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 w+ s6 Y# a9 R/ R# F" v-2

2 C' H  {! M4 i" c-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 =
; q; C9 Q* `" x5 E- P# v! @+ `  G1
% W# C1 G8 i$ C- ~$ k; N' L2
4
-1

2
% \0 y/ w0 y9 [0 b8 @4 u; K3
-1
1

- N: b6 V8 `6 j, n4 D1
0
1
1

-1
0
0
0

) L3 Y* W. O7 P, E1 r3 K9 V: r( i0
$ N7 r  i* P9 q8 [-1
) A8 t/ C6 b: `5 k) v+ Y9 r0
0

0
0
4 p3 Z4 n" g: N" r-1
0

' Q; Q- |6 H1 C* u" h0 o0
0
0
-1

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

b =
6

12

& W! E  H4 Z2 u) ~7 L% {1 G4

0

2 H! u  I: _: t& O5 t; \; F0

0

: e: H* k( F+ P4 |$ x6 [% {+ O5 J9 m; K0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
0.0000

; Y2 H1 K  }+ N2.6667

1 F8 {5 r( G8 j  N/ L# m8 ]-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
. c) Z6 T$ B- k4 s$ W3 m2

, A: H  o! M" z$ J7 cOBJECTIVE FUNCTION VALUE

8 s: ]0 [* l$ M: h2 q0 i0 P1)
# E* S4 `. p* _: H, _145.0000

VARIABLE
VALUE
REDUCED COST

; |' _9 r; N5 d" e4 Q. G* tX
; P. p% [' @3 q9 b7 _7 l2 o- _10.000000
, S4 N4 J7 q3 q0 p1 i0.000000

5 K$ _  u3 ]: |8 @0 D; V& Z& bY
; S( `- K- w: Y* c! E3.000000
2 }9 A5 s) P( x1 d& ~: Z  x9 A* A0.000000

1 K4 q' f* L$ [$ \! A9 cROW
SLACK OR SURPLUS
DUAL PRICES

# G9 T# E1 c$ a; ^2 X) Z, |2)
0.000000
# O6 }. L' Y4 r, m' d7 Z2.500000

3)
7 Z( L) d7 U0 l$ l9.000000
6 b! m, O# _1 o6 r. `4 ?# G0.000000

4)
9 P& _- N9 v. f8 x$ [# Y$ L- c0.000000
; W' v6 J, }5 k6 s% E7 l7.500000

* |$ f# \$ b8 t$ k5 B9 oNO. ITERATIONS=
, [$ A6 R, G! ^0 h  R- p2

RANGES IN WHICH THE BASIS IS UNCHANGED:

- E! i* t! k; b7 U9 _OBJ COEFFICIENT RANGES

VARIABLE
CURRENT
' q2 N$ P1 ~9 `) M, ~/ nALLOWABLE
ALLOWABLE

COEF
$ X" v' {/ r: A& d+ [. Q- c9 n% v' CINCREASE
DECREASE

X
10.000000
) l0 ?! b2 H+ L( O& j* lINFINITY
+ E) w6 ~9 |% j: G  y- S. E2.500000

7 o+ g, b- r8 h, t8 p6 TY
15.000000
5.000000
15.000000

2 F4 X. L) G  _( \$ m  NRIGHTHAND SIDE RANGES

: K) ^7 A. o( `ROW
4 q$ q# ~2 i+ T& g/ Y. CCURRENT
ALLOWABLE
5 m3 o/ h7 `) S. lALLOWABLE

RHS
INCREASE
  U5 {/ B, l+ n4 t5 a" zDECREASE

- `+ k* \, C4 r1 o# U+ U2
/ k2 z$ s7 D* E' K: K% [; b+ |* [10.000000
( I; J. x8 s" a0 {6.000000
10.000000

3
12.000000
INFINITY
4 O) d- |0 |+ M6 S. M4 X9.000000

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
, }- _0 j# y  E% ereport 窗口得到以下结果：

) }2 Q- ]0 s; j4 oGlobal optimal solution found at iteration:
3 i& Q+ j+ \. t; x& O* x6 M2

( E" Y2 ?/ V' KObjective value:
7.750000

; f) C+ W% m$ g: q2 ]+ ^/ _Variable
Value
$ q8 B& A1 l. Y$ a& _6 p0 {Reduced Cost

% p7 u. L, Q8 Y: U6 ~3 u3 u3 {3 T  bX1
# Y* L5 O% ?$ o$ l  k0.5000000
/ `6 Q) o, `7 Z6 n  J0 K: e' U% v0.000000

, T1 e) n* E, |X3
0 ?4 V: b% }  U" v4 {5 H6 Q) e! q# B0.2500000
; r1 O! t. q9 N$ f, \0 c; c; k0.000000

X2
% ]9 f. g. ^$ D0.000000
0.5000000

% V# i3 v) r$ z! A0 s% v, LX4
0.000000
, [/ {3 Z4 ~% ^0 N3 O1 w2.750000

  s) ]5 i4 S, t  O9 n- Y8 |. FX5
3 `$ ~) _6 Z/ D# G1 z7 N( C0.000000
2.250000

& \" q7 v7 Q; N5 {  S& vRow
Slack or Surplus
Dual Price

+ D, A: s! ]8 M; J( F& {1
% R# @& i! S) Y5 O/ l7.750000
& ^( X6 e% s. k& M8 Y: l( n" c-1.000000

2
0.000000
$ J; D" j. V7 A* |4 m7 q-2.750000

3
5 h% d. {6 W  J" E, R0 `' `0.000000
2 W3 ?4 |: t1 L$ L-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:
& }' S, N4 ?" a% `* N3 }# o2

+ b' ~% R1 \( E9 z" ]" ^+ C3 R  M1 B, GObjective value:

14500.00

Variable
4 H9 g2 [6 i  hValue
Reduced Cost

X
100.0000
" W$ ?1 U$ t4 f$ L) @, W2 H  l0.000000

Y
: |- ]( S8 w2 N# H! h30.00000
& Q, _( N5 f% @* |: z* v0.000000

& {- s& W! J' n8 z" N- ?  {1 FRow
- J* H0 }: r/ X! A% m4 lSlack or Surplus
* w, M! r  D5 uDual Price

! z( o& R* S% |, Z/ @7 F, h1
14500.00
1.000000

2
. L& ]8 j9 m3 m( u0.000000
9 \9 m3 `0 @8 y' s) S25.00000

3
90.00000
0.000000

4
0.000000
" H9 V. {- j- q+ D
( Z- {+ A4 [- x' C" j( D' g% t75.00000

loooog12

先下下来看看