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 =
# M8 b& h- \! f  Z7 U6 i0 o-2

! Z) Z$ c. k+ G-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 =
1
! D% r. R1 I9 j1 d; c  G* D2
4
: t/ `! B9 Q% ]6 {' K-1

2
3
-1
1

1
/ m% R0 h- F$ ?+ W3 Z9 l5 R0
1 @" y; U( R5 \; ~/ f1
! {0 T5 ~( N) @3 v7 _: {5 K1

-1
" C( G/ ^% M  j4 ?) ?: j+ h& m: ~0
0
  |* N, N" J3 m  V! t, p/ o0

7 p. J7 h, ~  P2 x! b0
1 n6 u0 |! b1 k# n2 x2 a& |" u-1
0
0

0
0
-1
. Z  K; G7 p$ Y7 q, [2 [, j0

3 _2 e; C. ^# F: n3 u6 d% E0
0
# w8 y6 h; f! a0
-1

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

b =
6

12

4

/ e& u( P5 R; I4 r" c  k% M# ]0 m2 G- c0

- K- r2 E; V% K+ n+ n0 c1 c0

5 }% m5 U" v9 K  L( R- R$ h( f0

" M$ I) V5 P: @' e7 G0

Optimization terminated successfully.

The optimization solution of the programming is:

x =
0 {# m7 u9 N5 p2 p3 B7 W0 q0.0000

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
( D. {+ N' K$ Y+ M6 @5 @！注释符号; 系统默认为自变量>0, 若不要求用free命令.

& \/ k* n0 h- F2 I; X！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

VARIABLE
' m! w0 K0 z* b1 @0 E# C# I/ JVALUE
& t& W  z) j& e! t1 n+ gREDUCED COST

X
# t0 l) W$ q( l/ g10.000000
0.000000

5 S+ e- \+ o3 o* M& G$ pY
3.000000
6 Q$ h: _" d! d, `/ Y6 V0.000000

ROW
SLACK OR SURPLUS
DUAL PRICES

* A- u) S: q0 Q  {: B2)
* \4 L! ?" e, K0.000000
2.500000

! f2 X- t1 e. P1 y. Y% d3)
' t5 ^+ [" @  h& E0 \! F9.000000
; J* K/ h+ w" I: z0.000000

4)
0.000000
7.500000

NO. ITERATIONS=
7 h0 i2 d9 |. S! {) u# J! L2 Y2

, c  i8 O6 l$ A. Q6 [. o$ ARANGES IN WHICH THE BASIS IS UNCHANGED:

/ J+ D8 {5 P4 r; N% K6 r/ nOBJ COEFFICIENT RANGES

VARIABLE
- \9 |  [- s! ]/ V2 MCURRENT
3 X9 ?. P( o' F: qALLOWABLE
# v+ v1 l% X9 f1 |  l4 ?8 K) `, PALLOWABLE

COEF
- @" C$ k/ H) T. G+ tINCREASE
DECREASE

X
& B  V( j. t# y0 Y- a10.000000
INFINITY
2.500000

5 h3 ^" c5 y5 H- d3 _: d( p+ DY
8 o; g- c( `* I1 I15.000000
/ W8 f/ J3 E8 |' }% K$ y% B# l! y5.000000
15.000000

RIGHTHAND SIDE RANGES

ROW
) f" T/ J) N* q# u- _- P, JCURRENT
. |% j: ~1 Y7 w8 UALLOWABLE
7 B) Q4 A8 Q0 F) ~) s9 e8 MALLOWABLE

& E8 ~$ _# M/ w. l! G9 l+ {! YRHS
INCREASE
DECREASE

) W# D% h# w+ V& l* D* p* B7 j2
0 O7 f* A' T; {9 f4 T3 i! |10.000000
6.000000
10.000000

" D. h3 i9 n+ D2 |& m; V3
7 q  r  C/ A6 B3 I: v: `12.000000
$ O$ ^2 h' G; t/ g+ IINFINITY
9.000000

4
, P  l3 k# l7 ?7 e$ f: D! [8 i. M16.000000
' v% V/ k- v5 {* T! e6 z3 Q18.000000
# z& C7 E5 ]4 S0 o1 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
. P; R) t# P1 R1 d9 }, Rreport 窗口得到以下结果：

( K/ A- h3 k# A' |0 JGlobal optimal solution found at iteration:
2 A  x% q7 y, G( @' v$ K# q5 v% L2

Objective value:
7.750000

Variable
Value
' j! D0 f, J* L4 CReduced Cost

X1
0.5000000
0.000000

+ g4 h7 C9 |; Q/ ~- e9 V1 WX3
# K* f$ [; ^+ M4 h) v5 u3 s0.2500000
0.000000

9 D- S& z' s; \* ~, U, A/ G9 F$ U. eX2
: P2 ?. [( m: {/ a. q- I- q0.000000
0.5000000

X4
0.000000
2.750000

+ }, C; j! Z% z- I1 D" {X5
0.000000
2.250000

  `  O" C* q- T5 h6 x# u- |4 aRow
Slack or Surplus
1 l" Z/ |; Q1 ^: q4 MDual Price

1
7.750000
2 r) j# a! |; I7 O0 d-1.000000

2
- C% q" P3 Q$ Y& g2 i, O! x0.000000
-2.750000

3
( v# h6 b3 v, i( C$ E0.000000
-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;
, d8 q/ g; E$ r' S: ?8 ~8 X; q! 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

# w% K8 c0 Q3 o0 G/ h; h2 F% v( Y. wObjective value:
' B* K# L+ Z0 D% F) M7 L
14500.00

Variable
Value
& ?, q: y$ M. A! cReduced Cost

X
: j" I( v$ u9 q0 N100.0000
9 x4 ]" c' n, N; ^; x7 z0.000000

Y
3 @' ?) u& B3 F, B30.00000
0.000000

Row
; O) V9 o6 e* M1 ?Slack or Surplus
Dual Price

1
14500.00
1.000000

  V' E( q7 H6 U  L0 J8 J0 U2
0.000000
25.00000

; g% G) ]. h0 Z7 |0 t% v' A9 C/ ^5 D3
  {* A, z& S& m90.00000
3 M/ C/ c$ v6 v2 b3 j7 n' j) ]* p; v0.000000

4
0.000000

75.00000

loooog12

先下下来看看