Table of Contents
Chapter 3 (Linear Programming: Formulation and Applications)
Super Grain Corp. Advertising-Mix Problem (Section 3.1)
Resource Allocation Problems (Section 3.2)
Cost-Benefit-Trade-Off Problems (Section 3.3)
Mixed Problems (Section 3.4)
Transportation Problems (Section 3.5)
Assignment Problems (Section 3.6)
3.2–3.5
3.6–3.16
3.17–3.22
3.23–3.28
3.29–3.33
3.34–3.37
Applications of Linear Programming with Spreadsheets (UW Lecture)
3.38–3.57
These slides are based upon lectures to first-year MBA students at the University of Washington
that discuss the application and formulation of linear programming models (as taught by one of the
authors).
McGraw-Hill/Irwin
3.1
© The McGraw-Hill Companies, Inc., 2008
Super Grain Corp. Advertising-Mix Problem
•
Goal: Design the promotional campaign for Crunchy Start.
•
The three most effective advertising media for this product are
– Television commercials on Saturday morning programs for children.
– Advertisements in food and family-oriented magazines.
– Advertisements in Sunday supplements of major newspapers.
•
The limited resources in the problem are
– Advertising budget ($4 million).
– Planning budget ($1 million).
– TV commercial spots available (5).
•
The objective will be measured in terms of the expected number of exposures.
Question: At what level should they advertise Crunchy Start in each of the
three media?
McGraw-Hill/Irwin
3.2
© The McGraw-Hill Companies, Inc., 2008
Cost and Exposure Data
Costs
Cost Category
Ad Budget
Planning budget
Expected number of
exposures
McGraw-Hill/Irwin
Each
TV Commercial
Each
Magazine Ad
Each
Sunday Ad
$300,000
$150,000
$100,000
90,000
30,000
40,000
1,300,000
600,000
500,000
3.3
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
B
3
4
5
6
7
8
9
10
11
12
13
14
15
Exposures per Ad
(thousands)
Ad Budget
Planning Budget
Number of Ads
Max TV Spots
McGraw-Hill/Irwin
C
TV Spots
1,300
300
90
TV Spots
0
<=
5
D
Magazine Ads
600
Cost per Ad ($thousands)
150
30
Magazine Ads
20
3.4
E
SS Ads
500
100
40
SS Ads
10
F
Budget
Spent
4,000
1,000
G
H
<=
<=
Budget
Available
4,000
1,000
Total Exposures
(thousands)
17,000
© The McGraw-Hill Companies, Inc., 2008
Algebraic Formulation
Let TV = Number of commercials for separate spots on television
M = Number of advertisements in magazines.
SS = Number of advertisements in Sunday supplements.
Maximize Exposure = 1,300TV + 600M + 500SS
subject to
Ad Spending:
300TV + 150M + 100SS ≤ 4,000 ($thousand)
Planning Cost:
90TV + 30M + 30SS ≤ 1,000 ($thousand)
Number of TV Spots:
TV ≤ 5
and
TV ≥ 0, M ≥ 0, SS ≥ 0.
McGraw-Hill/Irwin
3.5
© The McGraw-Hill Companies, Inc., 2008
The TBA Airlines Problem
•
TBA Airlines is a small regional company that specializes in short flights in
small airplanes.
•
The company has been doing well and has decided to expand its operations.
•
The basic issue facing management is whether to purchase more small
airplanes to add some new short flights, or start moving into the national
market by purchasing some large airplanes, or both.
Question: How many airplanes of each type should be purchased to maximize
their total net annual profit?
McGraw-Hill/Irwin
3.6
© The McGraw-Hill Companies, Inc., 2008
Data for the TBA Airlines Problem
Small
Airplane
Large
Airplane
Net annual profit per airplane
$1 million
$5 million
Purchase cost per airplane
5 million
50 million
2
—
Maximum purchase quantity
McGraw-Hill/Irwin
3.7
Capital
Available
$100 million
© The McGraw-Hill Companies, Inc., 2008
Violates Divisibility Assumption of LP
• Divisibility Assumption of Linear Programming: Decision variables in a
linear programming model are allowed to have any values, including fractional
values, that satisfy the functional and nonnegativity constraints. Thus, these
variables are not restricted to just integer values.
• Since the number of airplanes purchased by TBA must have an integer value,
the divisibility assumption is violated.
McGraw-Hill/Irwin
3.8
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Model
B
3
4
5
6
7
8
9
10
11
12
13
14
Unit Profit ($millions)
Capital ($millions)
Units Produced
Maximum Small Airplanes
McGraw-Hill/Irwin
C
Small Airplane
1
D
Large Airplane
5
Capital Per Unit Produced
5
50
Small Airplane
0
<=
2
3.9
Large Airplane
2
E
Capital
Spent
100
F
G
<=
Capital
Available
100
Total Profit
($millions)
10
© The McGraw-Hill Companies, Inc., 2008
Integer Programming Formulation
Let S = Number of small airplanes to purchase
L = Number of large airplanes to purchase
Maximize Profit = S + 5L ($millions)
subject to
Capital Available:
5S + 50L ≤ 100 ($millions)
Max Small Planes: S ≤ 2
and
S ≥ 0, L ≥ 0
S, L are integers.
McGraw-Hill/Irwin
3.10
© The McGraw-Hill Companies, Inc., 2008
Think-Big Capital Budgeting Problem
•
Think-Big Development Co. is a major investor in commercial real-estate
development projects.
•
They are considering three large construction projects
– Construct a high-rise office building.
– Construct a hotel.
– Construct a shopping center.
•
Each project requires each partner to make four investments: a down payment
now, and additional capital after one, two, and three years.
Question: At what fraction should Think-Big invest in each of the three
projects?
McGraw-Hill/Irwin
3.11
© The McGraw-Hill Companies, Inc., 2008
Financial Data for the Projects
Investment Capital Requirements
Year
Office Building
Hotel
Shopping Center
0
$40 million
$80 million
$90 million
1
60 million
80 million
50 million
2
90 million
80 million
20 million
3
10 million
70 million
60 million
Net present value
$45 million
$70 million
$50 million
McGraw-Hill/Irwin
3.12
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Net Present Value
($millions)
Now
End of Year 1
End of Year 2
End of Year 3
Participation Share
McGraw-Hill/Irwin
C
Office
Building
45
D
Hotel
70
E
Shopping
Center
50
Cumulative
Capital
Spent
25
44.757
60.583
80
Cumulative Capital Required ($millions)
40
80
90
100
160
140
190
240
160
200
310
220
Office
Building
0.00%
Hotel
16.50%
3.13
F
Shopping
Center
13.11%
G
H
<=
<=
<=
<=
Cumulative
Capital
Available
25
45
65
80
Total NPV
($millions)
18.11
© The McGraw-Hill Companies, Inc., 2008
Algebraic Formulation
Let OB = Participation share in the office building,
H = Participation share in the hotel,
SC = Participation share in the shopping center.
Maximize NPV = 45OB + 70H + 50SC
subject to
Total invested now:
40OB + 80H + 90SC ≤ 25 ($million)
Total invested within 1 year:
100OB + 160H + 140SC ≤ 45 ($million)
Total invested within 2 years: 190OB + 240H + 160SC ≤ 65 ($million)
Total invested within 3 years: 200OB + 310H + 220SC ≤ 80 ($million)
and
OB ≥ 0, H ≥ 0, SC ≥ 0.
McGraw-Hill/Irwin
3.14
© The McGraw-Hill Companies, Inc., 2008
Template for Resource-Allocation Problems
Activities
Constraints
Unit Profit
Level of Activity
McGraw-Hill/Irwin
profit per unit of activity
Resources
Used
SUMPRODUCT
(resource used per unit,
changing cells)
resource used per unit of act ivit y
Resources
Available
<=
Total Profit
SUMPRODUCT(profit per unit, changing cells)
changing cells
3.15
© The McGraw-Hill Companies, Inc., 2008
Summary of Formulation Procedure for ResourceAllocation Problems
1. Identify the activities for the problem at hand.
2. Identify an appropriate overall measure of performance (commonly profit).
3. For each activity, estimate the contribution per unit of the activity to the
overall measure of performance.
4. Identify the resources that must be allocated.
5. For each resource, identify the amount available and then the amount used per
unit of each activity.
6. Enter the data in steps 3 and 5 into data cells.
7. Designate changing cells for displaying the decisions.
8. In the row for each resource, use SUMPRODUCT to calculate the total
amount used. Enter <= and the amount available in two adjacent cells.
9. Designate a target cell. Use SUMPRODUCT to calculate this measure of
performance.
McGraw-Hill/Irwin
3.16
© The McGraw-Hill Companies, Inc., 2008
Union Airways Personnel Scheduling
•
Union Airways is adding more flights to and from its hub airport and so needs
to hire additional customer service agents.
•
The five authorized eight-hour shifts are
–
–
–
–
–
Shift 1:
Shift 2:
Shift 3:
Shift 4:
Shift 5:
6:00 AM to 2:00 PM
8:00 AM to 4:00 PM
Noon to 8:00 PM
4:00 PM to midnight
10:00 PM to 6:00 AM
Question: How many agents should be assigned to each shift?
McGraw-Hill/Irwin
3.17
© The McGraw-Hill Companies, Inc., 2008
Schedule Data
Time Periods Covered by Shift
Time Period
1
6 AM to 8 AM
√
8 AM to 10 AM
√
√
79
10 AM to noon
√
√
65
Noon to 2 PM
√
√
√
87
√
√
64
2 PM to 4 PM
2
3
4
5
Minimum
Number of
Agents Needed
48
4 PM to 6 PM
√
√
73
6 PM to 8 PM
√
√
82
8 PM to 10 PM
√
43
10 PM to midnight
√
Midnight to 6 AM
Daily cost per agent
McGraw-Hill/Irwin
$170
$160
$175
3.18
$180
√
52
√
15
$195
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Cost per Shift
Time Period
6am-8am
8am-10am
10am- 12pm
12pm-2pm
2pm-4pm
4pm-6pm
6pm-8pm
8pm-10pm
10pm-12am
12am-6am
Number Working
McGraw-Hill/Irwin
C
6am-2pm
Shift
$170
1
1
1
1
0
0
0
0
0
0
6am-2pm
Shift
48
D
8am-4pm
Shift
$160
E
Noon-8pm
Shift
$175
F
4pm-midnight
Shift
$180
Shift Works Time Period? (1=yes, 0=no)
0
0
0
1
0
0
1
0
0
1
1
0
1
1
0
0
1
1
0
1
1
0
0
1
0
0
1
0
0
0
8am-4pm
Shift
31
Noon-8pm
Shift
39
3.19
G
10pm-6am
Shift
$195
0
0
0
0
0
0
0
0
1
1
4pm-midnight
Shift
43
10pm-6am
Shift
15
H
Total
Working
48
79
79
118
70
82
82
43
58
15
I
J
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
Minimum
Needed
48
79
65
87
64
73
82
43
52
15
Total Cost
$30,610
© The McGraw-Hill Companies, Inc., 2008
Algebraic Formulation
Let Si = Number working shift i (for i = 1 to 5),
Minimize Cost = $170S1 + $160S2 + $175S3 + $180S4 + $195S5
subject to
Total agents 6AM–8AM:
S1 ≥ 48
Total agents 8AM–10AM:
S1 + S2 ≥ 79
Total agents 10AM–12PM:
S1 + S2 ≥ 65
Total agents 12PM–2PM:
S1 + S2 + S3 ≥ 87
Total agents 2PM–4PM:
S2 + S3 ≥ 64
Total agents 4PM–6PM:
S3 + S4 ≥ 73
Total agents 6PM–8PM:
S3 + S4 ≥ 82
Total agents 8PM–10PM:
S4 ≥ 43
Total agents 10PM–12AM:
S4 + S5 ≥ 52
Total agents 12AM–6AM:
S5 ≥ 15
and
Si ≥ 0 (for i = 1 to 5)
McGraw-Hill/Irwin
3.20
© The McGraw-Hill Companies, Inc., 2008
Template for Cost-Benefit Tradoff Problems
Activities
Constraints
Unit Cost
cost per unit of act ivit y
Benefit
Achieved
SUMPRODUCT
(benefit per unit,
changing cells)
benefit achieved per unit of activity
Level of Activity
McGraw-Hill/Irwin
Benefit
Needed
>=
Total Cost
SUMPRODUCT(cost per unit , changing cells)
changing cells
3.21
© The McGraw-Hill Companies, Inc., 2008
Summary of Formulation Procedure for
Cost-Benefit-Tradeoff Problems
1. Identify the activities for the problem at hand.
2. Identify an appropriate overall measure of performance (commonly cost).
3. For each activity, estimate the contribution per unit of the activity to the
overall measure of performance.
4. Identify the benefits that must be achieved.
5. For each benefit, identify the minimum acceptable level and then the
contribution of each activity to that benefit.
6. Enter the data in steps 3 and 5 into data cells.
7. Designate changing cells for displaying the decisions.
8. In the row for each benefit, use SUMPRODUCT to calculate the level
achieved. Enter >= and the minimum acceptable level in two adjacent cells.
9. Designate a target cell. Use SUMPRODUCT to calculate this measure of
performance.
McGraw-Hill/Irwin
3.22
© The McGraw-Hill Companies, Inc., 2008
Types of Functional Constraints
Type
Resource constraint
Benefit constraint
Fixed-requirement
constraint
Form*
Typical Interpretation
Main Usage
LHS ≤ RHS
For some resource,
Amount used ≤
Amount available
Resource-allocation
problems and mixed
problems
LHS ≥ RHS
For some benefit,
Level achieved ≥
Minimum Acceptable
Cost-benefit-trade-off
problems and mixed
problems
LHS = RHS
For some quantity,
Amount provided =
Required amount
Transportation
problems and mixed
problems
* LHS = Left-hand side (a SUMPRODUCT function).
RHS = Right-hand side (a constant).
McGraw-Hill/Irwin
3.23
© The McGraw-Hill Companies, Inc., 2008
Continuing the Super Grain Case Study
•
David and Claire conclude that the spreadsheet model needs to be expanded to
incorporate some additional considerations.
•
In particular, they feel that two audiences should be targeted — young children
and parents of young children.
•
Two new goals
– The advertising should be seen by at least five million young children.
– The advertising should be seen by at least five million parents of young children.
•
Furthermore, exactly $1,490,000 should be allocated for cents-off coupons.
McGraw-Hill/Irwin
3.24
© The McGraw-Hill Companies, Inc., 2008
Benefit and Fixed-Requirement Data
Number Reached in Target Category (millions)
Each
TV Commercial
Each
Magazine Ad
Each
Sunday Ad
Minimum
Acceptable
Level
Young children
1.2
0.1
0
5
Parents of young children
0.5
0.2
0.2
5
Contribution Toward Required Amount
Coupon redemption
McGraw-Hill/Irwin
Each
TV Commercial
Each
Magazine Ad
Each
Sunday Ad
Required
Amount
0
$40,000
$120,000
$1,490,000
3.25
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
C
TV Spots
1,300
D
Magazine Ads
600
E
SS Ads
500
Ad Budget
Planning Budget
300
90
Cost per Ad ($thousands)
150
30
Young Children
Parents of Young Children
1.2
0.5
Exposures per Ad
(thousands)
Coupon Redemption per Ad
($thousands)
Number of Ads
Maximum TV Spots
McGraw-Hill/Irwin
F
G
H
100
40
Budget Spent
3,775
1,000
<=
<=
Budget Available
4,000
1,000
Number Reached per Ad (millions)
0.1
0
0.2
0.2
Total Reached
5
5.85
>=
>=
Minimum Acceptable
5
5
Total Redeemed
1,490
=
Required Amount
1,490
TV Spots
0
Magazine Ads
40
SS Ads
120
TV Spots
3
<=
5
Magazine Ads
14
SS Ads
7.75
3.26
Total Exposures
(thousands)
16,175
© The McGraw-Hill Companies, Inc., 2008
Algebraic Formulation
Let TV = Number of commercials for separate spots on television
M = Number of advertisements in magazines.
SS = Number of advertisements in Sunday supplements.
Maximize Exposure = 1,300TV + 600M + 500SS
subject to
Ad Spending:
300TV + 150M + 100SS ≤ 4,000 ($thousand)
Planning Cost:
90TV + 30M + 30SS ≤ 1,000 ($thousand)
Number of TV Spots:
TV ≤ 5
Young children:
Parents:
1.2TV + 0.1M ≥ 5 (millions)
0.5TV + 0.2M + 0.2SS ≥ 5 (millions)
Coupons:
40M + 120SS = 1,490 ($thousand)
and
TV ≥ 0, M ≥ 0, SS ≥ 0.
McGraw-Hill/Irwin
3.27
© The McGraw-Hill Companies, Inc., 2008
Template for Mixed Problems
Activities
Unit Profit or Cost
profit/cost per unit of activity
Resources
Used
SUMPRODUCT
(resource used per unit,
changing cells)
Constraints
resource used per unit of act ivit y
Level of Activity
McGraw-Hill/Irwin
Resources
Available
<=
Benefit
Achieved
SUMPRODUCT
(benefit per unit,
changing cells)
benefit achieved per unit of activity
Benefit
Needed
>=
=
Total Profit or Cost
SUMPRODUCT(profit/cost per unit, changing cells)
changing cells
3.28
© The McGraw-Hill Companies, Inc., 2008
The Big M Transportation Problem
•
The Big M Company produces a variety of heavy duty machinery at two
factories. One of its products is a large turret lathe.
•
Orders have been received from three customers for the turret lathe.
Question: How many lathes should be shipped from each factory to each
customer?
McGraw-Hill/Irwin
3.29
© The McGraw-Hill Companies, Inc., 2008
Some Data
Shipping Cost for Each Lathe
To
Customer 1
Customer 2
Customer 3
From
Output
Factory 1
$700
$900
$800
12 lathes
Factory 2
800
900
700
15 lathes
Order Size
10 lathes
8 lathes
9 lathes
McGraw-Hill/Irwin
3.30
© The McGraw-Hill Companies, Inc., 2008
The Distribution Network
C1
10 lathes
needed
C2
8 lathes
needed
C3
9 lathes
needed
$700/lathe
12 lathe
produced
F1
$900/lathe
$800/lathe
$900/lathe
$800/lathe
15 lathes
produced
F2
$700/lathe
McGraw-Hill/Irwin
3.31
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
3
4
5
6
7
8
9
10
11
12
13
14
15
B
Shipping Cost
(per Lathe)
Factory 1
Factory 2
Units Shipped
Factory 1
Factory 2
Total To Customer
Order Size
McGraw-Hill/Irwin
C
D
E
Customer 1
$700
$800
Customer 2
$900
$900
Customer 3
$800
$700
Customer 1
10
0
10
=
10
Customer 2
2
6
8
=
8
3.32
Customer 3
0
9
9
=
9
F
Total
Shipped
Out
12
15
G
H
=
=
Output
12
15
Total Cost
$20,500
© The McGraw-Hill Companies, Inc., 2008
Algebraic Formulation
Let Sij = Number of lathes to ship from i to j (i = F1, F2; j = C1, C2, C3).
Minimize Cost = $700SF1-C1 + $900SF1-C2 + $800SF1-C3
+ $800SF2-C1 + $900SF2-C2 + $700SF2-C3
subject to
Factory 1:
SF1-C1 + SF1-C2 + SF1-C3 = 12
Factory 2:
SF2-C1 + SF2-C2 + SF2-C3 = 15
Customer 1: SF1-C1 + SF2-C1 = 10
Customer 2: SF1-C2 + SF2-C2 = 8
Customer 3: SF1-C3 + SF2-C3 = 9
and
Sij ≥ 0 (i = F1, F2; j = C1, C2, C3).
McGraw-Hill/Irwin
3.33
© The McGraw-Hill Companies, Inc., 2008
Sellmore Company Assignment Problem
•
The marketing manager of Sellmore Company will be holding the company’s
annual sales conference soon.
•
He is hiring four temporary employees:
–
–
–
–
•
Ann
Ian
Joan
Sean
Each will handle one of the following four tasks:
–
–
–
–
Word processing of written presentations
Computer graphics for both oral and written presentations
Preparation of conference packets, including copying and organizing materials
Handling of advance and on-site registration for the conference
Question: Which person should be assigned to which task?
McGraw-Hill/Irwin
3.34
© The McGraw-Hill Companies, Inc., 2008
Data for the Sellmore Problem
Required Time per Task (Hours)
Temporary
Employee
Word
Processing
Graphics
Packets
Registrations
Hourly
Wage
Ann
35
41
27
40
$14
Ian
47
45
32
51
12
Joan
39
56
36
43
13
Sean
32
51
25
46
15
McGraw-Hill/Irwin
3.35
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
C
D
E
F
G
H
I
J
Task
Required Time
(Hours)
Assignee
Ann
Ian
Joan
Sean
Word
Processing
35
47
39
32
Ann
Ian
Joan
Sean
Word
Processing
$490
$564
$507
$480
Graphics
41
45
56
51
Packets
27
32
36
25
Registrations
40
51
43
46
Packets
$378
$384
$468
$375
Registrations
$560
$612
$559
$690
Packets
1
0
0
0
1
=
1
Registrations
0
0
1
0
1
=
1
Hourly
Wage
$14
$12
$13
$15
Task
Cost
Assignee
Graphics
$574
$540
$728
$765
Task
Assignment
Ann
Ian
Joan
Sean
Total Assigned
Assignee
Demand
McGraw-Hill/Irwin
Word
Processing
0
0
0
1
1
=
1
Graphics
0
1
0
0
1
=
1
3.36
Total
Assignments
1
1
1
1
=
=
=
=
Supply
1
1
1
1
Total Cost
$1,957
© The McGraw-Hill Companies, Inc., 2008
The Model for Assignment Problems
Given a set of tasks to be performed and a set of assignees who are available to
perform these tasks, the problem is to determine which assignee should be
assigned to each task.
To fit the model for an assignment problem, the following assumptions need to be
satisfied:
1.
2.
3.
4.
5.
The number of assignees and the number of tasks are the same.
Each assignee is to be assigned to exactly one task.
Each task is to be performed by exactly one assignee.
There is a cost associated with each combination of an assignee performing a task.
The objective is to determine how all the assignments should be made to minimize
the total cost.
McGraw-Hill/Irwin
3.37
© The McGraw-Hill Companies, Inc., 2008
Formulating an LP Spreadsheet Model
•
Enter all of the data into the spreadsheet. Color code (blue).
•
What decisions need to be made? Set aside a cell in the spreadsheet for each
decision variable (changing cell). Color code (yellow with border).
•
Write an equation for the objective in a cell. Color code (orange with heavy
border).
•
Put all three components (LHS, ≤/=/≥, RHS) of each constraint into three cells
on the spreadsheet.
•
Some Examples:
–
–
–
–
–
Production Planning
Diet / Blending
Workforce Scheduling
Transportation / Distribution
Assignment
McGraw-Hill/Irwin
3.38
© The McGraw-Hill Companies, Inc., 2008
LP Example #1 (Product Mix)
The Quality Furniture Corporation produces benches and picnic tables. The firm
has a limited supply of two resources: labor and wood. 1,600 labor hours are
available during the next production period. The firm also has a stock of 9,000
pounds of wood available. Each bench requires 3 labor hours and 12 pounds of
wood. Each table requires 6 labor hours and 38 pounds of wood. The profit margin
on each bench is $8 and on each table is $18.
Question: What product mix will maximize their total profit?
McGraw-Hill/Irwin
3.39
© The McGraw-Hill Companies, Inc., 2008
Algebraic Formulation
Let B = Number of benches to produce,
T = Number of tables to produce.
Maximize Profit = $8B + $18T
subject to
Labor:
3B + 6T ≤ 1,600 hours
Wood:
12B + 38T ≤ 9,000 pounds
and
B ≥ 0, T ≥ 0.
McGraw-Hill/Irwin
3.40
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
3
4
5
6
7
8
9
10
11
12
B
C
D
Profit
Benches
$8
Tables
$18
Resources
Labor
Wood
Units Produced
McGraw-Hill/Irwin
Used per Unit Produced
3
6
12
38
161.90
185.71
3.41
E
Total
1,600
9,000
F
G
<=
<=
Available
1,600
9,000
Total Cost
$4,638.10
© The McGraw-Hill Companies, Inc., 2008
LP Example #2 (Diet Problem)
A prison is trying to decide what to feed its prisoners. They would like to offer
some combination of milk, beans, and oranges. Their goal is to minimize cost,
subject to meeting the minimum nutritional requirements imposed by law. The
cost and nutritional contents of each food, along with the minimum nutritional
requirements are shown below.
Milk
(gallons)
Navy
Beans
(cups)
Oranges
(large Calif.
Valencia)
Minimum
Daily
Requirement
Niacin (mg)
3.2
4.9
0.8
13.0
Thiamin (mg)
1.12
1.3
0.19
1.5
32
0
93
45
2.00
0.20
0.25
Vitamin C (mg)
Cost ($)
Question: What should the diet for each prisoner be?
McGraw-Hill/Irwin
3.42
© The McGraw-Hill Companies, Inc., 2008
Algebraic Formulation
Let x1 = gallons of milk per prisoner,
x2 = cups of beans per prisoner,
x3 = number of oranges per prisoner.
Minimize Cost = $2.00x1 + $0.20x2 + $0.25x3
subject to
Niacin:
3.2x1 + 4.9x2 + 0.8x3 ≥ 13 mg
Thiamin:
1.12x1 + 1.3x2 + 0.19x3 ≥ 1.5 mg
Vitamin C:
32x1 + 93x3 ≥ 45 mg
and
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
McGraw-Hill/Irwin
3.43
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
3
4
5
6
7
8
9
10
11
12
13
B
C
D
E
Cost
Milk
(gal.)
$2.00
Beans
(cups)
$0.20
Oranges
$0.25
Niacin (mg)
Thiamin (mg)
Vitamin C (mg)
Quantity
(per prisoner)
McGraw-Hill/Irwin
Nutritional Contents (mg)
3.2
4.9
0.8
1.12
1.3
0.19
32
0
93
0
2.574
3.44
0.484
F
Total
13
3.438
45
G
H
>=
>=
>=
Minimum
Requirement
13
1.5
45
Total Cost
$0.64
© The McGraw-Hill Companies, Inc., 2008
George Dantzig’s Diet
•
Stigler (1945) “The Cost of Subsistence”
– heuristic solution. Cost = $39.93.
•
Dantzig invents the simplex method (1947)
– Stigler’s problem “solved” in 120 man days. Cost = $39.69.
•
Dantzig goes on a diet (early 1950’s), applies diet model:
– ≤ 1,500 calories
– objective: maximize (weight minus water content)
– 500 food types
•
Initial solutions had problems
– 500 gallons of vinegar
– 200 bouillon cubes
For more details, see July-Aug 1990 Interfaces article “The Diet Problem”
McGraw-Hill/Irwin
3.45
© The McGraw-Hill Companies, Inc., 2008
Least-Cost Menu Planning Models in
Food Systems Management
•
Used in many institutions with feeding programs: hospitals, nursing homes,
schools, prisons, etc.
•
Menu planning often extends to a sequence of meals or a cycle.
•
Variety important (separation constraints).
•
Preference ratings (related to service frequency).
•
Side constraints (color, categories, etc.)
•
Generally models have reduced cost about 10%, met nutritional requirements
better, and increased customer satisfaction compared to traditional methods.
•
USDA uses these models to plan food stamp allotment.
For more details, see Sept-Oct 1992 Interfaces article “The Evolution of the Diet Model in Managing Food Systems”
McGraw-Hill/Irwin
3.46
© The McGraw-Hill Companies, Inc., 2008
LP Example #3 (Scheduling Problem)
An airline reservations office is open to take reservations by telephone 24 hours
per day, Monday through Friday. The number of reservation agents needed for
each time period is shown below. A union contract requires that all employees
work 8 consecutive hours.
Time Period
Number of
Agents Needed
12am – 4am
11
4am – 8am
15
8am – 12pm
31
12pm – 4pm
17
4pm – 8pm
25
8pm – 12am
19
Question: How many reservation agents should work each 8-hour shift?
McGraw-Hill/Irwin
3.47
© The McGraw-Hill Companies, Inc., 2008
Algebraic Formulation
Let x1 = agents who work 12am – 8am,
x2 = agents who work 4am – 12pm,
x3 = agents who work 8am – 4pm,
x4 = agents who work 12pm – 8pm,
x5 = agents who work 4pm – 12am,
x6 = agents who work 8pm – 4am.
Minimize Number of agents = x1 + x2 + x3 + x4 + x5 + x6
subject to
12am–4am: x1 + x6 ≥ 11
4am–8am:
x1 + x2 ≥ 15
8am–12pm: x2 + x3 ≥ 31
12pm–4pm: x3 + x4 ≥ 17
4pm–8pm:
x4 + x5 ≥ 25
8pm–12am: x5 + x6 ≥ 19
and
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0, x5 ≥ 0, x6 ≥ 0.
McGraw-Hill/Irwin
3.48
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
A
1
2
3
4
5
6
7
8
9
10
11
B
C
D
E
F
G
H
Shift
12am - 8am
4am - 12pm
8am - 4pm
12pm - 8pm
4pm - 12am
8pm - 4am
Total
Number
Working
0
15
16
17
8
11
67
Reservation Agents Scheduling Problem
Time
Period
12am Š 4am
4am Š 8am
8am Š 12pm
12pm Š 4pm
4pm Š 8pm
8pm ŠŹ12am
McGraw-Hill/Irwin
Number
Working
11
15
31
33
25
19
>=
>=
>=
>=
>=
>=
Minimum
Required
11
15
31
17
25
19
3.49
© The McGraw-Hill Companies, Inc., 2008
Workforce Scheduling at United Airlines
•
United employs 5,000 reservation and customer service agents.
•
Some part-time (2-8 hour shifts), some full-time (8-10 hour shifts).
•
Workload varies greatly over day.
•
Modeled problem as LP:
– Decision variables: how many employees of each shift length should begin at each
potential start time (half-hour intervals).
– Constraints: minimum required employees for each half-hour.
– Objective: minimize cost.
•
Saved United about $6 million annually, improved customer service, still in
use today.
For more details, see Jan-Feb 1986 Interfaces article “United Airlines Station Manpower Planning System”
McGraw-Hill/Irwin
3.50
© The McGraw-Hill Companies, Inc., 2008
LP Example #4 (Transportation Problem)
A company has two plants producing a certain product that is to be shipped to
three distribution centers. The unit production costs are the same at the two plants,
and the shipping cost per unit is shown below. Shipments are made once per week.
During each week, each plant produces at most 60 units and each distribution
center needs at least 40 units.
Distribution Center
Plant
1
2
3
A
$4
$6
$4
B
$6
$5
$2
Question: How many units should be shipped from each plant to each
distribution center?
McGraw-Hill/Irwin
3.51
© The McGraw-Hill Companies, Inc., 2008
Algebraic Formulation
Let xij = units to ship from plant i to distribution center j (i = A, B; j = 1, 2, 3),
Minimize Cost = $4xA1 + $6xA2 + $4xA3 + $6xB1 + $5xB2 + $2xB3
subject to
Plant A:
xA1 + xA2 + xA3 ≤ 60
Plant B:
xB1 + xB2 + xB3 ≤ 60
Distribution Center 1:
xA1 + xB1 ≥ 40
Distribution Center 2:
xA2 + xB2 ≥ 40
Distribution Center 3:
xA3 + xB3 ≥ 40
and
xij ≥ 0 (i = A, B; j = 1, 2, 3).
McGraw-Hill/Irwin
3.52
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
B
3
4
5
6
7
8
9
10
11
12
13
14
15
Cost
Plant A
Plant B
Shipment
Quantities
Plant A
Plant B
Shipped
Needed
McGraw-Hill/Irwin
C
D
E
Distribution
Center 1
$4
$6
Distribution
Center 2
$6
$5
Distribution
Center 3
$4
$2
Distribution
Center 1
40
0
40
>=
40
Distribution
Center 2
20
20
40
>=
40
Distribution
Center 3
0
40
40
>=
40
3.53
F
Shipped
60
60
Cost
G
H
Available
<=
60
<=
60
=
$460
© The McGraw-Hill Companies, Inc., 2008
Distribution System at Proctor and Gamble
•
Proctor and Gamble needed to consolidate and re-design their North American
distribution system in the early 1990’s.
–
–
–
–
50 product categories
60 plants
15 distribution centers
1000 customer zones
•
Solved many transportation problems (one for each product category).
•
Goal: find best distribution plan, which plants to keep open, etc.
•
Closed many plants and distribution centers, and optimized their product
sourcing and distribution location.
•
Implemented in 1996. Saved $200 million per year.
For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply
Chain”
McGraw-Hill/Irwin
3.54
© The McGraw-Hill Companies, Inc., 2008
LP Example #5 (Assignment Problem)
The coach of a swim team needs to assign swimmers to a 200-yard medley relay
team (four swimmers, each swims 50 yards of one of the four strokes). Since most
of the best swimmers are very fast in more than one stroke, it is not clear which
swimmer should be assigned to each of the four strokes. The five fastest
swimmers and their best times (in seconds) they have achieved in each of the
strokes (for 50 yards) are shown below.
Backstroke
Breaststroke
Butterfly
Freestyle
Carl
37.7
43.4
33.3
29.2
Chris
32.9
33.1
28.5
26.4
David
33.8
42.2
38.9
29.6
Tony
37.0
34.7
30.4
28.5
Ken
35.4
41.8
33.6
31.1
Question: How should the swimmers be assigned to make the fastest relay team?
McGraw-Hill/Irwin
3.55
© The McGraw-Hill Companies, Inc., 2008
Algebraic Formulation
Let xij = 1 if swimmer i swims stroke j; 0 otherwise
tij = best time of swimmer i in stroke j
Minimize Time = ∑ i ∑ j tij xij
subject to
each stroke swum:
∑ i xij = 1 for each stroke j
each swimmer swims 1: ∑ j xij ≤ 1 for each swimmer i
and
xij ≥ 0 for all i and j.
McGraw-Hill/Irwin
3.56
© The McGraw-Hill Companies, Inc., 2008
Spreadsheet Formulation
3
4
5
6
7
8
9
B
C
D
E
F
Best Times
Backstroke
37.7
32.9
33.8
37.0
35.4
Breastroke
43.4
33.1
42.2
34.7
41.8
Butterfly
33.3
28.5
38.9
30.4
33.6
Freestyle
29.2
26.4
29.6
28.5
31.1
Backstroke
0
0
1
0
0
1
=
1
Breastroke
0
0
0
1
0
1
=
1
Butterfly
0
1
0
0
0
1
=
1
Freestyle
1
0
0
0
0
1
=
1
Carl
Chris
David
Tony
Ken
G
H
I
10
11
12
13
14
15
16
17
18
19
Assignment
Carl
Chris
David
Tony
Ken
McGraw-Hill/Irwin
3.57
1
1
1
1
0
Time
<=
1
<=
1
<=
1
<=
1
<=
1
= 126.2
© The McGraw-Hill Companies, Inc., 2008