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 =
  F6 {. T$ J# f. `8 v0 _% B1 x-2

" q$ S; r  q! U% Y4 `-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 =
; T" ], Q' u1 [& O) G9 r1
2
/ I% r+ p/ ?& g. Q. s4
-1

2
3
-1
' @$ X& e7 u* h- B- f1

& m2 X3 P3 T2 j# Q0 s1
0
1
' l9 I3 W# }% V! b1

- y" G5 I5 W4 s! O( X6 z-1
7 I8 k" v6 Q( I6 h" p6 o0
4 }4 q9 E# i+ `6 }0
9 ?# m4 R7 i. G) e0

0
+ m/ e; V* k& t- K+ X3 u0 o-1
# @6 G  p( r# P3 V9 ^0
4 w: ?4 m) U/ L; t2 g! k0

/ u5 `& m. P  V7 ~6 k0
0 }7 P  v! T* s% j0 D3 r, N: Y) A0
# N' R9 s1 i- Z: D0 z1 m-1
0

/ P6 h# ^" A$ U# ?% l& h0
' j  C" F( t8 T' U$ G/ b8 G0
* N, e" D; S3 [. B; \0
-1

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

b =
4 u1 Y- ]; x* S" n6

& D# Z) n! b; D/ w12

7 z+ M* ^+ p6 a+ c( F* O2 d1 v# F0 ]4

4 N. I# r/ `" \: Y; I$ v5 K0 t) C0

' O6 }/ P2 y6 B$ O: j$ A+ {0

0

0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
) ]& {& x, {$ c( m# P0.0000

2.6667

' i% D1 @2 s& A" c+ S$ l8 z-0.0000

% d- P9 {( ]& [( [2 [& q4.0000

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.

1 X& B; w8 m. [5 ?/ O！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

( O# c3 m/ W6 e/ K9 yOBJECTIVE FUNCTION VALUE

- ~4 C: \, p5 u/ [1)
6 c- l( y2 b+ R5 E145.0000

VARIABLE
! _1 }" y/ |4 K+ s1 u5 g% g- D6 fVALUE
REDUCED COST

X
) @, m7 j( L; y! C- R; m1 Z10.000000
* e) ]$ b/ ~2 R0 V  w4 U0.000000

) M% b3 g4 ?, m; @4 yY
& Y6 Z! [' c1 B1 q) Y3 G3.000000
0.000000

+ ^( [  Y8 K  y& e( r- vROW
# I+ j8 m/ g- Q5 M3 cSLACK OR SURPLUS
DUAL PRICES

2)
0 i$ C4 W7 E4 i0.000000
2.500000

3)
9.000000
- f  V# r& D  h. e' _0.000000

4)
0.000000
  k  B/ @" `5 H7.500000

  H; O  q7 e6 z& f- s6 DNO. ITERATIONS=
2

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

8 N. T9 U' }% l4 g3 k9 h: gVARIABLE
2 M1 m& b; a6 d, G: dCURRENT
; q8 m) n+ n% G1 }+ L: L9 XALLOWABLE
7 c( E- B) b; C. qALLOWABLE

1 J1 m- n! B# y  sCOEF
; e% T% u) @( X5 y3 ~( AINCREASE
, Y8 E+ d8 s. P7 ~+ zDECREASE

& J; K4 F- L" ?' ^# u1 WX
. _; D* h  I0 ^0 ~. I/ U" g10.000000
INFINITY
$ L" [1 t4 `+ C$ {* o" h2.500000

7 ?" [. A* `- L" aY
15.000000
/ @9 D: M; q8 `% V2 \/ h5.000000
( r/ u: X% Y, ~. w15.000000

RIGHTHAND SIDE RANGES

ROW
! K! z! `8 r/ Z$ X. MCURRENT
ALLOWABLE
# u4 O7 P; E: c3 ], k& XALLOWABLE

RHS
INCREASE
DECREASE

6 }8 T- P: h4 Y; F6 K
, S# |- Z) H8 W0 B) ~" R3 x2
10.000000
6.000000
- ?) A, g' Z( C- L10.000000

3 }, G6 D/ j/ O: f1 F. y3
12.000000
INFINITY
9.000000

4
16.000000
  O3 I  y/ }/ r18.000000
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
* S( n8 F6 a. R" B! x, G5 j: Xreport 窗口得到以下结果：

# M% W" ]! n' S' S% ~5 L& Q8 a% @# pGlobal optimal solution found at iteration:
2

0 P0 s: G8 f! u& V( B' WObjective value:
7.750000

, ^# e/ h  Q2 P, a8 P  YVariable
Value
Reduced Cost

X1
0.5000000
0.000000

3 S" a1 B! A( U3 Z8 uX3
/ Q5 \. v* B9 I7 B9 X/ n& j/ X* N0.2500000
1 b. `9 a4 G9 H0.000000

X2
0.000000
+ X3 j* ?; F# H) z' G8 _# y- a0.5000000

X4
0.000000
! F5 O+ j  e  b2 f! E2 D" T6 t! ~2.750000

; N5 @/ o6 Q2 i% ZX5
0.000000
2.250000

Row
Slack or Surplus
Dual Price

/ n9 a6 U6 u6 s3 y6 _9 r1
4 k7 _0 u* A* f, l5 s6 k7.750000
-1.000000

2
7 ^5 d6 |6 Y$ N1 B. q% L0.000000
-2.750000

, d1 s- ~+ [+ |7 p3
1 s8 t. C/ J* I; x$ ^0 M0.000000
-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:
5 {1 S* t' K" n8 C3 i2

Objective value:
) Q3 o3 J/ c- ]+ N/ [0 v
14500.00

Variable
Value
Reduced Cost

X
2 v! O# v% Z% p2 F; {5 }100.0000
0 M2 _& _* w8 S& X5 Q0.000000

# |) a7 a& F* @! U& ~) k& FY
- }2 T9 F+ y( |0 ]1 Y; J4 i7 f( V1 ^30.00000
- ^0 q7 D8 h& B; k0.000000

8 B: ^; v/ z0 p! x, DRow
Slack or Surplus
Dual Price

+ h% W( d% r. F3 X/ a1
  V. K. J9 |  z4 ?14500.00
7 a/ \( ]1 Y$ j7 S1.000000

& ?* M$ f, ~, T: X: U1 s2
" `" ~4 ^! e* Z' X+ f4 H3 b0.000000
; F- {2 P# ~+ {3 A. ]25.00000

. S# Q; \' y2 }: r; F3
90.00000
# @8 J! u  Z& v; F3 `0.000000

4
& J; o) k& d2 K; F6 Z0.000000

+ c* a& K. B+ @/ e7 S0 ?3 o! Z75.00000

loooog12

先下下来看看