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 "Enamasti on n\[ADoubleDot]iteks koolimatemaatika \[UDoubleDot]lesandel ",
 StyleBox["\[UDoubleDot]ksainus",
  FontSlant->"Italic"],
 " lahend. Vahel on lahendeid kaks (ruutv\[OTilde]rrandil \
n\[ADoubleDot]iteks); harva on \[UDoubleDot]lesandel lahendeid rohkem. M\
\[OTilde]nel \[UDoubleDot]lesandel on aga",
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  FontSlant->"Italic"],
 ", m\[OTilde]nel koguni ",
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 "."
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Cell["Mida siis teha?.", "Text",
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Cell["\<\
K\[OTilde]igi lahendite hulgas leidub tavaliselt \[UDoubleDot]ks v\[OTilde]i \
mitu sellist lahendit, mis mingit t\[ADoubleDot]iendavalt \
p\[UDoubleDot]stitatud lisatingimust (n\[ADoubleDot]iteks odavust) paremini \
rahuldab.\
\>", "Text",
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Cell[TextData[{
 "Sellist lahendit nimetatakse ",
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 "."
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Cell["\<\
Optimaalseid lahendeid otsitakse paljudes situatsioonides. \
N\[ADoubleDot]iteks:\
\>", "Text",
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Cell["\<\
1. Kuidas korraldada tehases tootmist nii, et kasum oleks suurim?\
\>", "Text",
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Cell["\<\
N\[ADoubleDot]iteks: \[UDoubleDot]htede ja samade algkomponentide baasil on \
keemiakombinaat v\[OTilde]imeline tootma mitmeid erinevaid aineid. On teada \
algkomponentide hinnad ja erinevate l\[OTilde]pp-produktide hinnad. Kulutused \
t\[ODoubleDot]\[ODoubleDot]j\[OTilde]ule on samuti teada. Mida on kasulikum \
toota?\
\>", "Text",
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Cell["\<\
2. Kuidas peab juurdel\[OTilde]ikaja m\[ADoubleDot]rkima kangale \
l\[OTilde]iked, et j\[ADoubleDot]\[ADoubleDot]gid oleksid v\[ADoubleDot]himad?\
\>", "Text",
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Cell["\<\
3. Kuidas koostada n\[ADoubleDot]dala men\[UDoubleDot]\[UDoubleDot]d nii, et \
toit oleks tervislik ja odav?\
\>", "Text",
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Cell["4. Kuidas koostada koolis tunniplaani?", "Text",
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Cell["5. Kuidas katta mingi maakoht bussiliinidega?", "Text",
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Cell["\<\
6. Kuidas valida erinevate automarkide vahel ostjale k\[OTilde]ige sobivamat?\
\>", "Text",
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Cell["\<\
7. Orienteeruja peab l\[ADoubleDot]bima metsas olevad kontrollpunktid, mille \
asukohad ja omavahelised kaugused on teada. Kontrollpunktide \
l\[ADoubleDot]bimise j\[ADoubleDot]rjekord ei ole t\[ADoubleDot]htis. Millise \
marsruudi peab valima orienteeruja, et l\[ADoubleDot]bitava trassi pikkus \
oleks minimaalne?\
\>", "Text",
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Cell[TextData[{
 "K\[OTilde]ik need on k\[UDoubleDot]simused, kus tuleb valida paljude \
erinevate v\[OTilde]imaluste hulgast optimaalne, s.o. k\[OTilde]ige parem, k\
\[OTilde]ige \[ODoubleDot]konoomsem lahend. ",
 StyleBox["Teadust, mis tegeleb optimaalsete lahendite otsimisega ja kasutab \
selleks matemaatilisi meetodeid nimetatakse ",
  FontSlant->"Italic"],
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  FontWeight->"Bold",
  FontSlant->"Italic"],
 ". "
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Cell["\<\
Operatsioonianal\[UDoubleDot]\[UDoubleDot]s tekkis Teise maailmas\[OTilde]ja \
p\[ADoubleDot]evil s\[OTilde]jalistel vajadustel. P\[ADoubleDot]rast s\
\[OTilde]ja l\[OTilde]ppu hakati operatsioonianal\[UDoubleDot]\[UDoubleDot]si \
meetodeid rakendama majanduses, transpordis, tootmise juhtimises, kaubanduses \
ja mitmel pool mujal. \
\>", "Text",
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Cell[TextData[{
 "Operatsioonianal\[UDoubleDot]\[UDoubleDot]si abil t\[ODoubleDot]\
\[ODoubleDot]tatakse v\[ADoubleDot]lja ",
 StyleBox["parimate lahenduste leidmise viisid",
  FontSlant->"Italic"],
 ", operatsioonianal\[UDoubleDot]\[UDoubleDot]s ",
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  FontSlant->"Italic"],
 "."
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Cell[TextData[{
 "Operatsioonianal\[UDoubleDot]\[UDoubleDot]si alla kuulub ka ",
 StyleBox["matemaatiline planeerimine",
  FontSlant->"Italic"],
 ", mille \[UDoubleDot]heks lihtsamaks osaks on ",
 StyleBox["lineaarne planeerimine",
  FontWeight->"Bold"],
 ".\n"
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Cell[TextData[StyleBox["Lineaarse planeerimise sissejuhatav n\[ADoubleDot]ide",
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Cell[TextData[{
 "V\[ADoubleDot]ike tislerit\[ODoubleDot]\[ODoubleDot]koda \
\[OpenCurlyDoubleQuote]Aken ja Uks\[CloseCurlyDoubleQuote] toodab puidust \
aknaid ja uksi. Akna valmistamiseks kulub 4 m neljandilist h\[ODoubleDot]\
\[ODoubleDot]velpuitu, 1,2 ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["m", "2"], TraditionalForm]]],
 " klaasi ja 6 tundi tisleri t\[ODoubleDot]\[ODoubleDot]aega. Ukse \
valmistamiseks kulub 20 m neljakandilist h\[ODoubleDot]\[ODoubleDot]velpuitu \
ja 10 tundi tisleri t\[ODoubleDot]\[ODoubleDot]aega. T\[ODoubleDot]\
\[ODoubleDot]kojas on \[UDoubleDot]he kuu varu \
h\[ODoubleDot]\[ODoubleDot]velpuitu - 1400 m ja \[UDoubleDot]he kuu varu \
klaasi - 90 ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["m", "2"], TraditionalForm]]],
 ". Selles kuus v\[OTilde]ib arvestada 900 tislerit\[ODoubleDot]\[ODoubleDot] \
tunniga. Peale t\[ODoubleDot]\[ODoubleDot]meestele palga ja riigile maksude \
\[ADoubleDot]ramaksmist teenib t\[ODoubleDot]\[ODoubleDot]koja omanik iga \
akna pealt 10 \[Euro] kasumit, iga ukse pealt aga 30\[Euro] kasumit. Kui \
palju aknaid ja uksi tuleks valmistada, et t\[ODoubleDot]\[ODoubleDot]koja \
omaniku kasum oleks maksimaalne?"
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Cell["\<\
V\[OTilde]tame m\[OTilde]lemal pooltasandil m\[OTilde]ned punktid, n\
\[ADoubleDot]iteks nn. \:201e\[UDoubleDot]lemisel\" pooltasandil punktid (0; \
3), (2; 2), (4; 3) ja (3; 1); \:201ealumisel\" pooltasandil v\[OTilde]tame \
punktid (0; 0), ( - 1; 2), (1; 1), (2; 0) ja (3; - 1). \
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Asendades nende punktide koordinaadid v\[OTilde]rratusse 2x + 3y - 6 \
\[GreaterSlantEqual] 0, saame \[UDoubleDot]lemise pooltasandi punktide korral \
t\[OTilde]ese v\[OTilde]rratuse, alumise pooltasandi punktide korral aga v\
\[ADoubleDot]\[ADoubleDot]ra v\[OTilde]rratuse. \
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 "V\[OTilde]rratuse ax + by + c \[GreaterSlantEqual] 0 (v\[OTilde]i ax + by + \
c \[LessSlantEqual] 0) lahenditeks on \[UDoubleDot]he pooltasandi punktide \
koordinaadid. Pooltasandi rajasirge ax + by + c = 0 kuulub ka \
v\[OTilde]rratuse lahendite hulka. V\[OTilde]rratuse ax + by + c \
\[GreaterSlantEqual] 0 (v\[OTilde]i \nax + by + c \[LessSlantEqual] 0) \
graafiliseks lahendamiseks joonestame sirge ax + by + c = 0, mis eraldab \
kahte pooltasandit. Lahendite pooltasandi leidmiseks valime \[UDoubleDot]hel \
pooltasandil v\[ADoubleDot]ljaspool rajasirget mingi punkti ",
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 ") ja kontrollime, kas selle punkti koordinaadid rahuldavad \
v\[OTilde]rratust.Kui jah, siis on lahendite pooltasandiks see pooltasand, \
millelt valisime punkti ",
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 ". Kui ei, siis on lahendite pooltasandiks see pooltasand, millel punkt ",
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 " ei asu. Sagedasti v\[OTilde]etakse kontrollitavaks punktiks koordinaatide \
alguspunkt. N\[ADoubleDot]iteks v\[OTilde]rratusi x \[GreaterSlantEqual] 1  \
ja  \nx - 2y + 2 \[LessSlantEqual] 0 kujutavad piirkonnad on esitatud j\
\[ADoubleDot]rgmistel joonistel."
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Cell["\<\
Sissejuhatuses toodud tislerit\[ODoubleDot]\[ODoubleDot]koja \
n\[ADoubleDot]ites saime kahe muutujaga \
v\[OTilde]rratuses\[UDoubleDot]steemi. Kuidas selliseid v\[OTilde]rratuses\
\[UDoubleDot]steeme lahendada? \
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V\[OTilde]rratuses\[UDoubleDot]steemi lahendiks on selliste arvude hulk, mis \
rahuldab samaaegselt s\[UDoubleDot]steemi iga v\[OTilde]rratust. J\
\[ADoubleDot]relikult koosneb s\[UDoubleDot]steemi lahendite hulk s\
\[UDoubleDot]steemi k\[OTilde]ikide v\[OTilde]rratuste \[UDoubleDot]histe \
lahendite hulgast. Kui lahendame v\[OTilde]rratuses\[UDoubleDot]steemi, peame \
leidma k\[OTilde]igi v\[OTilde]rratuste lahendihulgad ja seej\[ADoubleDot]rel \
nende hulkade \[UDoubleDot]hisosa. \
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Kahest v\[OTilde]rratusest koosneva s\[UDoubleDot]steemi lahendamisel v\
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Cell["\<\
Piirkond, milles olevad punktid rahuldavad \[UDoubleDot]hte v\[OTilde]rratust \
kahest, on \[UDoubleDot]hekordse viirutusega. Piirkond, milles olevad punktid \
ei ole kummagi v\[OTilde]rratuse lahendid, on viirutamata. Piirkond, mille \
punktid on s\[UDoubleDot]steemi lahendid, on viirutatud kahekordselt. \
\>", "Text",
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\[OTilde]ib olla nurk, riba, pooltasand v\[OTilde]i t\[UDoubleDot]hi hulk. \
Viimasel juhul on v\[OTilde]rratused ",
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Cell["\<\
Kui meil on rohkem v\[OTilde]rratusi, siis leitakse s\[UDoubleDot]steemi \
lahendeid samal viisil kui kahe v\[OTilde]rratusega v\[OTilde]rratuses\
\[UDoubleDot]steemi korral. Lahendihulga kuju v\[OTilde]ib olla aga \
keerulisem. N\[ADoubleDot]iteks, kolmest lineaarv\[OTilde]rratusest koosneva \
s\[UDoubleDot]steemi lahendihulk v\[OTilde]ib moodustada j\[ADoubleDot]rgmisi \
kujundeid: \
\>", "Text",
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Cell[TextData[{
 "Edaspidi nimetame s\[UDoubleDot]steemi lahendite hulga poolt moodustatud \
kujundit (nii l\[OTilde]plikku kui ka l\[OTilde]pmatut) ",
 StyleBox["lahendite piirkonnaks",
  FontSlant->"Italic"],
 "."
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Cell[TextData[{
 "Vaatleme neid lahendite piirkondi. V\[OTilde]ttes suvalisest piirkonnast \
kaks vabalt valitud punkti, saame need punktid \[UDoubleDot]hendada sirgl\
\[OTilde]iguga, mis tervikuna kuulub samasse piirkonda. K\[OTilde]iki \
selliseid piirkondi nimetatakse ",
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  FontSlant->"Italic"],
 " piirkondadeks. "
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Cell[TextData[{
 "Punktihulka nimetatakse ",
 StyleBox["kumeraks hulgaks",
  FontWeight->"Bold"],
 ", kui selle hulga iga kahte punkti saab \[UDoubleDot]hendada \
sirgl\[OTilde]iguga, mis tervenisti kuulub samasse hulka. Kumerad hulgad on n\
\[ADoubleDot]iteks ring, pooltasand, kolmnurk. Ruumilistest kehadest p\
\[UDoubleDot]ramiid, kera, korrap\[ADoubleDot]rane prisma. Kumerad hulgad ei \
ole n\[ADoubleDot]iteks r\[OTilde]ngas, t\[ADoubleDot]htviisnurk, \
180\[Degree] suurema kesknurgaga sektor jmt. ",
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lineaarv\[OTilde]rratuses\[UDoubleDot]steemi lahendihulk on kumer. ",
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Cell[TextData[StyleBox["a) kui sihifunktsiooniga \
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optimaalne lahend;\n  b) kui sihifunktsiooniga \
m\[ADoubleDot]\[ADoubleDot]ratud sirged on paralleelsed piirkonna optimaalset \
tippu l\[ADoubleDot]biva l\[OTilde]iguga, siis on lahendeid l\[OTilde]pmata \
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Cell["\<\
Seega saab keemiatehas maksimaalse kasumi 14 000 $, kui toodetavad siset\
\[ODoubleDot]\[ODoubleDot]de ja \
v\[ADoubleDot]list\[ODoubleDot]\[ODoubleDot]de v\[ADoubleDot]rvi kogused x ja \
y on seotud v\[OTilde]rdusega x + 2 y = 7 ja v\[OTilde]rratusega 1\
\[LessSlantEqual] x \[LessSlantEqual] 3. (Need kaks tingimust koos annavad \
meile l\[OTilde]igu BC punktid.)\
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Cell["\<\
Kuidas peaks toimima tehase direktor, kui tal on valida l\[OTilde]pmatu hulga \
optimaalsete variantide hulgast? Kas valima juhuslikult \[UDoubleDot]he \
punkti sellest l\[OTilde]igust? Ei, sugugi mitte. Kogenud tehasedirektor m\
\[OTilde]istab kohe, et see mudel, mille meie koostasime, ei ole k\
\[UDoubleDot]llalt p\[OTilde]hjalik, sest see mudel arvestab ainult v\
\[ADoubleDot]rvi m\[UDoubleDot]\[UDoubleDot]gist saadavat tulu.\
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Tootmises on aga palju muid piiravaid tingimusi : erinev t\[ODoubleDot]\
\[ODoubleDot]viljakus erinevate toodete valmistamisel, kulutused t\
\[ODoubleDot]\[ODoubleDot]tasule, kuidas toota nii, et tooraine kulu oleks v\
\[ADoubleDot]him jne.\
\>", "Text",
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Cell["\<\
Kui direktor seab eesm\[ADoubleDot]rgiks toota nii, et komponente A ja B j\
\[ADoubleDot]\[ADoubleDot]ks v\[OTilde]imalikult palju j\[ADoubleDot]rele, \
siis ta n\[ADoubleDot]eb, et kui toota 3 t siset\[ODoubleDot]\[ODoubleDot]de \
v\[ADoubleDot]rvi ja 2 t v\[ADoubleDot]list\[ODoubleDot]\[ODoubleDot]de v\
\[ADoubleDot]rvi, siis kulutame me \[ADoubleDot]ra kogu \
l\[ADoubleDot]htetooraine hulga :\
\>", "Text",
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 " Broileritibu vajab iga p\[ADoubleDot]ev v\[ADoubleDot]hemalt 15 \
\[UDoubleDot]hikut vitamiini B1 ja v\[ADoubleDot]hemalt 15 \[UDoubleDot]hikut \
vitamiini B2. Ta saab neid segudest H ja K. Segu H 1 g sisaldab 1 \
\[UDoubleDot]hiku B1, 5 \[UDoubleDot]hikut B2 ja maksab 1 sent. Segu K 1 g \
sisaldab 5 \[UDoubleDot]hikut B1, 1 \[UDoubleDot]hiku B2 ja maksab 3 senti. \
Koostame p\[ADoubleDot]evase toiduratsiooni segudest H ja K nii, et broileri \
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 " Tammesalu ja Viljaanni jahuveskid jahvatavad 250 ja 350 tonni jahu kuus. \
Neist veskitest veetakse jahu kolme leivatehasesse. Rukkivere tehas vajab 150 \
tonni, Ahjusaare tehas 240 tonni ja Leivam\[ADoubleDot]e tehas 210 tonni jahu \
kuus. Kuidas korraldada jahu vedu veskitest nii, et k\[OTilde]ik tehased \
saaksid oma koguse jahu k\[ADoubleDot]tte ja veokulud oleksid minimaalsed? \
Kaugused veskite ja tehaste vahel on j\[ADoubleDot]rgnevas tabelis olevate \
arvude kordsed."
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Veetagu Tammesalust Rukkiverre x tonni ja Ahjusaarde y tonni jahu. \
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Cell["\<\
Et Rukkivere vajab 150 t, siis Viljaannilt peab ta saama (150 - x) t jahu. \
\>", "Text",
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Cell["\<\
Et Ahjusaare vajab 240 tonni, siis Viljaannilt peab ta saama veel juurde (240 \
- y) tonni jahu. \
\>", "Text",
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Cell["\<\
Et Tammesalu suudab toota 250 tonni jahu kuus ja Rukkiverre ning Ahjusaarde \
kokku veetakse sellest x + y tonni, siis Tammesalust veetakse Leivam\
\[ADoubleDot]ele (250 - x - y) t jahu. \
\>", "Text",
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Cell["\<\
\[CapitalUDoubleDot]lej\[ADoubleDot]\[ADoubleDot]nud 210 - (250 - x - y) = x \
+ y - 40 tonni saab Leivam\[ADoubleDot]e Viljaannilt.\
\>", "Text",
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Cell["Seega n\[ADoubleDot]eks vedude plaan v\[ADoubleDot]lja nii: ", "Text",
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Cell["\<\
Et leida vedude kogumaksumust, selleks tuleb korrutada selles tabelis olevad \
arvud eelmises tabelis olevate vastavate arvudega ja tulemused liita. Saame \
avaldise \
\>", "Text",
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Cell["\<\
c = 4x + 3y + 5(250 - x - y) + 5(150 - x) + 6(240 - y) + 4(x + y - 40) = - 2x \
- 4y + 3280. \
\>", "Text",
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Cell["\<\
Selle avaldise minimaalseid v\[ADoubleDot]\[ADoubleDot]rtusi otsime lahendite \
piirkonnast. Et veskist leivatehasesse veetava vilja kogus ei saa olla \
negatiivne, siis saame v\[OTilde]rratused: \
\>", "Text",
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Cell["\<\
x \[GreaterSlantEqual] 0, y \[GreaterSlantEqual] 0, 250 - x - y \
\[GreaterSlantEqual] 0, 150 - x \[GreaterSlantEqual] 0, 240 - y \
\[GreaterSlantEqual] 0, x + y - 40 \[GreaterSlantEqual] 0. \
\>", "Text",
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Cell["\<\
Nende v\[OTilde]rratuste poolt m\[ADoubleDot]\[ADoubleDot]ratud tasandiosa \
ongi selle \[UDoubleDot]lesande lahendihulgaks: \
\>", "Text",
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Cell[CellGroupData[{

Cell["\<\
Arvutades avaldise - 2x - 4y + 3280 v\[ADoubleDot]\[ADoubleDot]rtuse selle \
piirkonna tippudes, saame 
c(0; 40) = 3120,   c(40; 0) = 3200,   c(0; 240) = 2320, 
c(150; 0) = 2980, c(150; 100) = 2580, c(10; 240) = 2300. \
\>", "Text",
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Cell[CellGroupData[{

Cell["\<\
Saadud v\[ADoubleDot]\[ADoubleDot]rtustest on v\[ADoubleDot]him 2300, see \
saadakse, kui x = 10 ja y = 240. 
Asendades x ja y v\[ADoubleDot]\[ADoubleDot]rtused viimasesse tabelisse, \
saame optimaalse vedude korralduse plaani: \
\>", "Text",
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