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

6 e; \% }2 T$ l) x-1

) p1 c5 z" F- h+ m% [7 D3

-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 z5 e' _# C( }  x& W1
* s2 B( ], p' h6 D" x2
+ G  B1 i+ C' l% A( |% u5 j4
& E& v$ U. \& B- b-1

' Q% ]1 w' Q4 T, W# |2
) [7 @7 @4 R# d3
5 V9 X  _. o) n0 y" H; Q9 v9 h0 T-1
1

" V" J; }; w* b( h" w- {# C& F1
0
1
1

-1
0
/ |) e" b9 @* y9 g$ [0
0

; V* K7 j, R' B4 V9 ?( B0
7 N9 ~7 t7 s0 }% U2 B" a6 X-1
7 _/ X3 [* R; X9 D/ r0
0

, J6 U* H) u8 g0
0
* I& H% R4 f% H" }3 a0 m5 r-1
% P2 ]% L, }( D& e, B+ c0

3 t& Q9 R0 y3 T3 }: u# n0
( K1 a6 F" N! B+ Y/ F: @0
. X5 d) W+ ?; ]0
-1

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

b =
7 d! |& [: N% y2 j6

  D" \+ r9 E0 p12

4

0

0

- q$ e. i0 _5 H+ \5 y0

0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
$ N, v, ?8 J, c% y4 B: p3 f0.0000

& z" C: Q+ }! i  p, r- y4 b" f, M2.6667

-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
# N$ X' T4 P* ?6 O7 |' ^6 G！注释符号; 系统默认为自变量>0, 若不要求用free命令.

& I* n. l6 h1 M0 Q3 {5 n！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 _' `7 B8 _! V1)
145.0000

0 e6 G- F' S2 l% NVARIABLE
% w1 C( b( L8 F' V! A% {VALUE
! o( R0 j$ o0 a# d. uREDUCED COST

! _' D' d" e' I, ?7 d: _. J4 pX
! n5 l" `. T) ^6 @! m- U10.000000
7 }" ]" E: l* A, t& r/ r. S0.000000

Y
8 [$ W* A5 t' B$ Y3.000000
0.000000

6 c* ]! M; g1 o0 AROW
SLACK OR SURPLUS
DUAL PRICES

2 C$ j, U7 K" d9 ~2)
0.000000
" }  G) ~! W& |" r2.500000

  n1 V& V. H4 Q/ J6 G1 U3)
9.000000
0.000000

4)
0.000000
* O2 `& W8 t8 A6 }! {7.500000

* [$ w  U: I/ }9 V% DNO. ITERATIONS=
2

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE
CURRENT
% J6 C* s0 d( GALLOWABLE
# U. _/ T  J  V7 g& n: q5 ^ALLOWABLE

) S" p2 x# u6 D7 Z& N1 X% wCOEF
/ Y8 p7 l6 c* M( A9 P* _8 S, ~INCREASE
0 U8 y* J0 e( b% a7 a( A  nDECREASE

X
10.000000
INFINITY
0 K  u' P$ f3 Q9 G2.500000

Y
15.000000
5.000000
, A3 [$ r0 u( H15.000000

. U6 R" h' E+ c6 n: |5 z# M; @RIGHTHAND SIDE RANGES

ROW
$ A# {; \  U1 L6 o7 t, z3 c6 |* CCURRENT
ALLOWABLE
# c& c, f; h; N$ O: LALLOWABLE

. g8 D( x9 p3 p# FRHS
INCREASE
$ H2 q; C6 j1 x: \DECREASE

: k8 i; a. f1 I8 P) o3 f( X
8 g7 `% ^. {. }7 y0 Y9 z; d- ~2
( v' S# n' R/ M& P$ v10.000000
6.000000
; i% U  N$ s* j6 G- n1 _10.000000

3
2 v( j0 l- u# o2 s12.000000
INFINITY
, r4 }) G. t5 @- ?9.000000

4
5 L( `- g+ c/ P3 p( Y3 I* X16.000000
, C  M, f: x3 u& j( l5 p18.000000
0 T8 [; _, Y. D, K! y# |2 D7 F/ a6.000000

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：

. F- N5 d0 k' N0 q) y1 mGlobal optimal solution found at iteration:
2

Objective value:
7.750000

Variable
# D1 `% F; N9 i8 `5 u& t9 TValue
Reduced Cost

X1
1 A& P0 F' U' _0.5000000
( G, v& s& y' q% D3 C. k; e0 N0.000000

6 z) p0 j8 q1 s4 Q+ q5 \* E2 SX3
* {7 u! u# z. M- O$ R6 y0.2500000
" p4 {# N# M; P. m0.000000

9 ~) i( ?5 e. V* K. AX2
4 M& A$ q$ m7 R- H& I7 R0.000000
, A, p6 w* K" m; {0.5000000

& }& E- A. M) E0 d& IX4
0.000000
2.750000

# ?9 m  N  {- ^4 ^X5
0.000000
6 J1 Y! i: G9 V% L- M# O" v2.250000

Row
Slack or Surplus
Dual Price

1
# u5 u+ e; A) Y" D7.750000
-1.000000

: `- Q; C+ f% F+ m2
0.000000
' o5 v* G$ c3 V7 B  n; G-2.750000

3
0.000000
6 {$ q- _& X( J7 ]0 b# A-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;
+ b$ s# \3 b1 \9 l8 h4 L! 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:
3 H8 v' p3 U5 J0 b, a. Q, A2 I6 n
14500.00

0 z' o3 j7 A- ~0 uVariable
) A0 e. @  A4 F+ M* b& i9 Q- nValue
. U+ V% T/ k* C, R7 c: jReduced Cost

! f3 f2 Y9 n3 A& JX
100.0000
0.000000

# _- z5 H' z& X) LY
30.00000
$ ?/ H$ o- L2 @& j5 |8 c0.000000

! S9 Z* x" c, k  a' F5 JRow
Slack or Surplus
Dual Price

1
+ c1 D, g% b* s5 j; f2 X* n- H14500.00
" ~7 M, j1 t3 N" G4 U/ {1.000000

: Y( h) _+ G) H2
0.000000
% h4 K1 V  }4 b2 T25.00000

4 X! k# R9 F0 ~7 E3
90.00000
* a' O: g3 L  P0 [' Z1 J0.000000

4
0.000000

75.00000

loooog12

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