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 =
+ T, i8 E7 n: A+ ~! G-2

-1

3 W$ w6 L( A3 t- g7 A9 Y3

3 U- b: m- l- R1 v! d1 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 =
' _. M! c& _  G6 S7 p6 ]& j( [% _- o1
2
7 |3 M+ p2 p5 x! F4 _7 t2 Y% G4
. o2 O0 L- b  G$ |( N-1

! k- b5 J1 x; P& q* N# _3 @2
3
-1
1

1
  A$ ]9 s+ X. s0
1
* q- {. ~" o5 Y1 K3 |  C  R' G/ ^9 z1

6 G* ^/ n' P+ j2 R# ^+ f' [-1
0
0
0 g6 ^: g( B, L: n2 b3 t* |0

0
-1
1 S, Y' Q. E0 s+ Z0
/ Z$ |0 I# d- k, }; e0

  ]$ `, _; ^/ r! k0
# g: A4 K( k( A+ W8 J8 n( [' E. ?0
-1
: D7 `* B2 Z6 X6 Z0

: t: n; F6 C! d0
' m& p8 M, h; B/ u, p! G  e8 R0
0
-1

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

b =
6

- v) Y4 I) o4 _# r! Z12

4

; l; S" @( Y! V( ?& a- t  |0

0 P6 X* `4 }/ I/ o( w, r0 p! ]! E0

0

, _& @. w- X$ s. }0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
: I# @) l( M% J" g  I0.0000

4 X. E4 Q# w% Z9 J+ Y2.6667

-0.0000

& O5 _$ ~+ v2 ?! c( c4.0000

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.

/ K% A* k$ ~% m$ @！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
* K: K( b3 B9 e+ b6 C2

) d5 v. C4 e, Z* t, p9 I' COBJECTIVE FUNCTION VALUE

1 h4 e5 C# L- Z4 N4 q1 E5 e- o1)
* E% j: O4 K* M$ n145.0000

/ p2 ]& k) z' c+ I, [# C' r2 KVARIABLE
; W7 v  \& n* ]8 z; aVALUE
8 @6 f( z# J0 k/ h- `REDUCED COST

  d$ o5 S8 N0 ^1 i7 B7 ]X
10.000000
0 I& o% g4 w0 |5 ^$ n0 h, ?0.000000

Y
3.000000
; ~# j/ `3 N' M# q5 G, I" l0.000000

ROW
SLACK OR SURPLUS
0 }5 d+ |. p7 D+ |* lDUAL PRICES

* V3 p3 i/ s/ i* {/ g) Z2)
0.000000
2.500000

3)
9.000000
0.000000

4)
! A8 @- e4 o2 p0 d1 f0.000000
7.500000

NO. ITERATIONS=
3 X, R: V% D1 F4 w, q* v% Y3 {2

" x* b& I& w+ u/ b: X- CRANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

# R/ a* O) B' B- kVARIABLE
0 H8 V: E( A& _* r8 S% t. j: e/ GCURRENT
ALLOWABLE
ALLOWABLE

COEF
4 u! Y6 r( A. o1 M+ zINCREASE
DECREASE

X
10.000000
. G2 A( G' y" y  l) P, aINFINITY
0 m( o: l( j( U+ o) G8 C2.500000

Y
15.000000
. w4 B* U7 \4 H5.000000
, H) x, n  ]) Q2 j7 @15.000000

3 a4 p1 F. P- H  `RIGHTHAND SIDE RANGES

ROW
CURRENT
2 ?/ i% J; ^" Z9 N7 v8 C! L6 V" k  NALLOWABLE
ALLOWABLE

- [! W. R. F3 h3 d- pRHS
8 x) d+ X0 K/ S  aINCREASE
DECREASE

4 C" Q- {6 a  U, C) `  h. T3 j
" t# J+ {7 o- d) H" {& {2
10.000000
6.000000
10.000000

3
12.000000
INFINITY
9.000000

# K4 n) B0 h- w% E4
16.000000
18.000000
* G( j% q3 v  z9 q6 h6.000000

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
3 M$ W+ X/ G% G1 U8 T3 \5 \report 窗口得到以下结果：

. ~, c8 Z% v! pGlobal optimal solution found at iteration:
2

% O- @! N8 f& q0 x% V8 S, xObjective value:
% }8 V+ k" {. J/ o" N1 f7.750000

Variable
7 {# R% N" n3 q* Z7 a( w7 J% dValue
Reduced Cost

X1
' ^- {- o' K+ U- n5 L0.5000000
0.000000

X3
6 I& k" k3 ?. o* o7 n" X' l- t3 ~0.2500000
0.000000

+ D/ e& C  I; b/ L1 OX2
0.000000
0.5000000

X4
( O8 t, f8 B, ?0.000000
4 ]- \  n2 c; W8 Z# g$ R2.750000

X5
0.000000
2.250000

Row
Slack or Surplus
3 Z# _* J4 C( W$ W. rDual Price

! V  Y! ]5 @7 j; M1 x1
7.750000
- p! w6 k- ]/ r# x5 I* i" m6 \- P5 F-1.000000

2
/ b9 J  t: [9 H* I0 C! D: V5 g0.000000
-2.750000

! g# t8 S  x* F6 |3
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;
& \% ?7 L  z1 H2 {3 F7 c' K! 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:
7 |; c8 @, Q. K  P" g5 {2

* w* l* |* F( C  t+ k5 F; g* LObjective value:
+ n* w! l0 `% o' Y
14500.00

4 d. L/ ]. w+ m4 ]4 D7 ~. DVariable
Value
Reduced Cost

# @% P& H+ J- _X
) F9 {; Q, s) u7 x$ N* P100.0000
0.000000

; S1 a7 `/ B% a7 {# @Y
+ I: h6 }6 U, i& Z30.00000
% {; F( i2 a$ E5 [, k9 m0.000000

1 b/ u) l2 E$ C, ?Row
1 ]* ~9 B% @5 h, _7 tSlack or Surplus
  m* l$ H5 l* o5 ~6 J3 QDual Price

/ [% Q: a7 N: R) S! I- J1
. ~3 n6 r& D4 _14500.00
2 }8 X9 F+ x+ b0 @1.000000

2
4 q) I- n0 Y# H" g2 U1 ~. h4 k0.000000
25.00000

3 T; _# N3 l. q/ r, p: f3
* @3 M# L' s2 I: U6 d7 u90.00000
% m4 J' D7 K/ O6 A: i5 R0.000000

4
0.000000

75.00000

loooog12

先下下来看看