整数线性规划word版

第三章 整数线性规划

本章, 我们介绍三种解决整数线性规划问题的软件:

第一种: MATLAB 中的optimization toolbox 中的若干程序;

第二种: LINDO 软件;

第二种: LINGO 软件.

1. MATLAB 程序说明

程序名: intprogram, L01p_e, L01p_ie, transdetobi, biprogram

intprogram 是利用分支定界法解决整数规划问题, 是全部的整数规划问题;

L01p_e 是利用枚举法解决0-1规划问题, 变量要求全部为0或者1;

L01p_ie 是利用隐枚举法解决0-1规划问题, 变量要求全部为0或者1;

Transdetobi 是枚举法和隐枚举法中利用到的将十进制数转化为二进制数的函数;

Biprogram 是MATLAB6.5以上版本中有的求解0-1规划的函数的程序.

intprogram 执行实例1：

12

121212max 2010s.t.5424

2513

,0, f x x x x x x x x =++≤+≤≥ 且为整数

在命令窗口的程序执行过程和结果如下:

>> c=[-20,-10]; %将最大转化为最小;

>> a=[5,4;2,5];

>> b=[24;13];

>> [x,f]=intprogram(c,a,b,[0;0],[inf;inf],[],0,0.0001) % c,a,b 之后[0;0] is the value of low bound;[inf;inf] is the value of up bound;[] is the initialization;0 is the number of the equation constraints; 0.0001 is the concise rate. x =

4.0000

1.0000

f =

-90

intprogram 执行实例2: 书中例题3.3.1

在命令窗口的程序执行过程和结果如下:

>> c=[-1,-1];

>> a=[-4,2;4,2;0,-2];

>> b=[-1;11;-1];

>> [x,f]=intprogram(c,a,b,[0;0],[inf;inf],[],0,0.0001)

x =

2 2

1 1

f =

-3

L01p_e 和L01p_ie 执行实例:

123

1231231223123max 325s.t.22

44

3

46

,,01

f x x x x x x x x x x x x x x x x =-++-≤++≤+≤+≤=　－ 或

在命令窗口的程序执行过程和结果如下:

>> c=[3,-2,5]; %将最大转化为最小;

>> a=[1,2,-1;1,4,1;1,1,0;0,4,1];

>> b=[2;4;3;6];

>> x1=L01p_e(c,a,b);x2=L01p_ie(c,a,b); %x1表示利用枚举法解决0-1规划问题，x2表示用隐% 枚举法解决问题, 结果是一样的

>> x1

x1 =

1

>> x2

x2 =

1

biprogram 执行实例: 1234

1234341324min ()9564s.t.63529

1

f x x x x x x x x x x x x x x x =---+++≤+≤+≤-+≤　－ -

在命令窗口的程序执行过程和结果如下：

the program is with the binary linear programming

Please input the constraints number of the programming m=4

m =

4

Please input the variant number of the programming n=4

n =

4

Please input cost array of the objective function c(n)_T=[-9,-5,-6,-4]'

c =

-9

-5

-6

-4

Please input the coefficient matrix of the constraints A(m,n)=[6,3,5,2;0,0,1,1; -1,0,1,0;0,-1,0,1]

A =

6 3 5 2

0 0 1 1

-1 0 1 0

0 -1 0 1

Please input the resource array of the program b(m)_T=[9,1,0,0]'

b =

9

1

Optimization terminated successfully.

x =

1

1

程序的相关知识:

Solve binary integer programming problems of the form

where f, b, and beq are vectors, A and Aeq are matrices, and the solution x is required to be a binary integer vector -- that is, its entries can only take on the values 0 or 1.

语法如下:

x = bintprog(f)

x = bintprog(f, A, b)

x = bintprog(f, A, b, Aeq, beq)

x = bintprog(f, A, b, Aeq, beq, x0)

x = bintprog(f, A, b, Aeq, beq, x0, options)

[x, fval] = bintprog(...)

[x,fval, exitflag] = bintprog(...)

[x, fval, exitflag, output] = bintprog(...)

解释:

x = bintprog(f) solves the binary integer programming problem

x = bintprog(f, A, b) solves the binary integer programming problem

x = bintprog(f, A, b, Aeq, beq) solves the preceding problem with the additional equality constraint.

x = bintprog(f, A, b, Aeq, beq, x0) sets the starting point for the algorithm to x0. If x0 is not in the feasible region, bintprog uses the default initial point.

x = bintprog(f, A, b, Aeq, Beq, x0, options) minimizes with the default optimization options replaced by values in the structure options, which you can create using the function optimset.

[x, fval] = bintprog(...) returns fval, the value of the objective function at x.

[x,fval, exitflag] = bintprog(...) returns exitflag that describes the exit condition of bintprog. See Output Arguments.