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

4 c. ^' y+ L: M. i-1

) g+ [: v6 n  n3

! _$ A9 r: A" @5 }3 G-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 ~0 f7 q# l  D  k/ p5 u' C4 O( S1
2
4
-1

( V- x  v8 m7 V- @: W, Y2
3
5 v! J' k4 l9 S  y  b) ?-1
) N- D. S6 w6 `8 S$ `1

1
- |0 {$ J# c' A- s$ a" I/ u0
1
" U5 w, ]7 r, M/ [1

-1
' j3 b6 u" z( r% |0
0
0

; Y% F; t7 m9 D3 _* i8 m, I) _& E0
-1
0
0

0
. k" m3 A* G7 P% s+ {; b0
-1
6 h% r0 r% T6 z8 b$ A0

+ S3 o  [( T- n, k2 [4 C0
: R" r4 Y4 H5 l9 z3 r4 n( i0
0
-1

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

b =
  C+ D' Q5 @; ~1 d! K6 g6

5 D1 y( G3 _, Q3 h12

- d, X+ Z8 p) J) H2 r4

" m* p! C* [: c* x$ _0

0

0

" D! ?  T; n- o- j4 J' f6 y  d7 r0

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
！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
' h" A, {' o  n/ ?+ N7 ?7 r7 {2

  h- q$ n* T$ x6 k! i* d4 yOBJECTIVE FUNCTION VALUE

1)
7 I* z6 \3 i3 ^% X" W145.0000

VARIABLE
VALUE
# m; e0 T6 q0 `! |5 @. LREDUCED COST

X
10.000000
6 [* K% K( @5 s  l0.000000

7 N. l9 S% b1 kY
8 G4 l, k6 N1 B. n3.000000
9 b5 o& w/ m7 |- l0.000000

/ K2 d( R! y* WROW
% D5 \- G( @+ ]SLACK OR SURPLUS
2 Q% n& [. ^% L+ u8 e6 PDUAL PRICES

1 d& a% n+ l- j6 a+ ?$ [2)
& t# C1 t0 F, j( b% ?! O0.000000
) S5 A* I/ |* T0 L0 ^3 V6 N; T2.500000

3)
4 I: _# |& b, y7 c/ |+ e+ [" n9.000000
0.000000

, `! E3 k2 g& O4 @- X: d. B" d4)
0.000000
7.500000

NO. ITERATIONS=
+ o$ {* t, M+ `2

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

: e+ a% i- P, ]- dVARIABLE
" L  r- ]8 x! m6 a* d- I  d6 S3 k! VCURRENT
ALLOWABLE
2 E3 Z0 ]- c$ g2 t6 ?ALLOWABLE

COEF
) M3 ]( s4 Q6 m/ Z; R4 }INCREASE
. ]. C" q3 }6 TDECREASE

& r, }" y: b' Q# {0 xX
10.000000
INFINITY
& F2 M$ B; m+ |) W! U- X2.500000

Y
15.000000
5.000000
$ P4 C" H* B% @/ m  _* i' H15.000000

RIGHTHAND SIDE RANGES

7 l1 C5 k  o% {% C) o' [  Q. T9 i+ g& RROW
% F/ `5 M; w% d# Y% ^CURRENT
- y) A  s1 D) B8 A8 a* QALLOWABLE
9 A6 `, a' }4 lALLOWABLE

% ]  c* Z& N8 F( q; j0 ]RHS
& v7 w; Y: }8 H* X1 l# J0 j7 l3 HINCREASE
! d( {5 G6 t5 o6 wDECREASE

2
" b- F3 v1 B, _, @10.000000
) e8 c9 ]' n) P/ x4 w6.000000
3 C+ M2 s$ j. n" C4 I3 E3 W5 l5 \+ K) H10.000000

3
% v8 V* x0 W4 h- M+ E9 H12.000000
$ e4 I0 {6 F/ y! Z# n6 N1 z3 BINFINITY
0 n5 I& P3 I. r" p6 v3 Q% {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：

; h9 C& i4 C$ v- M' B6 R- O8 `* hGlobal optimal solution found at iteration:
2

; I* A7 P: w3 v- P8 D1 x- LObjective value:
7.750000

Variable
  Z1 o7 u4 f6 U0 }/ r/ m6 RValue
Reduced Cost

) j4 m4 L6 T1 EX1
0.5000000
0.000000

X3
% @0 y, I* O: @$ z1 w0.2500000
9 h* n8 Z  y( m% [0.000000

; d5 f9 n- X1 X0 C
& u! e% n. e. n7 p6 p$ Y/ Z5 CX2
) g0 m+ v! I- {! z! ?- t4 P0.000000
) ~2 c# t/ T% q7 h% |+ y0.5000000

, [+ C/ R- E1 e/ z: xX4
0.000000
2.750000

6 S3 Z9 `* m, p% n8 \: ~X5
0.000000
! b& X. Q9 v2 X2.250000

Row
9 r: F" p6 H# DSlack or Surplus
Dual Price

1 j8 u, u: k6 O! _1
6 `0 N+ x' f9 J8 o7.750000
, z: ]& C* w" T' C/ G* ?-1.000000

2
3 m- i0 C4 f2 |1 a2 `; O0.000000
* P% B% X! O7 X# D* v% ?; y- _-2.750000

( F3 Y, y8 K5 ]; f6 I# ~3
0.000000
, b8 O5 P2 w! t% U- |-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;
  X' J9 L3 E$ e! |% ^! 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 G4 m, H! e- h; D+ }; TObjective value:

# W2 h" o7 O2 L, L) {8 g( v14500.00

Variable
Value
+ m* {" }, Y4 k- |3 {Reduced Cost

1 F; y+ P0 n, D; g: \X
* W% h6 f, Z5 d100.0000
* ^  L% ?3 f) x( _6 g) W% t0.000000

- {0 R4 e3 }2 j+ y. RY
30.00000
" x3 G1 P, B8 e  Y/ E: t/ J# J0.000000

Row
Slack or Surplus
$ \) Y9 V5 }, P$ MDual Price

1
14500.00
1.000000

5 r, N. W, Z8 v: z2
0.000000
25.00000

. @- Z7 ~- O! ]+ `( Z4 O/ I$ r( f, B3
8 \9 F& {: B3 Z4 A/ f- w' Z5 z# b/ K90.00000
0.000000

4
) j# x) y$ q+ L- R- i7 a& U0.000000

: J4 {* h0 R2 ?+ m75.00000

loooog12

先下下来看看