Engineering Optimization: Theory and Practice, Fourth Edition

624 Integer Programming

g 3 (X)= 1 −

0. 5 x 1 − 2 x 2

1. 5 x 1 x 2 +

√

2 x 2 x 3 + 1. 319 x 1 x 3

≥ 0

g 4 (X)= 1 +

0. 5 x 1 − 2 x 2

1. 5 x 1 x 2 +

√

2 x 2 x 3 + 1. 319 x 1 x 3

≥ 0

xi∈ { 0. 1 , 0. 2 , 0. 3 , 0. 5 , 0. 8 , 1. 0 , 1. 2 }, i= 1 , 2 , 3

The optimum solution of the continuous variable problem is given byf∗= 2. 7 336,

x 1 ∗ = 1. 1 549,x 2 ∗= 0. 4 232, andx 3 ∗= 0. 0 004. The optimum solution of the discrete

variable problem is given byf∗= 3. 0 414,x∗ 1 = 1. 2 ,x∗ 2 = 0. 5 , andx 3 ∗= 0. 1.

10.9 Solution of Binary Programming Problems Using MATLAB

The MATLAB functionbintprogcan be used to solve a binary (or zero–one) pro-

gramming problem. The following example illustrates the procedure.

Example 10.7 Find the solution of the following binary programming problem using

the MATLAB functionbintprog:

Minimizef (X)= − 5 x 1 − 5 x 2 − 8 x 3 + 4 x 4 + 4 x 5

subject to

3 x 1 − 6 x 2 + 7 x 3 − 9 x 4 − 9 x 5 ≤ − 10 , x 1 + 2 x 2 −x 4 − 3 x 5 ≤ 0

xibinary;i= 1 , 2 , 3 , 4 , 5

SOLUTION

Step 1: State the problem in the form required by the programbintprog:

Minimizef (x)=fTx ubject tos Ax≤bandAeqx=beq

Here

fT= {−^5 −^5 −^8 2 4}, x= {x 1 x 2 x 3 x 4 x 5 }T

A=

[

3 − 67 − 9 − 9

1 2 0 − 1 − 3

]

, b=

{

− 10

0

}

Step 2: The input is directly typed on the MATLAB command window and the program

bintprogis called as indicated below:

clear; clc;

f = [-5 -5 -8 2 4]′;

A = [3 -6 7 -9 -9; 1 2 0 -1 -3];

b = [-10 0]′;

x = bintprog (f, A, b,[])