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 =
7 J' n* ]# I4 r6 C+ |-2

-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
2
" i8 p7 I6 j1 m$ n4
-1

2 q& V  R0 V& q* v+ h% v/ q8 J2
3
, I5 T3 c4 a# L, o* k, ~4 t( o-1
1

1
% f3 d7 @3 R% d' [5 L$ l0
7 n+ |# u( I$ b1
1

6 ]1 T0 I& `. H  _0 f/ z9 M-1
0
0
0

0
9 v0 w, T! D4 ^% W" J-1
/ q6 o: F& {; P3 U! _: H0 N0
0

0
3 ]' L# m5 c- G! V! [2 q7 u0
-1
0

0
0
+ O/ Z( Y8 h6 h$ t) z, ]0
& M3 G" F' p1 x8 D6 e+ t( W-1

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

b =
6

12

% [7 j. _  C6 Z5 ~4

0

0 X  }2 T! o' X. v% o0

7 ~" H% |6 Q" J! w) \0

8 {& P; p, y0 g( F+ ~0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
0.0000

$ R  E, s" {' U( k/ \! n( M# X" h2.6667

-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.

+ ~$ C1 b/ o9 k% l$ |4 [% k！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

$ s) k. z, b1 NOBJECTIVE FUNCTION VALUE

/ q) w& j, h" V+ s1)
145.0000

1 i/ g" ^0 v; B% ], kVARIABLE
- f8 M. o% M3 uVALUE
REDUCED COST

& E. `" p# b8 a( w+ x1 [$ c6 XX
10.000000
* v* c. X& |, l0 d8 q+ m0 \0.000000

Y
8 R: p) v; i' t. ~( c9 y2 q& u3.000000
+ p/ F6 N6 K0 t0.000000

ROW
5 e; O5 k% G5 Q) }SLACK OR SURPLUS
1 \; @- T' v$ r$ |' N  DDUAL PRICES

- }/ j4 V, G4 o% B3 Z. Y) x6 Z5 I2)
0.000000
2.500000

; y! K$ ]8 r! W3)
9.000000
0.000000

) S3 }! q, I1 T4 _1 U/ i' E4)
5 n, Q" l- ?; N  u' Y! c# n0.000000
: t6 ~1 s% l+ J, k4 P( t7.500000

& v. w8 A# v7 P# Z8 bNO. ITERATIONS=
2

$ P- I& @7 Q3 DRANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE
CURRENT
- n# y4 h, C( W3 [ALLOWABLE
) J& [) N- |/ |$ _: Y3 ZALLOWABLE

) X0 D$ E, E- ^: d4 U6 B9 j/ D0 nCOEF
( T& E0 ?+ p9 FINCREASE
DECREASE

X
. j' ~; c; ?" [- f/ a6 w10.000000
, u3 S+ n0 d1 l- d$ \INFINITY
( S9 m7 ]& Y: B0 W! v9 l2 _2.500000

6 j  Q4 s9 ~0 g( g5 _Y
15.000000
) r) p6 w3 t2 l* |+ f$ u/ ~9 ]5.000000
  M, P# f% V1 X2 q15.000000

RIGHTHAND SIDE RANGES

ROW
CURRENT
ALLOWABLE
ALLOWABLE

0 V' h1 \# `9 h! l: J* _( `( f/ \RHS
- \% _' y' T4 [9 nINCREASE
DECREASE

1 }( E3 n/ I5 U5 q8 a
2
10.000000
6.000000
" R! Z) r4 m; i! z! c10.000000

% W  ^3 P3 j: A3 d( y- k$ V3
12.000000
! r- }* [. u+ W2 \# _* c8 mINFINITY
- X, T) _' v" r. c9.000000

4
' J' W$ {3 r2 I) i16.000000
* T# v; r5 r# k( {6 Z4 h! N18.000000
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：

  v5 u* @- d5 @" h0 ]% j% n/ hGlobal optimal solution found at iteration:
2

, {/ D/ o& E, M3 ?- O' c5 XObjective value:
1 ^& ~# P' b6 O# g, O7.750000

- P! O) {9 Q5 P2 H5 ^% ]Variable
/ f, c/ s  Y( V# \$ P* c' bValue
Reduced Cost

X1
0.5000000
0.000000

5 B7 G$ v$ y7 T. k% n9 p" V- r  B* W( BX3
/ N9 ~. K% g& w  H! b0.2500000
0.000000

X2
0.000000
0.5000000

X4
/ G4 M; w' b' u5 e( ?# {8 `2 X+ I0.000000
( J5 R2 k( l& Y3 i2.750000

X5
0.000000
2.250000

Row
* n- b0 @6 ~$ SSlack or Surplus
- L* Z8 A3 q' l& sDual Price

1
7.750000
-1.000000

2
0.000000
' }1 l- v% z8 |# Z. |4 s-2.750000

# R+ H, Z! @- T- O, ?& o. |3
0.000000
% N& ?8 _/ K9 W) D-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

Objective value:
  X/ \1 A- g, ]% L8 N) A
1 l1 |# I+ @; c7 {14500.00

Variable
9 f* O  R& L& Q3 Z( O: F" f3 p7 C' vValue
Reduced Cost

& [, U4 p9 y  q6 {( A3 Z6 }6 KX
  z# ?$ V7 E0 ]5 w( b0 T2 F100.0000
0.000000

Y
30.00000
/ T  N0 z' \& x  a" r# k' b0.000000

Row
Slack or Surplus
2 p, Z  q) E2 P  n8 z. ODual Price

1
5 A# e, K5 G& Z  w( u" b. Y14500.00
. y  c0 [3 a! o8 c# e7 k/ c4 t8 @. i1.000000

2
0.000000
0 M* e( O2 G" d1 n  F3 \/ V25.00000

9 w& |" G5 R6 W3
: g3 @. }! j6 x/ K- m- i90.00000
0.000000

4
0.000000

75.00000

loooog12

先下下来看看