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FORECASTING DEMAND
Chapter 2
1
The Importance of Forecasts
• Used as inputs for

budgeting
capacity planning
inventory management
staffing decisions
• Areas that use forecasts
– New facility planning
– Production planning
– Workforce scheduling
2
The Forecasting Criteria
1. Accuracy
– How well a forecasted condition mirrors
the actual condition?
2. Simplicity of computation
3. Flexibility to adjust the response rate
– How adaptable the forecast is to changing
conditions?
3
The Conceptual Forecasting
Framework
Historical
Data
Selection and
Initialization
of Model
Mathematical
Model
1. Determine the forecast
purpose
2. Establish the forecast
interval
3. Select a forecasting
technique
4. Collect and analyze data
5. Initialize the forecast
6. Generate the forecast
7. Monitor the forecast quality
Statistical
Forecast
Forecast of
Demand
Calculation
of Forecast
Error
4
Illustration of Forecasting process
Historical Demand Data
A Forecast
of Future
Events
Time
Building the Forecast Model
Test the Quality
of Model
Past
Forecast Initialization
Future
5
Measures of Forecast Accuracy
(Forecast Model Quality)
• Average differences between
– Observed values
– Forecasted values
• Forecast must be valid and lack bias
– Magnitude
– Pessimistic or optimistic model
performance
6
Forecasting measurement criteria
1. Mean error (BIAS)
7
Mean Error (Bias)
8
Example of Mean Error Calculation
– Model: two-period simple moving average
– Demand Data: 12 periods
– Forecast Data: 10 periods starting with period 3
9
Example of Mean Error (cont.)
• The magnitude of the bias is 4.1 units.
The ideal forecast bias is 0.0 units.
• The model forecast is pessimistic.
The ideal forecast should not be
consistently high or low.
10
• Magnitude of the individual forecast
errors
11
• Estimating Standard deviation from
12
– Model: two-period simple moving average
– Demand Data: 12 periods
– Forecast Data: 10 periods starting with
period 3
13
• The magnitude of the bias is 5.9 units.
The ideal forecast bias is 0.0 units
14
Forecasting Approaches
• Type of Forecasting Approaches
–Qualitative
–Quantitative
15
Qualitative Forecast Models
• Judgment or opinion forecasts
– Based on human factors
• executive opinions
• consumer surveys
• expert opinions
• Examples of approaches
– The Delphi method
– Survey of customers
– Market research
16
Quantitative Forecasting Models
• Time series forecasts
– most popular forecast models
– uncover the underlying patterns of the
time series data
• A time series:
– a set of numbers in which the order or
sequence of the numbers is important
– e.g. historical demand
17
Components of time series forecast
data
1. Trend:
18
Components of time series forecast
2. Seasonality
19
Components of time series forecast
3. Cyclical
20
Components of Time Series Forecast
4. Irregular variations
5.Random fluctuation
21
Quantitative Forecasting Techniques
A. Linear regression
B. Simple moving average
C. Weighted moving average
D. Exponential smoothing
i. Simple exponential smoothing
ii. Trend-enhanced exponential smoothing
iii. Seasonality-enhanced exponential
smoothing
iv. Trend and seasonality exponential
smoothing

Can forecast more than 1 period in the future
22
A. Linear Regression
• Fitting the available data to form a trend
line
– future demands would be around the trend
line
• Ordinary least squares (OLS) method
23
A. Linear Regression (cont.)
• Coefficient of determination (R2)
– Trend line’s fit to the data
– R2 =0.8 means
• 80 percent of the variation in the actual demand
data is explained by the model with the trend line
24
B. Simple Moving Averages
• Uses recent historical date to
forecast
– Smoothing of the random changes of
demand
• Characteristics
– A predetermined number of periods
– Only for a period immediately after the most
recent data
– When the demand is relatively stable and no
evidence of trend or seasonality
25
B. Simple Moving Averages (cont.)
• MAF(n)t : n-period moving-average
forecast made at the end of period t
– same as the forecast of the demand for period
t +1
26
Example of Simple Moving Averages
2 Period:
3 Period:
6 Period:
27
C. Weighted Moving Averages
• Applies weights to recent historical
data
– WMAF(n)t : n-period weighted movingaverage forecast made at the end of
period t
28
Example of Weighted Moving
Averages
• Assigning importance to more recent data
3 Period:
29
D. The Exponential Smoothing
Model
• The moving-average model assumes
– Actual demand data has no value after
n periods
– Excessive data availability
• Smoothing forecasting
– More elegant and parsimonious
30
Exponential Smoothing Techniques
i. Simple exponential smoothing
ii. Trend-enhanced exponential
smoothing
iii.Seasonality-enhanced exponential
smoothing
iv. Trend and seasonality exponential
smoothing
31
i. Simple Exponential Smoothing Model
• Uses past demands to forecast
future values as a weighted
decreasing average of all available
past actual demand data
¯
– 𝑭𝒕+𝟏 = 𝑭𝒕 = 𝑬𝑺𝑭𝒕
• forecast of the expected demand for
period (t+1) made at the end of the
period t
32
Simple Exponential Smoothing
Model Equation
𝐸𝑆𝐹𝑡 = 𝐸𝑆𝐹𝑡−1 + 𝛼 × 𝐷𝑡 − 𝐸𝑆𝐹𝑡−1
= 𝛼 × 𝐷𝑡 + 1 − 𝛼 × 𝐸𝑆𝐹𝑡−1
¯
𝐸𝑆𝐹𝑡 𝑜𝑟 𝐹𝑡 𝑜𝑟 𝐹𝑡+1 = 𝑓𝑜𝑟𝑒𝑐𝑎𝑠𝑡𝑒𝑑 𝑑𝑒𝑚𝑎𝑛𝑑 𝑓𝑜𝑟 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡 + 1
𝑎𝑡 𝑡ℎ𝑒
𝐷𝑡 = 𝐴𝑐𝑡𝑎𝑢𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 𝑑𝑢𝑟𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑
𝑡 𝑒𝑛𝑑 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑
𝛼 = 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 𝑠𝑚𝑜𝑜𝑡ℎ𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑜 ≤ 𝛼 ≤ 1
𝑡
33
ii. Exponential Smoothing Model
With Trend Enhancement
• Difference with Simple exponential
smoothing model
– Does not assume that demand is constant,
with random variation around the mean
– Use Trend Enhancement for data displaying
trend patterns
• 3 Step Procedure
34
Step1. Calculating the Base value for period t
Base valuet
¯
𝐹 = 𝛼 𝐴𝑐𝑡𝑢𝑙𝑎 𝐷𝑒𝑚𝑎𝑛𝑑𝑡 +
1 − 𝛼 𝐵𝑎𝑠𝑒 𝑣𝑎𝑙𝑢𝑒𝑡−1 + 𝑇𝑟𝑒𝑛𝑑𝑡−1
¯
Base valuet-1=𝑭𝒕−𝟏
35
Example of Step 1.
• Actual sales volume figures for May is
560
• α= 0.2 β=0.1
¯
• TrendApril=50, 𝑭𝑨𝒑𝒓𝒊𝒍 = 𝟓𝟎𝟎
Solution for the Base value at period May
36
Step2. Updating Trend Estimate
(Trendt)
𝑻𝒓𝒆𝒏𝒅𝒕 = 𝜷 𝑩𝒂𝒔𝒆 𝒗𝒂𝒍𝒖𝒆𝒕 − 𝑩𝒂𝒔𝒆 𝒗𝒂𝒍𝒖𝒆𝒕−𝟏
+ 𝟏 − 𝜷 × 𝑻𝒓𝒆𝒏𝒅𝒕−𝟏
37
Example of Step 2.
• Actual sales volume figures for May is
560.
• α= 0.2
¯
β=0.1 , 𝑭𝑨𝒑𝒓𝒊𝒍 = 𝟓𝟎𝟎
¯
• TrendApril=50, 𝑭𝑴𝒂𝒚 = 𝟓𝟓𝟐
Solution for the Trend at period May
38
Step 3. Calculating the trendenhanced forecast
𝑭𝒐𝒓𝒆𝒄𝒂𝒔𝒕𝒕+𝒏
= 𝑩𝒂𝒔𝒆 𝒗𝒂𝒍𝒖𝒆𝒕 + 𝒏 × 𝑻𝒓𝒆𝒏𝒅𝒕
• From Step 1: Base valuet
• From Step 2: Trendt
39
Example of Step 3.
• Actual sales volume for May is 560
• α= 0.2
¯
• TrendApril=50, 𝑭𝑨𝒑𝒓𝒊𝒍 = 𝟓𝟎𝟎
• From Steps 1 and 2
¯
– 𝑭𝑴𝒂𝒚 = 𝟓𝟓𝟐
– 𝑻𝒓𝒆𝒏𝒅𝑴𝒂𝒚 = 𝟓𝟎. 𝟐
Solution for the Forecast for period June
40
iii. Exponential Smoothing Model
With Seasonality
• The 5 Steps
1. Plot the data and visually determine any
obvious time-series characteristics
2. Determine if a significant seasonal trend
exists
3. Deseasonalize the data arithmetically
4. Develop a seasonality-enhanced forecast
model
5. Continuously update the model as new data
41
Seasonality
• Demand patterns
and Seasons
– Ex) Total 4 seasons
Per year
• s=1,2,3,4
625000
500000
375000
250000
125000
0
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
s=1
s=2
s=3
s=4
• Same seasons are repeated every cycle
– Ex) year
• The number of periods t within a cycle is the same
as number of seasons
– ex) If there are 2 cycles in a year cycle and it is currently
season (s) 1 in the 3rd year, t value is 5.
42
Seasonal Index
• Measure of the degree of seasonal
variation
– How much demand during a season will
be above or below the item’s average
demand?
– Used to adjust the Exponential
Smoothing Model forecast for seasonal
patterns
43
Determining the Seasonal Index
OLD INDEX
NEW
OLDINDEX
INDEX
S1
S2
S3
S4
Seasons within a cycle
NEW INDEX
Cycle
Cycle
Cycle
Cycle
Time
44
Notation for Seasonal Index
OLD INDEX
New Index
NEW INDEX
S1
S2
S3
S4
Current cycle
Old Index
S1-L
Cycle
Cycle
S2-L
S3-L
S4-L
Past cycle
Time
45
3 Steps of Seasonality enhanced
Forecasts
• At the end of period t which is season s
– Step 1: Generate a Base value forecast for 1
¯
– Step 2: Update the Seasonality Index (𝑺𝒕 ) for
the current period t which is season s
• Season s is a season among the many in the
cycle
– Step 3: Generate a Seasonality enhanced
forecast for the next period 𝑭𝒕+𝟏
• Period t+1
• Season s+1 or season 1 if season s is the last
in the cycle
46
Step 1: Generating a Base value
forecast
¯
𝑭𝒕 𝒐𝒓 𝑩𝒂𝒔𝒆 𝒗𝒂𝒍𝒖𝒆𝒕 = 𝜶 𝑨𝒄𝒕𝒖𝒂𝒍 𝒅𝒆𝒎𝒂𝒏𝒅𝒕 Τ× 𝑶𝒍𝒅 𝑰𝒏𝒅𝒆𝒙𝒕
+ 𝟏 − 𝜶 × 𝑩𝒂𝒔𝒆 𝒗𝒂𝒍𝒖𝒆𝒕−𝟏
¯
𝐹𝑡 = 𝐵𝑎𝑠𝑒 𝑣𝑎𝑙𝑢𝑒𝑡 = 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 𝑣𝑎𝑙𝑢𝑒 𝑓𝑜𝑟 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡 + 1 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑎𝑡 𝑡ℎ𝑒 𝑒𝑛𝑑 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡
¯
𝐹𝑡−1 = 𝐵𝑎𝑠𝑒 𝑣𝑎𝑙𝑢𝑒𝑡−1
= 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑
=seasonal
𝑎𝑡 𝑡ℎ𝑒 𝑒𝑛𝑑 𝑜𝑓 𝑙𝑎𝑠𝑡 𝑝𝑒𝑟𝑖𝑜𝑑
𝑡 − 1index value for season t of a full cycle in the past
𝐷𝑡 = 𝐴𝑐𝑡𝑎𝑢𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 𝑑𝑢𝑟𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡
47
Illustration of Base Value Forecast
Process
S1
S2
S3
S4
Seasons within a
cycle
Base value
Base value
Cycle
Cycle
Cycle
Cycle
Time
48
𝒕
Step 2: Updating the Index
𝒐𝒓 𝑺𝒕 = 𝜸 𝑨𝒄𝒕𝒖𝒂𝒍 𝒅𝒆𝒎𝒂𝒏𝒅𝒕 Τ𝑩𝒂𝒔𝒆 𝑽𝒂𝒍𝒖𝒆𝒕 + 𝟏 −
=seasonal index value for period t for the current cycle
¯
𝐹𝑡 = 𝐵𝑎𝑠𝑒 𝑣𝑎𝑙𝑢𝑒𝑡 = 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 𝑣𝑎𝑙𝑢𝑒 𝑓𝑜𝑟 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡 + 1 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑎𝑡 𝑡ℎ𝑒 𝑒𝑛𝑑 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡
𝐷𝑡 = 𝐴𝑐𝑡𝑎𝑢𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 𝑑𝑢𝑟𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡
=seasonal index value for season t of a full cycle in the past
49
Illustration of Seasonal Index
Updating Process
Update
S1
S2
S3
S4
Seasons within a cycle
Update
Cycle
….
Update
Cycle
Cycle
Cycle
Time
50
Step 3: Seasonality enhanced forecast
• Forecast for 1 period in the future
¯
at the end of period t (𝑭 𝒕,𝑺 𝒐𝒓 𝑭𝒕+𝟏 )
– Currently season s
– Forecasting for period t+1 in which the
season is either s+1 or 1
• depending on the cycle location of s
¯
𝐹𝑡,𝑠 = 𝐵𝑎𝑠𝑒 𝑣𝑎𝑙𝑢𝑒𝑡 × 𝑂𝑙𝑑 𝐼𝑛𝑑𝑒𝑥𝑠+1
51
Illustration of Seasonality enhanced
Forecasts
S1
Update
Forecast
S2
S3
S4
Seasons within a
cycle
Forecast
Cycle
Cycle
Cycle
Cycle
Time
52
Example
Generate a Seasonality enhanced forecast for
¯
quarter 7 at the end of quarter 6 (𝑭 𝟔,𝟐 = 𝑭𝟕,𝟑 )
– quarter 6 is season 2
– quarter 7 is season 3
– 𝜶 = 𝟎. 𝟐 𝜸 = 𝟎. 𝟑 𝑶𝒍𝒅 𝑰𝒏𝒅𝒆𝒙𝟐 = 𝑺𝟐−𝑳 = 𝟏. 𝟓
– 𝑶𝒍𝒅 𝑰𝒏𝒅𝒆𝒙𝟑 = 𝑺𝟑−𝑳 = 𝟎. 𝟖
𝑫𝟐 = 𝟗𝟎
¯
– 𝑭𝟏 = 𝟏𝟎𝟎
S1-L
S1
S2-L
S2
S3-L
S3
S4-L
S4
Past Year
Quarter 7
Current Year
53
Example
¯
Step 1: 𝑭𝟔 = 𝟎. 𝟐 ×
𝟗𝟎
𝟏.𝟓
Step 2: 𝑺𝟐 = 𝟎. 𝟑 ×
𝟗𝟎
𝟗𝟐
+ 𝟎. 𝟖 × 𝟏𝟎𝟎=92
+ 𝟎. 𝟕 × 𝟏. 𝟓 =
𝟏. 𝟑𝟒
Step 3: 𝑭𝟕 = 𝟗𝟐 × 𝟎. 𝟖 = 𝟕𝟑. 𝟔
54
Exponential Smoothing Model With
Seasonality and Trend
Step 1: Base value
Step 2: Seasonal Index
Step 3: Trend
Step 4: Forecast for n periods in the future
at the end of period t
55
Initializing Exponential Smoothing
Models
• Base Forecast Value at end of period n
𝒏
1
𝐹𝒏 = ෍ 𝑑𝑖
𝒏
¯
𝑖=1
• Trend Value at end of period n
𝒏−𝟏
1
𝑇𝒏 =
෍ 𝑑𝑖+1 − 𝑑𝑖
𝒏−𝟏
𝑖=1
56
Initializing Exponential Smoothing
Models (cont.)
• Seasonality Index for period t at
end of period n
𝑺𝒕 =
𝒅𝒕
¯
𝑭𝒏
– period t and n are in the initial cycle
57
Questions?
58
MASTER PRODUCTION
SCHEDULING
Chapter 4
1
Master Production Schedule (MPS)
• An anticipated schedule of demand
– Tool for controlling product availability
• Disaggregation of the aggregate plan
– More detail than aggregate plan
– Focusing on distinct end items
– Finer time intervals (e.g. weeks)
2
Master Production Schedule (MPS) (cont.)
• Input
– beginning inventory
– sales forecast for a particular end item
• Output
– Production per period needs to meet
anticipated customer demand
• trade-off between costs and product
availability
3
The MPS Interfaces
Marketing
Planning
Demand
Management
Financial
Planning
Planning
S&OP
Supply chain
planning
Aggregate
Plan
Production
planning
Capacity
Planning
MPS
Material
requirements
planning (MRP)
4
Example of Disaggregating the
Aggregate Plan
• Products: Tractors
• Aggregate Plan
– In January, produce a total of 100 tractors
– In February, produce a total of 400 tractors
• Disaggregation
– Total 8 weeks (4 Weeks per Month)
– Two Models (A & B)
– Demand composition of Models
(A:40%,B:60%)
5
Example Continue…
• Aggregate Plan
• Disaggregation into a forecast of
MPS needs =100
=400
6
The Planning Horizon
• The length of the MPS plan
– Covers at least the lead time required to
fabricate the MPS items
– Months, Quarters
• Includes

Production time
Procurement time
Engineering time for custom environments
Delivery-to-customer response times
7
Bill of material (BOM)
• List of the materials needed to
produce the end product
J750 Engine
Assembly
Turbine Housing A
Used: 1 per engine
LEVEL 1
LEVEL 0
Turbine Housing B
Used: 1 per engine
LEVEL 1
Turbine Assembly
Used: 2 per housing
LEVEL 2
8
Phases of Master Schedule
(MPS)
1. Designing the MPS
2. Creating the MPS
3. Controlling the MPS
9
1. Designing the MPS
• Steps

Select the items
Organize the MPS by product groups.
Determine the planning horizon
Select the available-to-promise (ATP)
method
10
2. Creating the MPS
• Steps
– Obtain the necessary informational
inputs
– Prepare the initial MPS draft
– Rough-cut capacity requirements plan
– If required, increase capacity or revise
the initial draft of the MPS
11
3. Controlling the MPS
• Activities
– Track actual production and
determine if the planned MPS
quantities and delivery promises are
being met
– Calculate the ATP
– Calculate the projected on-hand
12
A Time-phased MPS Record
• MPS Record Items
– Forecast Demand
– Available
• (MPS + On hand – Forecast)
– MPS
• MPS Record Initial factors
– On hand Inventory
– Safety Stock (SS)
– MPS Quantity
13
Example
• MPS Quantity: 130 units with 1 Quarter lead
time
• On hand:80 units
Planning Horizon
QUARTER
1
2
3
4
Forecast
70
120
120
230
Available
10
130
20
30
60
130
260
MPS
On hand
80
(130+10-120)=20
(0+80-70)=10
(260+30-230)=60
(130+20-120)=30
14
Organizational Environments
• Make to Stock
• Make to Order
• Assemble to Order
15
Final Assembly Schedule
Make to Order
Order
Acceptance
Order
Acceptance

Make to Stock
Time
Order
Acceptance
Only Raw Materials
Final subassemblies
Components
Assemble to Order
Minor subassemblies
16
Final Assembly Schedule (cont.)
• Beginning of the production lead time
– Raw materials, components, minor
subassemblies
– Major subassemblies not yet committed to a
specific end item configuration
• Final assembly schedule (FAS)
– A time fence marking the start of
committing final assemblies to a specific
configuration
– New orders cannot be accepted in the FAS
17
Final Assembly Schedule (cont.)
• FAS length is determined by the firm
and technology
– Depends on how well the engineering
department can design the parts and of the
technology available to the final assembly
operations
18
Order Acceptance Ratio
• Ratio of the time between
– order acceptance
– beginning of the FAS to the production lead
time less FAS time
• If this ratio is near unity, close to make
to order
19
Available to Promise (ATP)
• MPS’ functions as a control
– A standard for evaluating actual
production
– To calculate Available-to-promise (ATP)
– To estimate supply conditions through
projected on-hand inventory
20
ATP Value for Immediate Time
Period
• At the end of the period
– On-hand inventory + Scheduled
production that is not yet promised to
actual customer orders
21
ATP Value for Subsequent Time
Periods
• At the end of the period
– Scheduled production – Quantities
allocated to specific customer orders
MPS of the week
– cumulative COt up to the next MPS receipt
22
Projected On-hand Inventory
• At the end of the period t
𝑰𝒕 = 𝑰𝒏𝒗𝒆𝒏𝒕𝒐𝒓𝒚 𝒍𝒆𝒗𝒆𝒍𝒔 𝒂𝒕 𝒕𝒉𝒆 𝒆𝒏𝒅 𝒐𝒇 𝒑𝒆𝒓𝒊𝒐𝒅 𝒕
𝑴𝑷𝑺𝒕 = 𝑶𝒓𝒅𝒆𝒓𝒔 𝒑𝒍𝒂𝒄𝒆𝒅 𝒂𝒕 𝒕𝒉𝒆 𝒃𝒆𝒈𝒊𝒏𝒏𝒊𝒏𝒈 𝒐𝒇 𝒑𝒆𝒓𝒊𝒐𝒅 𝒕
𝑭𝒕 = 𝑭𝒐𝒓𝒆𝒄𝒂𝒔𝒕𝒆𝒅 𝒐𝒓𝒅𝒆𝒓𝒔 𝒇𝒐𝒓 𝒑𝒆𝒓𝒊𝒐𝒅 𝒕
𝑪𝑶𝒕 = 𝑪𝒖𝒔𝒕𝒐𝒎𝒆𝒓 𝒐𝒓𝒅𝒆𝒓𝒔 𝒇𝒐𝒓 𝒑𝒆𝒓𝒊𝒐𝒅 𝒕
23
Example: MPS Record with ATP
Week
1
2
3
4
5
6
Forecast
5
5
5
5
5
5
Customer
Orders
5
3
2
Available
15
10
5
30
25
20
ATP
10
MPS
On hand
7
30
30
20
24
MPS complexity
• With many end items with complex
bill of materials
– All components need to be physically
available at the beginning of the FAS
– The complexity in managing and
planning the MPS is high
• ATP time
– The time waiting for the arrival of all
necessary items to begin the FAS
25
MPS techniques for ATO
environment
• The MPS techniques used for
managing MPS planning for large
number of end items
1. Superbills
2. Covering set
26
1. Superbills
• When Large combinatorial number of end items
– Need to Focus on Options rather than end items
– Represents the MPS units in t …