1 1 © 2003 Thomson  /South-Western Slide Slides Prepared by JOHN S. LOUCKS St. Edward’s University

2 2 © 2003 Thomson  /South-Western Slide Chapter 8 Integer Linear Programming n Types of Integer Linear Programming Models n Graphical and Computer Solutions for an All-Integer Linear Program n Applications Involving 0-1 Variables n Modeling Flexibility Provided by 0-1 Variables

3 3 © 2003 Thomson  /South-Western Slide Types of Integer Programming Models n An LP in which all the variables are restricted to be integers is called an all-integer linear program (ILP). n The LP that results from dropping the integer requirements is called the LP Relaxation of the ILP. n If only a subset of the variables are restricted to be integers, the problem is called a mixed-integer linear program (MILP). n Binary variables are variables whose values are restricted to be 0 or 1. If all variables are restricted to be 0 or 1, the problem is called a 0-1 or binary integer linear program.

4 4 © 2003 Thomson  /South-Western Slide Example: All-Integer LP n Consider the following all-integer linear program: Max 3 x x 2 Max 3 x x 2 s.t. 3 x 1 + x 2 < 9 s.t. 3 x 1 + x 2 < 9 x x 2 < 7 x x 2 < 7 - x 1 + x 2 < 1 - x 1 + x 2 < 1 x 1, x 2 > 0 and integer x 1, x 2 > 0 and integer

5 5 © 2003 Thomson  /South-Western Slide Example: All-Integer LP n LP Relaxation Solving the problem as a linear program ignoring the integer constraints, the optimal solution to the linear program gives fractional values for both x 1 and x 2. From the graph on the next slide, we see that the optimal solution to the linear program is: x 1 = 2.5, x 2 = 1.5, z = 10.5

6 6 © 2003 Thomson  /South-Western Slide Example: All-Integer LP n LP Relaxation LP Optimal (2.5, 1.5) Max 3 x x 2 Max 3 x x 2 - x 1 + x 2 < 1 x2x2x2x2 x1x1x1x1 3 x 1 + x 2 < x x 2 < 7 x x 2 < 7

7 7 © 2003 Thomson  /South-Western Slide Example: All-Integer LP n Rounding Up If we round up the fractional solution ( x 1 = 2.5, x 2 = 1.5) to the LP relaxation problem, we get x 1 = 3 and x 2 = 2. From the graph on the next slide, we see that this point lies outside the feasible region, making this solution infeasible.

8 8 © 2003 Thomson  /South-Western Slide Example: All-Integer LP n Rounded Up Solution LP Optimal (2.5, 1.5) Max 3 x x 2 Max 3 x x 2 - x 1 + x 2 < 1 x2x2x2x2 x1x1x1x1 3 x 1 + x 2 < ILP Infeasible (3, 2) ILP Infeasible (3, 2) x x 2 < 7 x x 2 < 7

9 9 © 2003 Thomson  /South-Western Slide Example: All-Integer LP n Rounding Down By rounding the optimal solution down to x 1 = 2, x 2 = 1, we see that this solution indeed is an integer solution within the feasible region, and substituting in the objective function, it gives z = 8. We have found a feasible all-integer solution, but have we found the OPTIMAL all-integer solution? The answer is NO! The optimal solution is x 1 = 3 and x 2 = 0 giving z = 9, as evidenced in the next two slides.

10 © 2003 Thomson  /South-Western Slide Example: All-Integer LP n Complete Enumeration of Feasible ILP Solutions There are eight feasible integer solutions to this problem: x 1 x 2 z x 1 x 2 z optimal solution optimal solution

11 © 2003 Thomson  /South-Western Slide Example: All-Integer LP ILP Optimal (3, 0) Max 3 x x 2 Max 3 x x 2 - x 1 + x 2 < 1 x2x2x2x2 x1x1x1x1 3 x 1 + x 2 < x x 2 < 7 x x 2 < 7

12 © 2003 Thomson  /South-Western Slide Example: All-Integer LP n Partial Spreadsheet Showing Problem Data

13 © 2003 Thomson  /South-Western Slide Example: All-Integer LP n Partial Spreadsheet Showing Formulas

14 © 2003 Thomson  /South-Western Slide Example: All-Integer LP n Partial Spreadsheet Showing Optimal Solution

15 © 2003 Thomson  /South-Western Slide Special 0-1 Constraints n When x i and x j represent binary variables designating whether projects i and j have been completed, the following special constraints may be formulated: At most k out of n projects will be completed: At most k out of n projects will be completed:  x j < k  x j < k j j Project j is conditional on project i : Project j is conditional on project i : x j - x i < 0 x j - x i < 0 Project i is a corequisite for project j : Project i is a corequisite for project j : x j - x i = 0 x j - x i = 0 Projects i and j are mutually exclusive: Projects i and j are mutually exclusive: x i + x j < 1 x i + x j < 1

16 © 2003 Thomson  /South-Western Slide Example: Metropolitan Microwaves Metropolitan Microwaves, Inc. is planning to expand its operations into other electronic appliances. The company has identified seven new product lines it can carry. Relevant information about each line follows on the next slide.

17 © 2003 Thomson  /South-Western Slide Example: Metropolitan Microwaves Initial Floor Space Exp. Rate Initial Floor Space Exp. Rate Product Line Invest. (Sq.Ft.) of Return Product Line Invest. (Sq.Ft.) of Return 1. Black & White TVs$ 6, % 2. Color TVs 12, Large Screen TVs 20, VHS VCRs 14, Beta VCRs 15, Video Games 2, Home Computers 32,

18 © 2003 Thomson  /South-Western Slide Example: Metropolitan Microwaves Metropolitan has decided that they should not stock large screen TVs unless they stock either B&W or color TVs. Also, they will not stock both types of VCRs, and they will stock video games if they stock color TVs. Finally, the company wishes to introduce at least three new product lines. If the company has $45,000 to invest and 420 sq. ft. of floor space available, formulate an integer linear program for Metropolitan to maximize its overall expected rate of return.

19 © 2003 Thomson  /South-Western Slide Example: Metropolitan Microwaves n Define the Decision Variables x j = 1 if product line j is introduced; = 0 otherwise. = 0 otherwise. n Define the Objective Function Maximize total overall expected return: Max.081(6000) x (12000) x (20000) x 3 Max.081(6000) x (12000) x (20000) x (14000) x (15000) x (2000) x (14000) x (15000) x (2000) x (32000) x (32000) x 7

20 © 2003 Thomson  /South-Western Slide Example: Metropolitan Microwaves n Define the Constraints 1) Money: 1) Money: 6 x x x x x x x 7 < 45 6 x x x x x x x 7 < 45 2) Space: 2) Space: 125 x x x x x x x 7 < x x x x x x x 7 < 420 3) Stock large screen TVs only if stock B&W or color: 3) Stock large screen TVs only if stock B&W or color: x 1 + x 2 > x 3 or x 1 + x 2 - x 3 > 0 x 1 + x 2 > x 3 or x 1 + x 2 - x 3 > 0

21 © 2003 Thomson  /South-Western Slide Example: Metropolitan Microwaves n Define the Constraints (continued) 4) Do not stock both types of VCRs: x 4 + x 5 < 1 5) Stock video games if they stock color TV's: 5) Stock video games if they stock color TV's: x 2 - x 6 > 0 6) At least 3 new lines: 6) At least 3 new lines: x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 > 3 7) Variables are 0 or 1: 7) Variables are 0 or 1: x j = 0 or 1 for j = 1,,, 7 x j = 0 or 1 for j = 1,,, 7

22 © 2003 Thomson  /South-Western Slide Example: Metropolitan Microwaves n Partial Spreadsheet Showing Problem Data

23 © 2003 Thomson  /South-Western Slide Example: Metropolitan Microwaves n Partial Spreadsheet Showing Example Formulas

24 © 2003 Thomson  /South-Western Slide Example: Metropolitan Microwaves n Solver Parameters Dialog Box Solver Parameters Set Target Cell: Equal To: By Changing Cells: Reset All X ? Help Close Solve Options Max Min Value of: $B$13:$H$13 Subject to the Constraints: Delete Change Guess Add $B$13:$H$13 = binary $D$17:$D$18 <= $H$17:$H$18 $D$19 >= $H$19 $D$20 <= $H$20 $B$13:$H$13 = binary $D$17:$D$18 <= $H$17:$H$18 $D$19 >= $H$19 $D$20 <= $H$20

25 © 2003 Thomson  /South-Western Slide Example: Metropolitan Microwaves n Solver Options Dialog Box Select Assume Linear Model Select Assume Linear Model Select Assume Non-negative Select Assume Non-negative Set Tolerance equal to 0 Set Tolerance equal to 0

26 © 2003 Thomson  /South-Western Slide Example: Tom’s Tailoring Tom's Tailoring has five idle tailors and four custom garments to make. The estimated time (in hours) it would take each tailor to make each garment is shown in the next slide. (An 'X' in the table indicates an unacceptable tailor-garment assignment.) Tailor Tailor Garment Garment Wedding gown Wedding gown Clown costume X Clown costume X Admiral's uniform X 9 Admiral's uniform X 9 Bullfighter's outfit X Bullfighter's outfit X

27 © 2003 Thomson  /South-Western Slide Example: Tom’s Tailoring Formulate an integer program for determining the tailor-garment assignments that minimize the total estimated time spent making the four garments. No tailor is to be assigned more than one garment and each garment is to be worked on by only one tailor This problem can be formulated as a 0-1 integer program. The LP solution to this problem will automatically be integer (0-1).

28 © 2003 Thomson  /South-Western Slide Example: Tom’s Tailoring n Define the decision variables x ij = 1 if garment i is assigned to tailor j x ij = 1 if garment i is assigned to tailor j = 0 otherwise. = 0 otherwise. Number of decision variables = Number of decision variables = [(number of garments)(number of tailors)] - (number of unacceptable assignments) - (number of unacceptable assignments) = [4(5)] - 3 = 17

29 © 2003 Thomson  /South-Western Slide Example: Tom’s Tailoring n Define the objective function Minimize total time spent making garments: Minimize total time spent making garments: Min 19 x x x x x x 21 Min 19 x x x x x x x x x x x x x x x x x x x x x x x x x x x x 45

30 © 2003 Thomson  /South-Western Slide Example: Tom’s Tailoring n Define the Constraints Exactly one tailor per garment: 1) x 11 + x 12 + x 13 + x 14 + x 15 = 1 1) x 11 + x 12 + x 13 + x 14 + x 15 = 1 2) x 21 + x 22 + x 24 + x 25 = 1 2) x 21 + x 22 + x 24 + x 25 = 1 3) x 31 + x 32 + x 33 + x 35 = 1 3) x 31 + x 32 + x 33 + x 35 = 1 4) x 42 + x 43 + x 44 + x 45 = 1 4) x 42 + x 43 + x 44 + x 45 = 1

31 © 2003 Thomson  /South-Western Slide Example: Tom’s Tailoring n Define the Constraints (continued) No more than one garment per tailor: 5) x 11 + x 21 + x 31 < 1 6) x 12 + x 22 + x 32 + x 42 < 1 7) x 13 + x 33 + x 43 < 1 8) x 14 + x 24 + x 44 < 1 9) x 15 + x 25 + x 35 + x 45 < 1 Nonnegativity: x ij > 0 for i = 1,..,4 and j = 1,..,5

32 © 2003 Thomson  /South-Western Slide End of Chapter 8