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 =
; V2 V2 K, U* W  H6 X2 a4 x0 e-2

-1

- @7 e+ I. J: |/ f- b: }' E3

  F3 P- a: a  v- Q1 S4 Z) B-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 =
. X' D$ T6 u" `) K; B/ m1
- `" o& ?( n5 @! T, x2
. t! g, w! s5 x  P. i8 m4
9 e3 T7 N2 _5 k-1

2
2 R+ _# h4 j8 ^+ D; g$ J, }- K; N) l/ n3
1 p3 K) H4 u4 s" @; {! X) d-1
1

7 V$ V6 o& y# U* ?; [3 T3 P1
0
  `2 h$ P  w) [- l# e1
1

$ V6 ?) F: S$ E- ]. }) ~/ G-1
0
( A% @" K  h, J; C( c0
+ z) [6 j: }$ N9 e% x* Z& H; v5 f0 h( G0

/ }6 Y' @/ D' Y/ g( q8 S4 N- }0
-1
9 ~' w0 b0 U1 E+ F# I0
. G0 g9 C( U, x0 D* G& c0

0
0
9 v1 ]# s: b& P/ K/ u) ]-1
0

0
3 T; i9 k) Q5 \3 t  @/ v0
6 Z+ r2 q8 E4 Q# h0
  [# W. B3 N! X5 r5 M# B3 J-1

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

b =
$ v/ ]- [. ?5 x: w+ X) v- ^4 K6

" D2 _5 Y, g" S' }2 ~12

2 r9 }; D- v5 a8 M- I6 W* Y4

0

% q+ X/ L" U1 W' x" X* B0

0

% N# H% j6 a, S5 a& |# ?$ G# o* j0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
0.0000

5 m& ]0 B) \( t5 ?6 q! U2.6667

-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
; I! Q2 X7 _" Q: w$ C！注释符号; 系统默认为自变量>0, 若不要求用free命令.

/ Q0 s' }0 k$ C) u4 r/ }  @8 a8 s！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

, Z6 e8 n2 i4 wOBJECTIVE FUNCTION VALUE

' M( f( ^" m1 y1 ?5 V+ ?  P1)
145.0000

$ b7 h3 g& |; ^9 h. SVARIABLE
VALUE
; S) G- L: i7 l& n8 }REDUCED COST

X
10.000000
0.000000

$ D$ a" ?/ S' I/ _+ a0 [Y
3.000000
* f# d- V2 m$ {! W0.000000

! c7 T3 o# L2 N" ]: h2 YROW
SLACK OR SURPLUS
7 E, h. x, |' w: qDUAL PRICES

2)
0.000000
( i& Q0 M; Z# o9 m2.500000

. \) A/ p' V( B4 C- J5 N3)
9.000000
0.000000

  M1 [# J7 O( I1 {4 X7 w$ O4)
0.000000
2 E; c6 S' S# Q7.500000

NO. ITERATIONS=
2

RANGES IN WHICH THE BASIS IS UNCHANGED:

0 H7 V' \0 \  @- \6 EOBJ COEFFICIENT RANGES

VARIABLE
CURRENT
6 ]% I+ p5 F9 vALLOWABLE
3 T9 a3 \+ @" [/ p- @6 x, GALLOWABLE

' f: J7 L) ~7 A1 L+ B- s& ?COEF
INCREASE
DECREASE

+ }2 `- i" K0 d* }! g$ m) DX
4 q' G/ M& F& S3 R% o8 D  Y: j10.000000
INFINITY
8 N! V5 h4 d- \0 d: @0 x2.500000

Y
15.000000
5.000000
$ i5 a( }+ q# F15.000000

/ f& Y4 ~; H  w7 N) fRIGHTHAND SIDE RANGES

ROW
CURRENT
( z$ L& h" X) M1 l7 S. {7 R/ AALLOWABLE
' m1 X0 g# O( |3 X5 g, k. }ALLOWABLE

RHS
7 h( Q4 R8 O4 H5 Z& k" WINCREASE
DECREASE

. }9 D; ^0 A, [! P( X, s5 m2
( J- P3 D. Y5 m, q1 y6 Q% z$ C10.000000
6.000000
" h* J8 ^- X+ Q1 o1 N10.000000

3
; J& d' U  i. H6 A+ ?12.000000
+ p8 t. J$ W) P) v; f9 mINFINITY
9.000000

4
# C! j2 a) Q. b  J, b& ~- Q16.000000
18.000000
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：

8 U7 K4 {8 c( LGlobal optimal solution found at iteration:
& j5 D, k# @0 ~  ~0 T2

6 _: k4 F7 A7 B! {7 i2 aObjective value:
7.750000

Variable
Value
Reduced Cost

X1
0.5000000
0.000000

# Z" t8 h7 |* ?( N- O5 u; t9 |. BX3
0.2500000
0.000000

X2
8 k- g- u7 A9 m8 N$ d5 R0.000000
0.5000000

( d# E2 f# _! m/ uX4
1 c& N8 L$ ]9 L1 b: q; n3 p3 J7 [+ C0.000000
2.750000

8 Y2 |( |7 S( P6 ZX5
# d8 U5 z4 ]- g( J  G7 `+ b/ `0.000000
2.250000

* U& p) H: P7 S& |4 Q8 {Row
2 y  {: X& ]. |& v( I$ SSlack or Surplus
" ]; \- G. B! |/ \6 M' W6 ~3 YDual Price

1
7.750000
-1.000000

! f1 {' o' u% n1 V2
! Y' d, x, c9 z' P% Q) G0.000000
: M0 d3 }# s4 q& @0 o7 x-2.750000

3
0.000000
6 U0 R5 o6 k& I8 F) z-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:
, C* E- ~. V% }  H  b5 O7 i, |2

Objective value:
$ i0 }- ]2 M# i! T. m: H& b7 G
14500.00

Variable
Value
) a" @+ L( I* J) h+ [Reduced Cost

X
100.0000
0.000000

Y
30.00000
0.000000

/ A1 @# @* D) d3 v: kRow
+ g9 E# |4 V0 N" ~5 GSlack or Surplus
Dual Price

1
14500.00
1.000000

2
1 m8 y2 P' G7 t  Q' y6 d0.000000
25.00000

, C, U% l5 G0 \2 T- N' [* y; p% n) {  e) y3
90.00000
0.000000

4
9 X0 {9 ?, R& X( {- q7 j0.000000
, B% [' C0 q& V
3 L8 L' \" v) M; J& A# n$ r' q$ ~75.00000

loooog12

先下下来看看