1. Chapter 5 Integer Programming

2. What is an integer program (IP)? IP is a linear program in which all or some variables can only take integral values. A satisfiable solution of IP cannot be obtained by “rounding” the corresponding LP solution. Solution method of IP is different from, and more difficult than, solution method of LP.

3. Example, p.182 (185) Machine SpaceUnitProfit required Pricepredicted Press15 ft 2 /unit$8,000$100/day Lathe30 ft 2 /unit$4,000$150/day There are 200 ft 2 and $40,000 available. How many presses and lathes to purchase to maximize daily profit?

4. Example: Investment CostProfitAvailable opportunityper unitper unit (units) Condominium$50,000$9,000 4 Land$12,000$1,500 15 Bond$ 8,000$1,000 20 There are $250,000 available. Where to invest to maximize the profit?

5. Types of IP models Total IP model – All variables must be integral. 0-1 IP model – Variables can be 0 or 1 only. Mixed IP model – variables can be 0-1, integral or non-integral.

6. Logical Representations by Using 0-1 variables 0-1 variables is used to represent various logical relationships in integer programming formulations.

7. 0-1 variables Let i=1, 2, 3, …

8. Mutually exclusive relation. Either X 1 or X 2 (but not neither, not both): X 1 +X 2 =1

9. Contingency relation Either X 1 or X 2, or neither (but not both): X 1 +X 2 <=1 At least one of X 1 and X 2 : X 1 +X 2 >=1

10. Co-requisite Relation X 1 and X 2 must be ‘on’ or ‘off’ together: X 1 =X 2

11. Mutually exclusiveness on more than two variables Select exactly one of {X 1, X 2, X 3 } (Multiple- choice relation): X 1 +X 2 +X 3 =1

12. Contingency relations on more than two variables Select no more than one from {X 1, X 2, X 3 }: X 1 +X 2 +X 3 <=1 Select no more than two from {X 1, X 2, X 3 }: X 1 +X 2 +X 3 <=2 Select at least one of {X 1, X 2, X 3 }: X 1 +X 2 +X 3 >=1

13. Conditional relation (if … then …) If X 1 is ‘on’, then X 2 must be ‘on’, (and if X 1 is ‘off’ then X 2 can be either ‘on’ or ‘off’.) X 1 <=X 2

14. Example, p.183 (186) Facility UsageCostLand (acres) consideredpeople/day $ required Swimming pool 30035,000 4 Tennis center 9010,000 2 Athletic field 40025,000 7 Gymnasium 15090,000 3 There are 12 acres and $120,000 available. (continued on next page)

15. Example, p.183 (186) (cont.) Additional restriction on selection of the facilities: (1) Between swimming pool and tennis center, only one can be constructed. (2) Between athletic field and tennis center, at least one must be built. (3) If athletic field is built, then swimming pool must be built. (4) Among the four, at least two must be built. Which facilities should be constructed to maximize the daily usage?

16. Example p.200 (205) A Set Covering Problem APS wants to build package distribution hubs to cover 12 cities. A hub can cover cities within 300 miles, as shown on p.201 (205). Which cities should be selected as hubs so that number of hubs to be built is minimized?

17. Example p.200 (205) Define Variables x i = 0 if city i is not selected as a hub, and x i = 1 if city i is selected as a hub; where i = 1, 2, 3, …, 12 such that 1 for Atlanta,2 for Boston3 for Charlotte 4 for Cincinnati5 for Detroit6 for Indianapolis 7 for Milwaukee8 for Nashville9 for New York 10 for Pittsburgh11 for Richmond12 for St. Louis

18. Example p.200 (205) Set up Integer Program Minimize total number of hubs to build For each of the 12 cities: It must be covered by at least a hub within 300 miles; i.e., there must be at least a hub within 300 miles of it. The complete integer program is on p.201 (206).

19. Solution Methods of IP: Solving IP is more complicated than solving LP. Two main solution methods of IP: Branching and bound method Cutting plain method

20. Integer Programming (IP), Discrete Optimization, and NP Complete An integer program has finite number of feasible solutions. IP is a typical problem of discrete optimization that selects the best from a finite number of alternatives. IP is a computationally hard problem (NP complete problem). That is, no method has been found to solve IP “efficiently”.