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 =
; F( j6 ~4 |; ?5 Q# B-2

/ J* d' z3 d% f2 d( f" f2 x-1

' z& @+ Y, c8 A; u# s3

' n. F( F1 B; E/ u# J3 @) ]-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 =
: V; w. `) a/ ?7 Y# ~: T" W1
* Y* g# ^6 |  a6 G( H2
4
7 p7 ^  Y! \8 Z6 k9 _-1

2
/ u. x5 j0 {* y* l! U# N3
% c2 g( E& N/ S* U-1
8 H2 m, \' V5 `, v1

1
0
1
1

-1
0
% f5 ?% f3 O+ e( w5 v2 S0
0

) {& ]" d6 S0 B$ n0
-1
0
0

' R+ V4 O! w9 R) i7 s4 Q0
0
-1
0

& L$ ?- f. C6 j. S& t+ `% n0
0
2 V+ U/ c* o9 L% {  Y3 j/ W0
% L, l+ j( ~( V8 x6 {  y) ?% A: i8 Q-1

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

b =
4 `, i$ j; ]2 }+ q" a6

1 z2 x' \8 ~- Q2 K6 S12

4

0

0

0

& p- ]0 K1 F; l; M  ~0 T0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
. _6 N7 ]/ a1 `6 X8 E* |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
+ j5 F8 T9 w" G！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

OBJECTIVE FUNCTION VALUE

6 \; h) Z: g5 z$ i5 U6 I; P2 R1)
4 i  v3 k: Y+ A- t# G145.0000

3 T$ k! y4 e8 q. `VARIABLE
9 G7 Z. g. p! O2 C" O' Z# {& UVALUE
REDUCED COST

' _8 t5 r, _" k; f3 K$ lX
2 ^: V8 x0 H% ?4 i$ x  O10.000000
+ ]3 C5 G- u: t/ r0.000000

Y
3.000000
0.000000

/ j; r7 g1 ?! ~3 r% e: v3 ~ROW
SLACK OR SURPLUS
DUAL PRICES

3 N( r+ L% k* {/ q! w2)
0.000000
1 g4 S- x* h) S8 ~8 e2.500000

3)
3 n& B5 z  o- |; l* t2 |, Z( Z; P$ M9.000000
0.000000

4 G, |  L9 c: Z2 }4)
0.000000
7.500000

( s0 t- s* ?; R* g) q8 FNO. ITERATIONS=
1 P% h4 h* H; m! D2

4 @) X6 T# u2 Q* n  l2 r/ e' iRANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

" g. Y* ^& }; v1 eVARIABLE
CURRENT
0 i& _  u$ ^& g' L! sALLOWABLE
6 i3 e+ |8 ^$ V+ b7 Z: v1 ?+ Z6 QALLOWABLE

9 E' c: t& S/ b  MCOEF
2 d5 e- ^4 V" e2 C2 J5 ?+ cINCREASE
DECREASE

X
10.000000
1 f6 d; o' Q8 y  jINFINITY
7 ?& j5 k& k6 U$ G2.500000

Y
15.000000
5.000000
15.000000

6 _! J8 T: r6 b3 jRIGHTHAND SIDE RANGES

ROW
CURRENT
' k9 w3 e5 i* a/ c9 j5 CALLOWABLE
  C4 R8 Z5 I- P+ B, KALLOWABLE

RHS
INCREASE
DECREASE

2
10.000000
9 u* b' H, j* f3 x( @6.000000
7 W0 y, `# ^8 e10.000000

4 O! @3 w8 ]( ~: ^- G8 x3
9 b7 U& D0 w8 p- q! Y, K12.000000
/ Z- l& G9 z) D; BINFINITY
9.000000

4
16.000000
; g. b, S0 F( d1 F! v18.000000
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：

: x1 K2 V" B5 r8 Y  q! `- j2 O. R/ G& EGlobal optimal solution found at iteration:
& A6 _  s, b) y% h: f2

. E% ^6 }& N7 b* d: Z# x. w4 ^. |Objective value:
( {/ R7 q+ `$ r8 o/ X0 T- V) g7.750000

Variable
Value
Reduced Cost

( J% i- k5 v6 J- {1 a5 E1 LX1
) F; r6 P+ w7 j( T! [  d( [0.5000000
0.000000

X3
. V5 n  k. ?6 a. H0.2500000
0.000000

X2
0.000000
0.5000000

. R3 s; c4 H+ }0 S0 LX4
. b. g  Z3 t9 d1 X/ [9 S0.000000
% `& w8 V1 @# _/ u. M2.750000

X5
8 C1 m, Z" u$ ^( O  q6 c0.000000
2.250000

Row
Slack or Surplus
2 F! N( O2 k* Y$ H- v4 ^Dual Price

1
7.750000
-1.000000

2 e  x/ T* h; p& v* B$ ]6 g2
* F+ \3 b$ ^: y$ j8 I7 B0.000000
-2.750000

3
0.000000
3 s  H6 @5 e6 G* L% X% B& H: q-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;
8 ?/ Z$ ~- W. b! 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

8 M6 W5 K6 d) aObjective value:
; L7 t/ k, `- p* \2 a3 T9 [
, S! Q6 {3 d0 m5 |5 f' r0 a14500.00

Variable
Value
Reduced Cost

( G0 z( Q4 a- LX
1 J% ?) Q7 U5 {( Y5 W! T: }5 w100.0000
0.000000

Y
30.00000
  r; W" J; E0 U7 o( y3 _% B0.000000

& ]) ~/ A% a& `0 g/ R0 j% JRow
Slack or Surplus
Dual Price

" V  F2 L! I; c. h6 K' _3 O5 t) O4 L1
) a" t+ y( Y# ~! G+ e7 S14500.00
: s, f5 [3 L- y" R1.000000

2
0.000000
) Z; }' J8 S6 v  M25.00000

- x& X! x) B7 A  |  @3
90.00000
0.000000

4
% Z# D$ x5 P  T# `0.000000

75.00000

loooog12

先下下来看看