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LINDO (Linear INteractive and Discrete Optimizer)

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Title: LINDO (Linear INteractive and Discrete Optimizer)

1
????(?)
2

LINDO (Linear INteractive and Discrete
Optimizer)
?? ?? ??? ?? ??? ????

T????? ??

Max. Z 40X1 30X2 s. t. 4X1 5X2
200 2X2 50 6X1
3X2 210 X1, X2 0
LINDO ????
3
LINDO ????
4

LINGO ??? ??? ?? ? ?? ???? ??? ????

LINGO ????
LINGO ????
5

?? ?? D????? 1?? ????? ?? ??
?????(portfolio) ????
????? ?????

???? ?????
A?? 7.3
B?? 10.3
M?? 6.4
N?? 7.5
? ? 4.5
6

????
?????? 50??? ????? ???? ???.
?? ???? ??? ???? ???? 25??? ??? ??.
B??? ?????? ??? ????? ??? ????? ? 60???
???? ???.

???? (????? ????, ?? ??)     X1 A??? ???
??     X2 B??? ??? ??     X3 M??? ??? ??
    X4 N??? ??? ??     X5 ??? ??? ??
????(?????? ???) Max. Z 0.073X1
0.103X2 0.064X3 0.075X4 0.045X5

????
    s. t. X1 X2 X3 X4 X5     10,000
  (?????)            X1 X2
                         5,000   (?????? ??)
                         X3 X4
5,000   (?????? ??)           -0.25X3
- 0.25X4 X5 0      (???? ??)
-0.6X1 0.4X2
0     (B???? ???? ??)
X1, X2, X3, X4, X5 0
??? B????? ?? ???? ??? X2/(X1 X2)
0.6?? ??? ???.

D??????? ?? ??

?? ? ???? ??? ??/????? ?????
X1 2,000 0 1 0 0.069
X2 3,000 0 2 0 0.022
X3 0 0.011 3 1,000 0
X4 4,000 0 4 0 -0.024
X5 1,000 0 5 0 0.030
????? ????? 800 800 800 800
10

?? ????
A??, B??, N??, ??? ?? 2,000, 3,000, 4,000,
1,000(??)? ??,
????? 800?? (?? 8? ?????? ??)

?? ??
X3 0??, ? ????? 0.011 ? M???? ???? ?????
0.011?? ??
? ?? ???? ???? ?? 1,000 ? ?????? ????? 5,000??
1,000? ? ???? ?? ????? ??
? ?? ???? ???? ? 0.022 ? ?? ????? ?? ??? ??
?? ??? 0.022? ??? ?? ? ??? ??
? ?? ???? ?????? -0.024 ? ??? ???? ??? ?????
??? ??? 0.024? ?? ??? ??
?? ?? ?????? ???? ? 0.03
? B????? ?? ???? ??? ????? ?? 60??
(1/?????)?? ????? 0.03? ????? ??? ? ???
??

????? ??

X1 ??? ???? ?? 0.018 0.103 ? A??? ?? ????
7.3??, ??? 1.8??? ????? 10.3?? ? ??? ????
??? ??
? ?? ???? ?????? ?? 3,750 10,000 ? ??????
??? ?? 3,750?? ????? 10,000?? ?? ? ??? ????
???????

?? ?? E??? ???? ????
???? ??(???? ??)

??? ????? ??? ??????
??????? 9.0 1.0 2.5
????? 7.0 1.8 3.0
??????? 8.4 0.8 3.0

??? ??? 500??, ????? 1,200 ??, ?? ???? ? 600?
??? ??? ?????? ??? 100?? ???? ?? ??

????(? ???? ??? ??) X1 ?????????? ????
X2 ????????? ???? X3 ??????????
????
????(3???? ???? ???)
Max. Z 9X1 7X2 8.4X3
????
   s. t.     X1 1.8X2 0.8X3 500
(??? ??)            2.5X1      3X2 3X3
1,200 (???? ??)                 X1
X2       X3 600  (???? ??)
                     X2
100 (????? ?????)
X1, X2, X3 0

E????? ?? ??

?? ? ???? ??? ??/????? ?????
X1 240 0 1 0 6.0
X2 100 0 2 0 1.2
X3 100 0 3 160 0
????? ????? 3,700 4 0 -7.4
16

E????? ?? ??

?? ???? ?? ?? ??? ?? ??
X1 9.0 7.0 10.5 1 420 540
X2 7.0 -8 14.4 2 1,100 1,500
X3 8.4 7.2 10.2 3 440 8
4 33.33 180

?????? ???????, ?????, ???????? ?? 240, 100,
100 ? ??? ???? ??, ? ???? ?? ?? ???? 240, 180,
80, ????(??)? 600, 300, 300 ??, ??????? 3,700??.

????? ??

?? ??
? ?? ???? ???? ? 160 ?? ??? 600-160 440?
??, ? ????? 73(440/600) ???? ???? ??.

?? ??
? ?? ???? ?????(???? ????) 6 ???? 1?? ????,
???? 6??? ??? ??? ??
???? ?? ?? ???? 540?? ??? ??? ????? 6? ?? ???
??
???? 1,100???? 1,500????? ?? ????? 1.2? ??? ??
??????? ?????(?? 100) ????? ??? 7.4?? ?? ??

?? ?? T????? ???? ?? ?? ?? ?? A, B? 1kg,
2kg ???? ?? 1kg ?? ???? ?? ? ????? 1kg? ??,
?????? ?? ?? ???? ??? ??? ???? ??
1. ??? ?? A? ?? 1kg? ???? ???? K? ????. 2. ???
?? B? ?? 1kg? ???? ???? M? ????. 3. ??? ??????.
? ??? ??kg? ?? ???? 2,100, ???? K? M? ??
-100, 150
????? ???? ??? 250?
? ??? ???? 5,000kg? 7,000kg

????(?????? ??????, ?? kg)
X1 ??? ??? X2 ???? K? ??? X3
???? M? ??? X4 ????? ??? ?
????(???, ?? ?)
Max Z 2,100X1 - 100X2 150X3 - 25X4

????
s. t. X1 0.5X2                   5,000
(?? A? ??? ??)         2X1
0.5X3       7,000 (?? B? ??? ??)           -X1
0.5X2 0.5X3 X4 0      (???? ??)
X1, X2, X3, X4 0
??A, B? ??? ?? ???? K? M? 1kg ??? ?? 0.5kg?
?????
???? ?? ??? ???? ??? ??? ?(X1)? 3?? ????
???? ??? ?(0.5X2 0.5X3 X4)? ???? ???

T???? ??? ?? ??

?? ? ???? ??? ??/????? ?????
X1 3,500 0 1 0 50
X2 3,000 0 2 0 900
X3 0 -175 3 0 -250
X4 2,000 0 ????? ????? 6,550,000

????? ??

?? ?? ?? A 3,500kg, ?? B 7,000kg?? ?????
3,500kg ?? ? ?? 3,500kg ?, 2,000kg? ????,
1,500 kg? ?? A? 11? ???? ???? K? 3,000kg
??
?????? ? ??? ??? ????, ? ????? kg? ?? 50??
900??
???? ??? ??? 1,400 ?? ?????, ???? M? ??? ???
325?? ??? ??? ???? ???? ?
??A, B? ?? ??? ?? 5,000? 7,000??, ? ?? ??? ??
?? 50, 900? ????? ??? ? ???, ? ???? ? 7,000?
10,000???

?? ?? M???? ???? ?? ?? ?? ?? ??
????? ?? ?????? ??
??? ? ??? ??? ?? ??? ??
? ?? ??? ?? ?? ??? ??
1000?? ??? ?? ???? ?? ??
1. ??? ? ??? ??? 400?? ??, 2. ??? ?? ??? ???
400?? ??, 3. ??? ???? ???? ??? ?? ??, 4. ??? ?
?? ? ??? 40? ???, 5. ??? ?? ?? ? ??? 60?
??? ???? ??

??? ??(?? ??)

? ? ??? ????
?? ?? ?? 2 2.5
?? ?? ?? 1.8 2
26

????(? ??? ??? ???) X1 ??? ? ???? ????
????? ??? X2 ??? ? ???? ????? ????? ???
X3 ??? ?? ???? ???? ????? ??? X4 ??? ??
???? ????? ????? ???
????(??? ???, ?? ??) Min. Z 2X1
2.5X2 1.8X3 2X4

???? s. t. X1 X2 X3 X4 1000
(? ??? ??)           X1 X2
      400   (1? ?? ??)
             X3 X4 400   (2? ?? ??)
        -X1 X2 - X3 X4 0    (3? ??
??)     -0.4X1 0.6X2              0
(4? ?? ??)               -0.6X3 0.4X4
0    (5? ?? ??)             X1, X2, X3, X4
0
???, 4? ??? ?? ??? ??? ? ?? ? ??? ????
???(X2)? ??? ? ??? ? ??? ?? 40, ?, 0.4(X1
X2)????? ???, X2 0.4(X1 X2)? ???? ??

M??? ??? ?? ??

?? ? ???? ??? ??/????? ?????
X1 240 0 1 2 3 4 5 6 0 0 -200 -40 0 0 -1.92 -0.28 0 0 -0.50 -0.20
X2 160 0 1 2 3 4 5 6 0 0 -200 -40 0 0 -1.92 -0.28 0 0 -0.50 -0.20
X3 240 0 1 2 3 4 5 6 0 0 -200 -40 0 0 -1.92 -0.28 0 0 -0.50 -0.20
X4 360 0 1 2 3 4 5 6 0 0 -200 -40 0 0 -1.92 -0.28 0 0 -0.50 -0.20
????? ????? 2,032 1 2 3 4 5 6 0 0 -200 -40 0 0 -1.92 -0.28 0 0 -0.50 -0.20
29

????? ?? ?????? ??? 480??, ???? 520??,
?? ??????? ??? ?? ?? 400??, ?? ?? ??
600??? ???? ?? ???? 2,032?? ??
1, 4, 5?? ???? ???? ? ?? ??? ??? ?? ??
0.28(??), 0.5(??), 0.2(??) ? ???? ?? ??
????? ???? ?? ? ???? ??? ??? ??? ? ???
???? ??

????? ??
1? ??? ??, ??? ?? ??? ?? ??? ??(?? 400)?
1??? ?? ??? 0.28? ??? ?????, ? ???? 500????
?

?? ?? A??? ??? ?? ?? ?? 24?? ??, 8??? ??
?? ??
???? ?? ????

???(?) 2-6 6-10 10-14 14-18 18-22 22-02
????(?) 5 20 30 15 50 10
32

????(? ???? ??? ???? ???? ?)
X1 2?? ?????? ???? ?    X2 6?? ??????
???? ?    X3 10?? ?????? ???? ?    X4
14?? ?????? ???? ?    X5 18?? ?????? ???? ?
   X6 22?? ?????? ???? ?

????(? ??????? ???)
Min. Z X1 X2 X3 X4 X5 X6
???? s. t.    X1 X6    5 ( 2- 6 ????
???? ??)             X1 X2 20 ( 6-10
???? ???? ??)             X2 X3 30
(10-14 ???? ???? ??)             X3 X4
15 (14-18 ???? ???? ??)            X4 X5
50 (18-22 ???? ???? ??)             X5
X6 10 (22-2 ???? ???? ??)
X1, X2, X3, X4, X5, X6 0

A?? ??? ?? ??

?? ? ???? ??? ??/????? ?????
X1 5 0 1 0 -1
X2 30 0 2 15 0
X3 0 0 3 0 -1
X4 40 0 4 25 0
X5 10 0 5 0 -1
X6 0 0 6 0 0
????? ????? 85
35

?? ???? ??

50
??????
? ? ? ?
40
??????
30
20
10
2
6
10
14
18
22
02
???
36

?? ??
? ?????? ? 85?, ?????? ? 6-10???? 15?,
14-18??? 25?
1, 3, 5?? ???? ?????? -1 ? ???? ??? ???? ??
1? ? ???? ????? (? ??????)? 1? ????? ?? ??
2, 4?? ???? ?????? 0 ? ????? ???? ?? ???? ?
????? ???? ??? ??

?? ?? C??? ?????? ?? ????
??? ? ?????(?? ?)

? ? 4? 5? 6?
??A 2,000 5,000 3,000
??B 4,000 2,000 1,000

3?? ?? ??A, ??B? ???? 500?, 400??? 6?? ??
??? ???? ???? ????? ?
?? ????? ??? ??? ???? ? 2,000?
?? 1?? ??? ??? ????? ?? 0.05??? 0.04??
3? ??? ?????? ?? 300??, 500??, 200??
?? ???(?? ??) ???? ???? ????????

??? ??????

? ? 4? 5? 6?
??A 30 20 40
??B 20 15 30

? ??? ??? ????? 0.5? 0.4
???????? ????? ??? 0.3, ????? 0.2
3? ????? 5,000 ? ?? 3??? ????

????(?? ??? ???, ???, ???????, ?? ?)
  X1, X2, X3 ??A? 4, 5, 6? ???    Y1,
Y2, Y3 ??B? 4, 5, 6? ??? IX1, IX2, IX3
??A? 4, 5, 6?? ??? IY1, IY2, IY3 ??B? 4,
5, 6?? ???    I1, I2,   I3 4, 5, 6??? ??
???    D1, D2, D3 4, 5, 6??? ?? ???
Ii ? Di? ? ??? ??? ??? ??? 0? ??? ??? Ii -
Di? ?? ? ???? ?? ???? ??

????(??? ???, ?? ??)
Min. Z 30X1 20X2 40X3 20Y1 15Y2
30Y3 0.5(IX1 IX2
IX3) 0.4(IY1 IY2 IY3)
0.3(I1 I2 I3) 0.2(D1 D2 D3)
????
   X1 - IX1 1,500          (??A? 4? ??)
    Y1 - IY1 3,600          (??B? 4? ??)
    IX1 X2 - IX2 5,000   (??A? 5? ??)
IY1 Y2 - IY2 2,000   (??B? 5? ??)     IX2
X3 - IX3 3,000   (??A? 6? ??)     IY2 Y3 -
IY3 1,000   (??B? 6? ??)    IX3 500
               (??A? 6?? ???)     IY3
400                   (??B? 6?? ???)

????
    X1 IY1 2,000          (4?? ???? ??)
IX2 IY2 2,000         (5?? ???? ??)     IX3
IY3 2,000         (6?? ???? ??)     0.05X1
0.04Y1 300 (4?? ??????)     0.05X2 0.04Y2
500 (5?? ??????)     0.05X3 0.04Y3 200 (6??
??????)     X1 Y1 - I1 D1 5,000 (4?? ????
??)     X2 Y2 - X1 - Y1 - I2 D2 0 (5??
???? ??)     X3 Y3 - X2 - Y2 - I3 D3 0
(6?? ???? ??)
?? ??? ???
??? ???? ?? ????, ???? ???(Xi Yi)?? ? ???
???(Xi-1 Yi-1, 4?? ??? 5,000)? ?? ?? ?? ??
?? ?????(Ii - Di)? ?? ?? ???.

C?? ??? ?? ?? - 1

?? ? ???? ??? ??/????? ?????
X1 1500 0 1 0 -30.0
X2 7000 0 2 0 -20.0
X3 1500 0 3 0 -20.5
Y1 3600 0 4 0 -15.5
Y2 2000 0 5 0 -39.8
Y3 1400 0 6 0 -40.3
IX1 1400 0 7 0 -30.2
IX2 1400 0 8 0 0
43

C?? ??? ?? ?? - 2

?? ? ???? ??? ??/????? ?????
IX3 500 0 9 2,000 0
IY1 0 4.9 10 0 18.8
IY2 0 4.9 11 1,100 0
IY3 400 0 12 81 0
I1 100 0 13 70 0
I2 3900 0 14 69 0
I2 0 0.5 15 0 0.3
D1 0 0.5 16 0 0.3
D2 0 0.5 17 0 -0.2
D3 6100 0 ????? ????? 392,830
44

????? ??

??????

?? 4? 4? 5? 5? 6? 6? ??
?? ??A ??B ??A ??B ??A ??B ??
????? 500 400 0 0 2,000 0 ?????
??? 1,500 3,600 7,000 2,000 1,500 1,400 2,000?
??? 2,000 4,000 5,000 2,000 3,000 1,000 3????
????? 0 0 2,000 0 500 400 5,000?
???? 75 144 350 80 75 56 ??????
?????? 300 300 500 500 200 200 ??A0.05
???? 81 81 70 70 69 69 ??B0.04
46

????
????? ?? ???(9-11? ???)? ???? 0, 18.8, 0
5? ????(??? 2,000)? ??? ??? ??? 18.8?? ????
? ??? ??
????? ?? ????? ?? 0 ?? ????? 81, 70, 69???
???? ??
???? ???? (??? ??? ???)(??? ?? ??) ? ??
???? ???, 4? 5,100-5,000 100,
5? 9,000-5,100 3,900, 6?
2,900-9,000 -6,100 ???, ???? 0.3(100
3900) 0.26100 2,400
? ?? ?????(??A? 4? ??)? ???? -30 4? ???
???? ????? ??? ???? ??? 30 ?? ??? ???? ??
(?? 1400 3120)