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  • TA的每日心情
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    2018-12-4 08:49
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    发表于 2009-12-31 14:14 |只看该作者 |正序浏览
    |招呼Ta 关注Ta |邮箱已经成功绑定
    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 =
    - K+ Q% V+ |# `- Y-2

    6 H$ E/ u* e+ z" D. Q
    -1

    ' u, ]# o! X: N; d% Y# m
    3


    " f9 ]: U8 T3 B9 ]3 l) M-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 =
    6 x7 K. L- h0 S) v1( C+ o( R' Y3 Y: {: `5 v
    2
    $ p& Y2 l: j0 n4- _5 Y6 L5 R3 g/ W3 n9 }- b/ i% R
    -1


    / @. B; T: q8 m8 |6 M2
    6 ~1 j; t3 `; W! V3
    : S$ I- k- w& m-1
    . ^0 S0 z6 b5 D! |7 b" p1

    , M4 x; l" M1 s  N& i  I: k
    1
    5 q4 U3 A" U' v: g7 K/ L0/ l' O, t. f: P6 v2 L
    17 O7 m. J1 W% \# g* e4 b3 b, Z
    1


    - J& q6 P5 F/ Y-14 E- B9 G; M/ J0 s0 U. K
    0! q, G2 L- C6 {: h- A" s( Z% D5 x
    02 ]& T9 h% B9 ]
    0

    , ]' }* {5 W$ o. }- K, X4 f
    0
    8 c8 ~4 u5 {1 k1 O& X-1
    ' x3 z( y+ {" T! c$ X( t$ b04 Q$ o+ R  u7 f4 f" Q+ v
    0


    2 I( }- Q6 K& O6 R0 Y0 y; n9 K0
    ! Z$ B( ?3 E% t6 Y: U0
    6 e1 b! C( U% A3 Q- ]5 r-16 {; E) V. o5 {) I! H5 q
    0


    . d$ a  {; u. U# s+ N2 z2 Y/ e) X" i0  o9 ?; R; ^3 }, r8 t3 L* O
    0) x7 k9 G4 {% j3 k" f6 Y/ P8 j' B( _
    0
    " @% h; F( S/ W4 L-1

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

    b =
    9 L' R* S! l3 V7 |6


    " h2 k% A! ]: M: I+ ]- D! l12


    ) |! s7 T% [/ k: [7 {% z& T4


    ' M$ a6 V$ E6 z5 A, @0

    2 k! x2 U" a8 B  c3 T
    0

    : x, ]$ ]5 I! j8 [9 ]. h  N+ r3 b
    0


    % X' c8 g1 m+ _+ K( R( }0

    Optimization terminated successfully.

    The optimization solution of the programming is:

    x =  Q/ ~* @; ~0 e9 D4 b
    0.0000


    ' b* ]) ]1 m+ R/ Y6 u2.6667


    : d4 A$ z  w4 I-0.0000


    & y4 O! h0 v; t) y0 B4.0000

    The optimization value of the programming is:

    opt_value = -22.6667

    : 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.

    2LINDO 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
    # F' O% j' R& x. y* a4 @5 g: b9 V!注释符号; 系统默认为自变量>0, 若不要求用free命令.

    ( x3 q1 g& J( f) D
    !在出来report windows之前可选择显示对此规划进行灵敏度分析等

    solve, reports window中出现以下内容:

    LP OPTIMUM FOUND AT STEP" M! F2 k; }9 O7 b" c, d
    2


    $ K8 @8 J1 {6 t6 J, O# }* U( QOBJECTIVE FUNCTION VALUE


    8 j5 q- Z( A8 u1)+ _7 g, I- [+ D1 a# Y
    145.0000


    # w, I2 f/ H' gVARIABLE
    0 O2 l* |, s, Z1 C* V8 D  Z. xVALUE
    9 F$ c, z. @# lREDUCED COST


    # ~" k' h5 H4 z- PX. f$ l4 J6 F1 X2 \
    10.0000005 W5 D$ W; U$ q9 m2 p" O
    0.000000


      @' K+ p- z# iY
    0 Y2 f9 O  e" ?' y4 l) L3 n3 ~( [3.000000% w2 Q, M( W- q2 c1 i) f! D
    0.000000

    3 F% g/ \: M- X& K$ b4 A; W
    ROW7 [) W1 B; R2 U
    SLACK OR SURPLUS/ L7 D! X) m3 o6 J2 G; F- T
    DUAL PRICES


    6 q" P/ ~6 N# u2)( ~7 p! O4 e. a, ?: I8 l
    0.000000& K  z9 N  V5 m' M+ b6 s  O6 U0 L7 C
    2.500000


    , B" Z' a/ H3 y, u5 y; `7 ~3)
    4 H6 L9 q' V+ W3 A# c9.0000002 O9 y. E) R7 _; ^* K2 e% P% k: x+ q
    0.000000


    ! f& f( K$ U; m* a& B' l, g4)
    & E3 c/ _6 ^# T0 h0.000000
    / [4 {7 E, J2 Z7.500000

    9 Y% g9 B, u0 v8 c: p
    NO. ITERATIONS=$ V; F4 t: n( Y# F
    2


    " h" d/ r9 Q* E6 ~RANGES IN WHICH THE BASIS IS UNCHANGED:


    + C+ @0 z# }& d# yOBJ COEFFICIENT RANGES


    ; O) w, @/ x' ^; V: y" Z6 eVARIABLE
    + Q" F# R& d" }1 J: u' f; pCURRENT
    + m6 [/ i! p0 b6 u7 p5 ?2 RALLOWABLE1 Q; E5 Z/ {% _3 ]* t' X0 d
    ALLOWABLE

    4 |+ ?$ T5 Q' M) r
    COEF; ]; H8 B" {* {
    INCREASE/ S; o6 |; |# `; X2 f% g
    DECREASE

    0 P2 ~! ^9 ?8 X* X
    X
    ' \& Z+ S1 `- J) k& Q0 ^1 H10.000000
    ! f, x6 @1 c- d/ @8 P3 jINFINITY* y( P$ j# o* p& s! j* r& \' \
    2.500000


    ; H0 K' U1 k# n6 s' \Y
    4 G( @! C( s! q2 v15.000000$ p( y1 p+ x2 Z2 }6 {+ u0 u! ~! r/ \- [
    5.000000
    " n6 S2 t  V8 Z% a( n15.000000


    . ^* B" f# ^5 Z( W4 \& l' J  rRIGHTHAND SIDE RANGES

    : T) T, N2 W' s0 {0 t
    ROW: S- B5 G8 n! n, ^. a  ^
    CURRENT2 q- ~& m1 c$ G8 B
    ALLOWABLE
    $ E6 s, r! L7 n: l7 F: ]ALLOWABLE


    & J" m' r% X+ I6 LRHS
    . ?! m, s  |4 h: ^; I) VINCREASE# }* p3 i1 Y6 [7 |
    DECREASE


    : z: n4 E! }7 u) s
    : \( F$ A8 |3 a) L% [2
    - i+ J' J" w* L. Z" Q- N10.000000* z0 ^2 L" t; o3 V+ e0 W
    6.000000
    8 N: C$ L, Z! ^- j10.000000


    0 p2 y# R8 x% V37 k3 t! }+ A* U8 O
    12.000000
    " c% i# L$ c' s3 oINFINITY
    % G! S: V. p9 s3 {! F) r* J& ^9.000000

    0 a* U2 v9 y0 R
    4
    0 P; Q$ c4 i5 k5 v" t  f  W16.000000
    2 o7 u* Y( b8 I18.000000
    ' h4 L9 h+ s( V9 b7 F: i9 e" t* A' f6.000000

    3LINGO 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

    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" C# R2 W2 A/ W- ^0 ~
    report
    窗口得到以下结果:

    # p8 y4 Y% g1 M+ Z/ i8 [& O
    Global optimal solution found at iteration:
    1 e2 m  p0 C# q. y- t9 S  [' Z2

    $ s' n2 Y$ N0 g" y
    Objective value:
    , H$ y% F/ i4 C: R$ G7.750000

    9 H( i2 j; k" ?- Q3 g* g& ?
    Variable
    - @; A( E; D* M, x& r" D( K" }Value: [* U) B0 L! d: P4 |; ~- h8 d
    Reduced Cost

    9 s3 O5 a' p% c& E& t: h- b
    X1
    - e' _+ Q+ v0 p7 B1 b* O0.5000000
    8 [1 r/ Q% f* n8 Y+ |* g4 \: t0.000000

    4 a' l0 Z5 c) D3 A
    X3# M9 G+ p; d5 P& X& }3 q
    0.2500000
    2 r# o  `' ]- P0.000000


    - {" _$ b& o1 B" Q
    & ?7 o/ n9 s, n" a5 `1 d8 JX29 F4 Q$ `9 U% ?2 N" {5 F
    0.000000' B) A6 v. M7 s( s4 D3 E- Q; U
    0.5000000


    ) c0 `" {' R) g( p  |3 cX4
    1 x; Z; n, U& t0.000000
    ; _( M8 G& b4 B2 e& C& s2.750000

    # R# T7 E. A) O$ e6 ?
    X5
    # l7 y6 n7 p8 f4 L! F+ e0.000000
    3 [: B+ ?: a/ H0 [; c4 C2.250000

    ! O- B$ x2 S$ a5 O9 W
    Row# j. U0 F3 {/ D6 \2 J8 e5 ]: `
    Slack or Surplus
    9 Q: v$ t! v& bDual Price


    4 d6 w, z+ d5 c1 ]* t! B" h' }1
    ; b+ h. A  s! @) p7.750000
    8 r0 s$ N- _" G  P  O  ~-1.000000

    . \6 ~, M& P' A" v/ v
    2( O7 G/ d! \4 O1 j2 u# g/ L# a
    0.000000  Z- v9 a- a6 \  U# W* }
    -2.750000


    % Z1 d) S) s% ]7 N- B3
    8 L' S: z5 L4 f7 b0.000000
    3 H2 {  [; ?2 ^# f( J  o! D-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

    model window中输入以下语句:

    max=100*x+150*y;
    1 i8 g/ J/ j: s# d* a+ {! 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:9 o7 Q: O) d9 k4 B( m
    2

    9 j: {3 s% o# a  n
    Objective value:
    ( }% n0 l1 P. ?. C* L2 W. I0 M9 \
    ) m: W& C1 g: g: Z" T! A  B: ^14500.00

    / O1 C2 G6 G% g' z/ B" u) h, X, g
    Variable
    : O5 A; O2 D" b2 O) Q1 dValue  ^# E3 @% d/ V: q5 I
    Reduced Cost


    6 J" |- t. {0 o8 pX' u+ j3 g3 ~7 ~, k0 V' h& `9 \
    100.0000$ x6 h5 j7 q7 U
    0.000000

    8 s6 w8 O* N5 f: Y2 ^8 c. K
    Y7 f) |$ d! U3 O- z4 `- |; L
    30.00000
    + l+ x0 b+ ?9 u0.000000

    ' x1 Y- H4 e) d1 p" L$ y5 s
    Row- {- h5 q4 t; I( W8 y3 T" t
    Slack or Surplus8 b' X% ~# @& Q3 f2 X) s
    Dual Price


    ( a  ]1 I; m2 P, Z) z* B10 [# j( l9 b0 b& b# N
    14500.00
    1 R$ n3 ~( z1 T1.000000


    % y) N% o& J, Z! U$ r: N, P2
    ! W1 T& N9 D: d" X5 k/ E$ x0.000000
    8 P" m( B4 o- A# Q3 x& E" b25.00000


    8 t! X+ G8 J; {1 o3
    ( s0 P/ ^. M) D90.00000, \9 P1 F9 R: r, F& J* c* Z
    0.000000

    43 [- w8 b: t: R$ w
    0.000000- m5 A+ q* U! k/ F# b7 M/ {

    1 q$ h/ R* [& @' d75.00000

    第二章 线性规划.doc

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    2013-8-16 10:51
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