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 =
. t6 t5 I0 V/ a-2

-1

& @( w+ d0 t( w3

-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 =
9 ?8 z; ^8 G' V/ ?# h. [1
2
4
-1

) ^9 S$ l9 E" U" V2
3
. S7 I# z( r+ F+ ~2 N5 y-1
1

1
, e' b0 j# p* O0 }, T0
0 \- k0 o3 t" k' m& z8 f1 e1
6 E, W$ n! E. O$ Q( C- F4 w6 I9 m1 o1

-1
+ A4 P- G( E/ J( O! r0
" s  n: }! i% Y* W6 O- s0
$ L; a) t3 o5 g4 F: F# m, j4 k: ~# T; N: U0

0
-1
0
0

0
0
$ a  f7 z- U* Z/ x5 R  [) p$ t6 m-1
0

+ q; E2 [) k" P/ }; C0 {0
0
/ R! i( r. G# M; s. @9 D0
) w) I7 U; |/ }' O1 y& u+ _-1

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

b =
# Q* g( D# S2 \2 M6

12

4

% b( o+ J. K: m! g: K0

0

; s( k$ T- u0 O$ ?8 w5 j0

0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
* G4 T. Z: r8 q9 u1 [0.0000

& I8 {  |0 Q/ J2.6667

-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
, l. i9 b+ g0 b9 U  E0 s2

OBJECTIVE FUNCTION VALUE

+ H6 m0 t( X! N1)
1 {, k2 ?8 d4 X0 B: j+ x145.0000

VARIABLE
VALUE
REDUCED COST

- c8 E# }6 K) m2 l6 UX
10.000000
0.000000

Y
3.000000
0.000000

ROW
6 x5 O& M  g5 z+ e! D) N) bSLACK OR SURPLUS
DUAL PRICES

2)
0.000000
' L2 h4 p! r5 l7 T8 T2.500000

3)
9.000000
0.000000

4)
$ H6 p9 m! ?7 g0 o6 _9 I/ d1 F( N0.000000
. ?; @* n7 }. T% |/ n! O7.500000

NO. ITERATIONS=
. Y$ D6 ~; j3 [, L2

8 b6 b$ j5 {4 F" ERANGES IN WHICH THE BASIS IS UNCHANGED:

5 x1 S: ~' z2 l1 c6 M- G! gOBJ COEFFICIENT RANGES

9 w+ \' }+ D9 }5 OVARIABLE
2 X. |3 |$ X$ A* n5 J9 H/ |CURRENT
; U+ O* Z  ~2 p8 t) lALLOWABLE
$ a  g+ N' E/ _, N1 X# W3 BALLOWABLE

# ~0 z4 _6 _( p$ G- {' |COEF
INCREASE
DECREASE

X
10.000000
, n. m$ k# L. V! l. E  @; \INFINITY
2.500000

Y
4 `: M- B/ w  r* P: i+ ~15.000000
5.000000
15.000000

% T$ b" G: Y7 |, X2 ~0 A* lRIGHTHAND SIDE RANGES

ROW
CURRENT
: J: M& W0 }  j9 d, v# p- CALLOWABLE
- e. W  c6 b2 S- \+ r/ ?) aALLOWABLE

! |+ r1 r5 o7 K0 m4 l6 c: M3 h% KRHS
/ i! a8 @0 B* B; D4 _4 f& ^9 a- IINCREASE
1 U- A+ S9 x! @9 MDECREASE

' d+ Y4 ?! h! J( e
2
- h" @' v+ ^2 h; e3 T- [10.000000
6.000000
' d% {8 `, @9 }+ x2 j* n10.000000

3
9 c. |/ j% e* `* M$ K. a6 H12.000000
INFINITY
  \% U9 g: P7 g  n4 T% x2 x9.000000

" Q3 J; j6 n$ z8 E4
, m* u7 i+ {+ Q' g9 n16.000000
" E: N$ O# L  k- }2 V18.000000
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：

2 H  u+ W' K. A+ ?Global optimal solution found at iteration:
2

Objective value:
! r, O3 d3 X* `1 j- L' z7.750000

Variable
Value
: X5 z( Z: \4 d* f9 b4 n; c& Z' v* OReduced Cost

4 h; B' l4 i5 X- r: @0 B  FX1
0.5000000
: t, @# n- O3 ?* D- Y% d0.000000

) U/ Z% [) T0 E  |- L4 SX3
0.2500000
& m6 E/ a3 e% D% J6 z  c( J0.000000

X2
/ Q4 A& Y9 x# n! s5 g( A0.000000
0.5000000

X4
: R" I  y( s* d  a6 k. Y0.000000
2.750000

3 g  F+ Q* K3 n+ D' O2 iX5
/ @/ D$ {5 w% K% _* j! E) E( _0.000000
2.250000

Row
Slack or Surplus
Dual Price

+ v; E$ K% N2 w7 n, w+ l& ~) I$ j1
7.750000
- x& i$ w; Z7 O4 P3 a' V-1.000000

/ X! A3 i& R, z" W4 O. u2
$ }7 _* Z- Z9 N0 }0.000000
-2.750000

# ]# S2 W8 p' o8 a) t! w  T  p3 U3
0.000000
7 O7 A: M- g# M+ C+ {-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:
5 p' D5 p* q( ~9 R; f4 U2

$ C7 ~  r( G+ \5 Q0 w4 L5 e, _. DObjective value:
! z, X7 I* ?- Z3 U  |% q! `
14500.00

2 E7 d7 \! o: r6 o9 ?/ s! b+ fVariable
Value
Reduced Cost

; z# {8 ^1 ?  M, f% t  v! vX
100.0000
0.000000

9 b& P) l! I+ G+ n  l9 s+ O7 ?4 cY
30.00000
0.000000

Row
$ @, y  I5 J, R2 b. S( i7 Z$ ~Slack or Surplus
" S  N" v1 U: F% sDual Price

) w0 C2 e$ |. E% q. D1
* g( C% g7 M, _5 W  R: m7 G, r14500.00
1.000000

9 {. {# K, a9 L- J: R3 r' V8 l% z2
+ q2 |9 J: y( n8 n+ S. M8 c1 V0.000000
25.00000

3
; S; k, i  O4 }$ f( ^90.00000
0.000000

4
0.000000
8 J8 m) h. G) @6 F! F
/ \7 ~# d0 u9 T$ S7 F9 k( _2 q1 [75.00000

loooog12

先下下来看看