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

5-2 Copyright © 2013 Pearson Education Integer Programming Models Types 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.

5-3 Copyright © 2013 Pearson Education 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:

5-4 Copyright © 2013 Pearson Education A Total Integer Model Integer Programming Model: Maximize Z = $100x 1 + $150x 2 subject to: $8,000x 1 + 4,000x 2  $40,000 15x x 2  200 ft 2 x 1, x 2  0 and integer x 1 = number of presses x 2 = number of lathes

5-5 Copyright © 2013 Pearson Education 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 Integer Model

5-6 Copyright © 2013 Pearson Education A Integer Model

5-7 Copyright © 2013 Pearson Education ■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). A Integer Model 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.

5-8 Copyright © 2013 Pearson Education Integer Programming Model: Maximize Z = 300x x x x 4 subject to: $35,000x ,000x ,000x ,000x 4  $120,000 4x 1 + 2x 2 + 7x 3 + 3x 4  12 acres x 1 + x 2  1 facility x 1, x 2, x 3, x 4 = 0 or 1 x 1 = construction of a swimming pool x 2 = construction of a tennis center x 3 = construction of an athletic field x 4 = construction of a gymnasium A Integer Model

5-9 Copyright © 2013 Pearson Education 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.

5-10 Copyright © 2013 Pearson Education 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

5-11 Copyright © 2013 Pearson Education 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

5-12 Copyright © 2013 Pearson Education 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.

5-13 Copyright © 2013 Pearson Education 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…..

5-14 Copyright © 2013 Pearson Education Cities Cities within 300 miles 1. AtlantaAtlanta, Charlotte, Nashville 2. BostonBoston, New York 3. CharlotteAtlanta, Charlotte, Richmond 4. CincinnatiCincinnati, Detroit, Indianapolis, Nashville, Pittsburgh 5. DetroitCincinnati, Detroit, Indianapolis, Milwaukee, Pittsburgh 6. IndianapolisCincinnati, Detroit, Indianapolis, Milwaukee, Nashville, St. Louis 7. MilwaukeeDetroit, Indianapolis, Milwaukee 8. NashvilleAtlanta, Cincinnati, Indianapolis, Nashville, St. Louis 9. New YorkBoston, New York, Richmond 10. PittsburghCincinnati, Detroit, Pittsburgh, Richmond 11. RichmondCharlotte, New York, Pittsburgh, Richmond 12. 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 Examples Set Covering Example

5-15 Copyright © 2013 Pearson Education x i = city i, i = 1 to 12; x i = 0 if city is not selected as a hub and x i = 1 if it is. Minimize Z = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 10 + x 11 + x 12 subject to:Atlanta:x 1 + x 3 + x 8  1 Boston:x 2 + x 9  1 Charlotte:x 1 + x 3 + x 11  1 Cincinnati:x 4 + x 5 + x 6 + x 8 + x 10  1 Detroit:x 4 + x 5 + x 6 + x 7 + x 10  1 Indianapolis: x 4 + x 5 + x 6 + x 7 + x 8 + x 12  1 Milwaukee:x 5 + x 6 + x 7  1 Nashville: x 1 + x 4 + x 6 + x 8 + x 12  1 New York:x 2 + x 9 + x 11  1 Pittsburgh:x 4 + x 5 + x 10 + x 11  1 Richmond: x 3 + x 9 + x 10 + x 11  1 St Louis: x 6 + x 8 + x 12  1 x ij = 0 or 1 0 – 1 Integer Programming Modeling Examples Set Covering Example