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 =
. F( m5 z; r2 W# M-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
6 h+ b" a1 }7 ?! s% b2 m2
3 s8 R. |- \$ ]/ H4
1 x- @/ u5 h4 O2 |5 _) d7 ^: @. k-1

. \  ^5 v1 Y& _6 S$ y2
3
) U7 C8 H1 t1 {* ^) ?-1
1

1
0
5 A8 q) {& F6 A5 w: W, k# V1
1

2 [& n* d3 a, w+ K6 p* G9 c-1
* i6 x) `" F# u. c* \, |  n0
0
; {* [. F1 F: T* T0 n8 [0

7 G+ t. z5 ]1 h7 L' c# g, E0
* c' ]+ _4 i7 q; b. @( K  b4 }' R-1
3 p$ V6 N8 t' P; m8 O$ A* C9 W0
0

/ P# V# g. R2 a% u1 G; K& x0
  x8 ]( }- S2 x# c0
5 A3 \* J/ Y& i: U2 {# e: I-1
0

) N7 [0 H) T$ w: Q0
0
( x! K! q+ H# @8 i+ `2 \# N0
; H6 U. w' R- U5 {* |' {-1

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

b =
4 ~% i3 U8 E: o7 g5 M; L; l) {* e6

12

8 y" M+ u- G- R: U1 N4

0

1 \9 ~4 F4 Z$ ~. _  c# ]0

# [# R3 t' `9 Q3 j. M0

0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
! V/ t  Q1 D& \; m+ G+ i) ~0.0000

2.6667

-0.0000

- O/ U3 f( J, M+ x. K  v7 ^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
2 ~4 ?% Y# b  P6 C7 }/ T* ^) r9 Z; @# P！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
) S  b& q( Q6 @4 \" Q2

, Q" {7 n# b' t. o/ dOBJECTIVE FUNCTION VALUE

1)
145.0000

VARIABLE
VALUE
REDUCED COST

3 h: T4 ]% v/ K  QX
10.000000
/ [4 `0 `8 Y% D0 T2 J/ ~0.000000

" E4 y0 Q% O" c8 @9 UY
) i( \( o' |& g) p. r2 y3.000000
0.000000

ROW
" o+ [; J% I5 e# j: oSLACK OR SURPLUS
DUAL PRICES

2)
: w0 Q% N% S1 d0.000000
2.500000

$ s( S: s3 i+ c9 P3)
9.000000
% i3 a7 ], H# h3 @" b! N# s* `; M0.000000

4)
0.000000
7.500000

( p. C# v4 R; h% C+ H# `( oNO. ITERATIONS=
4 |6 Z% L' {7 o3 }2

RANGES IN WHICH THE BASIS IS UNCHANGED:

" b# i. O  ?* x; O4 COBJ COEFFICIENT RANGES

" o; L/ N& F4 D: J8 }VARIABLE
CURRENT
ALLOWABLE
- b. A# b6 w7 d4 G9 oALLOWABLE

3 z# Y7 L3 _5 \2 L; {. ~9 RCOEF
) z; m7 t/ v% i0 D& {; ~5 k4 c7 R, _6 OINCREASE
DECREASE

X
, s- B) c( A! b, }10.000000
) ?+ A0 g. k( P; U' X( xINFINITY
" v0 k9 C; c4 k2 O+ c2.500000

+ O0 g; K9 i+ AY
15.000000
5.000000
15.000000

RIGHTHAND SIDE RANGES

; b/ _7 h) o0 ]* }- K' n# h$ BROW
CURRENT
, F6 x$ c- }$ n; u: ~( iALLOWABLE
ALLOWABLE

RHS
INCREASE
DECREASE

+ V$ F% `  P) f" s
$ Q8 ~; _! e+ V) G' N, a2
10.000000
6.000000
. h" w8 S1 D$ b. M8 `5 X10.000000

" m1 k! B5 z6 b" L4 V3
12.000000
INFINITY
9.000000

  a$ t/ j2 d2 i: O1 z4
16.000000
7 B3 w% t  g. X# U18.000000
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：

Global optimal solution found at iteration:
' L; D: z. c8 v6 f: w2

Objective value:
7.750000

$ b  N# A) S0 w, K6 IVariable
Value
! p) A9 i, A) N# F3 o  `Reduced Cost

X1
- F, r5 Z$ Z+ w0.5000000
0.000000

X3
7 N1 v" Z; q5 d7 F- a0 `2 y" B0.2500000
4 k0 f' q/ Y& l+ n& |2 L0.000000

% e8 B; W7 _5 b0 E7 L4 R* Y
X2
0.000000
0.5000000

  O% w6 V9 C9 F$ E$ HX4
  v, d+ r5 u, }* r! W$ ~% u0.000000
2 {1 d' T; {3 l9 q( p3 j! w2.750000

, K% l- b. Y, a! AX5
0.000000
" F5 P- L; i+ T% C2 B9 J0 ?7 h2.250000

Row
Slack or Surplus
Dual Price

1
( L: y5 X) _6 `# |& K7.750000
-1.000000

2
( s7 `% \. u! r8 ?7 Q( {" \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;
- |% \! U3 W' o5 Y/ b1 j% X! 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

2 X( Y0 O! H( K' ~. n7 I! nObjective value:
4 _4 k; G- _3 Y
( r. X4 g5 [- F- S3 M14500.00

Variable
Value
$ H. |+ |# t: i; t( e. {Reduced Cost

X
! L3 h' t$ w2 R100.0000
8 _. y9 b, R' h. o2 h; A: Z/ B6 w0.000000

5 L" o. n# A  d/ x5 NY
" x0 l8 d* E1 d30.00000
( F3 ~, [' l/ b0.000000

# D4 y5 C( V2 WRow
9 j, S& D" Z1 C% hSlack or Surplus
; R0 s3 j, B5 GDual Price

2 n! H- \- Y- b* d, D0 f3 I. t( h7 T1
14500.00
9 ?/ o7 a$ @/ q5 R' A1.000000

" V& l7 |0 a) D4 G1 h2
' `# P, n0 _) m0.000000
( ^2 s: G( U4 k3 E25.00000

3
90.00000
0.000000

4
0.000000
3 D1 I' I% M6 _! \% c
75.00000

loooog12

先下下来看看