A.A.Ee?ueyiia
CAAeA*A ?AAIIAANEss
Eaeoeee ii iaoaiaoe/aneei iaoiaeai iee?iyeiiiieee
Eaoaae?a aunoae iaoaiaoeee. N.IaOYO, 1996
YEIIIIE*ANEIA AAAAeAIEA
Aeaaa ia?aay. CAAeA*E ?AAIIAANIIAI OI?AAEAIEss
. . . ii naiie naiae i?e?iaea iaoaiaoe/aneea iaoiaeu
ia iiaoo i?eeaaaoueny iaiin?aaenoaaiii e aeaenoaeoaeueiinoe,
a oieueei e iaoaiaoe/aneei iiaeaeyi oiai eee eiiai e?oaa yaeaiee.
E.A.Eaioi?iae/ e A.A.Ai?noei [ , c.6].
NIAeA?AEAIEA IA?AIE AEAAU
2
1.1. Caaea/a cao?ao
1. Eeanneoeeaoeey caaea/.
2. Aaeoi?iua iaicia/aiey.
3. Oaaee/iia i?aaenoaaeaiea.
4. Eiee/anoaaiiay /anoue caaea/e cao?ao.
7
1.2. Oeaiiaay /anoue caaea/e cao?ao
1. Ioeaieaaiea ecaeaeee.
2. Oeaiiaua oneiaey ?aaiiaaney.
3. ?aaiiaaniua oeaiu ecaeaeee.
4. I?aaeea aeaienoaaiiiai niioaaonoaey.
5. O?ainiiie?iaaiea.
11
1.3. Caaea/a auionea
1. Oaaee/iia i?aaenoaaeaiea.
2. Eiee/anoaaiiay /anoue caaea/e auionea.
3. Oeaiiaay /anoue caaea/e auionea.
4. Eaiiie/aneay ia?a caaea/.
16
1.4. Caaea/a ?aaiiaaney
Oece/aneia niaea?aeaiea caaea/e ?aaiiaaney.
1.5. Enoi?ey e eeoa?aoo?a
1.1. Caaea/a cao?ao
1.Eeanneoeeaoeey caaea/. Ia/iai eco/aiea caaea/e ?aaiiaaney n i?inouo
yeiiiie/aneeo i?eia?ia.
?anniao?eaay ianniaia i?iecaiaenoai eaeeo-ieaoaeue iau/iuo ecaeaeee,
iai?eia? – no?ieoaeuenoai aeeeuo aeiiia (i?iecaiaenoai aaoiiiaeeae,
eiiiuethoa?ia e o.i.),- iu oaeaeei: anyeia oaeia aeaei ieacuaaaony
ninoiyuei ec aeaoo acaeiinaycaiiuo i?iecaiaenoa: i?iecaiaenoaa
no?ieoaeueiuo iaoa?eaeia (aaoiiiaeeueiuo aa?aaaoia, iee?inoai e i?i/.) e
nianoaaiii no?ieoaeuenoaa (nai?i/iiai i?iecaiaenoaa). I?e yoii,
i?iecaiaenoai no?ieoaeueiuo iaoa?eaeia i?aaenoaaeyao niaith i?ioeann
?aceiaeaiey neiaeiiai i?e?iaeiiai nu?uey a ?yae i?inouo ecaeaeee,
iai?eia?: e?oaeiai eana a aeinee noaiaea?oiuo ?acia?ia,- e iaiai?io:
no?ieoaeueiia i?iecaiaenoai anoue i?ioeann nai?ee ec i?inouo
no?ieoaeueiuo iaoa?eaeia ?acee/iuo neiaeiuo iino?iae. Aeey ian caeanue
aaaeii oi, /oi a ?acaeoii ia?iaeiii oicyenoaa iaa yoe i?iecaiaenoaa – e
i?iecaieueiue eaniieeueiue caaiae, e eaeay-ieaoaeue no?ieoaeueiay
a?oaeue – aeaenoaotho ia ?acee/iuo ?uieao: a iaoai neo/aa – ia ?uiea
ieeiiaoa?eaeia e ia ?uiea no?ieoaeueiuo oneoa,- e yaeythony, aiiaua
aiai?y, iacaaeneiuie ae?oa io ae?oaa. A oa?ieiao ia?iaeioicyenoaaiiie
iiaeaee “cao?aou-auione” Eaiioueaaa (ni.1.5.1) caaea/a ?aceiaeaiey
nu?uey yaeyaony caaea/ae cao?ao, a caaea/a nai?ee ecaeaeee – caaea/ae
auionea.
E?iia oiai: anyeee oi?aaeythuee i?iiuoeaiiui i?iecaiaenoaii, iacaaeneii
io oiai, aeaenoaoao ee ii a ia?a?aaaouaathuae eee nai?i/iie iaeanoyo
i?iiuoeaiiinoe, o/anoaoao ai aiaoiae ?uii/iie aeayoaeueiinoe aeaiyeei
ia?acii: e eae iio?aaeoaeue, iieoiathuee nu?uea aeey naiaai
i?iecaiaenoaa, e eae i?iecaiaeeoaeue, i?iaeathuee i?iecaaaeaiiua ei
ecaeaeey. Iieoiea nu?uey ninoaaeyao aai ?anoiae, a i?iaeaaea ecaeaeee –
aeioiae. Ii yoie i?e/eia, caaea/a ?acoiiiai oi?aaeaiey i?iiuoeaiiui
i?aaei?eyoeai ieacuaaaony aeey iaai ninoiyuae ec aeaoo caaea/: caaea/e
ieieiecaoeee ?anoiaeia e, iaeiia?aiaiii, – caaea/e iaeneiecaoeee
aeioiaeia oiai aea naiiai i?iiuoeaiiiai i?iecaiaenoaa. Oaeay ia?a caaea/
iacuaaaony acaeiii aeaienoaaiiie.
A eoiaa, iiiaeanoai caaea/ iao/iiai i?iecaiaenoaaiiiai oi?aaeaiey
ia?acoaony ec caaea/ /aou?ao aeaeia: ec caaea/e ?aceiaeaiey nu?uey e
caaea/e nai?ee ecaeaeee, eaaeaeay ec eioi?uo, a naith i/a?aaeue,
?aniaaeaaony a ia?o i?yiie e ae aeaienoaaiiie iiaecaaea/:
i?yiay iiaecaaea/a;
Caaea/a cao?ao:
aeaienoaaiiay iiaecaaea/a.
i?yiay e
Caaea/a auionea:
aeaienoaaiiay iiaecaaea/e.
Eo oi/iie iiaeaeueiie iinoaiiaea e iinayuaia ia?aay aeaaa iaoeo eaeoeee.
2.Aaeoi?iua iaicia/aiey. E i?iiuoeaiiia nu?uea, e ecaeaeey ec iaai
yaeythony oiaa?aie, e eae anyeea oiaa?u iienuaathony ia?ie
acaeiinaycaiiuo aaee/ei: eiee/anoaii q (io quantity) e oeaiie p (io
price). Iiyoiio iienaiea i?iecaiaenoaa eae i?aia?aciaaiey nu?uey a
ecaeaeey eiaao aeaei n aeaoiy eo naycaiiuie ia?aie: eiee/anoaaie e
oeaiaie nu?uey, e eiee/anoaaie e oeaiaie ecaeaeee. Aeey oaeianoaa
?acee/aiey yoeo aaee/ei oa ec ieo, eioi?ua ioiinyony e nu?ueaaui eee
ia?ae/iui oiaa?ai, iu aoaeai niaaaeaoue ia?aui cia/eii “1”, a
ioiinyueany e i?iecaiaeeiui eee aoi?e/iui oiaa?ai – cia/eii “2”,
iai?eia?: q 1 e p1, q 2 e p2 .
I?e eniieueciaaiee m aeaeia nu?uey aeey i?iecaiaenoaa n aeaeia
ecaeaeee: m, n = 1, 2, (, eae eo eiee/anoaa, oae e oeaiu noaiiayony
iiiaieiiiiiaioiuie eee aaeoi?iuie aaee/eiaie. A iao?e/iii en/eneaiee eo
i?aaenoaaeytho iaeiinoieaoeiauie eee iaeiino?i/iuie iao?eoeaie,
?acee/aiea eioi?uo naycaii n ianeiiao?e/iinoueth caeiia iao?e/iiai
oiiiaeaiey ii i?aaeeo “no?iea ia noieaaoe”. Iai aoaeao oaeiaii ia?aua
cia/ee eiee/anoaaiiui aaeoi?ai i?eienuaaoue naa?oo e eo ninoaaeythuea q
11 , (, q 1m e q 21 , (, q 2n a iao?e/iii i?aaenoaaeaiee caienuaaoue
a aeaea iaeiinoieaoeiauo m ( 1 e n ( 1 iao?eoe niioaaonoaaiii:
q 1 = q 11
(
q 1m
; q 2 = q 21
(
q 2n
;
a oa aea ia?aua cia/ee oeaiiaui aaeoi?ai iu aoaeai i?eienuaaoue nieco:
p1 e p2 , e eo ninoaaeythuea p1 1 , (, p1 m e p2 1 , (, p2 n
caienuaaoue a aeaea iaeiino?i/iuo 1 ( o e 1 ( n iao?eoe:
?1 = ( p1 1 ( p1 m ) ; ?2 = ( p2 1 ( p2 n).
Eiathuea iaeie e oa aea i?ino?ainoaaiiua ?acia?iinoe eiee/anoaaiiue e
oeaiiaue aaeoi?u iaeiiai e oiai aea iaai?ia oiaa?ia iu aoaeai iacuaaoue
acaeiii-aeaienoaaiiuie aaeoi?aie. Iie iaeaaeatho oai naienoaii, /oi eo
iao?e/iia i?iecaaaeaiea ii i?aaeeo “no?iea ia noieaaoe”, iai?eia?:
p1 q 1 = ( p1 1 ( p1 m)
q 11
(
q 1m
= p1 1 q 11 + ( + p1 m q 1m ( ( p1 , q 1 (,
aeaao iaeiieeaoi/ioth 1 ( 1 iao?eoeo eee “neaey?” (/enei) ( p1 , q 1
( – noiio iieiiiiiaioiuo i?iecaaaeaiee ia?aiiiaeaaiuo aaeoi?ia,
iacuaaaioth eo neaey?iui i?iecaaaeaieai eee, ei?ioei, naa?oeie yoeo
aaeoi?ia.
Ia i?ioyaeaiee anao iaoeo eaeoeee noi?i/iua eaoeineea aoeau n aeaoiy
cia/eaie aoaeoo iaicia/aoue iaeiiia?iua aaee/eiu eee /enea, oa aea aoeau
n iaeiei cia/eii – niioaaonoaothuea aaeoi?u, a aoeau aac cia/eia –
iao?eoeu eee iia?aoi?u. I?e/ai anaaaea ieaeiee cia/ie iao?e/iuo
ninoaaeythueo aoaeao ioia?iaaoue no?iee, a aa?oiee – noieaoeu.
3.Oaaee/iia i?aaenoaaeaiea. Caaea/a cao?ao i?aaenoaaeyao niaith
caaea/o ia?a?aaioee m acaeiicaiaiyaiuo aeaeia “neiaeiiai” nu?uey a n
aeaeia “i?inouo” ecaeaeee. A eeiaeiii neo/aa aa oaoiieiaey caaeaaony n(
m oaaeeoeae iaio?eoeaoaeueiuo /enae a1 1, (, an m :
al k [eiee/anoai l-ecaeaeee / ia aaeeieoeo k-nu?uey] ( 0 ;
l = 1, ( , n; k = 1, ( , m; m, n = 1, 2, ( ,
ninoaaeythueo iao?eoeo auionea a. A oeaeii, aianoa n aeaoiy ia?aie
aaeoi?ia q 1 e p1 , e q 2 e p2 anao naieo oiaa?ia, caaea/a cao?ao
iienuaaaony m(n+2(m+n) aaee/eiaie e anoanoaaiii i?aaenoaaeyaony a
neaaeothuai oaaee/iii aeaea:
q 11 ( q 1m
p2 1
(
p2 n a1 1 ( a1 m
( ( (
an1 ( an m q 21
(
q 2n
p11 ( p1 m
Anyeia i?iecaiaenoai, aoaeue oi ?aceiaeaiea nu?uey eee nai?ea ecaeaeee,
yaeyaony i?aia?aciaaieai nu?uey a ecaeaeey eae a ioiioaiee eo eiee/anoa,
oae e oeai:
q 1; p1 a
(
q 2; p2 ,
– e iiyoiio ec 2m+2n aai eiee/anoaaiiuo e oeaiiauo aaee/ei iaeia eo
iieiaeia i?aaeii?aaeaeyao ae?oaoth. Oae, a caaea/a cao?ao iai caaeaaony
?uii/iue ni?in ia auioneaaiua ecaeaeey (ieai eo i?iecaiaenoaa) a aeaea
iaio?eoeaoaeueiiai aaeoi?a ni?ina ecaeaeee q2 n n ninoaaeythueie:
q 2l [eiee/anoai. l-ecaeaeee] ( 0; l = 1, ( , n,
a aeiiieieoaeueiue aio aaeoi? q 1 ni?ina ia iio?aaeyaiia nu?uea
iiaeeaaeeo ii?aaeaeaieth a oneiaeyo caaeaiiuo oeai – iaio?eoeaoaeueiiai
aaeoi?a caeoii/iuo oeai nu?uey p1 n m ninoaaeythueie
p1 k [?oaee / ca aaeeieoeo k-nu?uey] ( 0; k = 1, ( , m.
Caaeaiiua iinoiyiiua caaea/e iacuaathony, oaeaea, aa ia?aiao?aie, a
eneiiua iaecaanoiua – ia?aiaiiuie. Aeey ioee/aiey ia?aiao?ia caaea/e io
aa ia?aiaiiuo iu aoaeai niaaaeaoue ia?aiao?u aeiiieieoaeueiui cia/eii –
iieeeii “ ( “ naa?oo.
4.Eiee/anoaaiiay /anoue caaea/e cao?ao. I?aaeeiaeaiea ecaeaeee. A
i?yiie /anoe caaea/e cao?ao ioiineoaeueii caaeaiiuo oeai p1 ia
iio?aaeyaiia nu?uea euaony iaeiaiaa ?anoiaeiia cia/aiea aai aaeoi?a
ni?ina q 1 . Ii yoie i?e/eia i?yiay /anoue caaea/e i?iecaiaenoaaiiiai
oi?aaeaiey iacuaaaony, oaeaea, aa eiee/anoaaiiie /anoueth.
Auioneay al k aaeeieoe l-ecaeaeee ec eaaeaeie cao?a/eaaaiie aaeeieoeu
k-nu?uey, ec q 11 , ( , q 1m aaeeieoe nu?uey anao m aeaeia
ecaioiaeytho q 21 , ( , q 2n :
q 21 = a 1 1 q 11 + ( + a 1 m q 1m ;
(
q 2n = a n 1 q 11 + ( + a n m q 1m ,
aaeeieoe ecaeaeee eaaeaeiai aeaea. Eiee/anoaa i?aaeeaaaaiuo ecaeaeee
eaaeaeiai aeaea i?aaenoaaeythony eeiaeiuie ooieoeeyie q 2l = q 2l
(q 1):
q 2l = q 2l (q 1) = ( a l , q 1 ( ; l = 1, ( , n ,
eiee/anoa cao?a/eaaaiiai nu?uey a aeaea neaey?iuo i?iecaaaeaiee (a l ,
q 1( m-ia?iiai noieaoeiaiai aaeoi?a q 1 cao?ao nu?uey n m-ia?iuie
no?i/iuie aaeoi?aie a1 , ( , a n iao?eoeu cao?ao a:
a1 = ( a1 1 ( a 1 m ) ,
(
an = ( an 1 ( a n m )
– aaeoi?aie auionea ecaeaeee eaaeaeiai aeaea ec anaai anni?oeiaioa
iio?aaeyaiiai nu?uey.
A iau/iuo iao?e/iuo iaicia/aieyo iaai? eeiaeiuo ooieoeee q 2l = q 2l
(q 1) ia?acoao n-ia?iue noieaoeiaue aaeoi? i?aaeeiaeaiey ecaeaeee q 2.
Iao?e/iia i?aaenoaaeaiea iieo/aiiuo aaeainiauo niioiioaiee:
q 2 = a1 1 ( a1 m
( ( (
an1 ( an m
q 11
(
q 1m
= a q1
iienuaaao inouanoaeyaiue m(n iao?eoeae auionea a eeiaeiia i?aia?aciaaiea
m eiee/anoa iio?aaeyaiiai nu?uey anao aeaeia a n eiee/anoaa
i?iecaiaeeiuo ec iaai ecaeaeee.
5.Iiiaeanoai aeiionoeiuo ieaiia. Aeiionoeiuie yaeythony oaeea caeoiee
nu?uey q 1, i?e eioi?uo i?aaeeiaeaiea i?iecaiaeeiuo ec iaai ecaeaeee
q 2 oaeiaeaoai?yao caaeaiiiio ia ieo ni?ino q 2:
q 2 = a q 1 ( q 2 ,
eee: i?aaeeiaeaiea oaeiaeaoai?yao ni?in.
Iieo/aiiua ia?aie/aiey:
a 1 1 q 11 + ( + a 1 m q 1m ( q 21 ;
(
a n 1 q 11 + ( + a n m q 1m ( q 2n ,
yaeythony i?yiuie eee eiee/anoaaiiuie iaiaoiaeeiuie oneiaeyie
?aaiiaaney. Eo ?aoaiey iacuaathony iiiaeanoaii aeiionoeiuo ieaiia
caaea/e.
Eae iu oaeaeei iicaeiaa (ni. ), iiiaeanoai ?aoaiee iieo/aiiie
nenoaiu ia?aaainoa, aiiaua aiai?y, iaiaeiicia/ii, aeiioneay ethaia
iaio?eoeaoaeueiia ia?ai?iecaiaenoai ecaeaeee (q 2 :
(q 2 ( q 2 ( q 2 ( 0 .
6.?aaiiaaniia iio?aaeaiea nu?uey. Ecaea?aeee aeaiiiai i?iecaiaenoaa, oi
anoue noieiinoue i?eia?aoaaiuo ii caaeaiiui caeoii/iui oeaiai p1 1 , (
, p1m iio?aaiuo eiee/anoa q 11 , ( , q 1m anao aeaeia nu?uey, ia?acoao
eo eeiaeioth ooieoeeth L(q 1):
L(q 1) = p1 1 q 11 + ( + p1m q 1m = ( p1 , q 1( ,
iacuaaaioth ooieoeeae noieiinoe, a oaeaea oeaeaaie ooieoeeae
?anniao?eaaaiie caaea/e. Eiee/anoaaiiay /anoue caaea/e ?aaiiaaniiai
oi?aaeaiey ninoieo a iouneaiee ia iaeanoe aeiionoeiuo ieaiia caeoiie
nu?uey ieai caeoiie q 1 iaeiaiueoae noieiinoe L(q 1):
q 1 : ( p1 , q 1( = min ( p1 , q 1(
q1 ( a q 1 ( q 2 .
Ieieiece?othuaa ooieoeeth noieiinoe caaea/e aeiionoeiia cia/aiea
eneiiiai aaeoi?a q 1 iacuaaaony aai ?aaiiaaniui cia/aieai eee, aua,
iioeiaeueiui ieaiii caaea/e, a iieo/aiiay caaea/a – caaea/ae
?aaiiaaniiai (eee, /oi oi aea naiia – iioeiaeueiiai) i?iecaiaenoaaiiiai
oi?aaeaiey. A iauai neo/aa o?aaiaaiea ieieiecaoeee noieiinoe
iaania/eaaao aaeeinoaaiiinoue aa ?aoaiey.
1.2. Oeaiiaay /anoue caaea/e cao?ao
1.Ioeaieaaiea ecaeaeee. A oneiaeyo oiai aea naiiai i?iecaiaenoaa:
q 11 ( q 1m
p2 1
(
p2 n a1 1 ( a1 m
( ( (
an1 ( an m q 21
(
q 2n
p11 ( p1 m
– iaeiia?aiaiii n aauanoaii nu?uey ia auioneaaiua ec iaai ecaeaeey
ia?aiineony e aai noieiinoue e aicieeaao aeaienoaaiiay caaea/a ioeaiee
nu?uey oeaiaie i?iecaiaeeiuo ec iaai ecaeaeee, iacuaaaiay, oaeaea,
oeaiiaie /anoueth caaea/e cao?ao.
Aeaenoaeoaeueii, ecaioiaeaiea ec aaeeieoeu nu?uey aeaea k: k=1, ( , m,
al k oooe ecaeaeee eaaeaeiai aeaea l: l=1, ( , n, ii oeaiai p2 l ca
oooeo niiauaao nu?ueth noieiinoe p1 k:
p1 1 = p2 1 a1 1 + ( + p2 n an 1 = ( p2 , b 1( ;
. . .
p1 m = p2 1 a1 m + ( + p2 n an m = ( p2 , b m(.
a aeaea eeiaeiuo ooieoeee
p1 k = p1 k (p2) = ( p2 , b k(
oeai i?iecaiaeeiuo ec ieo ecaeaeee, a niaieoiiinoe ia?acothueo m-ia?iue
no?i/iue aaeoi? oeaiiinoe nu?uey p1. Eiyooeoeeaioiuie aaeoi?aie yoeo
eeiaeiuo ooieoeee neoaeao noieaoeu b1 , ( , bm oie aea naiie
iao?eoeu cao?ao a:
b 1 =
a1 1
(
an 1
; . . . , b m = a1 m
(
an m
– aaeoi?u auionea anni?oeiaioa ecaeaeee ec nu?uey eaaeaeiai aeaea.
Iieo/aiiua oeaiiaua aaeainiaua niioiioaiey:
p1 = ( p1 1 ( p1 1)
a1 1 ( a1 m
( ( (
an1 ( an m
= p 2 a,
yaeythony eeiaeiui i?aia?aciaaieai p 2 a= p 1 oeai auioneaaiuo
ecaeaeee a i?iecaiaenoaaiiua oeaiiinoe iio?aaeyaiiai nu?uey,
aeaienoaaiiui inouanoaeyaiiio oie aea iao?eoeae auionea ecaeaeee a
eiee/anoaaiiiio eeiaeiiio i?aia?aciaaieth q 2 = a q 1 , nu?uey a
ecaeaeey.
2.Oeaiiaua oneiaey ?aaiiaaney. A oneiaeyo naiaiaeiiai aeinooia eae
i?iecaiaeeoaeae, oae e iio?aaeoaeae oiaa?ia e nu?ueth e oaoiieiaeyi,
i?iaeaaea anyeiai aioiaiai ecaeaeey aai i?iecaiaeeoaeai noaiiaeony
aiciiaeiie eeoue i?e oneiaee oiai, /oi i?eia?aoaiea aioiaiai ecaeaeey
iio?aaeoaeai ieacuaaaony aeey iaai ia aei?iaea aai naiinoiyoaeueiiai
ecaioiaeaiey. Ii yoie i?e/eia aeiionoeiuie yaeythony oaeea i?iaeaaeiua
oeaiu p2 auioneaaiuo ecaeaeee, i?e eioi?uo i?iecaiaenoaaiiua oeaiiinoe
p1= p1(p2) nu?uey ia i?aauoatho aai caeoii/iuo oeai p1 :
p1 = p2 a ( p1 .
Iieo/aiiua oneiaey i?iaeaae yaeythony aeaienoaaiiuie eee oeaiiauie
iaiaoiaeeiuie oneiaeyie ?aaiiaaney. Iie au?aaeatho oio iao
iio?aaeoaeueneee iiuo, a niioaaonoaee n eioi?ui oiaa?u ianniaiai
i?iecaiaenoaa i?e i?i/eo ?aaiuo oneiaeyo eiatho naienoai i?eia?aoaoueny
oai ioioiaa, /ai ieaea eo oeaia.
Iiiaeanoai ?aoaiee oeaiiauo ia?aie/aiee iacuaaaony iiiaeanoaii
aeiionoeiuo oeai.
3.?aaiiaaniua oeaiu ecaeaeee. Aeioiae i?iecaiaenoaa, aeaaaaiue
noieiinoueth i?iaeaaaaiuo ii oeaiai p2 1, ( , p2 n o?aaoaiuo eiee/anoa
q 21 ,( , q 2n auioneaaiuo ecaeaeee ia?acoao eeiaeioth ooieoeeth
Ldual(p2) yoeo oeai:
Ldual(p2) = p2 1 q 21 + ( + p2 n q 2n = ( p2 , q 2(,
iacuaaaioth ooieoeeae noieiinoe oeaiiaie /anoe caaea/e. Eae e anyeee
aeioiae ii no?aieony auoue iaeneiece?iaaiiui naiei iieo/aoaeai, e ii
yoie i?e/eia aeaienoaaiiay /anoue caaea/e oi?aaeaiey ninoieo a iouneaiee
ia iiiaeanoaa aeiionoeiuo oeai ecaeaeee eo iaeaieaa aeioiaeiuo cia/aiee
p2 :
p2 : ( p2 , q 2( = max ( p2 , q 2(
p2 ( p2 a ( p1
.
Iaeneiece?othuea ooieoeeth noieiinoe caaea/e aeiionoeiua oeaiu ecaeaeee
iacuaathony eo ?aaiiaaniuie oeaiaie, a naia caaea/a – aeaienoaaiiie eee
oeaiiaie /anoueth caaea/e ?aaiiaaniiai oi?aaeaiey.
4.I?aaeea aeaienoaaiiiai niioaaonoaey. Eoae, aeey iaeiie e oie aea
caaea/e cao?ao:
q 1
p2 a q 2 ,
p1
iu iieo/eee aa i?yioth e aeaienoaaiioth /anoe:
q 1 : min (p1 , q 1( i?e a q 1 ( q 2
e
p2 : max (p2 , q 2( i?e p2 a ( p1 .
Iaa iie, ianiio?y ia ?acee/iua “nii?yaeaiiua” iaai?u eneiiuo
iaecaanoiuo: a iaeiie q 1, a a ae?oaie p2 ,- iauaaeeiaiu iaeieie e oaie
aea iaai?aie ia?aiao?ia a, q 2 e p1 e iaeaaeatho ii?aaeaeaiiie
aeaienoaaiiie neiiao?eae, iicaieythuae ii iaeiie /anoe caaea/e
ainoaiiaeoue ae aeaienoaaiioth /anoue e iaiai?io.
Aeaenoaeoaeueii, n?aaieaay iaaeaeo niaie iaa iiaecaaea/e, iu iiaeai
onoaiiaeoue i?aaeea niioaaonoaey iaaeaeo ieie. Yoe i?aaeea ninoiyo a
caiaia
1) ciaea ia?aie/aiee n ( ia ( ,
2) aeaenoaey iioeiecaoeee ooieoeee noieiinoe c min ia max ,
3) ia?aiao?ia ia?aie/aiee ia ia?aiao?u ooieoeee noieiinoe c q 2 ia
p1 ,
4) eiee/anoaaiiuo ia?aiaiiuo ia ei nii?yaeaiiua oeaiiaua: c q 1 ia
p2 , e iaiai?io,
e iicaieytho ii ecaanoiie iaeiie /anoe caaea/e ooo aea iaienaoue ae
aeaienoaaiioth.
Caiaoei , oaeaea, /oi “nii?yaeaiiua” eiee/anoaaiiua q 1 e oeaiiaua p2
ia?aiaiiua iaaeo iiaecaaea/ ioiineoaeueii eiee/anoa oiaa?ia eiatho
acaeiii ia?aoiua eiee/anoaaiiua ?acia?iinoe oooe e ia?aoiuo oooe oiaa?a:
[ q 1k ] = oooee e [ p2 l] = ?oaee / oooee,
e eo aaeainiaua niioiioaiey acaeiii ia?aoiu a oii niunea, /oi a i?yiuo –
eiee/anoaa nu?uey i?aia?acothony a eiee/anoaa ecaeaeey, a a
aeaienoaaiiuo – iaiai?io: oeaiu ecaeaeee i?aia?acothony a oeaiu nu?uey:
q 2 = a q 1 e p2 a = p1 .
5.O?ainiiie?iaaiea. Niaethaeaaiia iaie ai acaeiii aeaienoaaiiuo
iiaecaaea/ao ?acee/aiea no?i/iuo e noieaoeiauo aaeoi?ia ono?aiyaony
aeaenoaeai o?ainiiie?iaaiey. O?ainiiie?iaaieai iao?eoeu iacuaaaony
aeaenoaea caiaiu aa no?ie noieaoeaie eee, /oi oi aea naiia,- noieaoeia
no?ieaie, e iau/ii iaicia/aaony cia/eii “t” naa?oo:
a t =
a1 1 ( a1 m
( ( (
an1 ( an m t
( a1 1 ( an 1
( ( (
a1 m ( an m
.
A /anoiinoe:
(q 1) t =
q 11
(
q 1m t
= ( q 11 ( q 1m) e (p1) t = ( p1 1 ( p1 m) t = p1 1
(
p1 m
.
O?ainiiie?iaaiea i?iecaaaeaiey iao?eoe aeiii?aaeaeyaony i?iecaaaeaieai
o?ainiiie?iaaiiuo iao?eoe, acyouo a ia?aoiii ii?yaeea:
(a c )t = (c )t (a )t;
a /anoiinoe:
( p2 a ) t = a t (p2) t e (a q 1) t = (q 1) t a t ,
a oaeaea
((p1 , q 1() t = ((q 1) t, (p1) t( .
Oaia?ue, aeaienoaaiiay /anoue caaea/e ?aaiiaaniiai oi?aaeaiey,
iieo/aiiay iaie a no?i/iuo aaeoi?ao p1 e p2 n oiiiaeaieai ia
iao?eoeo a ni?aaa:
p2 : max (p2 , q 2( i?e p2 a ( p1 ,
a o?ainiiie?iaaiiii aeaea caienuaaaony iiaeiaii naiae i?yiie /anoe
q 1 : min (p1 , q 1( i?e a q 1 ( q 2
a noieaoeiauo aaeoi?ao (p1)t e (p2)t n oiiiaeaieai ia
o?ainiiie?iaaiioth iao?eoeo a t neaaa:
(p2 )t : max ((q 2)t, (p2)t( i?e a t (p2) t ( (p1 )t.
1.3. Caaea/a auionea
1.Oaaee/iia i?aaenoaaeaiea. Caaea/a auionea yaeyaony “ia?aoiie” ii
ioiioaieth e i?aaeuaeouae caaea/a cao?ao caaea/ae ?aaiiaaniiai
i?iecaiaenoaaiiiai oi?aaeaiey. I?ioeannii i?iecaiaenoaa a iae yaeyaony
i?ioeann nai?ee ?yaea acaeiicaiaiyaiuo neiaeiuo ecaeaeee ec ianeieueeeo
aeaeia i?inoiai nu?uey. I?eia?aie caaea/e auionea yaeythony caaea/e
iioeiaeueiiai ieaie?iaaiey nai?ee ecaeaeee ec ianeieueeeo aeaeia
eiiieaeoothueo oceia, a /anoiinoe:
– no?ieoaeuenoaa ec ianeieueeeo aeaeia no?ieoaeueiuo iaoa?eaeia
– a?aiaie ?aaiou ianeieueeeo aeaeia i?iiuoeaiiiai iai?oaeiaaiey,
– a?aiaie ?aaiou ?aai/eo ianeieueeeo niaoeeaeueiinoae,
e ei iiaeiaiua caaea/e.
I?e eniieueciaaiee m aeaeia nu?uey aeey i?iecaiaenoaa n aeaeia ecaeaeee
ai anao caaea/ao auionea i?ioeann i?iecaiaenoaa iienuaaaony iao?eoeae
cao?ao c, ninoaaeythuea eioi?ie
ci j [eiee/anoai i-nu?uey / ia aaeeieoeo j-ecaeaeey] ( 0 ,
eiatho ia?aoiua eiee/anoaaiiua ?acia?iinoe ii ioiioaieth e
eiee/anoaaiiui ?acia?iinoyi iao?eoeu auionea a : [ aj i] = eiee/anoai
j-ecaeaeee / ia aaeeieoeo i-nu?uey.
A oneiaeyo caaeaiiiai aaeoi?a i?aaeeiaeaiey nu?uey q 1 e caaeaiiuo
oeai p2 ia i?iecaiaeeiua ecaeaeey a eiee/anoaaiiie (i?yiie) /anoe
ia?aoiie caaea/e euaony iaeaieaa aeioiaeiia i?aaeeiaeaiea (ieai
i?iecaiaenoaa) ecaeaeee q 2 , a a oeaiiaie (aeaienoaaiiie) /anoe –
iaeiaiaa ?anoiaeiua oeaiu p1 iio?aaeyaiiai nu?uey:
q 21 ( q 2n
p1 1
(
p1 m c1 1 ( c1 n
( ( (
cm1 ( cm n q 11
(
q 1m
p21 ( p2 n
Oi?iaeueiui ioee/eai i?eaaaeaiiie oaaeeoeu io oaaeeoeu i?aaeuaeouae
caaea/e yaeyaony, eae iu aeaeei, caiaia nu?ueaauo ia?aiaiiuo
“ecaeaeueiuie” e iaiai?io.
2.Eiee/anoaaiiay /anoue caaea/e auionea. A oneiaeyo cao?ao ci j
aaeeieoe i-nu?uey ia eaaeaeoth aaeeieoeo i?iecaiaeeiiai j-ecaeaeey, ia
auione q 21 , ( , q 2n aaeeieoe ecaeaeee anao n aeaeia iio?aaoaony
q 11 , ( , q 1m :
q 11 = c1 1 q 21 + ( + c1 n q 2n ( (c1 , q 2( ;
. . .
q 1m = cm 1 q 21 + ( + cm n q 2n ( (cm , q 2( ,
aaeeieoe nu?uey eaaeaeiai aeaea. n-ia?iua no?iee iao?eoeu cao?ao,
neoaeauea eiyooeoeeaioaie aaeainiauo niioiioaiee:
c1 = ( c1 1 ( c1 n );
. . .
cm = ( cm 1 ( cm n ),
anoue aaeoi?u cao?ao nu?uey eaaeaeiai aeaea ia aanue anni?oeiaio
i?iecaiaeeiuo ec iaai ecaeaeee. Iao?e/iia i?aaenoaaeaiea iieo/aiiuo
aaeainiauo niioiioaiee:
q 1 = q 1(q 2) = c q 2 ,
iienuaaao eeiaeiue i?ioeann ia?an/aoa i?aaeeiaeaiey auioneaaiuo ecaeaeee
a ni?in ia iio?aaeyaiia aeey eo i?iecaiaenoaa nu?uea.
Aeiionoeiui yaeyaony oaeia i?aaeeiaeaiea ecaeaeee, i?e eioi?ii ni?in ia
iio?aaeyaiia nu?uea ia i?aainoiaeeo aai i?aaeeiaeaiey:
q 1 = c q 2 ( q 1.
Aeioiae oaeiai i?iecaiaenoaa, au?aaeaaiue noieiinoueth M(q 2)
i?iaeaaaaiuo ii oeaiai p2 i?aaeeaaaaiuo eiee/anoa ecaeaeee:
M(q 2) = p2 1 q 21 + ( + p2 n q 2n ( (p2 , q 2( ,
iacuaaaony ooieoeeae noieiinoe eiee/anoaaiiie /anoe ia?aoiie caaea/e.
Naia aea caaea/a ninoieo a oii, /oiau ia iiiaeanoaa aa aeiionoeiuo
ieaiia i?iecaiaenoaa iaeoe ieai iaeaieueoae noieiinoe:
q 2 : ( p2 , q 2( = max ( p2 , q 2(
q 2 ( c q 2 ( q 1
.
A nouiinoe, ana caaea/e ?aaiiaaniiai oi?aaeaiey yaeythony ii?aaeaeaieyie
?aaiiaaniuo cia/aiee naieo eneiiuo iaecaanoiuo.
3.Oeaiiaay /anoue caaea/e auionea. Iaeiia?aiaiii, cao?aou ia eaaeaeoth
aaeeieoeo j-ecaeaeey ci j aaeeieoe nu?uey anao m aeaeia ii oeaiai
p1 i: i=1, ( , m, niiauatho auioneaaiui ecaeaeeyi oeaiu p2 1 , ( , p2 n
:
p2 1 = p1 1 c1 1 + ( + p1 m cm 1 ( (p1 , d 1( ;
. . .
p2 n = p1 1 c1 n + ( + p1 m cm n ( (p1 , d n( .
m-ia?iua noieaoeiaua aaeoi?u iao?eoeu cao?ao:
d 1 (
c1 1
(
cm 1
, ( , d n ( c1 n
(
cm n
,
anoue aaeoi?u cao?ao nu?uey ia auione ecaeaeey eaaeaeiai aeaea. Oeaiiaua
aaeainiaua niioiioaiey
p2 = p2(p1) = p1 c
iienuaatho inouanoaeyaiia iao?eoeae cao?ao aeaienoaaiiia eeiaeiia
i?aia?aciaaiea oeai iio?aaeyaiiai nu?uey a oeaiu i?iecaiaeeiuo ec ieo
ecaeaeee.
I?e caaeaiiuo i?iaeaaeiuo oeaiao ecaeaeee aeiaeaiiia a ieo nu?uea
i?eia?aoaao oeaiiinoue, ia iaiueooth oeaiiinoe auioneaaiuo ec iaai
ecaeaeee:
p2 = p1 c ( p2 .
Eae e a caaea/a cao?ao iieo/aiiua oeaiiaua oneiaey ?aaiiaaney au?aaeatho
iaiaoiaeeiia oneiaea i?iaeaae: iieoiea aioiauo ecaeaeee ia aeieaeia
auoue aei?iaea eo naiinoiyoaeueiiai ecaioiaeaiey.
Noieiinoue ?anoiaeoaiiai nu?uey:
Mdual(p1) = p1 1 q 11 + ( + p1 m q 1m ( (p1 , q 1( ,
ninoaaeyao ?anoiae i?iecaiaenoaa. Euoony aeiionoeiua oeaiu nu?uey,
niiauathuea aai noieiinoe iaeiaiueoaa cia/aiea:
p1 : ( p1 , q 1( ( min ( p1 , q 1(
p1 ( p1 c ( p2 .
4.Eaiiie/aneay ia?a caaea/. Eoae, iu iienaee ana /aou?a eeiaeiua
noaoe/aneea caaea/e ?aaiiaaniiai i?iecaiaenoaaiiiai oi?aaeaiey:
q 1
– ia?o caaea/ cao?ao: p2 a q 2 :
p1
n i?yiie caaea/ae iioeiaeueiiai ieaie?iaaiey caeoiie nu?uey:
q 1 : min (p1 , q 1( i?e a q 1 ( q 2 ,
e aeaienoaaiiie ae caaea/ae iioeiaeueiiai ieaie?iaaiey oeai auioneaaiuo
ecaeaeee:
p2 : max (p2 , q 2( i?e p2 a ( p1 ;
q 2
– e ia?o caaea/ auionea: p1 n q 1 :
p2
n i?yiie caaea/ae iioeiaeueiiai ieaie?iaaiey auionea ecaeaeee:
q 2 : max ( p2 , q 2( i?e c q 2 ( q 1 ,
e ae aeaienoaaiiie caaea/ae iioeiaeueiiai ioeaieaaiey nu?uey:
p1 : min ( p1 , q 1( i?e p1 c ( p2 .
Eae iu aeaeei, iaa caaea/e iaeaaeatho “ia?ae?anoiie” neiiao?eae e
oi?iaeueii, oi anoue aacioiineoaeueii e yeiiiie/aneiio niaea?aeaieth,
i?yiay e ia?aoiay ia?u caaea/ oiaeaeanoaaiiu ae?oa ae?oao n oi/iinoueth
aei – 1)- ia?aiaicia/aiey naieo aaee/ei e -2)- ia?anoaiiaee iaaeaeo
niaie eo acaeiii-aeaienoaaiiuo /anoae:
min ( p1 , q 1( i?e a q 1 ( q 2 max ( p2 , q
2( i?e c q 2 ( q 1,
max ( p2 , q 2( i?e p2 a ( p1 min ( p1 , q
1( i?e p1 c ( p2 .
Oi/iay acaeiicaiaiyaiinoue caaea/ aeinoeaaaony:
– caiaiie oaoiieiae/aneeo iao?eoe:
c ( a ,
– e ia?aiaicia/aieai eiee/anoaaiiuo e oeaiiauo aaeoi?ia:
(p1; 2 )t ( q 1; 2 .
I?e yoii i?yiay /anoue caaea/e cao?ao noaiiaeony ?aaiineeueiie
aeaienoaaiiie /anoe caaea/e auionea, a aeaienoaaiiay /anoue ia?aie –
i?yiie /anoe aoi?ie.
Aoaeai iacuaaoue acaeiii-aeaienoaaiioth ia?o caaea/ i?yiiai
(cao?aoiiai) aeaea n i?yiie (eiee/anoaaiiie) /anoueth ia ieieioi e
aeaienoaaiiie (oeaiiaie) /anoueth ia iaeneioi:
q 1
q 1 : min ( p1 , q 1( i?e a q 1 ( q 2 ,
p2 a q 2 :
p1
p2 : max ( p2 , q 2( i?e p2 a ( p1 .
– eaiiie/aneie ia?ie eeiaeiuo caaea/ noaoe/aneiai ?aaiiaaney, a eo
ia?aiaiiua q 1 e p2 – eaiiie/anee nii?yaeaiiuie ia?aiaiiuie.
1.4. Caaea/a ?aaiiaaney
Oece/aneia niaea?aeaiea caaea/e ?aaiiaaney. A o?aoia?iii neo/aa: m, n (
3, iaoa caaea/a eiaao i?inoia oece/aneia enoieeiaaiea. Ai aiaoiai
neeiaii iiea iinoiyiiie ai a?aiaie e i?ino?ainoaa iai?yaeaiiinoe p1
neaey?iay eeiaeiay ooieoeey eii?aeeiao L(q 1):
L(q 1) = (p1 , q 1( ,
yaeyaony iioaioeeaeueiie yia?aeae iaoiaeyuaainy a oi/ea q 1 i?iaiiai
oaea aaeeie/iie iannu (ca?yaea). Ana iaeaaaaiua ia ia?aiauaiey i?iaiiai
oaea aeiiieieoaeueiua ia?aie/aiey iacuaathony a iaoaieea naycyie.
Ia?aie/aiey iaoae caaea/e
q 1: a q 1 ( q 2
caaeatho a i?ino?ainoaa aa ia?aiaiiie q 1 auioeeoth iiiaia?aiioth
iaeanoue aeiionoeiuo ia?aiauaiee. A eoiaa, eaiiie/aneay caaea/a
iioeiaeueiiai i?iecaiaenoaaiiiai oi?aaeaiey:
q 1: min ( p1 , q 1( i?e a q 1 ( q 2 – ?
– oece/anee i?aaenoaaeyao niaith caaea/o au/eneaiey a ia?aie/aiiie
iaeanoe i?ino?ainoaa eii?aeeiao q 1 oi/ee iaeiaiueoae iioaioeeaeueiie
yia?aee L(q 1) i?iaiiai oaea aaeeie/iie iannu a iinoiyiiii aiaoiai
neeiaii iiea p1 .
Oi/ea iaeiaiueoae iioaioeeaeueiie yia?aee iacuaaaony oi/eie
noaoe/aneiai ?aaiiaaney e caaea/a aa ii?aaeaeaiey – caaea/ae
noaoe/aneiai ?aaiiaaney. Ii yoie i?e/eia eeiaeioth caaea/o iioeiaeueiiai
i?iecaiaenoaaiiiai ieaie?iaaiey iu aoaeai iacuaaoue oae, eae ia yoii
cayaeaii a iacaaiee, a eiaiii – eeiaeiie caaea/ae noaoe/aneiai
?aaiiaaney.
Iniaaiiinoueth eeiaeiuo caaea/ yaeyaony iacaaeneiinoue eo naienoa io
aaiiao?e/aeeo ?acia?iinoae eo aaee/ei. Yoi ianoiyoaeuenoai eniieuecoaony
aeey ?ani?ino?aiaiey o?aoia?iie oa?ieiieiaee ia eeiaeiua caaea/e
?aaiiaaney ethaie i?ino?ainoaaiiie ?acia?iinoe.
Aicueiai a ea/anoaa i?iaiiai oaea eaeaaeueiue iaeaiueeee oa?ee (oi
anoue oa?ee, n aeeaiao?ii, iaiueoei aeeeiu naiiai ei?ioeiai ?aa?a
aeiionoeiie iaeanoe, aac o?aiey iieiy ia?aeaouaathueeny iaaeaeo anaie aa
oaeiauie oi/eaie) e iiianoei aai a ia?acoaioth nenoaiie ia?aie/aiee
auioeeoth iiiaia?aiioth iaeanoue. Iniiaiua naienoaa caaea/e ?aaiiaaney
noaiiayony oece/anee i/aaeaeiuie naienoaaie aai iiaaaeaiey a yoeo
oneiaeyo.
Oae, oneiaea iaaueaouaaiey oa?eea ec iaeanoe ia?aie/aiee iiae
aeaenoaeai i?eeiaeaiiie e iaio aiaoiae neeu yaeyaony i?eciaeii
nouanoaiaaiey ?aoaiey caaea/e ?aaiiaaney. Aaiiao?e/anee ii ninoieo a
oneiaee i?eiaaeeaaeiinoe aaeoi?a neeu p1 auioeeie iaiei/ea
eiyooeoeeaioiuo aaeoi?ia anao ia?aie/aiee.
Oi/ea ?aaiiaaney, anee iia nouanoaoao, ?aniieaaaaony ia a?aieoea
iaeanoe aeiionoeiuo ia?aiauaiee e, aieaa oiai, – a iaeiie ec oaeiauo
oi/ae a?aieoeu.
Auioeeay iaeanoe eiaao auioeeoth a?aieoeo e iaiai?io. Oece/anee, yoi
ianoiyoaeuenoai ?aaiineeueii oneiaeth naiaiaeiiai ia?aiauaiey oa?eea ii
a?aieoea a iieneao oi/ee naiaai ?aaiiaaney. Niinia iineaaeiaaoaeueiiai
i?eaeeaeaiey e oi/ea ?aaiiaaney iin?aaenoaii aeaeaeaiey ii ?aa?ai
a?aie/iie iiaa?oiinoe iacuaaaony “neiieaen-iaoiaeii” ?aoaiey caaea/e
eeiaeiiai i?ia?aiie?iaaie. Caaea/a iioeiecaoeee caaeaiiie ooieoeee ia
caaeaiiie iiaa?oiinoe iacuaaaony a iaoaieea caaea/ae oi?aaeaiey.
A?aie oi/ee ?aaiiaaney iacuaathony ?aaiiaaniuie a?aiyie. A oi/ea
?aaiiaaney ni noi?iiu eaaeaeie ?aaiiaaniie a?aie ia oa?ee aeaenoaoao
neea ?aaeoeee iii?u, iai?aaeaiiay i?yiioaieueii yoie a?aie aaeieue
aaeoi?a aa ii?iaee. I?eciae ?aaiiaaney au?aaeaao niaith niaea?aeaiea
o?aoueaai caeiia Iuethoiia, ii eioi?iio a oi/ea ?aaiiaaney aan i?iaiiai
oaea o?aaiiaaoeaaaony noiiie nee ?aaeoeee iii?. ?aaiiaaniua oeaiu
auioneaaiuo ecaeaeee yaeythony eiyooeoeeaioaie p2 yoiai ?aceiaeaiey.
Anee iaeioi?ay a?aiue yaeyaony ?aaiiaaniie, oi iia i?ioiaeeo ia ioeaaii
?annoiyiee io oi/ee ?aaiiaaney e, iioiio, n aa noi?iiu ia oa?ee
aeaenoaoao iaioeaaay neea ?aaeoeee iii?u; anee aea a?aiue ia?aaiiaania,
oi iia ?aniieaaaaony ia no?iai iieiaeeoaeueiii ?annoiyiee io oi/ee
?aaiiaaney e, iioiio, neea ?aaeoeee n aa noi?iiu ?aaiyaony ioeth. A
oai?ee caaea/e ?aaiiaaney yoa ia?a naienoa iieo/eea iacaaiea
aeiiieiythuae iaaeanoeinoe.
Ionoonoaea au?iaeaeaiey a aeaea i?yiioaieueiinoe aaeoi?a iai?yaeaiiinoe
neeiaiai iiey iaeiie ec ?aaiiaaniuo a?aiae neoaeeo i?eciaeii
aaeeinoaaiiinoe ?aoaiey caaea/e ?aaiiaaney. I?e iai?a?uaiuo cia/aieyo
ia?aiao?ia oi/iay i?iii?oeeiiaeueiinoue eii?aeeiao aaeoi?a p1 e
eaeiai-oi aaeoi?a al ii?iaee a?aie iaaa?iyoia e iiaeao auoue eeoue
neaaenoaeai ie?oaeaiey /eneaiiuo cia/aiee eo eii?aeeiao. Oaeia
au?iaeaeaiea caaea/e iacuaaaony neo/aeiui e eaaei nieiaaony iaeuie
eciaiaieyie eee “oaaaeaieai” ia?aiao?ia. Ioiioaiey, nio?aiythueany i?e
oaaaeaiee eo ia?aiao?ia, iacuaathony neo/aai iauaai iieiaeaiey eee,
ii-i?inoo, – iauei neo/aai.
Iniiaiay eeoa?aoo?a
1. E.A.Eaioi?iae/. Yeiiiie/aneee ?an/ao iaeeo/oaai eniieueciaaiey
?ano?nia. I., 1960
2. Aeae.Aeaioeea. Eeiaeiia i?ia?aiie?iaaiea, aai i?eiaiaiey e
iaiauaiey. I., “I?ia?ann”, 1966
3. Ae.A.THaeei e A.A.Aieueooaei. Eeiaeiia i?ia?aiie?iaaiea: oai?ey,
iaoiaeu e i?eeiaeaiey. I., “Iaoea”,1969
4. I.Eio?eeeaaoi?. Iaoaiaoe/aneeea iaoiaeu iioeiecaoeee e yeiiiie/aneay
oai?ey. I., “I?ia?ann”, 1975
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Aeaaa ia?aay. CAAeA*E ?AAIIAANIIAI OI?AAEAIEss
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1.1. Caaea/a cao?ao.
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