# y - Athene

```Chapter 7 (Binary Integer Programming)
Including two CD-rom supplements
McGraw-Hill/Irwin
9.1
© The McGraw-Hill Companies, Inc., 2003
Assumptions of linear programming
Proportionality: the gross margin and resource requirements per unit of activity are
assumed to be constant regardless of the level of the activity use
Additivity: no interaction effects between activities
Homogeneity: all units of the same resource or activity are identical
Continuity: resources can be used and activities produced in any fractional units
Deterministic coefficients: all coefficients in the model are known with certainty
Optimization: One proper objective function to be maximized or minimized
Finiteness: only a finite number of activities and constraints is considered
McGraw-Hill/Irwin
9.2
© The McGraw-Hill Companies, Inc., 2003
Types of Integer Programming Problems
•
Pure integer programming problems are those where all the decision
variables must be integers.
•
Mixed integer programming problems only require some of the variables
(the “integer variables”) to have integer values so the divisibility assumption
holds for the rest (the “continuous variables”).
•
Binary variables are variables whose only possible values are 0 and 1.
•
Binary integer programming (BIP) problems are those where all the
decision variables restricted to integer values are further restricted to be binary
variables.
– Such problems can be further characterized as either pure BIP problems or mixed
BIP problems, depending on whether all the decision variables or only some of
them are binary variables.
McGraw-Hill/Irwin
9.3
© The McGraw-Hill Companies, Inc., 2003
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
9.4
© The McGraw-Hill Companies, Inc., 2003
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
9.5
Capital
Available
\$100 million
© The McGraw-Hill Companies, Inc., 2003
Linear 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.
McGraw-Hill/Irwin
9.6
© The McGraw-Hill Companies, Inc., 2003
Graphical Method for Linear Programming
McGraw-Hill/Irwin
9.7
© The McGraw-Hill Companies, Inc., 2003
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
9.8
© The McGraw-Hill Companies, Inc., 2003
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
9.9
© The McGraw-Hill Companies, Inc., 2003
Graphical Method for Integer Programming
•
When an integer programming problem has just two decision variables, its
optimal solution can be found by applying the graphical method for linear
programming with just one change at the end.
•
We begin as usual by graphing the feasible region for the LP relaxation,
determining the slope of the objective function lines, and moving a straight
edge with this slope through this feasible region in the direction of improving
values of the objective function.
•
However, rather than stopping at the last instant the straight edge passes
through this feasible region, we now stop at the last instant the straight edge
passes through an integer point that lies within this feasible region.
•
This integer point is the optimal solution.
McGraw-Hill/Irwin
9.10
© The McGraw-Hill Companies, Inc., 2003
Graphical Method for Integer Programming
McGraw-Hill/Irwin
9.11
© The McGraw-Hill Companies, Inc., 2003
B
3
4
5
6
7
8
9
10
11
12
13
14
Unit Prof it (\$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
9.12
Large Airplane
2
E
Capital
Spent
100
F
G
<=
Capital
Av ailable
100
Total Prof it
(\$millions)
10
© The McGraw-Hill Companies, Inc., 2003
The default Tolerance field on the Solver
Options dialog (relevant only for ILP
models) is 5%.
This means that the
Solver ILP optimization
procedure is continued
only until the ILP solution
OV is within 5% of the
ILP’s optimum OV.
A higher Tolerance speeds
up Solver at the risk of a reported solution
further from the true ILP optimum.
Setting Tolerance to 0% forces Solver to
find the ILP optimum but with much longer
9.13
solution times.
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
Applications of Binary Variables
•
Since binary variables only provide two choices, they are ideally suited to be
the decision variables when dealing with yes-or-no decisions.
•
Examples:
– Should we undertake a particular fixed project?
– Should we make a particular fixed investment?
– Should we locate a facility in a particular site?
McGraw-Hill/Irwin
9.14
© The McGraw-Hill Companies, Inc., 2003
California Manufacturing Company
•
The California Manufacturing Company is a diversified company with several
factories and warehouses throughout California, but none yet in Los Angeles
or San Francisco.
•
A basic issue is whether to build a new factory in Los Angeles or San
Francisco, or perhaps even both.
•
Management is also considering building at most one new warehouse, but will
restrict the choice to a city where a new factory is being built.
Question: Should the California Manufacturing Company expand with
factories and/or warehouses in Los Angeles and/or San Francisco?
McGraw-Hill/Irwin
9.15
© The McGraw-Hill Companies, Inc., 2003
Data for California Manufacturing
Decision
Number
Yes-or-No
Question
Decision
Variable
Net Present
Value
(Millions)
Capital
Required
(Millions)
1
Build a factory in Los Angeles?
x1
\$8
\$6
2
Build a factory in San Francisco?
x2
5
3
3
Build a warehouse in Los Angeles?
x3
6
5
4
Build a warehouse in San Francisco?
x4
4
2
Capital Available: \$10 million
McGraw-Hill/Irwin
9.16
© The McGraw-Hill Companies, Inc., 2003
Binary Decision Variables
Decision
Number
Decision
Variable
Possible
Value
1
x1
0 or 1
Build a factory in
Los Angeles
Do not build
this factory
2
x2
0 or 1
Build a factory in
San Francisco
Do not build
this factory
3
x3
0 or 1
Build a warehouse in
Los Angeles
Do not build
this warehouse
4
x4
0 or 1
Build a warehouse in
San Francisco
Do not build
this warehouse
McGraw-Hill/Irwin
Interpretation
of a Value of 1
9.17
Interpretation
of a Value of 0
© The McGraw-Hill Companies, Inc., 2003
Algebraic Formulation
Let x1 = 1 if build a factory in L.A.; 0 otherwise
x2 = 1 if build a factory in S.F.; 0 otherwise
x3 = 1 if build a warehouse in Los Angeles; 0 otherwise
x4 = 1 if build a warehouse in San Francisco; 0 otherwise
Maximize NPV = 8x1 + 5x2 + 6x3 + 4x4 (\$millions)
subject to
Capital Spent:
6x1 + 3x2 + 5x3 + 2x4 ≤ 10 (\$millions)
Max 1 Warehouse:
x3 + x4 ≤ 1
Warehouse only if Factory:
x3 ≤ x1
x4 ≤ x2
and
x1, x2, x3, x4 are binary variables.
McGraw-Hill/Irwin
9.18
© The McGraw-Hill Companies, Inc., 2003
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
B
NPV (\$millions)
Warehouse
C
LA
6
D
SF
4
Factory
8
5
Capital Required
(\$millions)
Warehouse
LA
5
SF
2
Factory
6
3
Build?
Warehouse
Factory
McGraw-Hill/Irwin
LA
0
<=
1
Total NPV (\$millions)
SF
0
<=
1
E
Capital
Spent
9
Total
Warehouses
0
F
G
<=
Capital
Av ailable
10
<=
Maximum
Warehouses
1
13
9.19
© The McGraw-Hill Companies, Inc., 2003
Sensitivity Analysis with Solver Table
23
24
25
26
27
28
29
30
31
32
33
34
35
36
B
Capital Av ailable
(\$millions)
5
6
7
8
9
10
11
12
13
14
15
McGraw-Hill/Irwin
C
Warehouse
in LA?
0
0
0
0
0
0
0
0
0
0
1
1
D
Warehouse
in SF?
0
1
1
1
1
0
0
1
1
1
0
0
9.20
E
Factory
in LA?
1
0
0
0
0
1
1
1
1
1
1
1
F
Factory
in SF?
1
1
1
1
1
1
1
1
1
1
1
1
G
Total NPV
(\$millions)
13
9
9
9
9
13
13
17
17
17
19
19
© The McGraw-Hill Companies, Inc., 2003
Management’s Conclusion
•
Management’s initial tentative decision had been to make \$10 million of
capital available.
•
With this much capital, the best plan would be to build a factory in both Los
Angeles and San Francisco, but no warehouses.
•
An advantage of this plan is that it only uses \$9 million of this capital, which
frees up \$1 million for other projects.
•
A heavy penalty (a reduction of \$4 million in total net present value) would be
paid if the capital made available were to be reduced below \$9 million.
•
Increasing the capital made available by \$1 million (to \$11 million) would
enable a substantial (\$4 million) increase in the total net present value.
Management decides to do this.
•
With this much capital available, the best plan is to build a factory in both
cities and a warehouse in San Francisco.
McGraw-Hill/Irwin
9.21
© The McGraw-Hill Companies, Inc., 2003
Some Other Applications
•
Investment Analysis
– Should we make a certain fixed investment?
– Examples: Turkish Petroleum Refineries (1990), South African National Defense
Force (1997), Grantham, Mayo, Van Otterloo and Company (1999)
•
Site Selection
– Should a certain site be selected for the location of a new facility?
– Example: AT&T (1990)
•
Designing a Production and Distribution Network
– Should a certain plant remain open? Should a certain site be selected for a new
plant? Should a distribution center remain open? Should a certain site be selected
for a new distribution center? Should a certain distribution center be assigned to
serve a certain market area?
– Examples: Ault Foods (1994), Digital Equipment Corporation (1995)
McGraw-Hill/Irwin
9.22
© The McGraw-Hill Companies, Inc., 2003
Some Other Applications
•
Dispatching Shipments
– Should a certain route be selected for a truck? Should a certain size truck be used?
Should a certain time period for departure be used?
– Examples: Quality Stores (1987), Air Products and Chemicals, Inc. (1983),
Reynolds Metals Co. (1991), Sears, Roebuck and Company (1999)
•
Scheduling Interrelated Activities
– Should a certain activity begin in a certain time period?
– Examples: Texas Stadium (1983), China (1995)
•
Scheduling Asset Divestitures
– Should a certain asset be sold in a certain time period?
– Example: Homart Development (1987)
•
Airline Applications:
– Should a certain type of airplane be assigned to a certain flight leg? Should a
certain sequence of flight legs be assigned to a crew?
– Examples: American Airlines (1989, 1991), Air New Zealand (2001)
McGraw-Hill/Irwin
9.23
© The McGraw-Hill Companies, Inc., 2003
Wyndor with Setup Costs (Variation 1)
Suppose that two changes are made to the original Wyndor problem:
1. For each product, producing any units requires a substantial one-time setup
cost for setting up the production facilities.
2. The production runs for these products will be ended after one week, so D and
W in the original model now represent the total number of doors and windows
produced, respectively, rather than production rates. Therefore, these two
variables need to be restricted to integer values.
McGraw-Hill/Irwin
9.24
© The McGraw-Hill Companies, Inc., 2003
Graphical Solution to Original Wyndor Problem
W
Production rate
for windows
8
Opt imal solution
6
4
(2, 6)
Fe asible
Re gion
P = 3,600 = 300 D+ 500 W
2
0
McGraw-Hill/Irwin
4
6
2
Production rate for doors
9.25
8
10 D
© The McGraw-Hill Companies, Inc., 2003
Net Profit for Wyndor Problem with Setup Costs
Net Profit (\$)
Number of
Units Produced
McGraw-Hill/Irwin
Doors
Windows
0
0(300) – 0 = 0
0 (500) – 0 = 0
1
1(300) – 700 = –400
1(500) – 1,300 = –800
2
2(300) – 700 = –100
2(500) – 1,300 = –300
3
3(300) – 700 = 200
3(500) – 1,300 = 200
4
4(300) – 700 = 500
4(500) – 1,300 = 700
5
Not feasible
5(500) – 1,300 = 1,200
6
Not feasible
6(500) – 1,300 = 1,700
9.26
© The McGraw-Hill Companies, Inc., 2003
Feasible Solutions for Wyndor with Setup Costs
McGraw-Hill/Irwin
9.27
© The McGraw-Hill Companies, Inc., 2003
Algebraic Formulation
Let D = Number of doors to produce,
W = Number of windows to produce,
y1 = 1 if perform setup to produce doors; 0 otherwise,
y2 = 1 if perform setup to produce windows; 0 otherwise .
Maximize P = 300D + 500W – 700y1 – 1,300y2
subject to
Original Constraints:
Plant 1:
D≤4
Plant 2:
2W ≤ 12
Plant 3:
3D + 2W ≤ 18
Produce only if Setup:
Doors:
D ≤ 99y1
Windows: W ≤ 99y2
and
D ≥ 0, W ≥ 0, y1 and y2 are binary.
McGraw-Hill/Irwin
9.28
© The McGraw-Hill Companies, Inc., 2003
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Unit Prof it
Setup Cost
Plant 1
Plant 2
Plant 3
Units Produced
Only If Setup
Setup?
McGraw-Hill/Irwin
C
Doors
\$300
\$700
D
Windows
\$500
\$1,300
Hours Used Per Unit Produced
1
0
0
2
3
2
Doors
0
<=
0
0
Windows
6
<=
99
1
9.29
E
Hours
Used
0
12
12
F
G
<=
<=
<=
Hours
Av ailable
4
12
18
H
Production Prof it \$3,000
- Total Setup Cost \$1,300
Total Prof it \$1,700
© The McGraw-Hill Companies, Inc., 2003
Wyndor with Mutually Exclusive Products
(Variation 2)
Suppose that now the only change from the original Wyndor problem is:
•
The two potential new products (doors and windows) would compete for the
same customers. Therefore, management has decided not to produce both of
them together.
–
At most one can be chosen for production, so either D = 0 or W = 0, or both.
McGraw-Hill/Irwin
9.30
© The McGraw-Hill Companies, Inc., 2003
Feasible Solution for
Wyndor with Mutually Exclusive Products
McGraw-Hill/Irwin
9.31
© The McGraw-Hill Companies, Inc., 2003
Algebraic Formulation
Let D = Number of doors to produce,
W = Number of windows to produce,
y1 = 1 if produce doors; 0 otherwise,
y2 = 1 if produce windows; 0 otherwise.
Maximize P = 300D + 500W
subject to
Original Constraints:
Plant 1:
D≤4
Plant 2:
2W ≤ 12
Plant 3:
3D + 2W ≤ 18
Auxiliary variables must =1 if produce any:
Doors:
D ≤ 99y1
Windows:
W ≤ 99y2
Mutually Exclusive: y1 + y2 ≤ 1
and
D ≥ 0, W ≥ 0, y1 and y2 are binary.
McGraw-Hill/Irwin
9.32
© The McGraw-Hill Companies, Inc., 2003
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Unit Prof it
Plant 1
Plant 2
Plant 3
Units Produced
Only If Produce
Produce?
McGraw-Hill/Irwin
C
Doors
\$300
D
Windows
\$500
Hours Used Per Unit Produced
1
0
0
2
3
2
Doors
0
<=
0
0
Windows
6
<=
99
1
E
Hours
Used
0
12
12
Total
Produced
1
F
G
<=
<=
<=
Hours
Av ailable
4
12
18
<=
Maximum
To Produce
1
Total Prof it
\$3,000
9.33
© The McGraw-Hill Companies, Inc., 2003
Wyndor with Either-Or Constraints
(Variation 3)
Suppose that now the only change from the original Wyndor problem is:
•
The company has just opened a new plant (plant 4) that is similar to plant 3, so
the new plant can perform the same operations as plant 3 to help produce the
two new products (doors and windows).
•
However, management wants just one of the plants to be chosen to work on
these new products. The plant chosen should be the one that provides the most
profitable product mix.
McGraw-Hill/Irwin
9.34
© The McGraw-Hill Companies, Inc., 2003
Data for Wyndor with Either-Or Constraints
(Variation 3)
Production Time Used for
Each Unit Produced (Hours)
Plant
Doors
Windows
Production Time
Available
per Week (Hours)
1
1
0
4
2
0
2
12
3
3
2
18
4
2
4
28
Unit Profit
\$300
\$500
McGraw-Hill/Irwin
9.35
© The McGraw-Hill Companies, Inc., 2003
Graphical Solution with Plant 3 or Plant 4
McGraw-Hill/Irwin
9.36
© The McGraw-Hill Companies, Inc., 2003
Algebraic Formulation
Let D = Number of doors to produce,
W = Number of windows to produce,
y = 1 if plant 4 is used; 0 if plant 3 is used
Maximize P = 300D + 500W
subject to
Plant 1:
D≤4
Plant 2:
2W ≤ 12
Plant 3:
Plant 4:
3D + 2W ≤ 18 + 99y
2D + 4W ≤ 28 + 99(1 – y)
and
D ≥ 0, W ≥ 0, y1 and y2 are binary.
McGraw-Hill/Irwin
9.37
© The McGraw-Hill Companies, Inc., 2003
B
C
Doors
\$300
D
Windows
\$500
3
4
Unit Prof it
5
6
7
Hours Used Per Unit Produced
8
Plant 1
1
0
9
Plant 2
0
2
10
Plant 3
3
2
11
Plant 4
2
4
12
13
Doors
Windows
14
Units Produced
4
5
15
16 Which Plant to Use? (0=Plant 3, 1=Plant 4)
McGraw-Hill/Irwin
9.38
E
Hours
Used
4
10
22
28
F
G
H
<=
<=
<=
<=
Modif ied
Hours
Av ailable
4
12
117
28
Hours
Av ailable
4
12
18
28
Total Prof it
\$3,700
1
© The McGraw-Hill Companies, Inc., 2003
Good Products Company Production Planning
•
The Research and Development Division of the Good Products Company has
developed three possible new products.
•
To avoid undue diversification of the company’s product line, management
has imposed the following restriction:
– From the three possible new products, at most two should be chosen to be
produced.
•
Each of these products can be produced in either of two plants. For
administrative reasons, management has imposed the following restriction:
– Just one of the two plants should be chosen to be the sole producer of the two new
products.
McGraw-Hill/Irwin
9.39
© The McGraw-Hill Companies, Inc., 2003
Data for the Good Products Company
Production Time Used for Each
Unit Produced (Hours)
Plant
Product 1
Product 2
Product 3
Production Time
Available per
Week (Hours)
1
3
4
2
30
2
4
6
2
40
Unit Profit
5
7
3
(\$thousands)
Sales potential
7
5
9
(units per week)
McGraw-Hill/Irwin
9.40
© The McGraw-Hill Companies, Inc., 2003
Algebraic Formulation
Let xi = Number of units of product i to produce per week (i = 1, 2, 3),
yi = 1 if product i is produced; 0 otherwise (i = 1, 2, 3),
y4 = 1 if plant 2 is used; 0 if plant 1 is used
Maximize Profit = 5x1 + 7x2 + 3x3 (\$thousands)
subject to
Auxiliary variables must =1 if produce any & Max Sales:
Product 1:
x1 ≤ 7y1
Product 2:
x2 ≤ 5y2
Product 3:
x3 ≤ 9y3
Either plant 1 (y4 = 0) or plant 2 (y4 = 1):
Plant 1:
3x1 + 4x2 + 2x3 + 99y4 ≤ 30
Plant 2:
4x1 + 6x2 + 2x3 + 99(1 – y4) ≤ 40
and
xi ≥ 0 (i = 1, 2, 3), yi are binary (i = 1, 2, 3, 4).
McGraw-Hill/Irwin
9.41
© The McGraw-Hill Companies, Inc., 2003
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Unit Prof it (\$thousands)
Plant 1
Plant 2
C
Product 1
5
D
Product 2
7
E
Product 3
3
Hours Used Per Unit Produced
3
4
2
4
6
2
Only If Produce
Maximum Sales
Product 1
5.5
<=
7
7
Product 2
0
<=
0
5
Product 3
9
<=
9
9
Produce?
1
0
1
Units Produced
Which Plant to Use? (0=Plant 1, 1=Plant 2)
McGraw-Hill/Irwin
9.42
1
F
Hours
Used
34.5
40
Total
Produced
2
G
H
I
<=
<=
Modif ied
Hours
Av ailable
129
40
Hours
Av ailable
30
40
<=
Maximum
To Produce
2
Total Prof it
(\$thousands)
54.5
© The McGraw-Hill Companies, Inc., 2003
Supersuds Corporation Marketing Plan
•
The Supersuds Corporation is developing its marketing plan for next year’s
new products.
•
For three of these products, the decision has been made to purchase a total of
five TV spots for commercials on national television networks.
•
Each spot will feature a single product.
Question: How should the five spots be allocated to these three products?
McGraw-Hill/Irwin
9.43
© The McGraw-Hill Companies, Inc., 2003
Data for the Supersuds Corp. Problem
Profit (Millions)
Number of TV Spots
Product 1
Product 2
Product 3
0
\$0
\$0
\$0
1
1
0
–1
2
3
2
2
3
3
3
4
McGraw-Hill/Irwin
9.44
© The McGraw-Hill Companies, Inc., 2003
Algebraic Formulation
Let yij = 1 if there are j TV spots for product i; 0 otherwise (i = 1, 2, 3; j = 1, 2, 3)
Maximize Profit = y11 + 3y12 + 3y13 + 2y22 + 3y23 – y31 + 2y32 + 4y33 (\$millions)
subject to
Mutually Exclusive:
Product 1:
Product 2:
Product 3:
Total available spots:
y11 + y12 + y13 ≤ 1
y21 + y22 + y23 ≤ 1
y31 + y32 + y33 ≤ 1
y11 + 2y12 + 3y13 + y21 + 2y22 + 3y23 + y31 + 2y32 + 3y33 ≤ 5
and
yij are binary (i = 1, 2, 3; j = 1, 2, 3).
McGraw-Hill/Irwin
9.45
© The McGraw-Hill Companies, Inc., 2003
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
B
Profit
(\$millions)
Number
of
Spots
C
D
E
F
1
2
3
Product 1
1
3
3
Product 2
0
2
3
Product 3
-1
2
4
Product 1
0
1
0
1
<=
1
Product 2
0
0
0
0
<=
1
Product 3
0
0
1
1
<=
1
2
0
3
Solution
Number
1
of
2
Spots
3
Total
Max Of One
Number of Spots
McGraw-Hill/Irwin
9.46
G
H
I
Total
Prof it
(\$millions)
7
Total
Spots
5
=
Required
Spots
5
© The McGraw-Hill Companies, Inc., 2003
Southwestern Airways Crew Scheduling
•
Southwestern Airways needs to assign crews to cover all its upcoming flights.
•
We will focus on assigning 3 crews based in San Francisco (SFO) to 11
flights.
Question: How should the 3 crews be assigned 3 sequences of flights so that
every one of the 11 flights is covered?
McGraw-Hill/Irwin
9.47
© The McGraw-Hill Companies, Inc., 2003
Southwestern Airways Flights
Seat tl e
(SEA)
San Francis co
(SFO)
Denver
(DEN)
Chi cago
ORD)
Los Angel es
(LAX)
McGraw-Hill/Irwin
9.48
© The McGraw-Hill Companies, Inc., 2003
Data for the Southwestern Airways Problem
Feasible Sequence of Flights
Flights
1
1. SFO–LAX
1
2. SFO–DEN
2
3
5
6
1
1
3. SFO–SEA
3
3
4
3
4
4
4
6
7
9.49
3
5
5
3
3
4
5
7
2
4
2
3
2
2
2
11. SEA–LAX
1
5
2
10. SEA–SFO
12
4
3
9. DEN–ORD
McGraw-Hill/Irwin
1
2
7. ORD–SEA
2
11
1
3
3
10
1
1
2
2
9
1
2
8. DEN–SFO
8
1
1
6. ORD–DEN
Cost, \$1,000s
7
1
4. LAX–ORD
5. LAX–SFO
4
8
5
2
4
4
2
9
9
8
9
© The McGraw-Hill Companies, Inc., 2003
Algebraic Formulation
Let xj = 1 if flight sequence j is assigned to a crew; 0 otherwise. (j = 1, 2, … , 12).
Minimize Cost = 2x1 + 3x2 + 4x3 + 6x4 + 7x5 + 5x6 + 7x7 + 8x8 + 9x9 + 9x10 + 8x11 + 9x12
(in \$thousands)
subject to
Flight 1 covered:
Flight 2 covered:
:
Flight 11 covered:
x1 + x4 + x7 + x10 ≥ 1
x2 + x5 + x8 + x11 ≥ 1
:
x6 + x9 + x10 + x11 + x12 ≥ 1
Three Crews:
x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 ≤ 3
and
xj are binary (j = 1, 2, … , 12).
McGraw-Hill/Irwin
9.50
© The McGraw-Hill Companies, Inc., 2003
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
B
C
D
E
Cost (\$thousands)
1
2
2
3
3
4
Includes Segment?
SFO-LAX
SFO-DEN
SFO-SEA
LAX-ORD
LAX-SFO
ORD-DEN
ORD-SEA
DEN-SFO
DEN-ORD
SEA-SFO
SEA-LAX
Fly Sequence?
McGraw-Hill/Irwin
1
0
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
0
0
0
2
0
0
0
1
0
0
0
0
0
0
1
0
3
1
F
G
Flight
4
5
6
7
1
0
0
1
0
1
0
1
0
0
0
4
1
0
1
0
0
0
1
0
1
1
0
0
5
0
H
I
J
Sequence
6
7
8
5
7
8
0
0
1
0
1
0
0
0
0
0
1
6
0
1
0
0
1
0
0
1
0
0
1
0
7
0
0
1
0
0
0
0
1
0
1
1
0
8
0
K
L
9
9
10 11 12
9
8
9
0
0
1
1
0
1
0
1
0
0
1
9
0
1
0
0
1
1
0
1
0
0
0
1
M
0
1
0
0
1
0
1
0
1
0
1
N
0
0
1
1
0
0
1
0
0
1
1
10 11 12
0
1
0
O
Total
1
1
1
1
1
1
1
1
1
1
1
Total
Sequences
3
P
Q
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
At
Least
One
1
1
1
1
1
1
1
1
1
1
1
<=
Number
of Crews
3
Total Cost (\$thousands)
9.51
18
© The McGraw-Hill Companies, Inc., 2003
Integer Programming
•
When are “non-integer” solutions okay?
– Solution is naturally divisible
• e.g., \$, pounds, hours
– Solution represents a rate
• e.g., units per week
– Solution only for planning purposes
•
When is rounding okay?
– When numbers are large
• e.g., rounding 114.286 to 114 is probably okay.
•
When is rounding not okay?
– When numbers are small
• e.g., rounding 2.6 to 2 or 3 may be a problem.
– Binary variables
• yes-or-no decisions
McGraw-Hill/Irwin
9.52
© The McGraw-Hill Companies, Inc., 2003
The Challenges of Rounding
•
Rounded Solution may not be
feasible.
•
Rounded solution may not be
close to optimal.
•
There can be many rounded
solutions.
– Example: Consider a problem
with 30 variables that are noninteger in the LP-solution.
How many possible rounded
solutions are there?
x2
5
4
3
2
1
1
McGraw-Hill/Irwin
9.53
2
3
4
5
x1
© The McGraw-Hill Companies, Inc., 2003
How Integer Programs are Solved
x2
5
4
3
2
1
1
McGraw-Hill/Irwin
2
3
9.54
4
5
x1
© The McGraw-Hill Companies, Inc., 2003
How Integer Programs are Solved
x2
5
4
3
2
1
1
McGraw-Hill/Irwin
2
3
9.55
4
5
x1
© The McGraw-Hill Companies, Inc., 2003
Applications of Binary Variables
•
Making “yes-or-no” type decisions
–
–
–
–
•
Build a factory?
Manufacture a product?
Do a project?
Assign a person to a task?
Set-covering problems
– Make a set of assignments that “cover” a set of requirements.
•
Fixed costs
– If a product is produced, must incur a fixed setup cost.
– If a warehouse is operated, must incur a fixed cost.
McGraw-Hill/Irwin
9.56
© The McGraw-Hill Companies, Inc., 2003
Example #1 (Capital Budgeting)
•
Norwood Development is considering the potential of four different
development projects.
•
Each project would be completed in at most three years.
•
The required cash outflow for each project is given in the table below, along
with the net present value of each project to Norwood, and the cash that is
available each year.
Project 1
Project 2
Project 3
Project 4
Cash
Available
(\$million)
Year 1
9
7
6
11
28
Year 2
6
4
3
0
13
Year 3
6
0
4
0
10
NPV
30
16
22
14
Cash Outflow Required (\$million)
Question: Which projects should be undertaken?
McGraw-Hill/Irwin
9.57
© The McGraw-Hill Companies, Inc., 2003
Algebraic Formulation
Let yi = 1 if project i is undertaken; 0 otherwise (i = 1, 2, 3, 4).
Maximize NPV = 30y1 + 16y2 + 22y3 + 14y4
subject to
Year 1:
9y1 + 7y2 + 6y3 + 11y4 ≤ 28 (\$million)
Year 2 (cumulative):
15y1 + 11y2 + 9y3 + 11y4 ≤ 41 (\$million)
Year 3 (cumulative):
21y1 + 11y2 + 13y3 + 11y4 ≤ 51 (\$million)
and
yi are binary (i = 1, 2, 3, 4).
McGraw-Hill/Irwin
9.58
© The McGraw-Hill Companies, Inc., 2003
A
1
2
3
4
5
6
7
8
9
10
11
12
13
B
C
D
E
F
G
H
I
<=
<=
<=
Cumulativ e
Av ailable
28
41
51
Norwood Development Capital Budgeting
NPV (\$million)
Y ear 1
Y ear 2
Y ear 3
Undertake?
McGraw-Hill/Irwin
Project 1
30
Project 2
16
Project 3
22
Project 4
14
Cumulativ e Outf low Required (\$million)
9
7
6
11
15
11
9
11
21
11
13
11
Project 1
1
Project 2
1
Project 3
1
9.59
Project 4
0
Cumulativ e
Outf low
22
35
45
Total NPV
(\$million)
68
© The McGraw-Hill Companies, Inc., 2003
(Logic and Dependency Constraints)
•
At least one of projects 1, 2, or 3
•
Project 2 can’t be done unless project 3 is done
•
Either project 3 or project 4, but not both
•
No more than two projects total
Question: What constraints would need to be added for each of these
McGraw-Hill/Irwin
9.60
© The McGraw-Hill Companies, Inc., 2003
Example #2 (Set Covering Problem)
•
The Washington State legislature is trying to decide on locations at which to
base search-and-rescue teams.
•
The teams are expensive, so they would like as few as possible.
•
Response time is critical, so they would like every county to either have a team
located in that county or in an adjacent county.
Question: Where should search-and-rescue teams be located?
McGraw-Hill/Irwin
9.61
© The McGraw-Hill Companies, Inc., 2003
The Counties of Washington State
McGraw-Hill/Irwin
9.62
© The McGraw-Hill Companies, Inc., 2003
Algebraic Formulation
Let yi = 1 if a team is located in county i; 0 otherwise (i = 1, 2, … , 37).
Minimize Number of Teams = y1 + y2 + … + y37
subject to
County 1 covered:
y1 + y2 ≥ 1
County 2 covered:
y1 + y2 + y3 + y6 + y7 ≥ 1
County 3 covered:
y2 + y3 + y4 + y7 + y8 + y14 ≥ 1
:
County 37 covered:
y32 + y36 + y37 ≥ 1
and
yi are binary (i = 1, 2, … , 37).
McGraw-Hill/Irwin
9.63
© The McGraw-Hill Companies, Inc., 2003
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
B
C
D
E
F
G
H
I
J
K
L
M
N
Team?
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0
# Teams
Nearby
2
1
1
1
3
1
1
1
1
1
1
1
1
2
1
1
1
1
1
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Search & Rescue Location
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
County
Clallam
Jef f erson
Gray s Harbor
Pacif ic
Wahkiakum
Kitsap
Mason
Thurston
Whatcom
Skagit
Snohomish
King
Pierce
Lewis
Cowlitz
Clark
Skamania
Okanogan
Team?
0
1
0
0
0
0
0
0
0
1
0
0
0
1
0
0
1
0
Total Teams:
8
McGraw-Hill/Irwin
# Teams
Nearby
1
1
2
1
1
1
1
1
1
1
1
1
2
2
2
1
2
1
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
9.64
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
County
Chelan
Douglas
Kittitas
Grant
Y akima
Klickitat
Benton
Ferry
Stev ens
Pend Oreille
Lincoln
Spokane
Whitman
Franklin
Walla Walla
Columbia
Garf ield
Asotin
© The McGraw-Hill Companies, Inc., 2003
Example #3 (Fixed Costs)
•
Woodridge Pewter Company is a manufacturer of three pewter products:
platters, bowls, and pitchers.
•
The manufacture of each product requires Woodridge to have the appropriate
machinery and molds available. The machinery and molds for each product
can be rented at the following rates: for the platters, \$400/week; for the bowls,
\$250/week; for the pitcher, \$300/week.
•
Each product requires the amounts of labor and pewter given in the table
below. The sales price and variable cost are also given in the table.
Labor
Hours
Pewter
(pounds)
Sales
Price
Variable
Cost
Platter
3
5
\$100
\$60
Bowl
1
4
85
50
Pitcher
4
3
75
40
130
240
Available
Question: Which products should be produced, and in what quantity?
McGraw-Hill/Irwin
9.65
© The McGraw-Hill Companies, Inc., 2003
Algebraic Formulation
Let x1 = Number of platters produced,
x2 = Number of bowls produced,
x3 = Number of pitchers produced,
yi = 1 if lease machine and mold for product i; 0 otherwise (i = 1, 2, 3).
Maximize Profit = (\$100–\$60)x1 + (\$85–\$50)x2 + (\$75–\$40)x3 – \$400y1 – \$250y2 – \$300y3
subject to
Labor:
3x1 + x2 + 4x3 ≤ 130 hours
Pewter:
5x1 + 4x2 + 3x3 ≤ 240 pounds
Allow production only if machines and molds are purchased:
x1 ≤ 99y1
x2 ≤ 99y2
x3 ≤ 99y3
and
xi ≥ 0, and yi are binary (i = 1, 2, 3).
McGraw-Hill/Irwin
9.66
© The McGraw-Hill Companies, Inc., 2003
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B
C
D
E
F
Bowls
\$85
\$50
\$250
Pitchers
\$75
\$40
\$300
G
H
<=
<=
Av ailable
130
240
Woodridge Pewter Company
Sales Price
Variable Cost
Fixed Cost
Constraint
Labor (hrs.)
Pewter (lbs.)
Lease Equipment?
Production Quantity
Produce only if Lease
McGraw-Hill/Irwin
Platters
\$100
\$60
\$400
Usage (per unit produced)
3
1
4
5
4
3
0
0
<=
0
1
60
<=
99
9.67
Total
60
240
0
0
<=
0
Rev enue
Variable Cost
Fixed Cost
Prof it
\$5,100
\$3,000
\$250
\$1,850
© The McGraw-Hill Companies, Inc., 2003
Applications of Binary Variables
•
Making “yes-or-no” type decisions
–
–
–
–
•
Build a factory?
Manufacture a product?
Do a project?
Assign a person to a task?
Fixed costs
– If a product is produced, must incur a fixed setup cost.
– If a warehouse is operated, must incur a fixed cost.
•
Either-or constraints
– Production must either be 0 or ≥ 100.
•
Subset of constraints
– meet 3 out of 4 constraints.
McGraw-Hill/Irwin
9.68
© The McGraw-Hill Companies, Inc., 2003
Capital Budgeting with Contingency Constraints
(Yes-or-No Decisions)
•
A company is planning their capital budget over the next several years.
•
There are 10 potential projects they are considering pursuing.
•
They have calculated the expected net present value of each project, along
with the cash outflow that would be required over the next five years.
•
Also, suppose there are the following contingency constraints:
– at least one of project 1, 2 or 3 must be done,
– project 4 and project 5 cannot both be done,
– project 7 can only be done if project 6 is done.
Question: Which projects should they pursue?
McGraw-Hill/Irwin
9.69
© The McGraw-Hill Companies, Inc., 2003
Data for Capital Budgeting Problem
Cash Outflow Required (\$million)
1
2
3
4
5
6
7
8
9
10
Cash
Available
(\$million)
Year 1
1
4
0
4
4
3
2
8
2
6
25
Year 2
2
2
2
2
2
4
2
3
3
6
25
Year 3
3
2
5
2
4
2
3
4
8
2
25
Year 4
4
4
5
4
5
3
1
2
1
1
25
Year 5
1
1
0
6
5
5
5
1
1
2
25
NPV
20
25
22
30
42
25
18
35
28
33
(\$million)
Project
McGraw-Hill/Irwin
9.70
© The McGraw-Hill Companies, Inc., 2003
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
B
C
D
E
F
G
H
I
J
K
L
Project
5
42
Project
6
25
Project
7
18
Project
8
35
Project
9
28
Project
10
33
M
N
O
<=
<=
<=
<=
<=
Cumulativ e
Av ailable
25
50
75
100
125
Capital Budgeting with Contingency Constraints
NPV (\$million)
Project
1
20
Project
2
25
Project
3
22
Cumulativ e Cash Outf low Required (\$million)
Y ear 1
1
4
0
Y ear 2
3
6
2
Y ear 3
6
8
7
Y ear 4
10
12
12
Y ear 5
11
13
12
Undertake?
Project
1
1
Contingency Constraints
Project 1,2,3
3
Project 4,5
1
Project 7
1
McGraw-Hill/Irwin
Project
4
30
4
6
8
12
18
4
6
10
15
20
3
6
8
11
16
2
4
7
8
13
8
11
15
17
18
2
5
13
14
15
6
12
14
15
17
Project
5
1
Project
6
1
Project
7
1
Project
8
0
Project
9
1
Project
10
1
Project
2
1
Project
3
1
Project
4
0
>=
<=
<=
1
1
1
Project 6
9.71
Cumulativ e
Total Outf low
22
44
73
97
117
Total NPV
(\$million)
213
© The McGraw-Hill Companies, Inc., 2003
Electrical Generator Startup Planning (Fixed Costs)
•
An electrical utility company owns five generators.
•
To generate electricity, a generator must be started up, and associated with this
is a fixed startup cost.
•
All of the generators are shut off at the end of each day.
Generator
Fixed Startup Cost
Variable Cost (per MW)
Capacity (MW)
A
B
C
D
E
\$2,450
\$1,600
\$1,000
\$1,250
\$2,200
\$3
\$4
\$6
\$5
\$4
2,000
2,800
4,300
2,100
2,000
Question: Which generators should be started up to meet the total
capacity needed for the day (6000 MW)?
McGraw-Hill/Irwin
9.72
© The McGraw-Hill Companies, Inc., 2003
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B
C
D
E
F
G
H
I
J
Total MW
6000
>=
MW Needed
6,000
Electrical Utility Generator Startup Planning
Generator A
\$2,450
\$3
2,000
Generator B
\$1,600
\$4
2,800
Generator C
\$1,000
\$6
4,300
Generator D
\$1,250
\$5
2,100
Generator E
\$2,200
\$4
2,000
Startup?
1
1
0
1
0
MW Generated
2,100
<=
2,000
3,000
<=
2,800
0
<=
0
900
<=
2,100
0
<=
0
Fixed Startup Cost
Cost per Megawatt
Max Capacity (MW)
Capacity
Fixed Cost
Variable Cost
Total Cost
McGraw-Hill/Irwin
9.73
\$5,300
\$22,800
\$28,100
© The McGraw-Hill Companies, Inc., 2003
Quality Furniture (Either-Or Constraints)
•
Reconsider the Quality Furniture Problem:
– 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.
•
Now suppose that they would not produce any fewer than 200 units of either
product (i.e., either produce 0 or at least 200).
Question: What product mix will maximize their total profit?
McGraw-Hill/Irwin
9.74
© The McGraw-Hill Companies, Inc., 2003
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
B
C
D
E
F
G
<=
<=
Resources
Av ailable
1,600
9,000
Quality Furniture (with either-or constraints)
Prof it
Min Production (if any )
Labor
Wood
Produce?
Min Production
Production Quantities
McGraw-Hill/Irwin
Max Production
Max Possible
Benches
\$8.00
200
Tables
\$18.00
200
Use of Resources
3
6
12
38
1
0
200
<=
533.33
<=
533
533
0
<=
0
<=
0
237
9.75
Resources
Used
1600
6400
Total Prof it
\$4,266.67
© The McGraw-Hill Companies, Inc., 2003
Meeting a Subset of Constraints
•
Consider a linear programming model with the following constraints, and
suppose that meeting 3 out of 4 of these is good enough
–
–
–
–
12x1 + 24x2 + 18x3 ≥ 2,400
15x1 + 32x2 + 12x3 ≥ 1,800
20x1 + 15x2 + 20x3 ≤ 2,000
18x1 + 21x2 + 15x3 ≤ 1,600
McGraw-Hill/Irwin
9.76
© The McGraw-Hill Companies, Inc., 2003
Meeting a Subset of Constraints
Let yi = 1 if constraint i is enforced; 0 otherwise.
Constraints:
y1 + y2 + y3 + y4 ≥ 3
12x1 + 24x2 + 18x3 ≥ 2,400y1
15x1 + 32x2 + 12x3 ≥ 1,800y2
20x1 + 15x2 + 20x3 ≤ 2,000 + M (1 – y3)
18x1 + 21x2 + 15x3 ≤ 1,600 + M (1 – y4)
where M is a large number.
McGraw-Hill/Irwin
9.77
© The McGraw-Hill Companies, Inc., 2003
Facility Location
•
Consider a company that operates 5 plants and 3 warehouses that serve
customers in 4 different regions.
•
To lower costs, they are considering streamlining by closing one or more
plants and warehouses.
•
Associated with each plant are fixed costs, shipping costs, and production
costs. Each plant has a limited capacity.
•
Associated with each warehouse are fixed costs and shipping costs. Each
warehouse has a limited capacity.
Questions:
Which plants should they keep open?
Which warehouses should they keep open?
How should they divide production among the open plants?
How much should be shipped from each plant to each warehouse, and from each
warehouse to each customer?
McGraw-Hill/Irwin
9.78
© The McGraw-Hill Companies, Inc., 2003
Data for Facility Location Problem
(Shipping + Production) Cost
(per unit)
Fixed
Cost
(per month)
WH #1
WH #2
WH #3
Capacity
(units per
month)
Plant 1
\$42,000
\$650
\$750
\$850
400
Plant 2
50,000
500
350
550
300
Plant 3
45,000
450
450
350
300
Plant 4
50,000
400
500
600
350
Plant 5
47,000
550
450
350
375
Shipping Cost (per unit)
Fixed Cost
(per month)
Cust. 1
Cust. 2
Cust. 3
Cust. 4
Capacity
(per month)
WH #1
\$45,000
\$25
\$65
\$70
\$35
600
WH #2
25,000
50
25
40
60
400
WH #3
65,000
60
20
40
45
900
250
225
200
275
Demand:
McGraw-Hill/Irwin
9.79
© The McGraw-Hill Companies, Inc., 2003
A
B
C
D
E
Warehouse 2
\$750
\$350
\$450
\$500
\$450
Warehouse 3
\$850
\$550
\$350
\$600
\$350
Warehouse 2
0
300
0
0
0
300
Warehouse 3
0
0
275
0
375
650
Customer 1
\$25
\$50
\$60
Customer 2
\$65
\$25
\$20
Customer 3
\$70
\$40
\$40
Customer 4
\$35
\$60
\$45
Customer 1
0
250
0
250
>=
250
Customer 2
0
0
225
225
>=
225
Customer 3
0
50
150
200
>=
200
Customer 4
0
0
275
275
>=
275
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Plant to Warehouse
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
Warehouse to Customer
Shipping + Production
Cost
Warehouse 1
Plant 1
\$650
Plant 2
\$500
Plant 3
\$450
Plant 4
\$400
Plant 5
\$550
Shipment
Quantities
Plant 1
Plant 2
Plant 3
Plant 4
Plant 5
Total Shipped
Shipping
Cost
Warehouse 1
Warehouse 2
Warehouse 3
Shipment
Quantities
Warehouse 1
Warehouse 2
Warehouse 3
Total Shipped
Needed
Warehouse 1
0
0
0
0
0
0
McGraw-Hill/Irwin
F
G
H
Fixed
Cost
\$42,000
\$50,000
\$45,000
\$50,000
\$47,000
Total
Shipped
0
300
275
0
375
<=
<=
<=
<=
<=
I
9.80
K
L
M
Capacity
400
300
300
350
375
Actual
Capacity
0
300
300
0
375
Open?
0
1
1
0
1
Fixed
Cost
\$45,000
\$25,000
\$65,000
Shipped
Out
0
300
650
J
<=
<=
<=
Shipping Cost (P-->W)
Shipping Cost (W-->C)
Fixed Cost (P)
Fixed Cost (W)
Total Cost
Total Costs
\$332,500
\$37,375
\$142,000
\$90,000
\$601,875
Capacity
600
400
900
Shipped
In
0
300
650
<=
<=
<=
Actual
Capacity
0
400
900
Open?
0
1
1
© The McGraw-Hill Companies, Inc., 2003
```

46 cards

27 cards

25 cards

24 cards

29 cards