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

6 y$ {$ v* {# r. O0 }/ s) I3

-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 =
; v7 K) K& U) |( S7 }; |+ b1
; q- N2 z& e; _2
# J/ f9 }6 J* p5 X7 @4
-1

2
$ _% o1 N& Q6 e- M9 j1 v, s, ?3
5 ]0 ^: t% o( U3 p$ w9 ~2 C-1
7 h& C* m" S( V. o1

1
% \4 n- t0 h6 l1 g" u. g* q0
1
1

-1
, p0 U/ o+ d( {& J5 n4 K0
0
4 w& o2 X( ^7 t5 q0

0
- \/ n0 @! N, W-1
$ H) e. |5 C8 b: ?/ s0
9 |9 u0 \6 d1 J  i5 `+ o9 M0

0
  h8 t* r; L! g+ r0 N2 x' t: r0
4 D: s0 ^2 q! t. u5 [$ ^9 T-1
0

0
" Z6 l# n8 Y) Q+ Y( M3 |; u& f0
0
-1

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

b =
6

12

4

0

0

0

/ W6 g- l, @3 ?0 ^$ {4 f0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
0.0000

0 H% S& m! r3 a* j9 X/ R: `2.6667

% V2 o& d! u+ E/ N2 O; n-0.0000

6 g2 X. h5 Z* q. F! w' @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.

) r  ]( w1 _# _' p6 H！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 e" o4 Q+ q6 P1 ^" H" GOBJECTIVE FUNCTION VALUE

. B0 G- p8 O* C; [1)
# }, v$ K* y6 r( x145.0000

9 g# e# B3 p) `, j  iVARIABLE
VALUE
REDUCED COST

. d) e6 c! s6 L' W$ G5 a2 LX
8 K: l6 H, N- H9 {7 e10.000000
5 g! B* \. E) l- v# d5 ]* \0.000000

$ y: g8 G9 Z& T0 tY
% ^$ F3 ~/ D3 g5 ]3.000000
0.000000

( K% L8 ^/ O* u2 n, b  LROW
: P/ t7 g' e& n$ o7 l' fSLACK OR SURPLUS
, z  o1 d" X- F% B% NDUAL PRICES

2)
0.000000
2.500000

6 t4 L2 y/ X3 Z+ m3 U8 `3)
9.000000
8 S* F) A% }8 V! t6 m0.000000

4)
0.000000
7.500000

8 z. Z- }6 I* j9 V/ F0 d+ XNO. ITERATIONS=
7 @1 v9 r( N  ~7 I+ s1 f; F1 a2

1 P7 f8 w! C) P6 nRANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE
& u  t7 t5 t" s/ n  {8 b0 d& WCURRENT
ALLOWABLE
ALLOWABLE

COEF
& l8 y1 U/ l8 Q7 @5 iINCREASE
DECREASE

X
10.000000
( m6 K+ f$ ?; x5 C3 ^INFINITY
2.500000

Y
15.000000
. q4 T; k( N; h6 \3 U5.000000
15.000000

RIGHTHAND SIDE RANGES

ROW
CURRENT
ALLOWABLE
  d: u' l' A5 r7 H8 n, QALLOWABLE

% @; A9 [/ ?+ E; ?* |1 S/ V; QRHS
INCREASE
DECREASE

% y1 Q" u& ~& M. o3 M; ]
2
10.000000
6.000000
10.000000

3
; \8 q" ~! i9 X+ c. ]+ b- u12.000000
INFINITY
  z; Z' B: X2 ?& O& u8 k4 T2 W' x9.000000

4
16.000000
) c, S4 J& H1 _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：

" k9 g5 I- t% t& `8 [$ Z4 oGlobal optimal solution found at iteration:
  c7 D$ z; ~- L7 z& u8 u- x2

6 |! S- L: _" f& ~8 A  e' M$ cObjective value:
- R  {5 U2 @9 [  t( ?7.750000

6 C6 g0 s+ _" |' o0 I& uVariable
! q6 ?8 _2 _* q7 w# n. n; {5 tValue
+ P" k1 J- g+ jReduced Cost

3 Y- v1 \& p9 O  |+ _4 ^X1
3 ~4 i' l/ c; p$ i! p0.5000000
0.000000

- f# I0 ?! r$ a1 Q7 }3 A: Y% q/ PX3
2 x1 e2 s# K; i2 I) O7 N0.2500000
0.000000

( _% r- }; l0 {/ J1 d) D+ |
$ p3 L0 O6 `) h9 C4 hX2
0.000000
, M! `; O7 H4 d4 {0.5000000

; g+ ?) ~7 g, Z! u$ k8 l$ K& {* GX4
0.000000
2.750000

# v& L8 Y! i+ W& ZX5
2 h. q9 C3 [  ~  }2 S6 R0.000000
) s2 B( F7 T9 I2.250000

9 x: m& P) f2 V( W8 a" N4 i8 BRow
Slack or Surplus
Dual Price

8 @7 H- ~8 p* G! {! @& t& {) H1
7.750000
-1.000000

" `7 T; N; w9 c5 v/ `5 d2
0.000000
-2.750000

0 T9 Y* U, `2 Z, O3
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;
! 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 \3 }' P  ?: a; |4 z2

) c# `4 a2 n: X' d$ b2 KObjective value:
6 B( g% `& {5 ~: x- c. A) e
7 v+ x  x- j% {7 L5 K5 Q14500.00

Variable
$ g5 Y# n& L0 ^( P9 V; ~Value
Reduced Cost

X
6 `0 n8 k, W( \$ l7 y8 D100.0000
5 |) i- ]6 E+ d& s; V+ R0.000000

" q1 }) H. d; |' OY
30.00000
5 V" D. H5 }4 L5 Y9 t1 x% [0.000000

+ a, N1 y' R3 a  O; p  I8 L$ CRow
1 R: i# U5 |3 e- ^5 ESlack or Surplus
Dual Price

0 M3 i( g+ D: I4 A5 F# T1
6 X8 |1 k" M8 D. X14500.00
1.000000

2
0.000000
" |/ Z* ~; o7 `) b5 W9 p25.00000

3
: r+ n) a$ |( @# `" n90.00000
0 [6 [" p5 H) K  k0.000000

4
* Q2 m+ l  B, `/ K- h0.000000
6 a4 }& u) I. h2 d+ g0 W3 M
" f) C8 a3 X" G& d75.00000

loooog12

先下下来看看