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 =
( M" t* |. o1 E' w-2

  V# t. T6 s4 I! w. v-1

' w: L1 A6 T7 m. n3

-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
* B& v' n8 Z( k. T1 m4 f/ ?* D% E2
4
7 q& S. G, V/ `, N-1

2
/ u& P1 W: y3 s8 |: W2 l# U/ K3
-1
1

( b) [! S- z0 ~4 M1
0
3 ^$ ?3 D( C' W* ~* z# Q, j6 [1 `) J% G1
1

-1
0
) ^1 `1 H! V4 d0
0

0
-1
0
2 K" W5 C' V- Q$ l7 S0

0
! c( ]* Q$ d  r0 [+ B, z/ x: L0 c' H0
-1
0

0
& a+ b& k( Y0 u9 W; J0
0
) e( e/ v" }6 `$ v-1

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

b =
1 M7 e/ a9 {% D0 y4 b3 b6

: E9 D( [1 D- C. b12

4

# a  p9 L9 D: x& S( |% c0

4 p+ {% `7 N% F0

! q  l9 h& w" J$ ], ]0

) x% O/ q+ \/ s* K& t! x0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
0.0000

2.6667

-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

OBJECTIVE FUNCTION VALUE

1)
145.0000

; E5 z' ^. @& H: a* h5 _0 s* WVARIABLE
* ?! x( w' ]) Y# w4 O9 ?$ fVALUE
REDUCED COST

X
10.000000
2 e" \' m  d8 }* _* r, ?/ l0.000000

0 a, ^7 Q- P; G- uY
3.000000
0.000000

ROW
, v' c- s, g( `% K/ `9 rSLACK OR SURPLUS
) g" y1 E' V0 p3 I% h( m. `DUAL PRICES

2)
# x- n% T8 c8 e/ ~2 P( `$ o2 l0.000000
2.500000

3)
  g8 F' ^" I, e+ k$ C2 ]! q: y* h" {9.000000
! ^' P! H  F: [) D6 N0.000000

8 R: u3 A% H1 D( D7 q5 ?. V4)
0.000000
7.500000

NO. ITERATIONS=
/ e; o  B& [: ]2

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

) B6 p5 ?- s( @1 O) Z/ w7 ~VARIABLE
CURRENT
7 f: q  p+ j3 ?+ ^$ PALLOWABLE
ALLOWABLE

COEF
INCREASE
DECREASE

X
5 ^* j0 t% K/ ^7 j1 f10.000000
INFINITY
. I4 [* u$ \6 L2.500000

' V! x& T9 \% g4 \8 f! o) x; pY
15.000000
9 p- @6 H& l  W$ j3 H8 \! l5.000000
15.000000

RIGHTHAND SIDE RANGES

0 o& E* o; B7 ~: }) k0 XROW
CURRENT
$ l& r8 W' p1 L' N$ q0 @5 V* OALLOWABLE
ALLOWABLE

) C( h: a# s3 [, P0 R# P% T( i8 sRHS
INCREASE
DECREASE

" g8 N& Y6 j$ H4 c3 P6 t/ m9 `
* z! k! Y7 M* O) Q  ]- K- ~8 B2
1 t. I# m5 E3 @1 t" h10.000000
% e6 r+ x* I/ v/ m1 b( z0 W' B6.000000
10.000000

3
12.000000
# F. r! X8 u2 X8 @) u- I: e; k) Z, AINFINITY
9.000000

4
: \$ z, }8 f3 ?& d2 J16.000000
8 O8 _: i  K- _8 c) |$ p4 `18.000000
; Y6 t- M3 a2 m. K3 x6.000000

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
% H' {8 C8 m: K/ A2 G9 [7 zreport 窗口得到以下结果：

: ~8 r) s, i3 I; p6 i) X; w9 |/ ~Global optimal solution found at iteration:
1 G: x; ?* G, X& r) o) T5 g8 p, {2

0 Z8 o: [/ Q0 C5 |Objective value:
, q7 e; i  ]2 [4 I/ a* Y7.750000

Variable
9 E# i$ s' r( Q0 @' X! U7 }  ^& wValue
! ?6 B4 o& B, g1 ]* LReduced Cost

X1
0.5000000
0.000000

X3
0.2500000
! J4 Y' \  W" T0 s0.000000

$ A2 b8 \4 m) [8 }: V
4 |- R0 s( \" k5 g! R2 cX2
5 x3 ^* L4 j) |5 D8 y0.000000
2 W; D1 T* }0 t0.5000000

X4
0.000000
* D3 n# S5 ~0 Z" [! W( y; O/ R( v2.750000

X5
+ O9 S, M$ @' ~  @- q0.000000
2.250000

Row
8 N& d" e, ?5 A1 K8 wSlack or Surplus
Dual Price

' {  E8 L8 N7 o% w1
7.750000
! L' u8 {2 M# J  U& s-1.000000

2
0.000000
-2.750000

3
0.000000
-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;
2 q, o% y) @- \& _5 [1 ~! 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:
; |3 Q/ p& j6 F0 r2

, A1 X( L+ _) R/ {. n" P% zObjective value:
2 z; `$ {2 w& R8 u% Y
14500.00

Variable
Value
/ `" L4 u3 U, zReduced Cost

X
100.0000
5 x/ V, X, W- h9 ?- Y. m0 v% c0.000000

) V9 `/ [+ A. I$ O) x' ?: ]' IY
30.00000
0.000000

% w' e+ C/ y* W2 F' O3 m6 F9 ~Row
Slack or Surplus
1 V4 i- ]. c3 E" d  S9 j; JDual Price

& D3 ]2 z) u0 E9 v1
14500.00
1.000000

2
0.000000
# B6 U/ c4 H6 m25.00000

3
, \( [1 c5 B* ?: j- `" B3 e90.00000
0.000000

4
5 u  d8 I6 t" l  k/ X$ q# P& m0.000000

75.00000

loooog12

先下下来看看