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Œ˘%ˇ£˘ 1nøı8I5yK{ 4 ˘, O¯˘E˘ Email:[email protected]

111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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Page 1: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

êÆï�Ø%Ï£��Æ��

111���nnnùùùµµµõõõ888III555yyy¯̄̄KKKïïï������{{{

4�ÀìÀ�Æ, O�Å�Æ�EâÆ�

Email:[email protected]

Page 2: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

�ùÌ�SN

1 õ8I5y¯KVã

2 õ8I5y�.���/ª

3 õ8I5y�.�)�½Â

4 õ8I5y�.�¦)�{

5 ï�Y~

Page 3: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

�!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���{{{.

Page 4: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

~~~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

Page 5: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

¯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

Page 6: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

�.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�¦���

Page 7: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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��`).

Page 8: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

�!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 ,

Page 9: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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.

Page 10: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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.

Page 11: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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.

Page 12: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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���:::.

Page 13: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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.

Page 14: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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)

Page 15: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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�.

Page 16: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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.

Page 17: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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���{��.

Page 18: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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���.

Page 19: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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 ��.

Page 20: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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 | .

Page 21: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

(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.

Page 22: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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§.

Page 23: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

|^{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.

Page 24: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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.

Page 25: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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���.

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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.

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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

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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 .

Page 29: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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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 .

Page 30: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

���...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¦).

Page 31: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

�.¦)

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

Page 32: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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���...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

Page 33: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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���...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

Page 34: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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:

Page 35: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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’)

Page 36: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

MATLAB ¦)(J

Page 37: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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 㥭�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 �����:.

Page 38: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,

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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Page 40: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 41: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 42: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 43: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 44: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 45: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 46: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 47: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 48: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 49: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 50: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 51: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 52: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 53: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 54: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
Page 55: 111łłłnnnøøøµµµııı888III yyyflflflKKKïïï˝˝˝’’’{{{ · o!ı8I˝.˙ƒ)’{rı8IflKz‘,«¿´eü8IflK?1ƒ)· ~^˙ƒ)da‘zflK˙’{. −ˆ5¿˙·, 3ı8I5y¥,
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