1 1 Slide Integer Linear Programming Professor Ahmadi

2 2 Slide Chapter 11 Integer Linear Programming n Types of Integer Linear Programming Models n Graphical Solution for an All-Integer LP n Spreadsheet Solution for an All-Integer LP n Application Involving 0-l Variables n Special 0-1 Constraints

3 3 Slide Types of Integer Programming Models n A linear program in which all the variables are restricted to be integers is called an integer linear program (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 program.

4 4 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 Slide Example: All-Integer LP n LP Relaxation LP Optimal (2.5, 1.5) Max 3x 1 + 2x 2 Max 3x 1 + 2x 2 -x 1 + x 2 < 1 x2x2x2x2 x1x1x1x1 3x 1 + x 2 < x 1 + 3x 2 < 7 x 1 + 3x 2 < 7

6 6 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 previous slide, we see that the optimal solution to the linear program is: x 1 = 2.5, x 2 = 1.5, z = 10.5

7 7 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 page, we see that this point lies outside the feasible region, making this solution infeasible.

8 8 Slide Example: All-Integer LP n Rounded Up Solution LP Optimal (2.5, 1.5) Max 3x 1 + 2x 2 Max 3x 1 + 2x 2 -x 1 + x 2 < 1 x2x2x2x2 x1x1x1x1 3x 1 + x 2 < ILP Infeasible (3, 2) ILP Infeasible (3, 2) x 1 + 3x 2 < 7 x 1 + 3x 2 < 7

9 9 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 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 Slide Example: All-Integer LP ILP Optimal (3, 0) Max 3x 1 + 2x 2 Max 3x 1 + 2x 2 -x 1 + x 2 < 1 x2x2x2x2 x1x1x1x1 3x 1 + x 2 < x 1 + 3x 2 < 7 x 1 + 3x 2 <

12 Slide Special 0-1 Constraints n When x i and 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 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 co-requisite for project j :Project i is a co-requisite 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

13 Slide Example: Chattanooga Electronics Chattanooga Electronics, Inc. is planning to expand its operations into other electronic equipment. The company has identified seven new product lines it can carry. Relevant information about each line follows: Initial Floor Space Exp. Rate Initial Floor Space Exp. Rate Product Line Investment (Sq.Ft.) of Return Product Line Investment (Sq.Ft.) of Return 1. Digital TVs $6, % 1. Digital TVs $6, % 2. HD TVs 12, HD TVs 12, Large Screen TVs 20, Large Screen TVs 20, DVDs 14, DVDs 14, DVD/RWs 15, DVD/RWs 15, Video Games 2, Video Games 2, PC Computers 32, PC Computers 32,

14 Slide Chattanooga Electronics - Continued n Define the Decision Variables xj = 1 if product line j is introduced; = 0 otherwise. Where the Product lines are defined as: 1. (X1) = Digital TVs 2. (X2) = HD TVs 3. (X3) = Large Screen TVs 4. (X4) = DVDs 5. (X5) = DVD/RWs 6. (X6) = Video Games 7. (X7) = Computers

15 Slide Example: Chattanooga Electronics Chattanooga Electronics has decided that: 1. they should not stock large screen TVs (X3) unless they stock either digital (X1) or HD TVs (X2). 2. also, they will not stock both types of DVDs (X4 & X5). 3. they will stock video games (X6) only if they stock HD TVs (X2). 4. the company wishes to introduce at least three new product lines. 5. If the company has $45,000 to invest and 420 sq. ft. of floor space available, formulate an integer linear program for Chattanooga Electronics to maximize its overall expected rate of return.

16 Slide Example: Chattanooga Electronics 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 7 or +.132(32000) x 7 or Max 486 x x x x x x x x x 7

17 Slide Example: Chattanooga Electronics n Define the Constraints 1) Money: 1) Money: 60 x x x x x x x 7 < x x x x x x x 7 < 45 2) Space: 125 x x x x x x x 7 < x x x x x x x 7 < 420

18 Slide Example: Chattanooga Electronics n Define the Constraints (continued) 3) Stock large screen TVs (X3) only if stock digital (X2) or HD (X2): 4) Do not stock both types of DVDs (X4 & X5): 5) Stock video games (X6) only if they stock HD TV's (X2): 6) At least 3 new lines: 7) Variables are 0 or 1: x j = 0 or 1 for j = 1,,, 7 x j = 0 or 1 for j = 1,,, 7

19 Slide Example: Mo’s Programming Mo's Programming has five idle Programmers and four custom Programs to develop. The estimated time (in hours) it would take each Programmer to write each Program is listed below. (An 'X' in the table indicates an unacceptable Programmer-Program assignment.) Programmer Programmer Program Program Java C X C X Assembler X 9 Assembler X 9 Pascal X Pascal X

20 Slide Example: Mo’s Programming Formulate an integer program for determining the Programmer-Program assignments that minimize the total estimated time spent writing the four Programs. No Programmer is to be assigned more than one Program and each Program is to be worked on by only one Programmer This problem can be formulated as a 0-1 integer program. The LP solution to this problem will automatically be integer (0-1).

21 Slide Example: Mo’s Programming n Define the decision variables x ij = 1 if Program i is assigned to Programmer j x ij = 1 if Program i is assigned to Programmer j = 0 otherwise. = 0 otherwise. Number of decision variables = Number of decision variables = [(number of Programs)(number of Programmers)] - (number of unacceptable assignments) - (number of unacceptable assignments) = [4(5)] - 3 = 17 n Define the objective function Minimize total time spent writing Programs: Minimize total time spent writing Programs: 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

22 Slide Example: Mo’s Programming n Define the Constraints Exactly one Programmer per Program: 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

23 Slide Example: Mo’s Programming n Define the Constraints (continued) No more than one Program per Programmer: 5) x 11 + x 21 + x 31 < 1 6) x 21 + x 22 + x 23 + x 24 < 1 7) x 31 + x 33 + x 34 < 1 8) x 41 + x 42 + x 44 < 1 9) x 51 + x 52 + x 53 + x 54 < 1 Non-negativity: x ij > 0 for i = 1,..,4 and j = 1,..,5

24 Slide The End of Chapter 6