Modeling Objectives
Goal Programming
Several Objectives
Minimum Regret Models
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Workforce Scheduling
Some days we will have too many workers
Excess Only concerned with the largest excess
Minimize the largest Excess
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Challenge
Formulate a Solver Model
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Formulating the LP
Scheduling Postal Workers
Shift
Day
Mon
Tues
Wed
Thurs
Fri
Sat
Sun
Mon - Tues - Wed - Thurs - Fri - Sat - Sun Fri
Sat
Sun Mon Tues Wed Thurs
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
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Demand
17
13
15
19
14
16
11
4
Enhancement
Given Goals:
Have the right number of workers each day
Cost for exceeding and falling short
|Actual Workers – Desired Number|
Challenge: Formulate a Linear Model
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Formulating the LP
Scheduling Postal Workers
Shift
Day
Mon
Tues
Wed
Thurs
Fri
Sat
Sun
Mon - Tues - Wed - Thurs - Fri - Sat - Sun Fri
Sat
Sun Mon Tues Wed Thurs
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
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Demand
17
13
15
19
14
16
11
6
Policy vs Physical Consts
Physical/Logical Constraints
Flow Conservation
Capacity
...
Policy Constraints
Meet service levels
...
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Small Violations
Can violate policy constraints a little
Sometimes can violate physical
constraints
Capacity
Raw materials
…
Handle with “deviation from targets”
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Conflicting Objectives
Cost vs Service
Price vs Quality
Revenue vs Market Share
Reach vs Frequency
Risk vs Return
...
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Other Examples
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Cost
Undominated Optima
Current Position
Dominating Positions
Service Level
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Getting to the Frontier
Associate a value with Service ($/Time)
Change this value of Time
Systematic change using Sensitivity Analysis
Remember the Retail Pricing Model
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Reach vs Frequency
Invest in Marketing
Radio Ads
TV Ads
Newspaper Ads
Want
Reach: Lots of people to see the ads
Frequency: People to see the adds lots of times
Generally conflicting
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Example
Budget of $500
Media
Radio
Cost
$15
Reach
100
Frequency
10
TV
News
$30
$10
500
10
2
100
What are the Frontier Solutions?
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The Frontier
6000
News
5000
Frequency
4000
Undominated Solutions
3000
2000
Solutions using
the full budget
1000
TV
Radio
0
0
1000
2000
3000
4000
5000
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Reach
6000
7000
8000
9000
15
Dollarize
Reaching one person worth $1 (normalize)
Frequency worth $f per repetition
Media
Radio
Cost
$15
Reach
100
Frequency
10
Value
$100 + 10f
TV
News
$30
$10
500
10
2
100
$500 + 2f
$10 + 100f
What happens to the answer as f increases?
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The Frontier
$120.00
Radio
Value/$
$100.00
TV
News
$80.00
$60.00
$40.00
$20.00
$0.00
0
1
2
3
4
F
5
6
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8
9
10
The Frontier
6000
News
5000
Frequency
4000
Undominated Solutions
3000
2000
Solutions using
the full budget
1000
TV
Radio
0
0
1000
2000
3000
4000
5000
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Reach
6000
7000
8000
9000
18
Multiple Scenarios
Uncertain Future
Make a decision today
Live with it tomorrow
Example
Singapore Electric
Production plan against FORECASTED orders
Uncertain Economic climate
HOT (130%)
Temperate (100%)
Cold (70%)
1/3 chance
1/3 chance
1/3 chance
Allow back-ordering
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Singapore Electric
Unit Costs
Prod. $
Inventory $
Prod. Quant.
Prod. Limits
Init. Inv.
Del. Reqmts
Ending Inv.
Prod. Cost $
Inv. Cost $
Total Cost $
Singapore Electric Generator Production
Jan
Feb
Mar
Apr.
28.00 $
27.00 $
27.80 $ 29.00
0.30 $
0.30 $
0.30 $ 0.30
0
60
15
58
(43)
0
62
-43
36
(79)
$
(4.20) $
(4.20) $
$
(18.30) $
(18.30) $
0
64
-79
34
(113)
May
0
66
-113
59 Minimum
(172)
7
$
(28.80) $ (42.75) Total
(28.80) $ (42.75) $
(94.05)
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Allow Back-Ordering
Defer demand at a cost of $3/unit/month
Challenge: Formulate a linear model
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With Backordering
Singapore Electric Generator Production
Economic Climate
100%
Unit Revenue $
50
Unit Costs
Jan
Feb
Mar
Apr.
Prod. $
28.00 $
27.00 $
27.80 $ 29.00
Inventory $
0.30 $
0.30 $
0.30 $
0.30
BackOrder
$3.00
$3.00
$3.00
$3.00
Prod. Quant.
Inventory
BackOrder
Net Inv.
Prod. Limits
Init. Inv.
Del. Reqmts
Ending Inv.
Prod. Cost $
Inv. Cost $
Total Cost $
31
0
0
0
60
10
58
(17)
868 $
$
868 $
25
0
0
0
62
(17.00)
36
(28)
675 $
$
675 $
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0
0
0
64
(28.00)
34
(38)
May
48
7
0
7
66
(38.00)
59 Minimum
(49)
7
667 $ 1,392
$
21 Total
667 $ 1,413 $
3,623
Net Rev.
5,727
22
Revised Model
No constraint on ending inventory
Charge $50 for any back orders in May
Maximize Net Revenue
$50 for each item sold –
Cost –
Lost Sales ($50 for back orders in May)
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Robust Answers
Minimum Regret: Max the Min Revenue
Regardless of the outcome, we get at least this
revenue
Idea of the model
One set of variables for production
Inventory variables for each scenario
Cost & Revenue calculations for each scenario
New Variable for Minimum Revenue
Maximize the Minimum Revenue
S.t. Minimum Revenue ≤ Revenue in Each Scenario
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Best Answers
Solution
Hot
Temper
Cold
Min Regret
Net Rev
$
5,627
$
4,442
$
3,604
$
2,221
Solution
Hot
Temper
Cold
Min Regret
$
$
$
$
Jan
Feb
60
51
36
60
Hot
5,627
1,452
(1,534)
2,221
Mar
62
62
62
62
Temper
$ 2,802
$ 4,442
$ 1,568
$ 4,150
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64
41
64
$
$
$
$
Apr.
47
1
Cold
873
2,513
3,604
2,221
25
Model (Messy)
Unit Costs
Prod. $
Inventory $
BackOrder
Unit Revenue $
50
Jan
Feb
Mar
Apr.
28.00 $
27.00 $
27.80 $ 29.00
0.30 $
0.30 $
0.30 $
0.30
$3.00
$3.00
$3.00
$3.00
60
Prod. Quant.
Scenario 1
Inventory
BackOrder
Net Inv.
Prod. Limits
Init. Inv.
Del. Reqmts
Ending Inv.
Prod. Cost $
Inv. Cost $
Total Cost $
Scenario 2
Inventory
BackOrder
Net Inv.
Prod. Limits
Init. Inv.
Del. Reqmts
Ending Inv.
Prod. Cost $
Inv. Cost $
Total Cost $
Scenario 3
Inventory
BackOrder
Net Inv.
Prod. Limits
Init. Inv.
Del. Reqmts
Ending Inv.
Prod. Cost $
Inv. Cost $
Total Cost $
62
Economic Climate
24
57
0
0
24
57
60
62
10
24.00
46
29
24
57
1,680 $
7 $
1,687 $
1,674 $
17 $
1,691 $
Economic Climate
12
38
0
0
12
38
60
62
10
12.00
58
36
12
38
1,680 $
4 $
1,684 $
1,674 $
11 $
1,685 $
Economic Climate
0
10
5
0
-5
10
60
62
10
(5.00)
75
47
(5)
10
1,680 $
15 $
1,695 $
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1,674 $
3 $
1,677 $
64
May
Minimum Net
0.981607 2221.43892
80%
94 47.981607
0
0
94 47.981607
64
66
57.00
94.00
27
47 Lost Sales
94
48 $
1,779 $
28
28 $
14 Total
1,807 $
43 $
5,229
Net Rev.
2,221
100%
68
0
68
64
38.00
34
68
9.981607
0
9.981607
66
68.00
59 Lost Sales
10 $
-
1,779 $
28
20 $
3 Total
1,800 $
31 $
5,200
Net Rev.
4,150
130%
30
0
0 46.018393
30 -46.01839
64
66
10.00
30.00
44
77 Lost Sales
30
(46)
4,601.84
1,779 $
28
9 $
138 Total
1,788 $
167 $
5,327
Net Rev.
2,221
26
Summary
Modeling Goals
Modeling Conflicting Objectives
The Efficient Frontier
Modeling Scenarios
Complicated models
Robust Solutions
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