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 =
, h; `6 ~4 R! x) h3 U-2

; {% D1 X" S4 r8 |-1

/ K7 X& H4 w7 o. p" D3

-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
2
- J* V+ w. ]  b" K4
-1

2
3
& x# N2 |1 D, `+ m-1
1

1
0
. z3 a# |* m+ J8 C/ q9 q/ |1
1

5 Y. q; d1 r4 |, [& J4 Y6 d-1
: Y/ O$ o$ U5 ~1 N7 U0
0
0

  ?  l* L; r( R3 a0
! e! c( P, U+ S( M-1
7 \: F, {- G4 Y+ O0
0

& g8 l$ J0 Z* e% _9 V7 I8 u* p0
0
-1
, _. C/ J  q+ q+ o1 N0 e& O0

! _% q( f; u. Q% L0
0
0
& X) f# H6 {9 k1 n9 i-1

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

b =
% Q6 p; L) Q8 v* J& b1 g; ^: H6

' D/ o, A3 V$ z0 c4 F( x& V12

( L) \/ k" p2 d6 ^" r. q( g% B# O4

. o# E" Z& l) Q+ z0

0 N' O& Y6 l" n7 ], [( K0

9 C) @( n' [) l9 Z: l. i0

0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
( Y& Y8 ^" c9 s  Q0.0000

* M) j9 ^' E& e/ z$ K% U, e2 J2.6667

7 G+ X6 C2 [' b-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
( Q1 a" X0 I  {0 e* z# ?！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
; _& B. O' i' d2 c2

OBJECTIVE FUNCTION VALUE

1)
! s/ j6 A) ?8 e145.0000

: h! H+ V4 o# ]VARIABLE
VALUE
REDUCED COST

: ~# Q+ y/ {/ ]% ]X
! D1 R, G- {0 V5 I) f5 K2 L; \10.000000
0.000000

Y
# O/ S% l: K1 T" D  ]5 o3.000000
+ r. r- E, ?4 |0.000000

$ |1 l; V( W$ u, `- Q( QROW
SLACK OR SURPLUS
) e* x# S7 c- x+ n" EDUAL PRICES

2)
5 I: M# L& J$ o: U( I2 c1 D0.000000
! q6 D1 H) t% ~& j; a2 ~) i; N, }2.500000

3)
9 E/ Y. v7 n4 B9 F9.000000
2 P: R. X7 B9 @0.000000

: ]5 R+ w, h! z( j, g4 j" _4)
* C! }8 T! H& A& q3 u+ k0.000000
  l1 f7 ~3 B7 j5 j% n7.500000

4 @0 ?2 z- {8 l( O3 NNO. ITERATIONS=
4 T/ q/ |) j! l% e1 ]- Z, g# a$ r2

RANGES IN WHICH THE BASIS IS UNCHANGED:

# p8 w3 K+ L% R0 X$ O9 WOBJ COEFFICIENT RANGES

VARIABLE
CURRENT
ALLOWABLE
5 z; @! e+ G/ w% q  j3 z" b; D0 hALLOWABLE

$ K" p# W; L' V! s' W  WCOEF
& M  T+ _, R" K5 q% U' ~( L0 JINCREASE
% ]8 h" J' [! T7 ?$ V; ]4 lDECREASE

X
10.000000
INFINITY
2.500000

- U5 w3 m+ w3 f; [% y" YY
15.000000
) f' O1 ]( I# ~9 i7 n' i5.000000
: t: c+ z* z+ ^- ^: R' ~5 H- K15.000000

RIGHTHAND SIDE RANGES

, ^1 F% W  N  \( }' oROW
CURRENT
  z+ ]! G& `* vALLOWABLE
ALLOWABLE

RHS
' a2 v+ r* R1 J% H  e$ b. QINCREASE
5 t* B' e+ A( D. @; [# d" P% BDECREASE

/ w/ I; X& K( e$ k7 u) q
% K4 R$ ^1 @7 M7 w! f2
10.000000
6.000000
: l6 t% j; {5 n' h  H4 j. O10.000000

6 @- p6 n; i2 U" O# A5 ^% @% d8 E3
  K' O  r, V/ G$ B; m12.000000
1 }, n" S8 J/ \+ O$ f; a& FINFINITY
9.000000

! i' W+ n" g1 S6 g4
7 E) |& J( Q3 O1 e7 L16.000000
4 t. q/ Z+ [8 Y/ R% [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
7 F( N: p3 Z/ Q. r2 b4 Nreport 窗口得到以下结果：

) r  F# Z! c2 M1 H+ t' k, pGlobal optimal solution found at iteration:
2

Objective value:
7.750000

" J5 w* l; p& v' J1 \Variable
Value
Reduced Cost

X1
" K& f: Q3 n1 w- k" L' o) f) X0.5000000
1 @1 ^6 |; T4 ]' s0.000000

X3
, X" D9 Y* o7 O- X! Y0.2500000
* B3 m' f: P/ E$ ]0 R9 a, b  o0.000000

! U% C" q; d" a( ~$ [6 w: q
X2
, ^6 A" ~$ f6 C2 {- N3 p( K0.000000
0.5000000

X4
# c) y# _. w1 }1 \0.000000
, P* {/ M1 z& W5 [6 ]2.750000

X5
( E# W: P! ^  P0.000000
2.250000

Row
Slack or Surplus
Dual Price

3 _" P6 Z# Z8 E' B$ k1
7.750000
. U% q! p, h/ M% U9 G& P-1.000000

8 Q8 Z% J2 [4 I# |/ W" l; E7 F6 o4 E2
; `7 P+ ~  F- }+ h. e! \: K0.000000
" n4 d3 ]1 O9 s. @$ }, J) S-2.750000

3
0.000000
% m, T1 H# h2 A1 z8 [-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:
2

/ X  c8 L; t1 v% A9 l+ Y) lObjective value:
7 X7 x' M  v2 t# n5 K8 I
14500.00

* v* L9 x8 j6 yVariable
3 @8 K: T' Y. B/ S0 z8 GValue
3 x" W9 _) [' c5 GReduced Cost

1 x; q1 E0 B9 E2 R0 I1 _X
100.0000
9 w* j: z/ v7 f4 }0.000000

Y
/ o" U9 j& C; F# l# e0 L% k' A30.00000
  \9 E: _" o1 w+ Y3 B( b0 P9 l0.000000

& C( h) R6 Z0 t6 J: e6 NRow
; c/ f6 l& G+ f  Z3 O3 mSlack or Surplus
Dual Price

1
14500.00
8 k8 O+ \0 ^/ E0 d; x1 w$ l6 J1.000000

2
0.000000
25.00000

- p  q) R  R5 u" ^5 S3
90.00000
0.000000

4
4 m$ p# R# {/ D( {0.000000

75.00000

loooog12

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