5-1 Copyright © 2013 Pearson Education Integer Linear Programming Models Chapter 5

5-1Copyright © 2013 Pearson Education

Integer Linear Programming Models

Chapter 5


Integer Programming ModelsTypes of Models

Total Integer Model:

All decision variables required to have integer solution values.

0-1 Integer Model:

All decision variables required to have integer values of zero or one.

Mixed Integer Model:

Some of the decision variables (but not all) required to have integer values.


Machine

Required Floor Space (ft.2)

Purchase Price

Press Lathe

15

30

$8,000

4,000

A Total Integer Model

■ Machine shop obtaining new presses and lathes.

■ Marginal profitability: each press $100/day; each lathe $150/day.

■ Resource constraints: $40,000 budget, 200 sq. ft. floor space.

■ Machine purchase prices and space requirements:


A Total Integer Model

Integer Programming Model:

Maximize Z = $100x1 + $150x2

subject to:

$8,000x1 + 4,000x2 $40,000

15x1 + 30x2 200 ft2

x1, x2 0 and integer

x1 = number of presses x2 = number of lathes


A community council must decide which recreation facilities to construct in its community. Four new recreation facilities have been proposed—a swimming pool, a tennis center, an athletic field, and a gymnasium. The council wants to construct facilities that will maximize the expected daily usage by the residents of the community, subject to land and cost limitations. The expected daily usage and cost and land requirements for each facility follow:

A 0 - 1 Integer Model


Recreation

Facility

Expected Usage (people/ day)

Cost ($)

Land Requirement (acres)

Swimming pool Tennis Center Athletic field Gymnasium

300 90 400 150

35,000 10,000 25,000 90,000

4 2 7 3



■ Recreation facilities selection to maximize daily usage by

residents.

■ Resource constraints: $120,000 budget; 12 acres of land.

■ Selection constraint: either swimming pool or tennis center (not both).


The community has a $120,000 construction budget and 12 acres of land. Because the swimming pool and tennis center must be built on the same part of the land parcel, however, only one of these two facilities can be constructed. The council wants to know which of the recreation facilities to construct to maximize the expected daily usage.


Integer Programming Model:

Maximize Z = 300x1 + 90x2 + 400x3 + 150x4

subject to:

$35,000x1 + 10,000x2 + 25,000x3 + 90,000x4 $120,000

4x1 + 2x2 + 7x3 + 3x4 12 acres

x1 + x2 1 facility

x1, x2, x3, x4 = 0 or 1

x1 = construction of a swimming pool x2 = construction of a tennis center x3 = construction of an athletic field x4 = construction of a gymnasium



In this model, the decision variables can have a solution value of either zero or one. If a facility is not selected for construction, the decision variable representing it will have a value of zero. If a facility is selected, its decision variable will have a value of one.

The last constraint, reflects the contingency that either the swimming pool or the tennis center can be constructed, but not both. In order for the sum of and to be less than or equal to one, either of the variables can have a value of one, or both variables can equal zero. This is also referred to as a mutually exclusive constraint.


If the community had specified that either the swimming pool or the tennis center must be built, but not both, then the last constraint would become….

…… a multiple-choice constraint


if the community council had specified that exactly two of the four facilities must be built, this constraint would be formulated as ……

If, alternatively, the council had specified that no more than two facilities must be constructed, the constraint would be


Another type of 0–1 model constraint is a conditional constraint. In a conditional constraint, the construction of one facility is conditional upon the construction of another……the tennis center is conditional upon construction of the swimming pool….

the tennis center cannot equal one (i.e., be selected) unless the pool equals one. If the pool equals zero (i.e., it is not selected), then the tennis center must also equal zero. However, this condition does allow the pool to equal one and be selected and the tennis center to equal zero and not be selected.


A variation of this type of conditional constraint is the corequisite constraint, wherein if one facility is constructed, the other one will also be constructed and vice versa. ……if the pool is accepted, the tennis center must also be selected and vice versa…..


Cities Cities within 300 miles 1. Atlanta Atlanta, Charlotte, Nashville 2. Boston Boston, New York 3. Charlotte Atlanta, Charlotte, Richmond 4. Cincinnati Cincinnati, Detroit, Indianapolis, Nashville, Pittsburgh 5. Detroit Cincinnati, Detroit, Indianapolis, Milwaukee, Pittsburgh 6. Indianapolis Cincinnati, Detroit, Indianapolis, Milwaukee, Nashville, St. Louis 7. Milwaukee Detroit, Indianapolis, Milwaukee 8. Nashville Atlanta, Cincinnati, Indianapolis, Nashville, St. Louis 9. New York Boston, New York, Richmond10. Pittsburgh Cincinnati, Detroit, Pittsburgh, Richmond11. Richmond Charlotte, New York, Pittsburgh, Richmond12. St. Louis Indianapolis, Nashville, St. Louis

APS wants to construct the minimum set of new hubs in these twelve cities such that there is a hub within 300 miles of every city:

0 – 1 Integer Programming Modeling ExamplesSet Covering Example


xi = city i, i = 1 to 12; xi = 0 if city is not selected as a hub and xi = 1 if it is.

Minimize Z = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12

subject to: Atlanta: x1 + x3 + x8 1Boston: x2 + x9 1Charlotte: x1 + x3 + x11 1Cincinnati: x4 + x5 + x6 + x8 + x10 1Detroit: x4 + x5 + x6 + x7 + x10 1Indianapolis: x4 + x5 + x6 + x7 + x8 + x12 1Milwaukee: x5 + x6 + x7 1Nashville: x1 + x4 + x6+ x8 + x12 1New York: x2 + x9+ x11 1Pittsburgh: x4 + x5 + x10 + x11 1Richmond: x3 + x9 + x10 + x11 1St Louis: x6 + x8 + x12 1 xij = 0 or 1

0 – 1 Integer Programming Modeling ExamplesSet Covering Example

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5-1 Copyright © 2013 Pearson Education Integer Linear Programming Models Chapter 5