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 =
- K+ Q% V+ |# `- Y-2

-1

3

" f9 ]: U8 T3 B9 ]3 l) M-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 =
6 x7 K. L- h0 S) v1
2
$ p& Y2 l: j0 n4
-1

/ @. B; T: q8 m8 |6 M2
6 ~1 j; t3 `; W! V3
: S$ I- k- w& m-1
. ^0 S0 z6 b5 D! |7 b" p1

1
5 q4 U3 A" U' v: g7 K/ L0
1
1

- J& q6 P5 F/ Y-1
0
0
0

0
8 c8 ~4 u5 {1 k1 O& X-1
' x3 z( y+ {" T! c$ X( t$ b0
0

2 I( }- Q6 K& O6 R0 Y0 y; n9 K0
! Z$ B( ?3 E% t6 Y: U0
6 e1 b! C( U% A3 Q- ]5 r-1
0

. d$ a  {; u. U# s+ N2 z2 Y/ e) X" i0
0
0
" @% h; F( S/ W4 L-1

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

b =
9 L' R* S! l3 V7 |6

" h2 k% A! ]: M: I+ ]- D! l12

) |! s7 T% [/ k: [7 {% z& T4

' M$ a6 V$ E6 z5 A, @0

0

0

% X' c8 g1 m+ _+ K( R( }0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
0.0000

' b* ]) ]1 m+ R/ Y6 u2.6667

: d4 A$ z  w4 I-0.0000

& y4 O! h0 v; t) y0 B4.0000

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
# F' O% j' R& x. y* a4 @5 g: b9 V！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

$ K8 @8 J1 {6 t6 J, O# }* U( QOBJECTIVE FUNCTION VALUE

8 j5 q- Z( A8 u1)
145.0000

# w, I2 f/ H' gVARIABLE
0 O2 l* |, s, Z1 C* V8 D  Z. xVALUE
9 F$ c, z. @# lREDUCED COST

# ~" k' h5 H4 z- PX
10.000000
0.000000

  @' K+ p- z# iY
0 Y2 f9 O  e" ?' y4 l) L3 n3 ~( [3.000000
0.000000

ROW
SLACK OR SURPLUS
DUAL PRICES

6 q" P/ ~6 N# u2)
0.000000
2.500000

, B" Z' a/ H3 y, u5 y; `7 ~3)
4 H6 L9 q' V+ W3 A# c9.000000
0.000000

! f& f( K$ U; m* a& B' l, g4)
& E3 c/ _6 ^# T0 h0.000000
/ [4 {7 E, J2 Z7.500000

NO. ITERATIONS=
2

" h" d/ r9 Q* E6 ~RANGES IN WHICH THE BASIS IS UNCHANGED:

+ C+ @0 z# }& d# yOBJ COEFFICIENT RANGES

; O) w, @/ x' ^; V: y" Z6 eVARIABLE
+ Q" F# R& d" }1 J: u' f; pCURRENT
+ m6 [/ i! p0 b6 u7 p5 ?2 RALLOWABLE
ALLOWABLE

COEF
INCREASE
DECREASE

X
' \& Z+ S1 `- J) k& Q0 ^1 H10.000000
! f, x6 @1 c- d/ @8 P3 jINFINITY
2.500000

; H0 K' U1 k# n6 s' \Y
4 G( @! C( s! q2 v15.000000
5.000000
" n6 S2 t  V8 Z% a( n15.000000

. ^* B" f# ^5 Z( W4 \& l' J  rRIGHTHAND SIDE RANGES

ROW
CURRENT
ALLOWABLE
$ E6 s, r! L7 n: l7 F: ]ALLOWABLE

& J" m' r% X+ I6 LRHS
. ?! m, s  |4 h: ^; I) VINCREASE
DECREASE

: z: n4 E! }7 u) s
: \( F$ A8 |3 a) L% [2
- i+ J' J" w* L. Z" Q- N10.000000
6.000000
8 N: C$ L, Z! ^- j10.000000

0 p2 y# R8 x% V3
12.000000
" c% i# L$ c' s3 oINFINITY
% G! S: V. p9 s3 {! F) r* J& ^9.000000

4
0 P; Q$ c4 i5 k5 v" t  f  W16.000000
2 o7 u* Y( b8 I18.000000
' h4 L9 h+ s( V9 b7 F: i9 e" t* A' f6.000000

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:
1 e2 m  p0 C# q. y- t9 S  [' Z2

Objective value:
, H$ y% F/ i4 C: R$ G7.750000

Variable
- @; A( E; D* M, x& r" D( K" }Value
Reduced Cost

X1
- e' _+ Q+ v0 p7 B1 b* O0.5000000
8 [1 r/ Q% f* n8 Y+ |* g4 \: t0.000000

X3
0.2500000
2 r# o  `' ]- P0.000000

- {" _$ b& o1 B" Q
& ?7 o/ n9 s, n" a5 `1 d8 JX2
0.000000
0.5000000

) c0 `" {' R) g( p  |3 cX4
1 x; Z; n, U& t0.000000
; _( M8 G& b4 B2 e& C& s2.750000

X5
# l7 y6 n7 p8 f4 L! F+ e0.000000
3 [: B+ ?: a/ H0 [; c4 C2.250000

Row
Slack or Surplus
9 Q: v$ t! v& bDual Price

4 d6 w, z+ d5 c1 ]* t! B" h' }1
; b+ h. A  s! @) p7.750000
8 r0 s$ N- _" G  P  O  ~-1.000000

2
0.000000
-2.750000

% Z1 d) S) s% ]7 N- B3
8 L' S: z5 L4 f7 b0.000000
3 H2 {  [; ?2 ^# f( J  o! D-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;
1 i8 g/ J/ j: s# d* a+ {! 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

Objective value:
( }% n0 l1 P. ?. C* L2 W. I0 M9 \
) m: W& C1 g: g: Z" T! A  B: ^14500.00

Variable
: O5 A; O2 D" b2 O) Q1 dValue
Reduced Cost

6 J" |- t. {0 o8 pX
100.0000
0.000000

Y
30.00000
+ l+ x0 b+ ?9 u0.000000

Row
Slack or Surplus
Dual Price

( a  ]1 I; m2 P, Z) z* B1
14500.00
1 R$ n3 ~( z1 T1.000000

% y) N% o& J, Z! U$ r: N, P2
! W1 T& N9 D: d" X5 k/ E$ x0.000000
8 P" m( B4 o- A# Q3 x& E" b25.00000

8 t! X+ G8 J; {1 o3
( s0 P/ ^. M) D90.00000
0.000000

4
0.000000

1 q$ h/ R* [& @' d75.00000

loooog12

先下下来看看