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 =
0 L7 c6 d- k4 w/ _, r) ^-2

-1

$ d  [) }: V2 C) m; H, m3

-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
, i4 S; t" l! h8 x* j2
7 A4 r+ I( U& W# ]4
-1

6 g* [+ n, ]# E9 w1 J, f7 Q7 |2
# T" i9 L  f) `: W' i! R2 d3
-1
1

1
9 r. i$ W& {( D; ]- k5 G% _8 T0
9 T  z) [2 F0 W3 M5 x8 v1
# a, b4 r% i1 B; t1

-1
4 _  ~: a, |# L6 _+ t, {1 L0
0
0

; R# {2 ^  a# g0
-1
0
4 O9 m" d7 R( Q3 Q6 M# \0

9 [' p' d4 l% N% |0
6 h. X! [8 i8 h- Z( x6 p0
; [& e1 Y, r6 c+ y-1
/ X9 l, o. j. F0

2 w; ]6 s, Q- Z( O- p0
0
0
% A" [5 ^$ x0 |; c-1

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

b =
6

1 o( i1 o0 x  d12

) }0 m, e$ C( d% O5 {- f: W4

' ^" |' k' p6 H7 I0

0

$ s+ a+ b8 E) E3 k0

0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
) A5 |/ k- D: O. L: [0.0000

2.6667

  c2 b& g, `  b% X/ j, r# ^) X# O1 P-0.0000

9 u9 l! t$ x& {4 }& W5 j' Y% l4.0000

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.

！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
4 w7 ^  L" P& V) \8 {* y0 S( {2

OBJECTIVE FUNCTION VALUE

1)
145.0000

VARIABLE
VALUE
* j+ M( L5 w  `* FREDUCED COST

X
10.000000
0.000000

7 P. e) e1 M: X* o: ~" @5 e! E" BY
# d7 _5 I+ `/ F- X7 \3.000000
0.000000

. K+ p* k& w* r- n0 d  Z$ }ROW
) G9 J; u$ s. }SLACK OR SURPLUS
DUAL PRICES

6 F+ K, _3 W8 L9 |6 u5 K2)
0.000000
3 k. T6 U( w7 J0 t; r2.500000

( M/ b2 ~% }/ L. i0 I/ {; p0 y3)
9.000000
0.000000

6 S4 P/ y- V* Z. _. Z: O4)
" s) n: k  x" r& @* U# }+ F0.000000
7.500000

9 S5 {' }& W/ s8 F5 N: ENO. ITERATIONS=
2

, H* S% N0 u/ K7 p+ rRANGES IN WHICH THE BASIS IS UNCHANGED:

6 X& {0 S9 v' WOBJ COEFFICIENT RANGES

VARIABLE
: A' Y+ _  Z; t' q. M$ wCURRENT
ALLOWABLE
ALLOWABLE

- y6 Q- x9 n) z2 Y0 l5 E/ U6 H; q( k$ W# CCOEF
% P+ p0 @( c( M! f6 IINCREASE
9 W0 c9 ~. @% {& ]DECREASE

" J5 O" C7 `$ M4 J0 MX
% P2 @$ p1 V. o% M10.000000
INFINITY
2.500000

3 Q! s, r. J% N( w) ?  k1 h" U0 ~" TY
! H4 ~+ }# V: }. k" R15.000000
5.000000
4 {* Z3 t1 }8 y% l* _# d15.000000

  _" q4 ^' ]4 N! n1 QRIGHTHAND SIDE RANGES

ROW
; g9 |9 _+ q! a4 G& {5 JCURRENT
) x+ E. n1 K' V/ ]6 q7 fALLOWABLE
ALLOWABLE

2 R4 T: S; x; q" jRHS
INCREASE
DECREASE

; ~8 Z" g* Y% P" h8 n7 @% Q' q
2
- I- v/ j2 n- A, q% Z2 n5 f10.000000
; w$ T2 l; C8 F' _6.000000
10.000000

2 Q' z, U; T4 h- o# ]# t3 ^3
12.000000
INFINITY
2 J6 O- I4 {- i  V9 p6 {8 T9.000000

2 Q2 a+ a3 ]4 L; u7 w4 _$ }4
16.000000
18.000000
4 Z6 Z3 y* z3 B8 S% G$ X6.000000

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：

0 N$ R- {9 s0 g4 j( bGlobal optimal solution found at iteration:
2

5 T2 L. a6 S: }& u6 yObjective value:
* u+ U) X7 Q6 z' m/ `2 H( M7.750000

! L$ [) Q6 ?) a' EVariable
Value
Reduced Cost

' |. y: K& E7 e  x% UX1
0.5000000
0.000000

' W0 v8 I' r3 a! uX3
0.2500000
0.000000

! `% i* u- ~, u/ A" Y/ r
X2
) m+ L: V5 H8 j/ H! Z7 Q$ t0.000000
+ @2 Z/ `4 C, o. X& g0.5000000

X4
0.000000
6 x& m, f% M# m4 x; b7 t/ \2.750000

X5
0.000000
# T- W: k7 Q) S9 S: q5 o: f2.250000

2 Q- ^& u1 w& w' l% C$ NRow
9 O" a% D! P; O+ X. z" B* oSlack or Surplus
; J% o8 W+ @! QDual Price

# ^" C/ \2 C! s; E! o8 c* h4 P+ D1
7.750000
! o0 J& ^3 l8 E-1.000000

2
0.000000
- ^2 c, }/ G# Y' W$ D# P8 ]-2.750000

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;
! 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:
7 k+ V' D7 U& y5 u
14500.00

5 K6 {. q4 d6 ~Variable
8 V& j1 u) x& Q! pValue
Reduced Cost

X
3 w0 I# O" P- }: E100.0000
0.000000

9 p4 d6 Q  B* }3 wY
30.00000
0.000000

Row
Slack or Surplus
Dual Price

) D1 A- z& x. G6 @! }: }) p" ]1
- U" {; T/ j3 B) }* k5 A7 U; E14500.00
1.000000

2
0.000000
6 i$ F6 I. z9 G& y: c25.00000

( O5 u" q" Y. g5 k2 @3
0 F" b: Q9 U& o2 N0 z9 D90.00000
0.000000

4
( w- Y; e' y. k" s3 M; I1 _0.000000
% O* O1 r5 ~: y8 q% y* O/ \
75.00000

loooog12

先下下来看看