Linear integer programming and its application

Transcription

1 Linear integer programming and its application Presented by Dr. Sasthi C. Ghosh Associate Professor Advanced Computing & Microelectronics Unit Indian Statistical Institute Kolkata, India

2 Outline Introduction to Integer programming Linear relaxation Rounding in linear relaxation Converting finite-valued integer variables to binary variable Implementing logical conditions using binary variable Converting product of binary variable to linear form Converting non-integer coefficients to integer coefficients Dealing with probabilistic constraints Solving integer programs: Branch and bound Implementation tool: CPLEX Conclusion 2017/7/9 2

3 Introduction Pure integer programming All variables are integer. Mixed integer programming Some variables are restricted to be integer but some are not. Binary integer programming When variables are restricted to 0 or 1. It could be pure or mixed. Maximize 8x x 2 + 6x 3 + 4x 4 Subject to 5x 1 + 7x 2 + 4x 3 + 3x 4 14 x j {0, 1} j = 1, 2,, /7/9 3

4 Linear relaxation Linear relaxation Can be solved in polynomial time. Polynomial in number of variables and constraints. Formed by dropping the integrality restrictions. Maximize 8x x 2 + 6x 3 + 4x 4 Subject to 5x 1 + 7x 2 + 4x 3 + 3x x j 1, j = 1, 2,, 4 It uses 0 x j 1 instead of x j {0, 1} 2017/7/9 4

5 Rounding in linear relaxation Rounding does not give optimal solution Maximize 8x x 2 + 6x 3 + 4x 4 Subject to 5x 1 + 7x 2 + 4x 3 + 3x 4 14 x j {0, 1} j = 1, 2,, 4 Solution of linear relaxation x 1 =x 2 =1, x 3 =0.5, x 4 =0 and optimal objective: 22 Rounding x 3 to 0 gives, optimal objective: 19 Rounding x 3 to 1 does not satisfy the constraint Optimal integer solution x 1 = 0, x 1 = x 3 = x 4 = 1 and optimal objective: /7/9 5

6 Rounding in linear relaxation Rounding may not be possible at all Maximize 8x x 2 + 6x 3 + 4x 4 Subject to 5x 1 + 7x 2 + 4x 3 + 3x 4 = 14 x j {0, 1} j = 1, 2,, 4 Solution of linear relaxation x 1 =x 2 =1, x 3 =0.5, x 4 =0 and optimal objective: 22 Rounding x 3 to 0 or 1 does not satisfy the constraint Optimal integer solution x 1 = 0, x 2 =x 3 =x 4 =1 and optimal objective: /7/9 6

7 Finite-valued integer variables Converting finite-valued integer variables to binary Assume a variable x j can only take a finite number of values: x j {p 1,..., p m }. Introduce variables z 1 j,..., z m j {0, 1} and add the constraint z 1 j z m j = 1. Substitute x j with p 1 z 1 j p m z m j in the objective function and all constraints. 2017/7/9 7

8 Finite-valued integer variables Converting finite-valued integer variables to binary Maximize 8x x 2 + 6x 3 + 4x 4 5x 1 + 7x 2 + 4x 3 + 3x 4 <= 14 x 1 {1, 2, 3} x j {0, 1}, j=2, 3, 4. Solution: x 1 = 2, x 2 = 0, x 3 = 1, x 4 = 0, Objective=22 Example: x 1 {1, 2, 3} can be modelled as z z z 3 1 = 1 where z 1 1, z 2 1, z 3 1 {0, 1} Substituting x 1 everywhere by 1 z z z 3 1 gives: Maximize 8 z z z x 2 + 6x 3 + 4x 4 5 z z z x 2 + 4x 3 + 3x 4 <= 14 z z z 3 1 = 1 Solution: z 1 1 = 0, z 2 1 = 1, z 3 1 =0, x 2 = 0, x 3 = 1, x 4 = 0, Objective=22 z 1 1 = 0, z 2 1 = 1, z 3 1 =0 x 1 =2 2017/7/9 8

9 Implementing logical conditions Implementing logical conditions using binary variable: At most one of A and B: a + b 1 At least one of A and B: a + b 1 If A then B: b a If A then not B: 1 - b a a + b 1 If not A then B: b 1 a a + b 1 If A then B, and if B then A: a = b If A then B and C: b a and c a If A then B or C: b + c a If B or C then A: a b and a c If B and C then A: a b + c /7/9 9

10 Implementing logical conditions Converting product of binary variables into linear form without any approximation c = ab a c, b c, and c a + b -1 If C then A and B: a c and b c (ab c) If A and B then C: c a + b 1 (c ab) d = abc a d, b d, c d and d a + b + c /7/9 10

11 Implementing logical conditions Implementing either/or logical conditions using binary variable: Maximize x 1 + 3x 2 subject to either x 1 + x 2 6 (c 1 ) or x 1 + 2x 2 7 (c 2 ) x 1 0 and x 2 0 Can be expressed as x 1 + x 2 - M 6 and x 1 + 2x 2 - M(1 - ) 7 {0, 1} and M is a large number =0: x 1 + x 2 6, x 1 + 2x 2 + c 1 is satisfied =1: x 1 + x 2 +, x 1 + 2x 2 7 c 2 is satisfied 2017/7/9 11

12 Implementing logical conditions Implementing either/or logical conditions using binary variable: Maximize 2x 1 + 3x 2 subject to either x 1 + 2x 2 6 (c 1 ) or 2x 1 + x 2 9 (c 2 ) x 1 0 and x 2 0 Can be expressed as x 1 + 2x 2 + M 6 and 2x 1 + x 2 + M(1 - ) 9 {0, 1} and M is a large number =0: x 1 + 2x 2 6, 2x 1 + x 2 - c 1 is satisfied =1: x 1 + 2x 2 -, 2x 1 + x 2 9 c 2 is satisfied 2017/7/9 12

13 Implementing logical conditions Implementing either/or but not both logical conditions using binary variable: Minimize x 1 - x 2 subject to x 1 + x 2 4 x 1 1 x 2 1 but not both x 1, x 2 > 1 x 1, x 2 0. This can be expressed as: Minimize x 1 - x 2 subject to x 1 + x 2 4 x 1 1 M x 2 1 M(1 ) x 1 1 +M(1 ) x 2 1 +M x 1, x 2 0, {0, 1} =0: x 1 1 and x 2 1 =1: x 1 1 and x 2 1 So, either x 1 1 or x 2 1 but not both x 1, x 2 > /7/9 13

14 Converting to integer coefficient Converting non-integer coefficients to integer coefficients: Minimize y = (1/3)x 1 (1/2)x 2 subject to (2/3)x 1 + (1/3)x 2 (4/3) (1/2)x 1 (3/2)x 2 (2/3) x 1, x 2 0, x 1, x 2 Z. Can be expressed as Minimize y = 2x 1 3x 2 (*6) subject to 2x 1 + x 2 4 (*3) 3x 1 9x 2 4 (*6) x 1, x 2 0, x 1, x 2 Z. 2017/7/9 14

15 Solving integer programs Solving integer programs (IP) Branch and bound: dividing the problem into a number of smaller problems. Cutting plane: adding constraints to force integrality Relationship to linear relaxation (LR) Since LR is less constrained than IP, the following is immediate: For maximization, the optimal objective of LR is greater than or equal to the optimal objective of IP. For minimization, the optimal objective of LR is less than or equal to the optimal objective of IP. If LR is infeasible, then so is IP If LR is optimized by integer values, then that solution is optimal for IP. 2017/7/9 15

16 Solving integer programs If objective function coefficients are all integers, then For maximization, the optimal objective for IP is less than or equal to the round down value of optimal objective of LR. For minimization, the optimal objective for IP is greater than or equal to the round up value of optimal objective of LR. Solving LR gives the following information: A bound on the optimal value and if lucky, may give the optimal solution to IP. Rounding the solution of LR will not in general give the optimal solution of IP. In fact, rounding may not be feasible for many cases. 2017/7/9 16

17 Solving integer programs Branch and bound: Solve the LR of the IP. If solution is integer, we are done. Otherwise, create two subproblems by branching on a fractional variable. Choose an active subproblem and branch on a fractional variable. Repeat untill there are no active subproblems. A subproblem is not active when any of the following holds: You used the subproblem to branch on, All variables in the solution are integer, The subproblem is infeasible, You can mark an subproblem not active by a bounding argument. Branching rule for binary variable: x=0 and x=1 for fractional value of x Branching rule for non-binary variable x 4 and x 4 for say x= /7/9 17

18 Solving integer programs Branch and bound 2017/7/9 18

19 Solving integer programs 2017/7/9 19

20 Solving integer programs 2017/7/9 20

21 Conclusion Integer programming is an effective means to solve many real world problems. Linear relaxation only gives bounds. Rounding produces non-optimal results and sometimes may not be possible at all. Finite-valued integer variable can be converted to binary. Non-integer coefficients can be converted to integer coefficients. Many logical conditions can be formulated using integer programming. Some form of non-linear constraints can be converted to equivalent linear constraints without any approximation while some others can be done so with approximation. 2017/7/9 21