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 =
3 b8 P# l. }' S7 j$ o2 W# s-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
! T5 Q9 e3 c( w, M. J; r3 T0 Q4
) v! J. l8 `4 A7 h6 D. X. ^  G- u-1

( [7 G$ `3 P3 Y5 S  }5 o1 |2
3
-1
1

8 |: D! j$ F9 {0 N1
& h6 y% R, O4 w' W7 y/ g$ M0
1
' s7 \! p8 G7 w4 [. M9 `  D6 T1

  t4 v4 ?/ B6 t+ _( v0 u4 X* @-1
) C$ _/ Z/ E3 w0
1 ~* Y9 w: _1 q# m  n0
( a# Q" I1 B0 X+ v  l& n0

0 E. ^1 @8 G! ]& |0 |0
-1
) x, q: y0 Z! F) E& h0
$ F% U+ Z) }% h8 _5 q6 T0

0
. r  a' }9 @- o- L0
-1
0

0
6 l6 |( a0 f7 U, ]5 P) i& n( t" H0
+ D5 E; k8 D+ Z2 V  V0 M0
4 c! i& y. t5 z) t-1

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

b =
, q& H) W! A8 p1 B6

12

( |9 u3 E$ z4 U9 X4

) r2 k& z% e% I9 z2 f2 W# ]! _0

$ a6 A+ f4 q6 |. m0

0

3 a+ Y  a& o2 K# e- }6 K0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
8 C6 N* C  d1 a4 `" K6 B& D0.0000

2.6667

-0.0000

4 v0 W+ ^1 N( G) _! F- T4 O8 a& a4.0000

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
" B) S5 I3 I2 W, Z5 W5 c  S！注释符号; 系统默认为自变量>0, 若不要求用free命令.

0 A, C3 }- b9 e( Y+ L$ I" u; `！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

OBJECTIVE FUNCTION VALUE

1)
145.0000

VARIABLE
VALUE
  \: }) T& z6 B% tREDUCED COST

X
: V  h8 ~7 I/ m9 V( |: @) S10.000000
  \0 @3 D" {! g; K0.000000

3 N# B& M" ^9 NY
3.000000
0.000000

ROW
7 ~/ Q( F; `  z% B2 `2 X1 v9 XSLACK OR SURPLUS
7 r( B% O+ O3 |DUAL PRICES

2)
5 g+ p$ V9 A# \6 i9 K/ }7 f0.000000
2.500000

3)
' n+ R" G! ?( E+ m' j9.000000
, ?1 _7 @  _! z: T6 a0.000000

4)
( }& p3 d" |$ c% p, R4 J0.000000
! K- j( I+ g# B2 M( {, s& Z6 O$ ]5 v& o7.500000

NO. ITERATIONS=
2

. o6 Z# q7 H$ a3 g! e  qRANGES IN WHICH THE BASIS IS UNCHANGED:

3 V  j3 E3 _% LOBJ COEFFICIENT RANGES

* j/ g! h5 {* H  T3 k2 a% ^$ eVARIABLE
6 u8 O; I& V1 b! x  _  RCURRENT
! e0 q) M/ {: x1 {ALLOWABLE
; ]  i7 ?* R! W+ N3 _ALLOWABLE

COEF
INCREASE
- {, c5 ]' `, [$ H, F  C5 CDECREASE

X
' N/ B8 |1 C. b5 |10.000000
INFINITY
2.500000

$ h  Q8 e, C/ H' L/ ZY
" l1 I! C; {" Z' _15.000000
5.000000
* p+ U. _$ s$ o. g7 V6 O/ ]15.000000

' A& k$ Q+ W. n2 r/ ZRIGHTHAND SIDE RANGES

: @$ S# ^$ R' Y" ~4 ^7 e, ?& iROW
! k$ ~1 x4 [, R# `/ gCURRENT
ALLOWABLE
ALLOWABLE

$ l* d  u2 T# K3 mRHS
* S6 ?1 M$ a! ^INCREASE
+ q1 E  Y( k. R$ x$ Z, bDECREASE

; w+ S. I, P- I% q, x% f2
: c: I2 B6 ^, }' F- i, e6 Z) Q( _10.000000
6.000000
& A. |& I0 P  U! w1 y& X# c: W( r10.000000

+ p' o3 P; r1 \8 j3
4 e: o% Q% h  Q( t' U  o: v: P12.000000
INFINITY
5 {; Y3 s5 U+ R( B& [9.000000

4
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
report 窗口得到以下结果：

& C! `3 K# j+ {, M7 BGlobal optimal solution found at iteration:
$ }/ q* a) K0 S& a9 x2

* H' ?. E% k7 y5 S$ d9 M, oObjective value:
7.750000

$ l* h. L8 F0 ]+ wVariable
Value
/ c. Y6 K  {! ~8 aReduced Cost

X1
0.5000000
0.000000

X3
. a; F4 ~" {. U9 F# f0.2500000
0.000000

/ _( D0 \5 I; l. i8 |- `5 h5 Q, A
* X, ]7 v6 d9 O* V# a) T9 y' qX2
0.000000
* x* ~) q. i0 B3 k* R0.5000000

6 T8 T! P) g$ |" t& q' x; ~2 J- S$ ?X4
0.000000
) m  S' }. D+ @* ^( l& p9 L2.750000

X5
" ]6 m$ |- v+ e' P0.000000
! ^9 J  O# ~) U/ G" M2.250000

Row
8 @$ @4 i* e3 A/ q$ QSlack or Surplus
+ t* d. G0 S) d8 VDual Price

. _5 `: I3 s, B# t  t1
7.750000
-1.000000

2
0.000000
. Y4 u$ f  F# S5 i-2.750000

. \( O+ d) N5 _6 }" H4 }9 m3
  X: d3 j' C7 j  t$ ^+ r* E  n. U0.000000
-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;
! 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

5 b, d/ k0 |# b; ZObjective value:
$ {: ]% ^+ o; n: Q
14500.00

* `2 }3 [* t" ~; E. a: oVariable
Value
Reduced Cost

X
7 V5 i" W& X- _+ J4 M0 M100.0000
0.000000

; `* H6 V2 i3 j+ `1 R( `Y
30.00000
0.000000

, U3 O8 n% H' a) I& [Row
Slack or Surplus
Dual Price

5 e! t: b: q( M/ k2 |0 e7 q! v1
8 J; k- i& L% v; s( t# P9 }14500.00
0 t1 _4 D, |. l4 t1.000000

2
# k4 T1 s& T7 L- b, K) o2 l0.000000
25.00000

& \! Y$ b$ D$ m, X9 n1 F- Y3
90.00000
$ n4 J! y3 p+ k9 c0.000000

4
8 V- C4 J: ]  x  X0.000000

75.00000

loooog12

先下下来看看