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 J% K9 X6 e1 e) }2 Y-2

) g; I% p' H& z% ?! t1 @# p-1

3

-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 =
1
  ^# c. w' [- `: V% `, o2
% ^9 _% N- l6 ?, @$ l! K1 i$ N4
-1

2
3
-1
1

1
0
4 Y# T0 N8 @" e: d( Z1
7 a+ b+ w7 k' q1

-1
0
( N* R' ^! ]* R" \, {0
4 ^9 ^2 I+ Y7 T& G) ~0

0
! E2 ]4 u% b" U: s, s7 x-1
0
0

2 H7 z6 V. @. Q2 t7 l: T9 [) i! s0
" q4 B4 J: C5 X* p1 @0
  F+ S' t1 m. K( T-1
0

# C# t! E( a" q, d1 J0 o% H0
0
0
: o% q2 V7 l' w" `" A0 b-1

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

b =
# n; q- I* F6 K* }  M" N# A& e. x6

3 E+ r8 `# u* n3 \, U9 j/ q12

' |- ~: }5 x/ u3 H" L4

2 s: r, G# U  u1 D% W" C0

0

& `5 C' U. D) q* a+ H0

  ?5 i/ d4 }6 e; M" K& f4 u& Y; ]0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
6 ~+ s& ]* Q$ o( R4 I0.0000

2.6667

3 f' i6 y- ^1 A  B2 q0 P-0.0000

4.0000

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
！注释符号; 系统默认为自变量>0, 若不要求用free命令.

！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

5 m0 S" D  y- m" ^OBJECTIVE FUNCTION VALUE

1)
145.0000

VARIABLE
4 U/ D* F2 K$ W* w& d2 F- vVALUE
! |, b; u) T# |2 Q$ XREDUCED COST

" }, |! K* n+ T0 `" kX
# L- }+ B. Y7 K4 s- N/ n10.000000
1 a& }, |3 I8 L- F* x5 G. E) Z  u0.000000

Y
3.000000
0.000000

ROW
0 j9 `# t& ^- k* [6 }- RSLACK OR SURPLUS
DUAL PRICES

9 p. x5 R/ M: n( N4 X9 E1 L2)
0.000000
& s7 Z* u" t4 B8 l2.500000

3)
9 E, u$ H3 F4 @5 _; |; _7 r4 a9.000000
0.000000

' D) l9 p4 j8 f. ]; A4)
6 V; e9 m0 _  Z1 ^0.000000
0 K" K+ f5 t: R+ F6 M5 ?7.500000

  U; }$ s; g! U3 R/ iNO. ITERATIONS=
6 C: Z  {2 ~$ {, Y! u& k3 v2

RANGES IN WHICH THE BASIS IS UNCHANGED:

/ f7 Y  a3 n0 G( p7 i, BOBJ COEFFICIENT RANGES

& A" o% j6 |0 I% |+ O( JVARIABLE
CURRENT
ALLOWABLE
ALLOWABLE

COEF
& D3 v! ?* U7 q8 m! EINCREASE
& m6 V: z. t9 l$ Y; xDECREASE

) ~/ C6 E$ e0 y- m0 DX
( l" {8 }0 m4 W" A, C+ e7 C+ ?10.000000
% t2 d& M) x' `0 D, s: ?6 N8 FINFINITY
; S) ]0 n4 X' F: N# [0 g, b: h2.500000

- O, \; ?& a) s' @; a( EY
! b  G' m0 \" D. t* g' p7 [15.000000
5.000000
+ X: h1 E/ S/ h7 {: f15.000000

4 J( l" `$ x) U6 LRIGHTHAND SIDE RANGES

3 K" [5 A# Y+ N& j& b  X/ @: I1 OROW
CURRENT
ALLOWABLE
ALLOWABLE

RHS
INCREASE
2 t% w7 {9 R- I5 q% zDECREASE

2
# _1 F3 v: }) }0 b2 ~+ T10.000000
& T$ ~# l, u- g$ L$ a6.000000
7 K6 H- t! |- z8 K5 [7 K10.000000

3
( e2 |7 S8 E& |, k9 w12.000000
: `+ B0 \  z  Z* v0 ]INFINITY
% Y; X/ i) n: ]- O9.000000

4
16.000000
18.000000
3 T- D$ J1 U+ r; o# _- v6.000000

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
& ]! y  D( T% F3 I. r9 ]" h: k( Ereport 窗口得到以下结果：

Global optimal solution found at iteration:
2

Objective value:
  T0 P% h8 L( Y- J! d: R( i7.750000

Variable
Value
( l  R6 K9 \& P1 \# S% bReduced Cost

! J8 n! w, i5 Y6 F+ ?7 K7 kX1
) E1 B; A# s5 r2 b" n0.5000000
0.000000

! `8 _7 `5 u. _1 U6 j" \+ {/ HX3
- Y0 ~: S4 L1 D- T0.2500000
0.000000

( x$ m" T4 |6 n+ N: y
8 X& S5 F5 V# `X2
2 ]) y9 i0 ]/ s2 s; l+ a/ p0.000000
0.5000000

X4
& u0 _1 C. g$ U; ~, w3 S7 g  [0.000000
. d) {; b4 x% R& J# _9 X  H2.750000

X5
0.000000
( {9 f/ e9 t9 C. {7 Z2.250000

% w, P3 D' o' ~, YRow
Slack or Surplus
Dual Price

7 {9 `/ }6 ~, G6 h" D: u1
) G. H8 Y6 [( A+ v7.750000
  \9 }" r+ J" k& D-1.000000

2
0 W5 _; v7 F# k& D# n7 c0.000000
- x$ t; i& z/ a6 w$ A" o8 D-2.750000

3
0.000000
" n5 ]* w, I2 z5 ~-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:
+ ~6 _/ s9 ~  r" M2 \7 Q6 \$ L; ^2

Objective value:

14500.00

Variable
Value
+ S; g% }! }4 o' l4 ~8 }Reduced Cost

+ ^( v) W: j5 m6 i- W, XX
100.0000
  @- z5 `$ r& q% f9 c0.000000

0 Y/ g; j3 X1 x% V! D- kY
30.00000
0.000000

Row
: C+ u) X2 f. T* X0 J# @  f0 V* g/ H( zSlack or Surplus
, a+ E! x4 @# {# d5 E0 i8 _/ S' @Dual Price

1
0 {3 F2 @2 R- X, F14500.00
1.000000

4 Y9 @# P; V2 n7 p+ \2 t2
2 d7 E! r: V0 [/ D3 h0 Z0.000000
25.00000

& t6 _# \7 o" d5 |3 E0 q3
( ^# Z; m" C! j8 X' }) d  g2 s. F90.00000
8 ~* K- |: ^; b! N4 S' g0.000000

4
3 G0 ?- J8 S! ^* r/ ~0.000000
, e* {, |, }9 y) L4 A( A
3 G- Y! i* [* ]) G* a, o; F# r+ `75.00000

loooog12

先下下来看看