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

-1

0 U& @  L8 y  v2 P. ?6 A/ K3

& ~" Y2 M$ |4 ?6 e-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
' j  Y# P* v) ~$ ?) E-1

2
3
) U+ g! Z$ L) F9 L-1
1

1
$ d3 v8 g* n' H1 }3 [- j5 b+ ~0
( Y  k* Z0 Q0 z% f1
7 P% c4 j5 ?7 Y) b% O; X5 O1

4 J8 P  y6 l) o" W  g6 ^$ K' o6 z-1
0
/ \) l1 a" m3 O8 h9 J. r0
0

0
-1
0
8 J5 S* Y# B4 U0

4 D. J+ `, ]" o3 Q6 p0
- O# P, I& O8 S9 `0
-1
0

# N3 W3 F, j: s6 M6 H0
0
1 n1 c5 }  z6 d5 d2 X; A0
, r6 K' m' ?: z: j* v& g' r) |-1

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

b =
6 q- d, t  p6 T& t6

12

3 L4 I, u; }. R7 i4

1 b+ q2 S& H3 I# L0

0

0

6 `5 e& N1 @. C" p' D9 M0 [- V" Z0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
) l- O; R( s4 S0.0000

2.6667

" z& u9 ~. i* X8 o) n. A) L-0.0000

8 n) z! e/ w2 o" y) b( M4.0000

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
1 Q! }$ m5 a: K3 r！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
2

) x( W) H, j* [8 \* |4 XOBJECTIVE FUNCTION VALUE

1)
1 H: u6 ~$ n( A" n* [1 s145.0000

- F! Z8 n9 w  h/ AVARIABLE
VALUE
" w, Y" g+ @& u5 a$ GREDUCED COST

$ ?9 U: P- X  g% \% C, q" DX
, w1 E, z1 D  V& b. G- `10.000000
  W9 g  p; K/ t0.000000

Y
3.000000
0.000000

$ A: V8 Z& ^7 q+ ZROW
SLACK OR SURPLUS
DUAL PRICES

- a! L9 c: v7 G2 A2)
; k+ w( y4 L& s) d5 H0.000000
2.500000

/ d* T; `$ |3 k2 w  j3)
9.000000
' _9 I! V, Q; \. b: Y8 r  R  b0.000000

4)
0.000000
7.500000

NO. ITERATIONS=
! C! S* u0 i3 Z2

RANGES IN WHICH THE BASIS IS UNCHANGED:

+ _2 Y/ j& }- |1 Q& T( u7 Z, BOBJ COEFFICIENT RANGES

8 d7 _$ R) {% T( qVARIABLE
& R" ~, z$ w8 v4 U9 o9 RCURRENT
ALLOWABLE
$ ]3 u% u8 D) `ALLOWABLE

4 s, s5 O# ]! g9 d7 sCOEF
, ?) u) h' B3 S& m1 n9 OINCREASE
9 U! h7 u) o  c, n# k/ r. n. L1 `- pDECREASE

X
1 J7 r; R3 H' p% L1 H0 X/ x10.000000
INFINITY
2.500000

* G1 ]. y7 u. P3 JY
1 M$ L- N+ d( m15.000000
2 n, \% t) p, B" {5.000000
15.000000

* p5 ?0 o; ]' i- q8 f" ?9 m4 FRIGHTHAND SIDE RANGES

ROW
  X3 e/ r% o5 U' Q; BCURRENT
  s* z# Q- i& wALLOWABLE
ALLOWABLE

2 }$ n% E) l$ z, }* ERHS
& Q. y  T/ O* v4 Q, h  {( M% rINCREASE
* J; R# N" z6 d8 D+ cDECREASE

5 t# H8 K6 w. w) L* d2 T
2
! t2 r$ u* |" c' H4 B0 X  ?+ ]& g10.000000
6.000000
10.000000

( o4 v7 f" F8 M& S. I) F/ V3
; V; q, z& ^' G12.000000
2 C* q6 v! ^0 ~! G( g" i8 {INFINITY
9.000000

4
9 X# C2 B6 v+ }% R3 b: v4 l16.000000
* G) [* {# c, d; l; m/ i6 O# {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
+ o& a' l  _: [8 r7 b5 L5 g. Sreport 窗口得到以下结果：

& [* f4 n& L7 q: E& N; m- M4 z4 |Global optimal solution found at iteration:
2

Objective value:
7.750000

6 k2 e3 _1 \! oVariable
Value
Reduced Cost

. S) t- u% c4 Y5 ?+ E, EX1
6 Z6 ~1 F, j1 z' U0 l* i" V0.5000000
0.000000

- d" I# X/ G  B2 E2 U* O. p5 ?. rX3
0.2500000
0.000000

0 d# ?  t) @( f6 b) S
) ^/ s& \) v: I  KX2
8 V5 N7 m7 @$ e7 C; E0 N3 D2 y, U0.000000
! m, P+ N" t6 b8 R2 \0.5000000

X4
0.000000
2.750000

9 @, G( i! H0 m- e# _( zX5
0.000000
  q1 d) J8 {3 L$ N( S2.250000

Row
Slack or Surplus
% t% D; z: h: I1 e/ W$ aDual Price

+ k5 ^6 F; L% Q8 B- V& x3 s3 F1
( e% N; Z2 r& R& p! d& o; E7.750000
, @$ |: Z1 V6 `! a. z-1.000000

2
# a% Z$ B4 R* d& [& D0 w1 b0.000000
- \  F6 Y1 H' o& ]# \-2.750000

5 g" F/ J3 g* y& m2 _# d/ i* V% c3 f( G: a3
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;
! 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:
; {4 m9 M( s' N% c& B" b2

/ X' V+ v5 K2 s* ^# L7 n9 ~Objective value:

14500.00

Variable
Value
Reduced Cost

X
! b' q2 {) O7 o100.0000
0.000000

Y
30.00000
0.000000

Row
8 p5 M0 S0 i# `# a( I% `Slack or Surplus
& s, e& G6 d8 F% w. |Dual Price

1
% h4 r9 L& G6 g# l. m) q+ T) U! E* W14500.00
$ D: ]/ ~$ k; `% @( ^; r6 P1.000000

9 v$ g4 T$ b0 W2
0.000000
' P6 A* S8 V) C25.00000

3
/ B2 L! ~* C6 a- o1 N90.00000
2 t) B  ?) d- ?; q: C. g0.000000

4
0.000000

75.00000

loooog12

先下下来看看