数学建模实验六

数学建模实验六

一、上机用Lindo 软件解决货机装运问题。

某架货机有三个货仓：前仓、中仓、后仓。三个货舱所能装载的货物的最大重量和体积都有限，如表所示，并且，为了保持飞机的平衡，货舱中实际装载货物的重量必须与其最大容许重量成正比例

三个货舱装载货物的最大容许重量和体积

四类装运货物的信息

应如何安排装运，使该货机本次飞行获利最大？ 解答过程： 模型建立：

决策变量：用x ij 表示第i 种货物装入第j 个货舱的重量（吨），货舱j=1、2、3分别表示前仓、中仓、后仓。

决策目标是最大化总利润，即Max Z=3100（x11+x12+x13）+3800（x21+x22+x23）+3500（x31+x32+x33）+2850（x41+x42+x43） 约束条件为：

1） 共装载的四种货物的总重量约束，即 x11+x12+x13<=18 x21+x22+x23<=15 x31+x32+x33<=23 x41+x42+x43<=12

2)三个货舱的重量限制，即 x11+x21+x31+x41<=10 x12+x22+x32+x42<=16 x13+x23+x33+x43<=8

3）三个货舱的空间限制，即

480x11+650x21+580x31+390x41<=6800 480x12+650x22+580x32+390x42<=8700 480x13+650x23+580x33+390x43<=5300

4）三个货舱装入重量的平衡约束，即

8

43

33231316423222121041312111x x x x x x x x x x x x +++=+++=+++

模型求解

将以上模型输入LINDO求解

程序如下：

max

3100x11+3100x12+3100x13+3800x21+3800x22+3800x23+3500x31+3500x32+3500x33+2850x4 1+2850x42+2850x43

st

x11+x12+x13<=18

x21+x22+x23<=15

x31+x32+x33<=23

x41+x42+x43<=12

x11+x21+x31+x41<=10

x12+x22+x32+x42<=16

x13+x23+x33+x43<=8

480x11+650x21+580x31+390x41<=6800

480x12+650x22+580x32+390x42<=8700

480x13+650x23+580x33+390x43<=5300

8x11+8x21+8x31+8x41-5x12-5x22-5x32-5x42=0

x12+x22+x32+x42-2x13-2x23-2x33-2x43=0

end

得到结果如下：

LP OPTIMUM FOUND AT STEP 7

OBJECTIVE FUNCTION V ALUE

1) 121515.8

VARIABLE V ALUE REDUCED COST

X11 0.000000 400.000000

X12 0.000000 57.894737

X13 0.000000 400.000000

X21 10.000000 0.000000

X22 0.000000 239.473679

X23 5.000000 0.000000

X31 0.000000 0.000000

X32 12.947369 0.000000

X33 3.000000 0.000000

X41 0.000000 650.000000

X42 3.052632 0.000000

X43 0.000000 650.000000

ROW SLACK OR SURPLUS DUAL PRICES

2) 18.000000 0.000000

3) 0.000000 300.000000

4) 7.052631 0.000000

5) 8.947369 0.000000

6) 0.000000 0.000000

7) 0.000000 5453.289551

8) 0.000000 0.000000

9) 300.000000 0.000000

10) 0.000000 3.421053

11) 310.000000 0.000000

12) 0.000000 437.500000

13) 0.000000 -1750.000000

NO. ITERATIONS= 7

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

V ARIABLE CURRENT ALLOWABLE ALLOWABLE

COEF INCREASE DECREASE X11 3100.000000 400.000000 INFINITY

X12 3100.000000 57.894844 INFINITY

X13 3100.000000 400.000000 INFINITY

X21 3800.000000 INFINITY 0.000000

X22 3800.000000 239.473801 INFINITY

X23 3800.000000 0.000000 239.473801

X31 3500.000000 0.000000 INFINITY

X32 3500.000000 2656.730713 122.222427

X33 3500.000000 239.473801 0.000000

X41 2850.000000 650.000000 INFINITY

X42 2850.000000 650.000000 110.000221

X43 2850.000000 650.000000 INFINITY

RIGHTHAND SIDE RANGES

ROW CURRENT ALLOWABLE ALLOWABLE

RHS INCREASE DECREASE

2 18.000000 INFINITY 18.000000

3 15.000000 3.000000 5.000000

4 23.000000 INFINITY 7.052631

5 12.000000 INFINITY 8.947369

6 10.000000 INFINITY 0.000000

7 16.000000 0.000000 1.000000

8 8.000000 INFINITY 0.000000

9 6800.000000 INFINITY 300.000000

10 8700.000000 580.000000 1700.000000