运筹学第三版（刁在钧）光盘中的内容

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第二章 线性规划

本章, 我们介绍三种解决线性规划问题的软件:

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

第二种: LINDO软件;

第三种: LINGO软件.

1. MATLAB程序说明程序名: lprogram执行实例：

file:///C:/DOCUME~1/ADMINI~1/LOCALS~1/Temp/msohtml1/01/clip_image002.gif

在命令窗口的程序执行过程和结果如下：

the program is with the linear programming

Please input the constraints number of the linear programming m=7

m =7

Please input the variant number of the linear programming n=4

n =4

Please input cost array of the objective function c(n)_T=[-2,-1,3,-5]'

c =
-2

-1

) ]$ K$ F9 R, m, h7 p3

-5

Please input the coefficient matrix of the constraints A(m,n)=[1,2,4,-1;2,3,-1,1;

1,0,1,1;-1,0,0,0;0,-1,0,0;0,0,-1,0;0,0,0,-1]

A =
+ ~; k7 E( b, W4 m' N1
7 ?1 Q# r/ j" t  q% A6 l* c* A$ H2
4
-1

2
1 m! K$ z1 p0 V" e! K. B3
! B  h4 p* j8 K) h" H! G-1
1

1
" R2 s0 k" x1 x" }/ Y; d$ ^" {9 K: {0
1
. w& l9 n" \( r) ]' R# `1

-1
+ E2 ?- ?) }) R; G( r+ O0
0
  `: V0 d4 h+ Y' z. T& g5 Z/ z5 c0

2 A' u" B0 ~; Q0 u5 ^0
-1
, ~* S" v8 {: I/ d/ J) a% {0
0 o$ f" B. M. I( w0 U0

5 ?: V2 l  o+ H/ A6 Q0
. y# e1 U7 B1 e# V( c0
-1
4 W1 @! ]  \& Z# D) z3 N& e0

0
0
0
1 ]; H! H8 V" w  a9 d! [' Z-1

Please input the resource array of the program b(m)_T=[6,12,4,0,0,0,0]'

b =
, j4 J/ b* |& p( `6

12

4

+ [4 L) |2 \1 [7 [3 A6 v" L0

0

- U  l) M9 D6 C+ A2 ]( S4 o9 ]0

; I# }: \3 R* x* z" N; u0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
0.0000

: u3 K1 P3 L8 z; ^$ [7 k; W2.6667

0 h& ?* c) z8 J-0.0000

$ {) p! Z8 o5 Y3 b: d4.0000

The optimization value of the programming is:

opt_value = -22.6667

注: 红色字表示计算机的输出结果.

程序的相关知识:

Solve a linear programming problem

file:///C:/DOCUME~1/ADMINI~1/LOCALS~1/Temp/msohtml1/01/clip_image003.gif

where f, x, b, beq, lb, and ub are vectors and A and Aeq are matrices.

相关的语法:

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

x = linprog(f,A,b,Aeq,beq,lb,ub)

x = linprog(f,A,b,Aeq,beq,lb,ub,x0)

x = linprog(f,A,b,Aeq,beq,lb,ub,x0,options)

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

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

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

[x,fval,exitflag,output,lambda] = linprog(...)

解释:

linprog solves linear programming problems.

x = linprog(f,A,b) solves min f'*x such that A*x <= b.

x = linprog(f,A,b,Aeq,beq) solves the problem above while additionally satisfying the equality constraints Aeq*x = beq. Set A=[] and b=[] if no inequalities exist.

x = linprog(f,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables, x, so that the solution is always in the range lb <= x <= ub. Set Aeq=[] and beq=[] if no equalities exist.

x = linprog(f,A,b,Aeq,beq,lb,ub,x0) sets the starting point to x0. This option is only available with the medium-scale algorithm (the LargeScale option is set to 'off' using optimset). The default large-scale algorithm and the **x algorithm ignore any starting point.

x = linprog(f,A,b,Aeq,beq,lb,ub,x0,options) minimizes with the optimization options specified in the structure options. Use optimset to set these options.

[x,fval] = linprog(...) returns the value of the objective function fun at the solution x: fval = f'*x.

[x,lambda,exitflag] = linprog(...) returns a value exitflag that describes the exit condition.

[x,lambda,exitflag,output] = linprog(...) returns a structure output that contains information about the optimization.

[x,fval,exitflag,output,lambda] = linprog(...) returns a structure lambda whose fields contain the Lagrange multipliers at the solution x.

2．LINDO 程序说明程序名：linear执行实例：

file:///C:/DOCUME~1/ADMINI~1/LOCALS~1/Temp/msohtml1/01/clip_image005.gif

在命令窗口键入以下内容：

max 10x+15y ！也可以直接解决min问题

subject to

x<10

y<12

x+2y<16

end
% o; E- w1 ]% ~& e! y& `！注释符号; 系统默认为自变量>0, 若不要求用free命令.

7 a: Z7 T6 n- ?: V+ v( [！在出来report windows之前可选择显示对此规划进行灵敏度分析等

按solve键, 在reports window中出现以下内容:

LP OPTIMUM FOUND AT STEP
2

OBJECTIVE FUNCTION VALUE

1)
145.0000

; r/ b/ l8 ~, xVARIABLE
+ N* Q7 n7 O+ {VALUE
REDUCED COST

X
10.000000
; o  c0 P, P$ _6 i; v4 x0.000000

Y
3.000000
" v* s  J! W. }. f0 @2 H; ~! e2 g0.000000

ROW
* S; s0 h0 M/ ]- ?& U. V0 g5 ySLACK OR SURPLUS
DUAL PRICES

2)
0.000000
2.500000

/ ~  b( H* R3 e! ^" y* H9 t3)
9.000000
$ i/ |0 a2 w  z# F+ F- {0.000000

4)
. B8 ~+ r7 X2 ?) s9 Y9 p' t; [! X0.000000
9 \  F- d6 p! h; x/ Y7.500000

NO. ITERATIONS=
7 f1 ?+ M8 k% h9 x2

- {% u! T/ C, pRANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE
CURRENT
1 M! Y5 j( p; g5 [7 l" o: \ALLOWABLE
ALLOWABLE

3 O( Z( p* e+ C0 Q1 i8 j+ hCOEF
INCREASE
0 S8 W: n6 p- h5 Z5 O( r) @DECREASE

X
% c) ~0 N+ M6 u; ~. {) T10.000000
INFINITY
2.500000

! Y* d- z: |7 J% W# ZY
15.000000
, n+ \* B. A- F& Z/ v' D5.000000
( N- V- J* @& K15.000000

/ Y/ C6 B& w5 `) @' gRIGHTHAND SIDE RANGES

ROW
7 q' w) f7 g0 K) D' B" @CURRENT
ALLOWABLE
. `+ w0 _* j+ s9 I! Z: |: l) QALLOWABLE

RHS
  [) ]. u: A* T3 D, S' I9 _INCREASE
DECREASE

2 P# {; D/ F* x( g1 _
2
* Q7 F$ g; y" D10.000000
  V9 N4 T% u$ |0 N6 d3 g( `6.000000
8 P1 i' A- \5 N! N5 j' H. E0 A1 k10.000000

3
12.000000
1 ]' j8 n% L: ^7 FINFINITY
9.000000

4
16.000000
* \$ b7 `/ S9 K4 \' k18.000000
6.000000

3．LINGO 程序说明3.1 程序名: linearp1（求极小问题）linearp1运行实例:

file:///C:/DOCUME~1/ADMINI~1/LOCALS~1/Temp/msohtml1/01/clip_image007.gif

在model window中输入以下语句：

min=5*x1+21*x3;

x1-x2+6*x3-x4=2;

x1+x2+2*x3-x5=1;

按运行按钮在solution
report 窗口得到以下结果：

; _, |& F" A9 s9 M  XGlobal optimal solution found at iteration:
" {0 q7 d( \* N8 b+ M: Y1 n2

+ D: l9 i1 j  A6 H; X; ?$ ]0 SObjective value:
& Q) P5 }7 B# P, l, T7.750000

Variable
' {+ v2 S& a9 Q2 f+ WValue
8 k1 M# S# L0 EReduced Cost

X1
6 }0 E7 K8 d- j' h0.5000000
$ ~# x1 J+ N5 `. r9 C0.000000

X3
$ A6 c6 h6 T, B/ d8 a6 ^5 t0.2500000
0.000000

$ q: R! G$ d# P: G' r9 E. ^
9 I. D9 ?. f1 r  T9 u- UX2
! d0 N, m4 Y4 R0.000000
9 r2 r: [+ D2 f" L0.5000000

% k- q0 f2 f* F- w5 R3 |. V1 I7 TX4
& N- C; ^- t. W( U- H$ @# v0.000000
" A1 A  m0 Q* q6 L0 A! K2.750000

2 G5 q7 `! C, g2 ~- sX5
0.000000
2.250000

Row
3 y2 P3 a& R9 I" m3 lSlack or Surplus
, b9 z+ ^$ }6 _Dual Price

1
7.750000
4 }" ^# v1 W( }. f. s* G' q-1.000000

2
, \+ p7 h6 c4 ^0 P6 f% G' s& N# ^0.000000
-2.750000

8 O# d* g& t+ @  A7 ?) t2 d! m3
+ T- Q7 A: m$ K0 a  X3 q0 `5 j0.000000
-2.250000

3.2 程序名: linearp2（求极大问题）linearp2运行实例：

file:///C:/DOCUME~1/ADMINI~1/LOCALS~1/Temp/msohtml1/01/clip_image009.gif

在model window中输入以下语句：

max=100*x+150*y;
! this is a commnent;

x<=100;

y<=120;

x+2*y<=160;

按运行按钮在solution report 窗口得到以下结果：

  Global optimal solution found at iteration:
2

# w* `2 }3 A' JObjective value:

14500.00

6 x, e) Q# g$ u5 R+ @/ L: c; [# BVariable
Value
Reduced Cost

  G0 N) L% T+ A: R- E' I8 gX
100.0000
0.000000

Y
2 b/ r; E' B9 x+ L7 v, }. K30.00000
0.000000

" l, p0 x" o/ b7 v. p& o4 ]* ]Row
1 ?( X( v9 z5 PSlack or Surplus
6 {$ m; W/ E3 k  v/ ~. }3 eDual Price

0 p+ P' ?/ ?4 S3 r& Q! \1
14500.00
; W0 ]  V. j/ p% R2 z2 t1.000000

: n9 e8 n. L" V6 s4 `2
! ^) J# L" N" b/ N  J, H0.000000
25.00000

2 ~8 t: ]9 B, ?' U3
90.00000
0.000000

4
$ [+ V7 e- h6 u. i0.000000
+ W9 u% j1 N) n" X$ O6 `
, r& s  R  U0 x1 x# {& J- p75.00000

loooog12

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