001, 369-370
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Silver-Meal Heuristic (S-M)
The principle
 Minimize average cost per period over T-period order horizon
 Define C(T) as the average holding and set-up cost per period
if the current order spans the next T periods
If we place an order in period 1, for…
 r 1:
 r 2:
 r 3:
C (1)  A
A  hr2
C ( 2) 
2
A  hr2  2 hr3
C (3) 
3
A  hr2  2hr3  ...  ( j  1) hrj
C ( j) 
j
Once Cj > Cj-1 stop, and set Q1 = r1 + r2 +…+ rj-1 and begin process
again starting at period j
 In general rn:
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e.g. Nahmias 2001, 368-369
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Wagner-Whitin Heuristic
The principle (one-way network, path enumeration)
 Every path through the network = a specific exact requirement policy
 Assign a value to each arc in the network
 Determine minimum cost production schedule = shortest path through
the network
Heuristic that determines the optimal lot size
 Based on dynamic programming and two lemmas
 Lemma 1: “Exact requirement policy”
 An optimal policy has the property that each value of order quantities (Q) is exactly a
sum of a set of future demands
 Lemma 2: If optimal to produce something during period t, then it-1< rt
 No production / ordering during period t, if enough inventory to satisfy the demand
1
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3
4
15
5
e.g. Hopp and Spearman 2000, 59-64
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Example 1
Comparison of Lot Sizing Policies
A component used in a manufacturing facility is ordered from an
outside supplier. Because the component is used in a variety of
end products, the demand is high.
Estimated demand (in thousands) over the next 10 weeks is:
Week
Demand
1
22
2
34
3
32
4
12
5
8
6
44
7
54
8
16
9
76
10
30
Cost per component is 0.65.
The interest rate used to compute holding costs is 0.5 % per week.
The fixed ordering cost is estimated to be 200.
 What ordering policy you recommend and why?
 Which method would result in the lowest-cost policy for this
problem?
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Example 1
Costs of the Different Lot Sizing Policies
Compared
Week
Requirements
LFL
EOQ
POQ
PPB
MOM
LUC
S-M
W-W
1
22
22
64
56
56
56
56
56
56
2
34
34
0
0
0
0
0
0
0
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32
32
64
44
52
52
44
52
52
4
12
12
0
0
0
0
0
0
0
5
8
8
0
52
0
0
106
0
0
6
44
44
64
0
98
98
0
114
44
17
7
54
54
64
70
0
0
0
0
70
8
16
16
0
0
92
16
92
0
0
9
76
76
64
106
0
106
0
106
106
10
30
30
64
0
30
0
30
0
0
Cost
2 000
2 305
1 442
1 624
1 475
1 891
1 379
1 352
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Example 1
Ranking of the Policies
1. Wagner-Whitin (W-W)
2. Silver-Meal (S-M)
3. Periodic Order Quantity (POQ)
1352
1379
1442
4. McLaren's Order Moment (MOM)
5. Part Period Balancing (PPB)
6. Least Unit Cost (LUC)
1475
1624
1891
7. Lot-for-Lot (LFL)
2000
8. Economic Order Quantity (EOQ)
2305
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Differences in Lot Sizing Policies
Lot-for-lot
 Cost estimation or calculation is not required
 Least likely to result in capacity problems
 Likely to cause high changeover costs (produced / ordered every period)
EOQ
 A simple calculation technique
 Likely to produce cost-wise inefficient solutions if demand is not stable
Wagner-Whitin
 Gives the optimal solution for static problems at one level of the product
structure
 Under some other conditions the optimality is lost
 Relatively much calculations required
S-M, LUC, PPB, MOM
 Similar methods that give a reasonable compromise between the simple LFL
scheduling and the W-W heuristic
 PPB is easiest in terms of calculations
 S-M seems to provide the most cost effective solutions on average, and it
involves less work than the W-W heuristic
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Elements of Marketing Planning
Capacity
Orders
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Forecasting
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Customers
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Manufacturing-Marketing Collaboration
Problem area
Marketing complaints
Manufacturing complaints
Capacity planning and long-range
sales forecasting
Insufficient capacity
Lack of accurate long-range sales
forecasts
Production scheduling and short-range
sales forecasting
Excessive lead times
Unrealistic customer commitments and
mercurial short-range sales-forecasts
Delivery and physical distribution
Insufficient inventory
Excessive inventory requirements
Quality assurance
Insufficient quality at excessive cost
Too many options offered with
insufficient customer interest
Breadth of product line
Insufficient product variety to satisfy
customer demand
Excessive product variety necessitating
short, uneconomical production runs
Cost control
Excessive costs which hamper
competitiveness
Unrealistic requirements on quality,
delivery time, product variety and
response to change
New product introduction
New products are important
Unnecessary design changes are
expensive
Adjunct services e.g. spare parts
inventory support, installation and
repair.
Field service costs are excessive
Products should not be used in ways for
which they were not designed
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Shapiro 1977, 105
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Why Orders Fall through the Cracks?
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Shapiro et al. 1992, 105
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Key Points
Lot sizing policies
 Consider the trade-off between holding inventory and
changeover
 By adjusting setup costs, the planner can trade inventory for capacity
 Simple methods are popular in practice
 People prefer to understand the solution
 Heuristics are good because those are relatively robust and intuitive
 Costs versus responsiveness
Order management
 Information sharing and incentive alignment important
 Separate orders from customers
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Abbreviations Used
EOQ
FOP
LUC
MOM
MRP
OMT
PPB
POQ
S-M
TBO
W-W
=
=
=
=
=
=
=
=
=
=
=
economic order quantity
fixed order period
least unit cost
McLaren’s order moment
material requirements planning
order moment target
part period balancing
periodic order quantity
Silver-Meal heuristic
time between orders
Wagner-Whitin heuristic
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