111łłłnnnøøøµµµııı888iii yyyflflflkkkïïï˝˝˝’’’{{{ ·...
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êÆï�Ø%Ï£��Æ��
111���nnnùùùµµµõõõ888III555yyy¯̄̄KKKïïï������{{{
4�ÀìÀ�Æ, O�Å�Æ�EâÆ�
Email:baodong@sdu.edu.cn
�ùÌ�SN
1 õ8I5y¯KVã
2 õ8I5y�.���/ª
3 õ8I5y�.�)�½Â
4 õ8I5y�.�¦)�{
5 ï�Y~
�!õ8I5y¯KVã
õ8Iûü¯K´�¬!+n�F~)¹¥²~���¯K, X
«�+nûü¥�²²²LLLuuuÐÐÐ����¸̧̧���ooo¯K;
Ý]ûü¥�ÂÂÂÃÃÃ�ºººxxx¯K;
J�¥�kkk ����ÅÅÅÀJ¯K;
p�W���¥�ÆÆÆ����;;;���¯K.
ùa¯KÏ~�±V)¤õõõ���888III���ûûûüüü¯K, ù888III���mmm~~´���ppp���^̂̂Úgggñññ�, XÛ²ïù8I, ÙûüL§�©E,, ûüöÏ~éJ�Ñ�ªûü. )ûùa¯K�ï��{Ò´õõõ888IIIûûûüüü���{{{.
~~~1 ,½OyuÐ�¬Süe��cÝ���85y§Oy�cSüoÝ]Ø�L8 ·�. ²LÐÏçÀÀ¥12 ��ø�ħz��8I�Ý]�êþ£ü Z��¤!ï¤��c|d£ü Z��¤!zc¢Ôü�þ£ü �ë¤Úu^�NÄå£ü Z<¤XeL¤«§��o�¸T½\¾��I?Ö§«ì#O¢ÔþØ�L20 �ë§l²L��Ý�¦|d¦�U�p§l�¬uÐ��Ýù�¦#OÒ�k ¦þõ§¯AXÛÀJÝ]�8º
�8 1 2 3 4 5 6 7 8 9 10 11 12Ý] 2 4 5.2 11 6.2 17 21 3.5 6.1 4.8 15 8.5 30|d 0.4 1 3 2 4 5 0.7 1.5 1.2 4 2.3 6¢ïÔ 0.3 2 3 3 3 5 1 0.5 1.4 2 2 4NÄå 0.6 1.1 2 2.8 1.5 2.6 0.7 1.5 1 2 2 1.2
¯K©Û
ûüCþµ´ÄÝ]º0 - 1 5y¯K§P
xi =
{1, Ý]1i ��8;0, ØÝ].
Pai , bi , ci , di ©O�1i ��8�Ý]!|d!¢ïÔ�Ñ!NÄåI¦, i = 1, 2, · · · , 12.Ý]µ
∑12i=1 aixi
|dµ∑12
i=1 bixi¢ïÔ:
∑12i=1 cixi
NÄå:∑12
i=1 dixi
�.ïá
���...£££ããã max12∑i=1
bixi
max12∑i=1
dixi
s.t.
12∑i=1
aixi ≤ 80,
12∑i=1
cixi ≤ 20,
xi = 0, 1, i = 1, 2, · · · , 12.���...AAA���©©©ÛÛÛµµµ
�å´�5�ê�å
8Ikü�§þ��5¼ê
Ñ�¦���
õ8I5y¯K�Ê�'���:
1 ûûûüüüCCCþþþ x = (x1, · · · , xn)T .2 888III¼¼¼êêê f(x) = (f1(x), · · · , fm(x))T ,m ≥ 2.
3 ���111)))888 ��/,
X ={x ∈ Rn |gi(x) ≤ 0, hj (x) = 0, i = 1, · · · , p, j = 1, · · · , q
}Ù¥ X ⊆ Rn .�X = Rn �,¡¯K´ÃÃÃ���ååå���.
4 ûûûüüü ÐÐÐ �Nûüö3�8I�m� ÐÐЧÝ, ½ö�5§Ý.
5 )))���½½½Â 3®�ûüö Ð��¹e, ½Â f 3X þ��`).
�!õ8I5y�.���/ª
minx
f(x) = (f1(x), · · · , fm(x))T
s .t .
gi(x) ≤ 0, i = 1, 2, · · · , p;hj (x) = 0, j = 1, 2, · · · , q ;x ∈ Rn ,
½
maxx
f(x) = (f1(x), · · · , fm(x))T
s .t .
gi(x) ≤ 0, i = 1, 2, · · · , p;hj (x) = 0, j = 1, 2, · · · , q ;x ∈ Rn ,
eP
X ={x ∈ Rn |gi(x) ≤ 0, hj (x) = 0, i = 1, · · · , p, j = 1, · · · , q
}��1�, Kþã�.��d/��
minx∈X
f(x) ½ maxx∈X
f(x).
��·�?Ø�õ8Iûü¯K�üüü<<<õõõ888IIIûûûüüü, k��ûüöØ���<�, ¡��õõõ<<<õõõ888IIIûûûüüü, �¡+++ûûûüüü¯̄̄KKK.
n!õ8I5y�.�)�½Â
õ8I¯K�õ�8I�m *dgñ, nØþ��Ø�3¦��8I¼êÓ����Ð�);
¢�¥, o�±GØ�²wØÜn½Ø�U�ÀJ��1);
Äu)�ÀJ½(½�{ØÓ, õ8I5y¥kX�«)�½Â, ù´õ8I`z¯K�ü8I`z¯K�m�����«O.
�BuLã, �ù�ÄXe���¯K��.:
minx∈X
f(x). (1)
Ù¥, x ∈ Rn �ûüCþ, X ⊂ Rn ��1�,f(x) = (f1(x), · · · , fm(x))T �8I¼ê�þ.
1. Pareto(ø\÷) k�)
P Y = {f(x)|x ∈ X} ��ûüCþ�H¤k�1�¥�:�, 8I¼ê�þ���N|¤�8Ü.½½½ÂÂÂ1 ¡ y ∈ Y �õ8I5y¯K(1)�Pareto rrrkkk���:::, eé ∀y ∈ Y, Ñk y ≤ y; éA/, ¡ x ∈ X �Paretorrrkkk���))), e f(x) ≤ f(x),∀x ∈ X.`̀̀²²²:
½Â¥/≤0L«/�u�u���k��©þ´î��u�0.
XJrk�)�3, K¯KÒ´¦z�8I¼ê fi(x)3 X þ��`).
du¢S¯K¥~~8I�m�pÀâ, ù«rk�)��´Ø�3�. ¤±·��a,��´±e�ü�Vg.
1. Pareto(ø\÷) k�)
½½½ÂÂÂ2 ¡ y ∈ Y �Pareto kkk���:::, eØ�3y ∈ Y ¦� y ≤ y; éA/, x ∈ X ¡�Pareto kkk���))), e f(x) ´Pareto kkk���:::.
½½½ÂÂÂ3 ¡ y ∈ Y �Pareto fffkkk���:::, eØ�3 y ∈ Y ¦� y < y; éA/,¡ x ∈ X�Pareto fffkkk���))), ef(x) � Pareto fffkkk���:::.
2.þþþ���)))
�8I¼êkn�: f∗ = (f ∗1 , · · · , f ∗m) —dûüö�â�< Ð(½.��/ f∗ 6∈ f(X).
rõ8I5y¯K=z�ü8I`z¯K
minx∈X
m∑i=1
ωi | fi(x)− f ∗i |p, , (2)
Ù¥ 1 6 p < +∞, �Xê ωi > 0 ÷vm∑i=1
ωi = 1.
2.þþþ���)))
½½½ÂÂÂ4 ¡ x ∈ X �þþþ���))), e x �¯K(2) ����`); ¡ y ∈ Y �þþþ���:::, e�3þ�) x , ¦� y = f(x).
þ�:�AÛ¿Âé²w, §Ù¢Ò´3,«ål�½Âeåln�:�C�:, k�·�¡da�{�nnn���))){{{.
¯K(2) ��±±Ù¦aq�¯K5�O, ~X±e�����¯K
min max1≤i≤m
ωi | fi(x)− f ∗i |s.t. x ∈ X
(3)
3 ÷¿)
��1)8� X , �¯K�¦ m �8I¼êfi ���Ð.
k�ûüö�Ï"�$, ¦�Ñ m �z�αi , =� x ∈ X ÷v fi(x) ≤ αi , i = 1, · · · ,m�, Ò@� x ´�±�É�!´÷¿�.
ù�� x Ò¡���÷÷÷¿¿¿))).
÷¿)�VgÌ�´lûüL§�Ý, �âûüö� Ð��¦ JÑ�.
o!õ8I�.�¦)�{
rõ8I¯Kz�,«¿Âeü8I¯K?1¦)´~^�¦)da`z¯K��{.
��5¿�´, 3õ8I5y¥, Ø�8I¼ê��´*dÀâ, �k,��A:: 8I¼ê�Ø�úÝ5.¤±Ï~3¦)c, Aké8I¼ê?1ý?n.
ý?n(½5�z)�SN�):(1) Ãþjz?n: z�8I¼ê�þjÏ~´Ø���, 3?1\�¦)�duþj�Ø�úÝ5, I�k?1Ãþjz?n.(2) êþ?�8�z?n: ���8I¼ê�êþ?�É���, N´Ñy�ê¯�êy�, =êþ?���8I3ûü©ÛL§¥N´Ó`, l K�ûü(J.
1 �5\�{
���555\\\���{{{: òõ8I¼ê�5\��ü8I`z¯Kµ
minm∑i=1
λi fi(x)
s.t. x ∈ X(4)
Ù¥, λ = (λ1, · · · , λm)T ���þ, ÷vm∑k=1
λk = 1.
��Xê λk ���¢Sþ�N8I fk(x) 3ûüö%8¥��é�§Ý, k = 1, 2, · · · ,m.
�5\�{´õ8I5y¯K¦)�^��2���{��.
2. ε �å{
ÄÄÄ���ggg���:�âûüö� Ð, ÀJ��Ì�'5�ë�8I, ~X fk(x), òÙ¦ m − 1 �8I¼ê�å���å^�¥�.
äN/, �.=z�:
min fk(x)
s.t.
{fi(x) ≤ εi (i = 1, · · · ,m, i 6= k)x ∈ X
(5)
Ù¥ëê εi , i = 1, 2, · · · , k − 1, k + 1, · · · ,m �ûüö¯k�½�.
555ºººµµµε �å{�¡ÌÌÌ���888III{{{½ëëë���888III{{{, ëêεi ��u´ûüöé1 i �8I ó�NNNNNN���ÉÉÉzzz���.
2. ε �å{
ε �å{�y1 k ��8I�|Ã, Ó�q·�ì�Ù¦8I, ù3Nõ¢Sûü¯K�¦)¥¹Éûüö� O.
õ8I¯K(1)�z�� Paretok�)Ñ�±ÏL·�/ÀJëê εi(i = 1, · · · ,m, i 6= k), ^ ε �å{¦�.
3¢SO�¥, XÛ(½ëê εi ,i = 1, 2, · · · , k − 1, k + 1, · · · ,m º
1 XJz� εi ��Ñé�, K¯K(5) ék�UÃ�1);2 XJ εi ����, K8I fk (x) ����UÒ��.3 ?nù�¯Kk�{, ~X, �ûüöJø
f ∗k , min{fk (x) | x ∈ X
}(k = 1, · · · ,m)
Ú,��1) x ?�8I�
(f1(x), · · · , fm(x))T ,
,�ûüö�â²�½�¦(½εi ��.
3 n�:{
Ä�g�: ¦lz��½�n�:
f = (f 1, · · · , f m)T
3,«ål¿Âeål�á��1), =3�1� X ¥,Ϧ¦� f(x) � f ����: x.~^5£ã ��¼êkµ(1) p ���¼¼¼êêê:
dp(f(x), f ;λ) =[ m∑k=1
λk | fk(x)− f k |p] 1
p .
Ù¥, 1 6 p 6 +∞; λ = (λ1, λ2, · · · , λm) �8I¼ê����þ.(2) 444��� ���¼¼¼êêê:
d+∞(f(x), f ;λ) = max16k6m
λk | fk(x)− f k | .
(3) AAAÛÛÛ²²²þþþ¼¼¼êêê:
d(f(x), f) =[ m∏k=1
| fk(x)− f k |] 1
m .
3þã¼ê½Â¥, ��þ¥��Xêλk > 0 ´¯k�½�.
3¢SO��, k��~�O�ó�þ, dp(f(x), f ;λ)Ú d(f(x), f) ~©Od±eü�¼ê
bp(f(x), f ;λ) =m∑k=1
λk | fk(x)− f k |p ;
Ú
b(f(x), f) =m∏k=1
| fk(x)− f k | .
5�O.
4 ÷¿Y²{
¢Sþ,kNõûü¯KûüöæB��YØ�½Ñ´Pareto k��.
duûü�¸�K�!�Y¢�¥�(J½öO�¤^�¡��Ä, ûüö �¿JÑ�|8IY²
f = (f 1, · · · , f m)T ,
XJ�Y÷vù|8IY², KæB§.
|^{ü÷¿Y²�{¦)õ8Iûü¯K(1)�O�Ú½�:
1111ÚÚÚ 4ûüö�½8IY² f = (f 1, · · · , f m)T .1112ÚÚÚ ¦)
minm∑k=1
fk(x)
s.t.
{x ∈ Xfk(x) ≤ f k , k = 1, 2, · · · ,m.
(6)
1113ÚÚÚ
e¯K(6)Ã�1), K?\e�Ú;
e¦�¯K(6)��`) x, KÑÑ x;
ÄK¯K(6)¥�8I¼êÃe., �Ù?��1)ÑÑ.
1114ÚÚÚ 4ûüö#�Ñ8IY² f , £�12Ú.
ÊÊÊ!!!ïïï���YYY~~~µµµÝÝÝ]]]���ÂÂÂÃÃÃÚÚÚºººxxx
I ¯̄̄KKK���JJJÑÑÑ
½|þk n «]� Si(i = 1, 2, . . . ,n) �±ÀJ��Ý]�8
y^ê�� M �����]7����Ï�Ý]
ù n «]�3ù��ÏSï Si �²þÂÃÇ� ri , ºx��Ç� qi .
Ý]�©Ñ, o�ºx��, oNºx�^Ý]� Si ¥�����ºx5Ýþ.
ï Si ��G�´¤( ¤Ç pi), �ï�Ø�L�½�ui�, �´¤Uï ui O�.
b½ÓÏÕ1�±|Ç´ r0(r0 = 5%), QÃ�´¤qúx.
ï�Y~
®� n = 4 ��'êâXe:
Si ri(%) qi(%) pi(%) ui(�)
S1 28 2.5 1 103
S2 21 1.5 2 198
S3 23 5.5 4.5 52
S4 25 2.6 6.5 40
Á�Túi�O�«Ý]|Ü�Y, =^�½�]7 M , kÀJ/ïeZ«]�½�Õ1)E, ¦ÀÀÀÂÂÂÃÃ榦���UUU��� , �oooNNNºººxxx¦¦¦���UUU���.
ï�Y~
II ÄÄÄ���bbb���ÚÚÚÎÎÎÒÒÒ555½½½ÄÄÄ���bbb���:
1 Ý]ê� M ���, �BuO�, �b�M = 1;
2 Ý]�©Ñ, o�ºx��;
3 oNºx^Ý]�8 Si ¥�����ºx5Ýþ;
4 n «]� Si �m´�pÕá�;
5 3Ý]�ù��ÏS,ri , pi , qi , r0 �½�, ØÉ¿Ï�K�;
6 ÀÂÃÚoNºx�É ri , pi , qi K�, ØÉÙ¦Ï�Z6.
ï�Y~
ÎÎÎÒÒÒ555½½½:Si — 1 i «Ý]�8, X�¦, Å ri , pi , qi — ©O�Si�²þÂÃÇ!�´¤Ç!ºx��Çui — Si��´½�r0 — ÓÏÕ1|Çxi — Ý]�8 Si �]7a — Ý]ºxÝQ — oNÂÃ∆Q — oNÂÃ�Oþ
ï�Y~
III ¯̄̄KKK©©©ÛÛÛ������...ïïïááá
ûûûüüüCCCþþþ: ^uï1 i «]��Ý]�Ýxi , i = 0, 1, 2, · · · ,n. Ù¥, i = 0 L«�Õ1.888III¼¼¼êêê:
(1) oNºx����: ^¤Ý]� Si ¥�����ºx5ïþ, = max{qixi |i = 1, 2, · · · ,n}.(2) oNÂÃ��: Ý]�Ý� xi �1 i «]�, ÙÂÃ�rixi , Ó�Uì5½, I�G�½�Ý�ÃY�´¤, ¤G�´¤´��©ã¼ê, =
�´¤ =
{pixi , xi > ui
piui , xi ≤ ui
dK8¤�½�½� ui( ü : �) �éoÝ] M é�,piui��, �±�ÑØO, ù�ï Si �ÀÂÃ�(ri − pi)xi .¯¢þ8I¼ê¥�~þé�`)üÑ¿ÃK�.¤±oNÂÃ�
max
n∑i=0
(ri − pi)xi .
ï�Y~
III ¯̄̄KKK©©©ÛÛÛ������...ïïïááá
�å^�:
(1)�K�å: é¤k�8�Ý]�ÝØA�K�, =xi ≥ 0, i = 0, 1, · · · ,n.(2)oÝ]�å: ¤k^uÝ]�o]7�Ý� M , 5¿�éÝ],���8 Si ó, ¢S¤^düÜ©|¤, �Ü©��)ÂÃ�X{Ý]xi , ÙÂÃ� rixi , ,�Ü©�þ���´¤, =ÃشļÃ, 7L�B�ÃY¤. Ù¤^���Ý]�Ýk', � pixi , Ïd¢S^u�8 Si �oÝ]�Ý�: xi + pixi = (1 + pi)xi . �oNÝ]�å�
n∑i=0
(1 + pi)xi .
���...ïïïááá:
÷vÀÂæ�U�, oNºx¦�U��êÆ�.£ã�:
8I¼ê
maxn∑
i=0(ri − pi)xi
min{max{qixi}}
�å^�
n∑
i=0(1 + pi)xi = M
xi ≥ 0, i = 0, 1, . . . ,n
ù´��õ8I5y�., ¢S¦)�Ï~ÏL,«�{z�ü8I¯K¦).
�.¦)
IV ���...{{{zzz:���...1µµµ �½ºxY², `zÂÃ/^ ε �å�{�g�, e`k�ÄÂÃ8I, K�±�½Ý
]ö�±«É�ºx��þ�K�, Ø�P� a, ¦�����ºx qixi/M ≤ a, ù�rõ8I5yC¤ü8I��55y¯K.
8I¼ê: Q = max
n∑i=0
(ri − pi)xi
�å^�:
qixiM ≤ a, i = 1, 2, · · · ,nn∑
i=0(1 + pi)xi = M ,
xi ≥ 0, i = 0, 1, . . . ,n
ï�Y~
���...2µµµ �½J|Y², 4�zºxerºx8I��Ì�8I, K�±�ÑÝ]öF"oJ|�
���Y² k��¹e, «É�ºx�������Ý]|Ü.
8I¼ê: R = min{max{qixi}}
�å^�:
n∑i=0
(ri − pi)xi ≥ k ,
n∑i=0
(1 + pi)xi = M ,
xi ≥ 0, i = 0, 1, . . . ,n
ï�Y~
���...3 �5\�¦Ú�{Ý]ö3�ï]�ºxÚýÏÂÃü�¡�, Ï~¬�âg�
�²L¢åÚéºx�«ÉUå, ÀJ��-gC÷¿�Ý]|Ü. Nyéºx!ÂÃ8I��ï�{, ´éùü�8I¼ê©OD�� λ Ú 1− λ, ¡ λ(0 < λ ≤ 1) �Ý] ÐXê.
8I¼ê: min λ{max{qixi}} − (1− λ)
n∑i=0
(ri − pi)xi
�å^�:n∑
i=0
(1 + pi)xi = M ,
xi ≥ 0, i = 0, 1, 2, . . . ,n
ï�Y~
V ���...1 ���¦¦¦))) éL¥�½�êâ, �.1 �:
min f = (−0.05,−0.27,−0.19,−0.185,−0.185)(x0 x1 x2 x3 x4)T
s.t.
x0 + 1.01x1 + 1.02x2 + 1.045x3 + 1.065x4 = 1
0.025x1 ≤ a
0.015x2 ≤ a
0.055x3 ≤ a
0.026x4 ≤ a
xi ≥ 0(i = 0, 1, . . . , 4)
du a ´?¿�½�ºxÝ, �.N��½vk��OK, ØÓ�Ý]ökØÓ�ºxÝ"Ïd·�l a = 0 m©, ±Ú�4a = 0.001 ?1Ì�|¢, ?�§SXe:
MATLAB ¦)§S
a=0:0.001:0.1;n=length(a);Q=zeros(1,n);for i=1:nc=[-0.05 -0.27 -0.19 -0.185 -0.185];Aeq=[1 1.01 1.02 1.045 1.065];beq=[1];A=[0 0.025 0 0 0;0 0 0.015 0 0;0 0 0 0.055 0;0 0 0 0 0.026];b=[a(i);a(i);a(i);a(i)];vlb=[0,0,0,0,0]; vub=[];[x,fval]=linprog(c,A,b,Aeq,beq,vlb,vub);x=x’; Q(i)=-fval;endplot(a,Q,’.’)xlabel(’a’),ylabel(’Q’)
MATLAB ¦)(J
ï�Y~
VI (((JJJ©©©ÛÛÛdO�(J9©Ûã, ��±e(Ø:
1 ºx�, ÂÃ��
2 �Ý]�©Ñ�, Ý]ö«ú�ºx��, ù�K¿��. =kx�Ý]ö¬Ñy8¥Ý]��¹, �Å�Ý]öK¦þ©ÑÝ]
3 ã¥�þ�?�:ÑL«TºxY²����UÂÃÚTÂÃ�¦���ºx. éuØÓºx�«ÉUå, ÀJTºxY²e��`Ý]|Ü
4 3a = 0.006 NCk��=ò:, 3ù�:�>, ºxO\é��, |dO�é¯; 3ù�:m>, ºxO\é��, |dO�é�ú. =U:�¦ ∆Q/Q �����:.
ï�Y~
éuºxÚÂÃvkAÏ Ð�Ý]ö5`, ATÀJ��$:���`Ý]|Ü, ��´a∗ = 0.6%,Q∗ = 20%, ¤éAÝ]�Y�:
ºxÝ ÂÃ x0 x1 x2 x3 x40.0060 0.2019 0 0.2400 0.4000 0.1091 0.2212
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