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 =
-2

2 g# J4 g) K5 G" p1 w% u* m" r-1

3

' o$ T, {% ^6 B% u2 U9 T& L-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
4
-1

2
& W2 ^. o" P2 t) c' `* J8 }3
-1
- c. B$ W6 o: V0 d9 o1

7 R1 d( H4 q6 ?9 e6 O; D9 W# k3 z1
0
; C" ]" N! Y/ ^' x) \; z9 }$ h1
1

-1
0
( p" M4 ?/ i& Z/ |/ h0
3 j- m( @, \) |; @$ ]* S9 c0

0
8 c4 |9 J0 L9 w-1
) S% i  n4 U; o9 b* e+ h0
4 @/ o  Y* a  j, t; [0

7 ~7 S* I0 r& q0 o! S! ]0
* @+ u- N& T5 o5 _6 ^- u9 \- J0
6 C" D9 A( ~; J2 j/ Z-1
0

0
. j% J1 ~6 J6 R5 i0 G. i0
0
-1

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

b =
6

8 J9 Y0 R1 d+ Z4 L6 p12

3 |- ?' `& k' f7 n4

2 Z8 @: x! G% ^% S( \# ~: Z& c0

7 {' I% @' }! p! g0

% W; o8 {# u, U) _0

$ ^2 ~- l6 c  w0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
; Y  ]7 G& j+ l: q) G+ w0.0000

& a0 c& U0 S. h; y, V  f2.6667

! F7 a/ ]! I- f( u! S+ M5 C-0.0000

1 R+ y, E$ |( |4 C0 J4.0000

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.

& p4 [3 Y: M7 R% d$ \; \( e！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

7 a( q+ u7 T* F  B" T. NOBJECTIVE FUNCTION VALUE

1)
145.0000

VARIABLE
! p& B/ D  e: `& X0 R7 [VALUE
+ Z8 J  E( ]! |6 m" V$ iREDUCED COST

X
6 W5 F/ R  x$ |+ q2 q4 H10.000000
0.000000

Y
8 L3 U! D+ V6 ]- k' {3.000000
0.000000

, k6 J1 a  J, IROW
SLACK OR SURPLUS
DUAL PRICES

& e6 v8 V- u1 R7 P" b- [0 Z! f2)
0.000000
2.500000

3)
) [! L( J1 l6 |+ d: v, ~$ N7 n9.000000
0.000000

4)
- I* f4 {- g5 F+ ]6 G/ \0.000000
+ @# ?2 ~3 i1 \+ Q7.500000

7 l6 M% l" B) c! C/ M/ G: @3 nNO. ITERATIONS=
2

RANGES IN WHICH THE BASIS IS UNCHANGED:

& x* ~) Z+ U/ a: K; BOBJ COEFFICIENT RANGES

2 h( i. C1 J* G8 [VARIABLE
CURRENT
ALLOWABLE
ALLOWABLE

COEF
INCREASE
DECREASE

' y: G( B8 D9 G$ zX
- {/ j* f" F) A. M4 L" h10.000000
INFINITY
2.500000

Y
% E" [3 @% M! M1 @8 y5 c# W% V" Q15.000000
4 d) ~. p7 U6 p5.000000
% S! }2 S" \& m8 |$ B- b7 }15.000000

RIGHTHAND SIDE RANGES

ROW
CURRENT
. m1 l% c7 j- L6 f/ R/ [ALLOWABLE
5 v% |- c9 _/ @ALLOWABLE

/ E8 d5 B7 o0 DRHS
INCREASE
DECREASE

9 E( |$ w# o  @, _1 I& I) b
% v1 j) U2 G9 }2
/ B# B% Y1 y: q10.000000
6.000000
! X0 e9 m6 D. T' P( F10.000000

3
, u2 g& J2 T9 ?5 Z: U12.000000
5 o) K) q# a" G4 l7 mINFINITY
  W, G- C3 i8 k0 m' L; y4 Q4 L9.000000

5 g% I5 s# g' ]' Q4
16.000000
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
" a2 U1 {' i) s! \! ]report 窗口得到以下结果：

Global optimal solution found at iteration:
2

/ s% \! ?! t" J0 I" d' lObjective value:
. r; k# k( s! Q3 l9 U* h9 ^7.750000

Variable
8 [$ M  p6 N1 f! S8 a% Y* V* w0 [  JValue
7 q- F7 S8 C9 m" VReduced Cost

X1
: m3 h( u4 c; `! o. |8 X0.5000000
6 d$ `2 r+ o9 z- ^0.000000

X3
$ e% G- Q7 Z: `  w0.2500000
2 H2 F/ i6 w! J/ c+ w2 X- U0.000000

3 h+ L7 I, W; M% g$ ]& V, @
X2
* O, B/ k2 q3 n8 B/ l7 Y0.000000
0.5000000

- m8 f* w5 p' ^5 OX4
8 U/ z; q+ \8 [& a; u$ O! {0.000000
2.750000

X5
0.000000
2.250000

0 |( r/ V1 i9 v$ I$ o  i8 ?Row
Slack or Surplus
Dual Price

1
9 B$ F' @2 t! R, d7.750000
-1.000000

( w0 }" K& `$ z+ \8 V2
+ ~" M, N, g4 ~7 p" Z) k0.000000
( W5 Y6 v, t9 Q! g. O7 e6 V-2.750000

) A) z3 h8 J; K( ~+ W3
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;
5 R" S1 E; k) w/ z: 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:
2

' W3 X2 G* G+ V. _! R4 QObjective value:
0 E1 J; t. y( V+ t, N0 e
14500.00

. n& W7 z$ w9 y0 nVariable
Value
Reduced Cost

X
! q& ^6 d" `( _9 R5 j100.0000
0.000000

Y
% V/ x9 N% b3 C. \30.00000
0.000000

+ K/ h1 E: k' d( [7 `Row
Slack or Surplus
- {4 W  J( j  o0 P5 Q6 ?Dual Price

. ~' ~/ A! p  t# I1
, i# f# x: `; W( L  ~14500.00
1.000000

& B! k1 ?6 U  {2
0.000000
25.00000

' a) ?. J- o# `8 h8 N7 X& {' r5 T3
5 I9 p5 z2 f6 E7 M4 y! S+ e: N90.00000
1 r1 S  l2 X: G0.000000

4
3 M# ^6 ~: e, f% g6 @1 `1 `& j2 o0.000000

: J3 G1 w( b; j# h, P" |  d75.00000

loooog12

先下下来看看