Session #6
Integer Programming
Page 1
Integer Programming
When are “non-integer” solutions okay?
• Solution is naturally divisible
• Solution represents a rate
• Solution only for planning purposes
When is rounding okay?
Session #6
Integer Programming
Page 2
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 non-integer in the
LP solution. How many possible rounded solutions are there?
Session #6
Integer Programming
Page 3
How Integer Programs are Solved
Session #6
Integer Programming
Page 4
Applications of Binary Variables
• Making “yes or no” type decisions
• Set-covering problems
• Fixed costs
Session #6
Integer Programming
Page 5
Example #1 (Capital Budgeting)
Mercer 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 Mercer, and the
cash that is available (from previous projects) each year.
Cash Outflow Required ($million)
Year 1
Year 2
Year 3
NPV
Project 1
10
8
8
35
Project 2
8
5
0
18
Project 3
6
4
6
24
Project 4
12
0
0
16
Cash
Available
($million)
30
15
12
Session #6
Integer Programming
Page 6
Spreadsheet Solution to Example #1
A
1
2
3
4
5
6
7
8
9
10
11
12
13
3
4
5
6
7
8
9
10
11
12
13
B
C
D
E
F
G
H
I
Mercer Development Capital Budgeting
NPV ($million)
Year 1
Year 2
Year 3
Undertake?
Project 1
35
10
8
8
Project 2
18
Project 4
16
Cash Outflow Required ($million)
8
6
5
4
0
6
Project 1
1
Project 2
1
B
C
D
E
F
Project 1 Project 2 Project 3 Project 4
NPV ($million) 35
18
24
16
Year 1 10
Year 2 8
Year 3 8
Project 3
24
Cash Outflow Required ($million)
8
6
12
5
4
0
0
6
0
Project 1 Project 2 Project 3 Project 4
Undertake? 1
1
0
1
Project 3
0
12
0
0
Total Outflow
30
<=
13
<=
8
<=
Available
30
15
12
Total NPV
($million)
69
Project 4
1
G
H
I
Total Outflow
=SUMPRODUCT($C$13:$F$13,C7:F7) <= 30
=SUMPRODUCT($C$13:$F$13,C8:F8) <= 15
=SUMPRODUCT($C$13:$F$13,C9:F9) <= 12
Available
Total NPV
($million)
=SUMPRODUCT(C13:F13,C4:F4)
Session #6
Integer Programming
Page 7
Additional Considerations
(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:
Session #6
Integer Programming
Page 8
Example #2 (Set Covering Problem)
Suppose the Washington State legislature is trying to decide on locations at which to
base search-and-rescue teams. The teams are expensive, and hence they would like as
few as possible. However, since response time is critical, they would like every county
to either have a team located in that county, or in an adjacent county. Where should the
teams be located.
Session #6
Integer Programming
Page 9
Spreadsheet Solution to Example #2
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
Search & Rescue Location
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
County
Clallam
Jefferson
Grays Harbor
Pacific
Wahkiakum
Kitsap
Mason
Thurston
Whatcom
Skagit
Snohomish
King
Pierce
Lewis
Cowlitz
Clark
Skamania
Okanogan
Total Teams:
# Teams
Team? Nearby
0
1
1
1
0
2
0
1
0
1
0
1
0
1
0
1
0
1
1
1
0
1
0
1
0
2
1
2
0
2
0
1
1
2
0
1
8
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
County
Chelan
Douglas
Kittitas
Grant
Yakima
Klickitat
Benton
Ferry
Stevens
Pend Oreille
Lincoln
Spokane
Adams
Whitman
Franklin
Walla Walla
Columbia
Garfield
Asotin
# Teams
Team? Nearby
0
2
0
1
1
1
0
1
0
3
0
1
0
1
0
1
1
1
0
1
0
1
0
1
0
1
0
2
1
1
0
1
0
1
1
1
0
1
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Session #6
Integer Programming
Page 10
Spreadsheet Solution to Example #2 (Formulas)
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
B
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
C
County
Clallam
Jefferson
Grays Harbor
Pacific
Wahkiakum
Kitsap
Mason
Thurston
Whatcom
Skagit
Snohomish
King
Pierce
Lewis
Cowlitz
Clark
Skamania
Okanogan
0
1
0
0
0
0
0
0
0
1
0
0
0
1
0
0
1
0
Total Teams:
=SUM(D5:D22,K5:K23)
I
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
D
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
Team?
J
County
Chelan
Douglas
Kittitas
Grant
Yakima
Klickitat
Benton
Ferry
Stevens
Pend Oreille
Lincoln
Spokane
Adams
Whitman
Franklin
Walla Walla
Columbia
Garfield
Asotin
K
E
# Teams
Nearby
F
G
=D5+D6
=D5+D6+D7+D10+D11
=D6+D7+D8+D11+D12+D18
=D7+D8+D9+D18
=D8+D9+D18+D19
=D6+D10+D11+D15+D16+D17
=D6+D7+D10+D11+D12+D17
=D7+D11+D12+D17+D18
=D13+D14+D22
=D13+D14+D15+D22+K5
=D14+D15+D16+K5
=D10+D15+D16+D17+K5+K7
=D10+D11+D12+D16+D17+D18+K7+K8
=D7+D8+D9+D12+D17+D18+D19+D21+K9
=D9+D18+D19+D20+D21
=D19+D20+D21
=D18+D19+D20+D21+K9+K10
=D13+D14+D22+K5+K6+K12+K15
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
L
# Teams
Team?
Nearby
0
=D14+D15+D16+D22+K5+K6+K7
0
=D22+K5+K6+K7+K8
1
=D16+D17+K5+K6+K7+K8+K9
0
=K6+K7+K8+K9+K11+K15+K16+K17
0
=D17+D18+D21+K7+K8+K9+K10+K11
0
=D21+K9+K10+K11
0
=K8+K9+K10+K11+K19+K20
0
=D22+K12+K13+K15
1
=K12+K13+K14+K15+K16
0
=K13+K14+K16
0
=D22+K8+K12+K13+K15+K16+K17+K18
0
=K13+K14+K15+K18
0
=K8+K15+K17+K18+K19
0
=K15+K16+K17+K18+K19+K21+K22+K23
1
=K8+K11+K17+K18+K19+K20
0
=K11+K19+K20+K21
0
=K18+K20+K21+K22
1
=K18+K21+K22+K23
0
=K18+K22+K23
M
N
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Session #6
Integer Programming
Page 11
Example #3 (Fixed Costs)
Gandy Cloth Company is capable of manufacturing three types of clothing:
shirts, shorts, and pants. The manufacture of each type of clothing requires
that Gandy have the appropriate type of machinery available. This
machinery can be rented at the following rates: shirt machinery,
$200/week; shorts machinery, $150/week; pants machinery, $100/week.
Each type of clothing requires the amount of cloth and labor given in the
table below. The sales price and variable cost are also given in the table.
Shirts
Shorts
Pants
Available:
Labor
Hours
3
2
6
150 hours
Cloth
(sq. yds.)
4
3
4
160 sq. yds.
Sales
Price
$12
$8
$15
Variable
Cost
$6
$4
$8
Session #6
Integer Programming
Page 12
Spreadsheet Solution to Example #3
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
B
C
D
E
Shirt
$12
$6
$200
Shorts
$8
$4
$150
Pants
$15
$8
$100
Constraint
Labor (hrs.)
Cloth (sq. yds.)
3
4
Usage (per unit produced)
2
3
6
4
Lease Equipment?
0
0
1
Production Quantity
Produce only if Lease
0
<=
0
0
<=
0
25
<=
100
Big Number
100
F
G
H
Total
150
100
<=
<=
Available
150
160
Revenue
Variable Cost
Fixed Cost
Profit
$375.00
$200.00
$100.00
$75.00
Gandy Cloth Company
Sales Price
Variable Cost
Fixed Cost
B
Sales Price 12
Variable Cost 6
Fixed Cost 200
Constraint
Labor (hrs.) 3
Cloth (sq. yds.) 4
C
Shirt
D
Shorts
8
4
150
E
Pants
Usage (per unit produced)
2
6
3
4
Lease Equipment? 0
0
1
Production Quantity 0
0
25
<=
<=
<=
Produce only if Lease =$C$18*C12 =$C$18*D12 =$C$18*E12
Big Number 100
F
G
Total
=SUMPRODUCT($C$14:$E$14,C9:E9)
=SUMPRODUCT($C$14:$E$14,C10:E10)
<=
<=
H
15
8
100
Available
Revenue
Variable Cost
Fixed Cost
Profit
150
160
=SUMPRODUCT(C14:E14,C4:E4)
=SUMPRODUCT(C5:E5,C14:E14)
=SUMPRODUCT(C12:E12,C6:E6)
=H13-H14-H15