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

( o9 Z1 W8 W6 l/ s: p% E-1

3

& q, n3 Y0 N' m  B0 M1 i-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 =
! C3 ?! g+ W5 g& g1
1 o- X9 g: ^! R3 q5 V2
6 h# Q1 f* X' [1 P8 R. o4
1 e6 C0 D; T/ _1 ^- X3 Z) Y) U4 w-1

  j1 l) Q3 l# [0 ^! P+ C6 N2 f2
# z9 c8 n: z5 h( g3
-1
1

: h9 `5 t( ^/ `9 o5 l1
0
1
1

% Z" ~. g) ]% m5 h! {" C+ M-1
' P6 Z1 }) p* ]* y' ]0
0
4 ]1 D% ~. j/ c& y5 u0

0
- d# k; ~/ ?7 X' x. F5 h  N) e5 G. g$ x-1
  g* E% _) J- t2 H- a0
) s1 C6 }* [8 K: Z+ i; ?1 W0 `0

+ ^/ \; {% P' U0
0
-1
! Z& E: [7 a, Q6 I' |0

+ L& d* q, N8 \  l6 s+ x0
8 D4 k* v: F2 |$ L, D9 P4 b3 e4 F2 c5 W0
6 x1 k  Y5 c0 U, x- w1 `0
. H9 W0 D+ Q! H! v  a0 Y/ ]-1

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

b =
6

12

4

% \- P9 f" H  ^2 n$ ]0

0

/ z) y! R  v- F% D* C8 T0

9 B& q8 ~0 J  u0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
0.0000

2.6667

! ^# o$ \) P4 h. M0 ?' Y% ]-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
2

OBJECTIVE FUNCTION VALUE

1)
145.0000

3 b" ?4 x" ~2 c0 S& k* FVARIABLE
VALUE
' R& j' k  E: oREDUCED COST

8 G5 m+ D$ S" d/ h. k0 d+ X; a. \: XX
) e) X0 V! G0 z' X" V% J10.000000
0.000000

! y/ f. ^& I  M1 R9 i) sY
8 D+ D/ K# A& }3.000000
$ o% l, ]7 ?% {0.000000

6 A& _; m+ h4 q  R  t) f" [8 nROW
SLACK OR SURPLUS
! l+ H9 B4 Q5 H3 z8 A! w  E7 v4 W3 PDUAL PRICES

2)
% j- p8 e$ f" F& a* |# x. Q$ O0.000000
% |$ n$ @! m9 D7 x% O% x" z2.500000

' z4 x1 T" L+ t% H! t0 \  i3)
9.000000
0.000000

4)
1 P" J2 Y2 M& v6 G0.000000
5 r) }7 u' l# f2 k7 e7.500000

NO. ITERATIONS=
2

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE
$ G- D! b, e2 m  zCURRENT
ALLOWABLE
( A6 _8 j0 v2 JALLOWABLE

COEF
, \$ _& [* F+ k" Q; k4 V/ GINCREASE
$ R! n: c) a0 O2 nDECREASE

3 |  u3 H6 Y' F8 YX
. x! L# |6 x7 i6 L  ?7 I% a10.000000
INFINITY
2.500000

5 [5 C! w6 V  G/ j# OY
9 H  \5 I" A; {' m. r. b  O8 }6 w( @15.000000
' C8 t6 ?7 h6 S0 ]) B: p2 S/ }9 f5.000000
15.000000

RIGHTHAND SIDE RANGES

ROW
) q3 _: }5 d: J* S( a) q5 f3 gCURRENT
3 r) B9 r8 e. g3 N; d( E" ^/ JALLOWABLE
ALLOWABLE

RHS
  o: Q. ~1 y# i6 }INCREASE
DECREASE

- R7 o; V' ?# @0 O& X/ ~2
10.000000
8 j* g9 P1 f0 S' g8 c4 t$ f9 k* H6.000000
; T4 P3 R, v3 @4 }( g# g10.000000

3
12.000000
INFINITY
9.000000

' s% p; u) G5 n: B1 v' K! I& o4
16.000000
18.000000
# d% ?) |- B$ p* c+ R8 Q% ]$ z6.000000

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
7 t. [0 T3 q" `. A. Rreport 窗口得到以下结果：

: j8 |% r( q) s9 HGlobal optimal solution found at iteration:
% ], ^  G. F) N: l# `) n; h2

: Z2 m1 @2 n; ^1 E) v! zObjective value:
, c& K& f9 n7 |8 Z& k) _7.750000

9 k/ Y' D4 U5 [2 M+ p; T+ BVariable
Value
' w: E8 ^: L. B/ E% D. h: a" zReduced Cost

/ l7 h) x6 C3 N& Y& R0 WX1
) [8 h$ g1 b- l& ^# ]: S0.5000000
0.000000

X3
1 u& H  s$ K' |  F0.2500000
0.000000

, y( j- n, _7 q/ K4 b
) R# d5 G3 _- @5 c  y; d# D) GX2
% u3 I5 g. k5 ^7 l0.000000
8 v2 z, {' P; @- p0.5000000

X4
0.000000
2.750000

X5
0.000000
2.250000

Row
0 ?, ^& n3 C1 C3 v* sSlack or Surplus
' A! {/ u, ^1 n: hDual Price

6 H+ M' x1 l6 X/ s/ |/ \( H1
2 b& C5 h) ~- @3 {, @7.750000
-1.000000

2
$ M+ X, \4 \% f1 I0 \& ^: A, V  S0.000000
/ F; _/ U+ h' P% H" V7 U-2.750000

3
0.000000
-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 b) O- y4 {) M4 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:
/ B2 O2 Y. i! r) t7 w! l' X+ V* J2

+ \  b# s; _& m' F$ q% VObjective value:
3 ~! e% ^9 H9 C: F7 q. O4 S5 F/ A
14500.00

- c  a: D4 Q5 ?: r2 _+ T2 }- |Variable
Value
Reduced Cost

/ m* _6 J* N/ u1 K$ VX
; ^, a! ?7 o/ h( |8 ~100.0000
0.000000

Y
30.00000
0.000000

! K& ?; a! Q7 g- U2 \  X+ @& oRow
) T2 y2 y( G7 s' Z5 l" o# @' bSlack or Surplus
/ `5 D- t8 {7 w( J$ @$ s- w( `Dual Price

1
14500.00
: E4 i  v# W: p; ?6 V, m1.000000

2
0.000000
25.00000

( U2 p) J$ H3 q2 U3
90.00000
: `2 y' v3 |: l: @/ e% B0.000000

4
9 h: x: o" o8 B# b) P. e, ]; N2 [7 q0.000000
; A. t1 O% Z" w7 O5 ^6 w3 D/ ?
75.00000

loooog12

先下下来看看