运筹学第三版（刁在钧）光盘中的内容

mnpfc

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

& @+ {' ?8 V- l+ O3 q7 _8 y5 p/ d& E  r-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
) l1 Q* N$ j9 A1 |; U2
9 ^9 f7 x  x: _$ j7 z" ?- ~4
* d# a) P3 P: k) R1 W3 b  H-1

2
1 v  E& v8 A# r* Y( K3 C* f0 O$ X8 R3
7 x8 Y2 I7 Q, ?5 X-1
0 P8 x$ I1 e4 r9 M9 N" ~1

1
0
1
' E6 j" V9 z  d1

  r3 Z) m" u5 }" y$ _2 i# P* `-1
0 u2 b4 {  I+ T0
0
0

4 B4 r4 f; {7 ]+ u2 a0
-1
3 b" K) F& @8 M; a- X' u% `0
! |5 x+ u3 B6 e0

$ _6 K7 k, d7 c+ i0
0
' b8 V9 @/ ~1 I5 y& ~-1
" m8 U) ]* s% |: c0

6 v2 m0 @6 V& I2 q% d% M0
5 \% j, X$ V+ m' b& J! U0
$ c+ q3 ~" L' I" |5 w4 }0
: J( m3 t2 _/ m-1

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

b =
6

4 M7 V3 P5 k# F12

: s& |" o' S  V6 g4

0

0

0

$ G# U* Q  W( i0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
0.0000

2.6667

-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
！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
- M) q9 s' G: a; L! F2

OBJECTIVE FUNCTION VALUE

1)
: G# b# O2 q( f0 c: o5 s/ Q145.0000

VARIABLE
VALUE
REDUCED COST

9 P2 a5 z  R- O2 o9 s9 Q. IX
10.000000
0.000000

3 i+ Y9 W- p1 gY
# [7 @, C, ?0 y: ]+ D  V0 o' T3.000000
0.000000

ROW
SLACK OR SURPLUS
DUAL PRICES

2)
0.000000
2.500000

  t9 }6 ^' ~/ V  L% Z3)
9.000000
# O3 [) d8 J5 T8 a+ k% z7 {0.000000

$ w$ M7 `6 G5 f) O, h7 L5 `% p  j4)
1 ?& t" o; W8 D& u0.000000
7.500000

NO. ITERATIONS=
; J& F0 ]2 W( ?3 K+ [# C% l2

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

: i' d. g3 w' B3 ]VARIABLE
- G1 f& E( _3 r. z% ICURRENT
ALLOWABLE
, R/ d( L3 v$ c+ w* q/ IALLOWABLE

COEF
INCREASE
6 O& s8 |! q5 r% SDECREASE

X
3 w0 c7 Z8 ^' @3 V7 s10.000000
INFINITY
4 o/ X1 W- X4 z2 N" w- D( l! w2 o2.500000

  q: O  M. p3 \" Z+ L$ ?! Y" O7 t4 RY
15.000000
' c, D! A2 U/ t- y' b# J5.000000
15.000000

RIGHTHAND SIDE RANGES

( [( a' R' I; W9 p5 l4 jROW
CURRENT
& I' O2 I8 f' G2 }- t8 _5 O4 \ALLOWABLE
1 `% h$ v3 o) s! L% K0 IALLOWABLE

RHS
INCREASE
* R( u/ j! U0 G; G! ZDECREASE

+ ~" F9 a+ a) h  U4 c2 B0 ]
% B" Q: T0 S5 S; Q/ |; K2 j$ |3 m2
0 q# S& o8 }' _* x10.000000
% o! z% x" R. _# Z6.000000
# d7 }5 E8 t6 j9 A10.000000

# o' h+ d" a* ^9 ]& a7 t9 j: b3
: U6 H4 z4 h' z( [+ A! w8 o12.000000
INFINITY
( P. ^# N% K9 }4 E6 ~- Q9.000000

4
3 T1 U. `! _2 R16.000000
8 C7 n$ Z2 u. R: q% p6 w18.000000
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：

/ _* w* ?2 t" }7 p, eGlobal optimal solution found at iteration:
2

& n* t, k0 J: T- eObjective value:
7.750000

Variable
Value
* \" D, g2 N8 a: U! {Reduced Cost

X1
& v' F/ c2 m. p! Z5 T0.5000000
# t) Q3 _( L' {- f: s) T0.000000

! }0 j" x. j* [! S& D( XX3
+ s/ o  w# ]2 m8 C# n6 j6 ~0.2500000
0.000000

* L! b( u5 ]" h1 S/ |8 bX2
$ ?7 U* Y1 S6 r9 E2 z& H- r0.000000
0.5000000

1 Z9 @: C9 G- o: c' s: J& GX4
, m( z8 X' q# `$ p0.000000
" r& J6 a- ~3 s3 S( W/ n2.750000

X5
0.000000
7 g$ O8 c# H- j! d9 w. \2.250000

5 ?3 e/ F! @8 `3 N  R' JRow
Slack or Surplus
Dual Price

1
7.750000
-1.000000

) {( x: }0 S( Q* m" r$ G2
9 c% s0 n8 f( I: Z& Z8 R0.000000
-2.750000

3
/ E1 l0 h" e) y: O& h% H9 @0.000000
- `3 u3 C. `, x1 z; K& n-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;
, U9 F4 b2 v8 {3 r# ~- V  Y/ B# h! 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 p0 t! ]( M5 D& m- O5 ?2

8 n  @) q+ ~  J) e. d5 k2 U6 MObjective value:
, z. R2 }( q1 O. L8 l$ D
14500.00

Variable
; M0 t# x3 M! T5 R4 o4 u( ^Value
Reduced Cost

X
100.0000
6 @# B5 i  r) |6 r* K% c! J0.000000

/ h6 m* j) _5 n( q0 ?9 W% jY
" g3 O- a" k1 R5 q5 I9 _& J  ?$ D30.00000
9 I& D" E% X$ R; o0.000000

Row
" Z1 m8 ^% z" @2 D7 LSlack or Surplus
Dual Price

1
14500.00
1.000000

2
0 u3 J4 s$ ^7 H7 A- P0.000000
25.00000

3
" i( K. d0 \( O3 i$ x  S90.00000
: ?* @. X4 R$ {3 E+ Y9 b6 ?0.000000

4
4 \5 ]5 \  C# s) `0.000000
( y- p4 W, ?7 W5 P
; O' a2 u+ y% F( d& H75.00000

loooog12

先下下来看看