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

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 =
$ \( T# ~, K- q$ ^1
6 C+ h1 [3 T3 Y! u% z2
4
! I- P/ j( ]2 a8 L8 [# K-1

2
3
% \0 p( q& T) S* ?. u  O7 J* `-1
1

1
9 i# x; l  L; a! i& i9 o0
" W+ c# P* Z" k( i# L% D5 h) V6 a1
. s, I9 h- `. }5 G1

0 j  Q* [- d2 B. ~6 t-1
0
0
0

( v0 n, g7 E; _+ O9 E0
( M1 y5 o, M. F6 L6 p/ A-1
8 B4 r& J+ C- D$ {0
0

0
0
-1
" \5 q/ M5 t; \7 A0 V. ]0

0
  E" |& ~  D& n1 i6 R! H# v" G0 t0
0
-1

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

b =
6

  I: [" P7 B  B" o9 }/ L12

4

8 ~% _* W2 k$ c0

6 ]: `3 S8 X3 g0

1 n% d  u1 R: U9 O1 E0

8 j* m' W2 ~" F6 {: P! Z; q: i0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
0.0000

2.6667

-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
! Q- e( Y9 _# v3 I！注释符号; 系统默认为自变量>0, 若不要求用free命令.

( D5 v7 X% f, s8 g2 z( i6 Y% v！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

8 p! k1 H3 ]+ @# d8 ]OBJECTIVE FUNCTION VALUE

' y9 x8 `% y7 {, i! t4 S1)
$ O; O% J+ f* e8 T8 M2 C2 t145.0000

' T/ ]$ ^7 f' n7 AVARIABLE
( N* _7 V; s! |- v) e* UVALUE
6 T5 @3 X8 n0 ]5 P: S+ y6 XREDUCED COST

X
10.000000
) m: ?+ c" r5 a# H( d: f0.000000

6 }: B2 n/ y) p6 l1 t6 UY
8 b3 P! M0 D* v5 F  C9 l% [, p- \3.000000
$ x" s* [8 l% J0 z( L0.000000

. `5 R+ [% M% j" |" wROW
SLACK OR SURPLUS
DUAL PRICES

0 u% D7 v! ]; O- d: x. m/ U2)
0.000000
0 g& p9 p/ D: V, o& u$ ^  c2.500000

3)
! a3 C. d6 a, n/ o" i9.000000
0.000000

$ i+ `4 n" i+ \4)
; G9 S' f8 V' ?& N; Y0.000000
$ ~3 C6 c# X) b, O; d9 b# L3 y" A8 f7.500000

NO. ITERATIONS=
& v# x" w+ P0 N, i3 a' F2

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE
# h% A0 Z1 I  p! a3 q) p4 U( yCURRENT
# |6 K2 e+ c/ |4 g! o' w7 bALLOWABLE
1 y+ }4 W% _/ tALLOWABLE

; O! D9 v+ I( g! I: _: ]- n- i8 XCOEF
! a: q8 U* T5 w! [- c' ]INCREASE
+ r9 a, y1 }  B9 s" l$ DDECREASE

X
10.000000
INFINITY
+ N( i4 B1 [. d3 L4 E4 s9 i2.500000

8 h" I4 m6 U) A$ j4 z# qY
15.000000
5.000000
15.000000

# z7 U+ w  {% r- t8 FRIGHTHAND SIDE RANGES

ROW
) B3 @# [+ V+ A) P8 N3 a2 pCURRENT
# y, J4 u6 G. S- yALLOWABLE
ALLOWABLE

  s, U4 f; ~4 {9 P% ^RHS
INCREASE
3 {1 \$ _* K2 IDECREASE

2
10.000000
6.000000
8 p% I7 n( a. M0 k  T; V3 d10.000000

% @( X8 f& K# u  f+ }3
12.000000
INFINITY
9.000000

, R) l$ Q5 ]- q2 S5 _4
4 g0 Q" g/ Q3 ~' `. M7 i4 \16.000000
6 J* i$ t7 f% b4 J2 |18.000000
5 w; c& s* Z1 |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：

; s; C7 k4 O9 P- zGlobal optimal solution found at iteration:
$ Y2 N) {3 G2 C  d& D* O6 ^$ k2

Objective value:
& ^9 l# Y( k: q2 E- h7.750000

Variable
Value
9 J& U6 z# u6 b8 I6 \! sReduced Cost

7 ]! ~) ^4 _$ Y2 }: QX1
, X3 T5 Y* j7 U3 g0.5000000
0.000000

2 h! T" E8 g  z! G; z. oX3
0.2500000
) F8 J' v: ^" ?! c5 l; i) E& g0.000000

* Z# J0 \- I5 ~5 U8 T
X2
0.000000
* r6 {  H2 N+ j: z0.5000000

X4
2 ?" z% ?) F  _0.000000
2.750000

X5
. x! n; i  M* ~6 r' t8 @( b0.000000
2.250000

+ m9 ?* l5 F. Q, g9 q! C- tRow
2 p% A* k0 ~8 A$ d) ~Slack or Surplus
7 J1 j8 s1 G5 _( C2 {5 A2 c" mDual Price

, X, g; j+ o  T$ ]- R& u" B1
: z$ i4 O! o0 M; d; ^" W/ @  I7.750000
5 ^1 A9 a/ C; R& v- H0 E-1.000000

2
' r% Y  t9 \0 N9 w1 L: l/ P0.000000
9 P+ T' l+ i* k- x/ e' O-2.750000

3
& @( M% M. Z& b+ e9 ?& b6 [0 F0.000000
-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:
1 ^+ y3 S. q  ^; h( N( _) B2

1 f- P% H& }# C; b5 c. i( MObjective value:

& i( Y# i5 |4 X  P14500.00

% h$ G8 t. c- J$ B* W1 m% |0 yVariable
Value
9 m5 f9 O9 n/ mReduced Cost

  R7 c0 V+ {8 hX
100.0000
9 j. T1 S: v4 j0.000000

; L: U8 z' j1 }& GY
30.00000
0.000000

Row
! o+ F1 N2 F. z: D' w; PSlack or Surplus
6 _# n; v. e  |$ JDual Price

1
14500.00
1.000000

2
0.000000
  ], B, y7 v9 W3 {& M, N0 N5 E25.00000

3
90.00000
0.000000

4
5 m( c7 z/ W, F* h6 Y& C1 e0.000000
6 d. R# W  L" `( l( v$ {# R. i9 m
6 W9 Q1 g$ x" W$ ?7 O6 |' I9 H75.00000

loooog12

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