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

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
2
4
-1

/ E+ G; e7 R5 T' I! h0 {3 A3 }2
3
-1
4 e# X8 D& D, L, W" [) o5 j1

1
8 B: H4 @8 U7 O! r! r0 q0
1
0 u8 V; J  w% A+ d8 _7 B1

; J( s4 V9 M) o% |# R) K-1
0
# @: e, P+ g" I# j7 o& O: ]" n3 M0
# G7 Z: e- h, Y0

0
-1
0
' L# w, g. H' ]' }+ x: p$ R0

0
0
-1
0

: b+ _/ Z% F3 }6 E) b0
0
0
: M$ z. Y3 ~% h: @7 s& Y0 h! R) [-1

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

b =
: d( ~" N; n6 p  N6

! f% A: ~+ b9 G. N. V12

  B8 A7 i- G1 ~3 ~4

) S$ v+ A! Z8 }0

! _: W5 ^9 v- y7 X6 E0

0

4 G" v" u! x$ s; h5 P- z* o& K0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
0.0000

2.6667

- [0 Q" Y5 }! [-0.0000

) F# ?! ^4 f0 f1 c. U4.0000

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
% {9 I7 C$ |* i！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
% ]: U5 r  \3 t4 c4 t2

4 ^8 @- S2 I( r3 |7 oOBJECTIVE FUNCTION VALUE

1)
145.0000

5 m4 [$ t; D8 S2 N/ l6 }! [7 dVARIABLE
' B  ]8 L  b* q$ K; FVALUE
' u$ Q1 z! {( n; qREDUCED COST

2 M  }1 T( c4 l2 U5 |X
10.000000
0.000000

Y
3.000000
" d4 x( }! b" u& _0.000000

ROW
7 ?/ F. Z  l% m$ S3 ySLACK OR SURPLUS
! K( n$ X, M9 G# {  IDUAL PRICES

2)
0.000000
2.500000

; y8 Q% b" v9 h3)
9.000000
( Z5 Z. i7 t3 T0 g7 d$ B0 a1 C1 {0.000000

4)
0.000000
7.500000

NO. ITERATIONS=
- y, g9 p. p; V! _5 }2

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE
; Y( M# d3 ~- I2 [9 ^8 `) HCURRENT
: K/ S7 s+ L( N$ x5 FALLOWABLE
" c$ l* ?' o& F  q. SALLOWABLE

5 _$ I2 m8 ~3 ^COEF
INCREASE
9 f! K. i7 M/ I0 gDECREASE

X
10.000000
% m4 Q# Q7 T1 K" b" Z, vINFINITY
2.500000

" j0 S( g( X: zY
15.000000
  r( y& W' ~; q) z% F$ x& \' r5 ]4 A! J5.000000
2 p  U' }0 n& u( b) d15.000000

RIGHTHAND SIDE RANGES

' y8 \2 O& s- _. h" m/ K9 MROW
7 `* F/ S0 Z/ ?9 DCURRENT
/ l6 M. O1 w% ]ALLOWABLE
$ h5 d& z  s: `ALLOWABLE

' D" y3 N" m# Y- h: S- j+ W5 SRHS
% x7 ~$ a+ U. q* _. |" Q, VINCREASE
DECREASE

# A0 g, f! u/ A: [) T
3 h6 o/ D! R$ K. T4 q- V2
! P$ }  p6 t3 ?% R; w7 n) l10.000000
6.000000
  \3 Q5 C/ j- t8 G9 g& e/ e) L) t10.000000

) n4 A/ g7 d, m' |6 l3
12.000000
* V( E8 J' k0 B2 m2 PINFINITY
9.000000

4
# {- D9 A. J2 ]16.000000
18.000000
! ]5 P4 [9 i2 `. S9 y/ L& Y6.000000

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
4 c6 V) f3 x9 ?/ C$ Dreport 窗口得到以下结果：

Global optimal solution found at iteration:
# f1 Y+ z2 Y" K& W" }3 M9 X2

Objective value:
: t9 y8 e% D$ E6 ]7.750000

1 g2 O/ D' b2 T. L7 lVariable
( ?5 B' V+ c9 f9 i2 |8 r" R" mValue
Reduced Cost

X1
0.5000000
0.000000

( z9 p. {5 @; y  `X3
  b& I9 U, H! _8 z, q& F) I0.2500000
- _/ c/ ?, u( @0 ~& {0.000000

8 Q) Z3 j/ `; |, I& g
4 ~* F6 _  Q* t* N7 }X2
0.000000
0.5000000

9 O1 u2 H( V$ N2 yX4
0.000000
1 S" i5 s: l9 h1 A( z* K2.750000

9 ?( ~5 ?! Y1 G. hX5
0.000000
2.250000

# S( d" o8 j9 T+ `4 ~Row
Slack or Surplus
+ x7 r' h* C; h9 s' h) [# T  HDual Price

1
7.750000
) d4 T1 T. V+ H-1.000000

, ]. W; s2 ?" r- y0 c8 u2
0.000000
3 P( H1 O( a! O4 G& {; Z1 J-2.750000

4 G1 t' y5 \# z2 E6 v" p3
0.000000
, o8 s) @- `: s9 N6 L-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

Objective value:

2 H; G* x& I7 ~# {  }9 t14500.00

% T: v* A& x( P* p0 \* H2 a+ hVariable
Value
; S' j8 R. A2 ]! P$ o: w/ z' SReduced Cost

) T& z  A6 K: O& O- hX
100.0000
0.000000

Y
30.00000
  J5 O( y% k/ g; [5 Q0.000000

Row
! |; q8 {/ u3 n- {2 bSlack or Surplus
" `/ A  E4 O# |! [# v/ `, U- eDual Price

1
) y4 P# G2 i# R, J# v14500.00
" Y! i& Z" v, R$ [: U2 [$ b1.000000

  o0 W$ c; T9 U: c' q& \9 U9 k9 N2
0 _+ B& K( s  U7 s+ R3 `( F0.000000
25.00000

3
9 \& q8 f. E6 f; F7 q8 P; c* x90.00000
  h; r( m: I# }/ F5 ]! x0.000000

4
0.000000
; ~* c$ P& \! p* M2 S
75.00000

loooog12

先下下来看看