y

1
??? ??????
4.1 ????????? 4.2 ?????????? 4.3
????????? 4.4 ?????????? 4.5 ????????? 4.6
????????
y
2
??????
????? ?????
gi(x)?0????
x????
f(x)????
??????n? ??????m??
????
???? ????? ????
????????
??????? ??????
??????????????
3
4.1 ?????????
??????
????
???????????????????????,????????????????
?????????????????????????,?????????????????
????
?????????????????????,??????????,?????????????
4
?1 ??????????
50???
??480??
????100??A1
??
??????,???????

• 35????1???,?????,????????

• ???????,??????????????

• A1?????? 30?/??,?????????

5
??480??
????100??A1
x1?????A1
x2?????A2
????
?? 243x1
?? 164 x2
????
????
??????(LP)
????
????
????
????
????
6
???????
??????
xi?????????xi?????
A1,A2?????????????????
???
xi?????????xi?????
???????A1,A2?????????????????
xi?????????xj????
A1,A2?????????????????
???
???????A1,A2?????????????????
xi?????????xj????
???
xi????
??A1,A2????????
7
????
???
????
zc (??) ???
?B(20,30)??????
??????????????
??????????????????
??????????????
???????????
8
????
????
LINDO 6.1
max 72x164x2 st 2)x1x2lt50 3)12x18x2lt480 4)3x1lt1
00 end
OBJECTIVE FUNCTION VALUE 1)
3360.000 VARIABLE VALUE
REDUCED COST X1 20.000000
0.000000 X2 30.000000
0.000000 ROW SLACK OR SURPLUS
DUAL PRICES 2) 0.000000
48.000000 3) 0.000000
2.000000 4)
40.000000 0.000000 NO.
ITERATIONS 2
DO RANGE (SENSITIVITY) ANALYSIS?
No
20?????A1, 30???A2,??3360??
9
????
OBJECTIVE FUNCTION VALUE 1)
3360.000 VARIABLE VALUE REDUCED
COST X1 20.000000
0.000000 X2 30.000000
0.000000 ROW SLACK OR SURPLUS DUAL
PRICES 2) 0.000000
48.000000 3) 0.000000
2.000000 4)
40.000000 0.000000 NO.
ITERATIONS 2
max 72x164x2 st 2)x1x2lt50 3)12x18x2lt480 4)3x1lt1
00 end
?????
????
?????
??????40
?? ???????????(????)
10
OBJECTIVE FUNCTION VALUE 1)
3360.000 VARIABLE VALUE REDUCED
COST X1 20.000000
0.000000 X2 30.000000
0.000000 ROW SLACK OR SURPLUS DUAL
PRICES 2) 0.000000
48.000000 3) 0.000000
2.000000 4)
40.000000 0.000000 NO.
ITERATIONS 2
????
????????1????????
????
????1??, ????48
????1??, ????2
???????????
35 lt48, ???!

• 35????1???,????

• ???????????????????

2?!
11
DO RANGE(SENSITIVITY) ANALYSIS?
??????????????????
Yes
RANGES IN WHICH THE BASIS IS UNCHANGED
OBJ COEFFICIENT RANGES VARIABLE
CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X1 72.000000 24.000000
8.000000 X2 64.000000
8.000000 16.000000
RIGHTHAND SIDE RANGES ROW
CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE
DECREASE 2 50.000000
10.000000 6.666667 3
480.000000 53.333332
80.000000 4 100.000000
INFINITY 40.000000
(??????)
x1????(64,96)
x2????(48,72)
x1???24 ?372???30?390,??????
??!

• A1????? 30?/??,????????

12
????
???????????????????
(??????)
RANGES IN WHICH THE BASIS IS UNCHANGED
OBJ COEFFICIENT RANGES VARIABLE
CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X1 72.000000 24.000000
8.000000 X2 64.000000
8.000000 16.000000
RIGHTHAND SIDE RANGES ROW
CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE
DECREASE 2 50.000000
10.000000 6.666667 3
480.000000 53.333332
80.000000 4 100.000000
INFINITY 40.000000
??????10
??????53
???10?!

• 35????1???,????????

13
?2 ??????????
??1??????
50???, 480??
??100??A1
??????,????????

• 30????1???,3????1????,????????150?,??????

• B1,B2??????10???,????????

14
??x1 ?? A1, x2 ?? A2,
????
X3?? B1, x4?? B2
x5?? A1??B1, x6?? A2??B2
????
??
????
????
????
????
????
????
15
????
OBJECTIVE FUNCTION VALUE 1)
3460.800 VARIABLE VALUE REDUCED COST
X1 0.000000 1.680000
X2 168.000000 0.000000 X3
19.200001 0.000000 X4
0.000000 0.000000 X5
24.000000 0.000000 X6
0.000000 1.520000 ROW SLACK OR SURPLUS
DUAL PRICES 2) 0.000000
3.160000 3) 0.000000
3.260000 4) 76.000000
0.000000 5) 0.000000
44.000000 6) 0.000000
32.000000 NO. ITERATIONS 2
????
LINDO 6.1
16
OBJECTIVE FUNCTION VALUE 1)
3460.800 VARIABLE VALUE REDUCED COST
X1 0.000000 1.680000
X2 168.000000 0.000000 X3
19.200001 0.000000 X4
0.000000 0.000000 X5
24.000000 0.000000 X6
0.000000 1.520000 ROW SLACK OR SURPLUS
DUAL PRICES 2) 0.000000
3.160000 3) 0.000000
3.260000 4) 76.000000
0.000000 5) 0.000000
44.000000 6) 0.000000
32.000000 NO. ITERATIONS 2
????
????168 ??A2?19.2 ??B1, ??3460.8(?)
8??????A1,42??????A2, ????24??A1?????B1
???????????
17
30????1???,3????1????,????????150?,??????
????
OBJECTIVE FUNCTION VALUE 1)
3460.800 VARIABLE VALUE REDUCED COST
X1 0.000000 1.680000
X2 168.000000 0.000000
X3 19.200001 0.000000 X4
0.000000 0.000000 X5
24.000000 0.000000 X6
0.000000 1.520000 ROW SLACK OR
SURPLUS DUAL PRICES 2) 0.000000
3.160000 3) 0.000000
3.260000 4) 76.000000
0.000000 5) 0.000000
44.000000 6) 0.000000
32.000000
??1????????3.161237.92
??1?????????3.26
??150???5???,???189.6??(???????????)
18
????
B1,B2????10???,???????
RANGES IN WHICH THE BASIS IS UNCHANGED
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE
DECREASE X1 24.000000
1.680000 INFINITY X2
16.000000 8.150000 2.100000
X3 44.000000 19.750002
3.166667 X4 32.000000
2.026667 INFINITY X5
-3.000000 15.800000 2.533334
X6 -3.000000 1.520000
INFINITY
B1????10,??X3 ??????
B2????10,??X4 ??????
????????
???????????x3?????39.6??,???????????
19
4.2 ??????????
????
?????????????????????,?????????????,?????
?????????,???????????,??????,?????,????????
20
?1 ?????
??900?/??
??
????450?/??

• ??????????,?????????

• ???????????,???????????

21
????
lt ????120180300
????160
???900?160144,000(?)
??900?/??
??
?????
????450?/??
????450?16072,000(?)
???????????
22
????
??3????4???????
????
??i ?j ??????? xij(x340)
????
????
??????(LP)
????
????
23
????
OBJECTIVE FUNCTION VALUE 1)
24400.00 VARIABLE VALUE REDUCED COST
X11 0.000000 30.000000 X12
50.000000 0.000000 X13
0.000000 50.000000 X14
0.000000 20.000000 X21
0.000000 10.000000 X22
50.000000 0.000000 X23
0.000000 20.000000 X24
10.000000 0.000000 X31
40.000000 0.000000 X32
0.000000 10.000000 X33
10.000000 0.000000
????? 24400(?)
?????-????-?????144000-72000-24400 47600(?)
24
????
??????????????
????(320) gt ????(300)
???????????
?? ??(900) ????(450) ?????
????
????
????????
B, C ????
25
??
OBJECTIVE FUNCTION VALUE 1)
88700.00 VARIABLE VALUE REDUCED COST
X11 0.000000 20.000000
X12 100.000000 0.000000 X13
0.000000 40.000000 X14
0.000000 20.000000 X21
30.000000 0.000000 X22
40.000000 0.000000 X23
0.000000 10.000000 X24
50.000000 0.000000 X31
50.000000 0.000000 X32
0.000000 20.000000 X33
30.000000 0.000000
??? 88700(?)
???????????? (Transportation Problem)
26
?2 ????
????????(?),????(?3)
????
????????????????????
????,??????????
27
????
????
????????????
???????????????????
????????,???????
????
xij--?i ??????j ??????(?) i1,2,3,4, j1,2,3
(??????????)
????
28
????
????
xij--?i ??????j ??????
????(??)
????
????
????
29
????
????
xij--?i ??????j ??????
????
????
????
30
????
????
OBJECTIVE FUNCTION VALUE 1)
121515.8 VARIABLE VALUE 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
??2??10,??5 ??3 ??13, ??3??4 ??3?
?????121516?
????? ?????
????
31
?1 ???????
4.3 ?????????
????????????,???????????????????,????????????

• ???????,?????????

• ??????????,??????80?, ???????????????

32
???????
????
???????????????????x1, x2, x3
??????(LP)
33
????
OBJECTIVE FUNCTION VALUE 1)
632.2581 VARIABLE VALUE REDUCED
COST X1 64.516129
0.000000 X2 167.741928
0.000000 X3 0.000000
0.946237 ROW SLACK OR
SURPLUS DUAL PRICES 2)
0.000000 0.731183
3) 0.000000
0.003226
?????,????
1)?????x164,x2167,???????z629,?LP???632.2581??
???
2)????x165,x2167x164,x2168?,?????z,?????????
????

• ????????????????????

3) ???????x1, x2, x3 ????,?????
34
????(Integer Programming,??IP)
????
IP??LINDO????
max 2x13x24x3 st 1.5x13x25x3lt600 280x1250x24
00x3lt60000 end gin 3
IP ????
OBJECTIVE FUNCTION VALUE 1)
632.0000 VARIABLE VALUE REDUCED COST
X1 64.000000 -2.000000
X2 168.000000 -3.000000 X3
0.000000 -4.000000
gin 3???3??????,??? gin x1 gin x2 gin x3
IP ????x164,x2168,x30,???z632
35
???????

• ???????,?????80?,??????

?
??1???8?LP???
??3???????,??????,???????,???????,????
?
?
x180,x2 150,x30,???z610
36

• ???????,?????80?,??????

??2??0-1??,??????
x10 ? ?80
M?????,??1000
OBJECTIVE FUNCTION VALUE 1)
610.0000 VARIABLE VALUE REDUCED COST
X1 80.000000 -2.000000 X2
150.000000 -3.000000 X3
0.000000 -4.000000 Y1
1.000000 0.000000 Y2
1.000000 0.000000 Y3
0.000000 0.000000
LINDO??0-1????? int y1 int y2 int y3
?????
37

• ???????,?????80?,??????

??3???????
x10 ? ?80
?????(Non- Linear Programming,??NLP)
NLP?????????????(?LINGO, MATLAB),???????????????
????,?????????????????????,??????????
38
?2 ???????

• ?????????1500????A
• ??????500??????10000?/?
• ?????500?????1000??,??500?? ??8000?/?
• ?????1000??,??1000????6000?/??

????????????? ?
39

• ????????? - ????A???
• ????A?????????????

????
????
??A????,??A, B?????,????
????
??(??)
c(x) ????A???
c(x)?????
40

• x? 500????10??/?
• 500?? x? 1000?,??500??8??/?
• 1000?? x? 1500?,??1000??6??/??

????
????
????
41
?????A?????
????

• ?????c(x)??????,??????
• ??????????c(x),??????????????????
• ????????,?????????

42
??1
????
x1 , x2 , x3 ???10, 8, 6(??/?)??A???
x x1x2x3, c(x) 10x18x26x3
x x1x2x3, c(x) 10x18x26x3
????

• 500?? x? 1000?,??500??8??/?

????10??/??????x1500(?)?,???8??/??????x2
???????,???LINGO??
43
??1LINGO??
Model Max 4.8x11 4.8x21 5.6x12 5.6x22
- 10x1 - 8x2 - 6x3 x11x12 lt x 500 x21x22
lt 1000 x11 - x21 gt 0 2x12 - 3x22 gt
0 xx1x2x3 (x1 - 500) x20 (x2 -
500) x30 x1 lt 500 x2 lt 500 x3 lt 500 x gt
0 x11 gt 0 x12 gt 0 x21 gt 0 x22 gt 0 x1 gt 0 x2
gt 0 x3 gt 0 end
Objective value 4800.000 Variable Value
Reduced Cost X11 500.0000
0.0000000E00 X21 500.0000
0.0000000E00 X12 0.0000000E00
0.0000000E00 X22 0.0000000E00
0.0000000E00 X1 0.1021405E-13 10.00000
X2 0.0000000E00 8.000000 X3
0.0000000E00 6.000000 X 0.0000000E00
0.0000000E00
????500???A?500???B?????,???????A,???4,800???
LINGO?????????,??????????
44
y1, y2 , y31 ???10, 8, 6(??/?)??A
??2
x1 , x2 , x3 ???10, 8, 6(??/?)??A???
????
y1,y2,y3 0?1
OBJECTIVE FUNCTION VALUE 1)
5000.000 VARIABLE VALUE REDUCED COST
Y1 1.000000 0.000000
Y2 1.000000 2200.000000
Y3 1.000000 1200.000000
X11 0.000000 0.800000
X21 0.000000 0.800000
X12 1500.000000 0.000000
X22 1000.000000 0.000000
X1 500.000000 0.000000
X2 500.000000 0.000000
X3 0.000000 0.400000
X 1000.000000 0.000000
0-1??????,??LINDO??
??1000???A,????500???A?1000???B??,?????,???5,000??
?
????1???
45
????????????c(x)
??3
b1? x?b2,x z1b1z2b2, z1z21,z1, z2?0, c(x)
z1c(b1)z2c(b2).
b1 b2 b3 b4
b2 ? x? b3,x z2b2z3b3, z2z31,z2, z3 ? 0,
c(x) z2c(b2)z3c(b3).
b3 ? x? b4,x z3b3z4b4, z3z41,z3, z4 ? 0,
c(x) z3c(b3)z4c(b4).
46
??k1,2,3
??3
bk?x?bk1 ,x zkbkz k1 bk1 zkzk1 1,zk, zk1
?0, c(x) zkc(bk)zk1 c(bk1 ).
bk?x?bk1?yk1,??,yk0
IP??,LINDO??,????????2??.
????????,??3?????
47
4.4 ??????????
????
???????????????,???????,????????????????????,?????
??????????,??????????
????????,??????????????????,?????????????,????????
??????,????????????
48
?1 ?????????
5?????????
????????4?100????????
?????????1152??????????575, ???????????????
?????????????5!120??
49
0-1????
cij(?)??i ?j ????????
?????i????j ???,?xij1, ???xij0
????
??????????
????????1?
????
50
????
??LINDO??
???x14 x21 x32 x43 1,
?????0 ???253.2(?)4132
MIN 66.8x1175.6x1287x1358.6x14
67.4x5171 x5283.8x5362.4x54 SUBJECT TO
x11x12x13x14 lt1 x41x42x43x44
lt1 x11x21x31x41x51 1
x14x24x34x44x54 1 END INT 20
? ????? ???? ???? ??.
51
???c43 69.6?75.2,????c5462.4 ? 57.5, ???????
??
??????
IP???????LP????????,LINDO???????????????????
c43, c54 ??????????,?LINDO??
???x21 x32 x43 x51 1, ???4177
? ???? ???? ???? ???
??(Assignment)??????????????,????????,????,??????
????.
52
?2 ????
????????????????????????
??????????,??????? ?
??????,??????,??????? ?
53
0-1????
????
xi1 ????i ???(xi0 ??)
????
????????
??2????,3?????, 2??????
????
54
0-1????
????
??????
x31??x1 x2 1
????(LINDO)
??? x1 x2 x3 x6 x7 x9 1,
???06???,???21
55
????????,?????,????????
????
????
???(???)??
??????????????????

• ????????,???????

• ????????,???????

56
?????

• ??????????????????

??? x1 x2 x3 x5 x7 x9 1,
???0????21??22?
????????!
??x9 1 ??x6 1
LINDO???????????????
57
?????

• ????????????????,?????

??? x1 x2 x3 x4 x5 x6 x7 x9
1, ???0???28?
58
?????
?????
?????10,?21?????????
?????11,?20?????????
59
4.5 ?????????

• ??????

???????
???????

• ??????

????????????
?????????????
60
?1 ???????????
????4?????????????
?????????? 0.2???

• ??????, ???????, ?4???????

• ?4??????????,????15??????,?????????5??,?????????

61
????

• ??4????????????????
• ????????
• ??????????

• ?????1????????
• ???????, ?4???????
• ???????????????
• ?????????????????

????
62
????
????
x1 x4?14?????
y1 y3?13?????
???0.2 (??/???)
????
??????????
????
????
????
63
????
LINDO??
??? x1 x415,40,25,20 y1
y3 0,15,5 .
4??????????528 (??)
64
????

• ?4??????????,????15??????,?????????5??,?????????

??????????
0-1??wt wt1 ??????t?(t1,2,3,4)
????
????
??????????????
65
????
??????
????????1?
0-1??wt wt1 ??????t?(t1,2,3,4)
LINDO??
??? w11, w2 , w3, w40 x1
x415,45,15,25 y1 y30,20,0 .
????528????527??
???????????????, ????????????????
66
?2 ?????????
???????????????????? ???????????, ????????8???
????4?????????????
?????????? 0.2???

• ??????, ???????, ?4???????

67
???????????
???????0
ct ??t ????(?/?) ht ??t (?)???(?/?) st ??t
?????(?) dt ??t ????(?) Mt ??t ????(?)?
??????, ????,??T??????????
??
????
xt ??t ??? yt ??t (?)??? wt 1 ??t ???? (wt
0 ???)?
??
68
???????????
??0-1????
?????????,?LINDO??
???x1 x415,40,45,0???554.0(??)
69
6 ????????
??????
????????????????,???????????
??????,??????,???????,?????
70
?1 ????
??1. ??????? ?
?????????
??2. ??????
????????????,??????????,??????????3??????????
71
????
????
????????????????????????
???????????????????????
72
??????
??????1
???????,?????????,?????????????,?????
????
1. ???????????
2. ???????????
73
????
xi ??i ????????????(i1,2,7)
??1(???)
??
????
???? xi ???
???x212, x515, ???0 ???27?
???2??12?,???5??15?,??27?
74
??????1
??2(???)
??????
???x215, x55, x75, ???0 ???25?
???2??15?,???5??5?,???7??5?,?25?,??35?
???1??????27?,??27? ??
?????8?,????2?
????????,???????????
75
??????2
??????5?10????????3??
??4???4?50?,5?10?,6?20?,8?15?,????????????,?????
??????,??????????????
????
xi ??i ????????????(i1,2,3)
r1i, r2i, r3i, r4i ?i ??????,????????4??5??6??8?
???????
76
??????2
????(???)
???????????3?
????
????
???? xi ,r1i, r2i, r3i, r4i (i1,2,3)???
?????????
77
????,?????,????
??????2
??4?50?,5?10?,6?20?,8?15?
???????19?
?????????
????????????? ??1???4?4???,?13? ??2???1?5??2?6
???,?10? ??3???2?8???,?8?? ?????????1310831
?????????
78
LINGO???????????
Local optimal solution found at iteration
12211 Objective value
28.00000 Variable Value Reduced Cost X1
10.00000 0.000000 X2
10.00000 2.000000 X3 8.000000
1.000000 R11 3.000000
0.000000 R12 2.000000
0.000000 R13 0.000000
0.000000 R21 0.000000
0.000000 R22 1.000000 0.000000
R23 0.000000
0.000000 R31
1.000000 0.000000
R32 1.000000 0.000000
R33 0.000000
0.000000 R41
0.000000 0.000000
R42 0.000000 0.000000
R43 2.000000
0.000000
??1?????????3?4??1?6???,?10? ??2?????????2?4??1
?5??1?6???,?10? ??3?????????2?8???,?8?? ????????
28??
79
?2 ?????
????1 ???,??24cm,5???
????2 ???, 32?28cm, 2???
???10cm,????????5cm?
????40??,???????0.10?,??????0.001? /
cm2(???????????????)
?????????
80
????
????????????
??1 ??? ??24cm
??????d5cm,???h10cm?
??1 ???? 242-10??d2/4 - ?dh222.6 cm2
81
????
??????????????????????
?????????????????
???????????40?? ??????1(??1
3)5??, ??2(??4)2??
??????????? ?
xi ???i ????????(i1,2,3,4) y1
?????????? y2 ???????? y3 ??????????
????
????
82
y1 ?????y2 ?????? y3 ????????
????
???????0.10?,????0.001? / cm2
?????dh157.1 cm2 ?????d2/419.6 cm2
??
????
(40??)
????
????
83
????
y1 ?????y2 ?????? y3 ????????
????
??xi?y1,y2,y3????,????????,?????????,??????????? ?
84
????
LINDO??????????????????,???????,?????????
?????????10000?(xi ??,yi ??)
??2??40125?, ??3??3750?, ??4??20000?, ?????160250?
(?????????), ????4298?
OBJECTIVE FUNCTION VALUE 1)
0.4298337 VARIABLE VALUE REDUCED COST
Y1 16.025000 0.000000 X1
0.000000 0.000050 X2
4.012500 0.000000 X3
0.375000 0.000000 X4
2.000000 0.000000 Y2
0.000000 0.223331 Y3
0.000000 0.036484
85
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