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7/31/2019 PhuongPhapToiUu

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1-1. NH NGHA V PHN LOI BI TON TI U

1. Bi ton ti u ho tng qut

Bi ton ti u ho tng qut c pht biu nh sau:

Cc i ho(cc tiu ho) hm f(x):

f(x)max (min) (1.1)Vi cc iu kin:

, , , 1, (1.2)

(1.3)

i i

n

g x b i m

x X R

Bi ton (1.1) (1.3) c gi l bi ton ti u hay bi ton quyhoch.

Hm f(x) c gi l hm mc tiu.


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Cc hm: , 1,ig x i m c gi l cc hm rng buc, ming thchocbtng thc trong h (1.2) cgi l mtrng buc.

Tp hp:

| , , , 1, (1.4)i iD x X g x b i m c gi l min rng buc (hay min chp nhn c).

Mi im: 1 2, ,..., nx x x x D c gi l mt phng nhay mt li gii chp nhn c.

Mt phng n: x D t cc i (hay cc tiu) ca hm

mc tiu, c th l:

( ),f x f x x D

( ),f x f x x D

(i vi bi ton max)

(i vi bi ton min)


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c gi l phng n ti u (hay li gii tiu). Khi gi tr f(x*) cgi l gi tr tiu

ca bi ton


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2. Phn loi cc bi ton ti u

Mt trong nhngphng php hin nhin nhtgii biton tiu l phng php imdin: Tnh gi tr hm mctiu f(x) trn ttc cc phng n, sau so snh cc gi trtnh c tm ra gi trtiu v phng n tiuca biton.

Thc hin theo phng php trn gp rt nhiu kh khnngay c khi kch thc ca bi ton(s bin n v s rngbuc m) l khng ln,bi v tp D thng thnggmmts

rt ln cc phn t, trong nhiu trng hp cn l khngmc.


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V vy,ngi ta nghin cuvmt l thuyt c th tch ra t bi ton tng qut thnh cclp bi ton dgii. Cc nghin cu l thuyt

thng l:- Nghin cu cc tnh cht ca cc thnh

phn bi ton(hm mc tiu, cc hm rngbuc, cc bins, cc hs);

- Cc iu kin tn ti li gii chp nhnc;- Cc iukincn v cacctr;- Tnh chtca cc itng nghin cu.

Da vo tnh chtca cc thnh phn bi ton vitng nghin cungi ta phn loi cclp bi ton tiu(hay bi ton quy hoch)nhsau:


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1. Quy hoch tuyn tnh (QHTT): Nu hm mc tiu f(x) v tt ccc hm rng buc gi(x), i=1,m l tuyn tnh. Mt trng hpring quan trng ca QHTT l Bi ton vn ti (BTVT);

2. Quy hoch tham s (QHTS): nu cc h s trong biu thc cahm mc tiu v ca cc rng buc ph thuc vo tham s;

3. Quy hoch ng (QH): nu i tng xt l cc qu trnh cnhiu giai on ni chung, hay cc qu trnh pht trin theo thigian ni ring;

4. Quy hoch phi tuyn (QHPT): nu hm mc tiu f(x) hoc c tnht mt trong cc hm g

i

(x) l phi tuyn hoc c hai trng hp cng xy ra;


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5. Quy hoch ri rc (QHRR): nu min rng buc D l tp rirc. Trng hp ring, khi cc bin ch nhn gi tr nguyn ta cQuy hoch nguyn (QHN). Mt trng hp ring ca QHN l quy

hoch bin booles khi cc bin s ch nhn gi tr 0 hoc 1;6. Quy hoch a mc tiu(QHMT): nu trn cng mt min rngbuc ta xt nhiu hm mc tiu khc nhau.


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1-2. M HNH TON HC

1. Xy dng m hnh ton hc cho mtvnthctVic m hnh ho ton hc cho mtvn thc t

c th chia ra lm bnbc:Bc 1: Xy dng m hnh nh tnh cho vn thc t, tc l xc nh cc yu t c ngha quantrng nht v xc lp cc quy lut m chng phi

tun theo. Ni mt cch khc l pht biu m hnhbng li v bng nhng biu , cc iu kin vkinh t,k thut,t nhin, x hi, cc mc tiu cntc.


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Bc 2: Xy dng m hnh ton hc chovn ang xt, tc l din t li di

dng ngn ng ton hc cho m hnh nhtnh. Khi c mth thng ta chn cc bins c trng cho cc trng thi ca hthng,thitlpmi lin hgia cc binsv cc hsiukhinhintng.Bc ny phi xc nhc hm mc tium gi troca n cng ln(hoc cng b)

cng tt v din tbng cc phng trnhhoc bt phng trnh ca cc iu kinm cc binsphi tun theo.


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Bc 3: S dng cc cng c ton hc kho st v gii quyt bi ton va hinmhf

thnh bpc 2. Tc l phi chnphng php gii cho ph hp v c thho phng php bng cc thut ton tiu.

V cc bi ton thc t thng c kchthc ln nn khng th gii bng tay mphi s dng my tnh. Do cn chngtrnh ho thut ton bng ngn ng lptrnh ph hp v cho chy th, in kt qu


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Bc 4: Phn tch v kim nh li cc kt qutnh tonthu c trong bc 3.

Trong bc ny cn phi xc nh mc phhpca m hnh v ktqu tnh ton bngthcnghim hoc p dng phng php phn tchchuyn gia. C th xy ra mt trong hai kh

nng sau:Kh nng 1: M hnh v kt qu tnh ton ph

hpvithct

Khi cn lp mt bn tng kt ghi r cch tvn , m hnh ton hc, thut ton ti u,chng trnh, sliua vo my tnh.


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Trng hp m hnh c s dng nhiu ln thphi xy dng h thng phn mm bo m sdng thun tin, khng i hi ngi s dngphi c trnh chuyn mn cao v ton v tinhc.

Kh nng 2:m hnh v cc kt qu tnh ton

khng ph hp vi thc t. Trong trng hpny cn phi xem xt cc nguyn nhn ca n.C th nu ra bn nguyn nhn sau:

Nguyn nhn1:Cc kt qu tnh ton trong bc3 cha c chnh xc cn thit. Khi cnphi xem li cc thut ton cng nh ccchng trnh tnh ton vit v s dng


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Nguyn nhn2:Cc s liu ban u (cc hs, thng s) khng phn nh ng thc

t gi c hoc chi ph trn th trng hoccc nh mc vt t, hoc cc s liu khcv cng sut, kh nng my mc, d trti nguynLc ny cn iu chnh li

mt cch nghim tc, chnh xc.Nguyn nhn3:M hnh nh tnh xy dng

cha phn nh c y hin tng

thc t. Nu vy cn r sot li bc1xem c yu t hoc quy lut no cn bb st khng?


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Nguyn nhn 4: Vic xy dng m hnhton hc bc 2 cha tho ng. Cn

phi xy dng li cho ph hp, mc tng dn t tuyn tnh n phi tuyn, ttnh n ng, t tt nh n ngu nhin


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2. Mt s m hnh thc t

a) Bi ton lp k hoch sn xutBi 1: C hai nh my nhit in 1 v 2 sn xut

in t cc loi than a v b. Sut tiu haonhin liu (Kg/Kwh) ca tng nh my ng

vi mi loi than c gi tr ghi trong bng 1-1.Hy xc nh lngin nng sn xut t hainh my sao cho tngin nng sn xut ratrong mt ngy m c gi tr ln nht, ng

thi phi tho mn iu kin lng than loi atiu hao trong trong mt ngy m c hainh my khng qu 24 tn v loi b khng qu12 tn.


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Loi than

Nh my

a b

1 0.4 0.4

2 0.6 0.2

Gi x1, x2l lng in nng(MWh) sn xut trong mt ngy m bi nh my1 v 2.

M hnh ton hc c dng:

Xc nh x1, x2 sao cho:

x1+x2max (1.4)

V tho mn cc bt ng thc:

0,4x1 +0,6x2 24

0,4x1+0,2x2 12 (1.5)

x1, x2 0 (1.6)


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Bi 2: Mt x nghip munsn xut hai loi snphm A v B bng cc

nguyn liu I, II, v III.Sut chi ph nguyn liu sn xut cc snphm trn cho trong bng

1-2.

Sn phm

Nguyn liu

A B

I 2 1

II 1 2

III 0 1

sn xut mt n v sn phm A cn hai n v nguyn liu I, mt n vnguyn liu II.

sn xut mt n v sn phm B cn mt n v nguyn liu I, 2 n vnguyn liu II v mt n v nguyn liu III

Cc nguyn liu I, II, III c d tr tng ng l 8, 7,3 n v.

Tin li mt n v sn phm A l 4 triu ng, mt n v sn phm B l 5 triung.

Cn lp k hoch sn xut thu c tin li ln nht trn c s hn ch vnguyn liu

G i l l h l i A


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Gi x1l lng sn phm loi A

x2l lng sn phm loi B

M hnh ton hc c dng:Xc nh x1, x2 sao cho:

4x1+5x2max (1.7)

V tho mn ccrng buc v nguyn liu:

Nguyn liu I: 2x1 +x2 8

Nguyn liu II: x1+2x2 7Nguyn liu III: x2 3 (1.5)

x1, x2 0 (1.6)


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Qua v d trn, ta c th xy dng bi ton lp khoch sn xut tng qut nh sau:

Gi s 1 x nghip sn xut n loi sn phm v s

dng m loi nguyn liu khc nhau, vi cc khiu:

Xjl lng sn phm loi J, J=1,nC

j

l tin li ca mt n v sn phm loi J, J=1,n.aij l sut chi ph nguyn liu i sn xut mt n

v sn phm i vi i=1,m; j=1,nBil lng d tr nguyn liu i, i=1,m.

Hy xc nh gi tr xjsao cho tng tin li (haytng gi tr sn lng hng ho) l ln nht vis nguyn liu hin c


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M hnh ton hc:1

ax (1.10)n

j j

j

c x m

Vi cc iu kin:1

, 1, (1.11)

0, 1, (1.12)

n

ij j i

j

j

a x b i m

x j n


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b) Bi ton vn tiNi dung ca bi ton vn ti nh sau:Tm phng n ti u vn chuyn hng ho t bt

k mt s ni pht n bt k mt s ni nhn saocho tng chi ph vn chuyn b nht.Bi ton c th m t nh sau:C m im pht (ni sn xut) Ai, i=1,m vi cc lng

hng ho tng ng a1, a2,, amV n a im nhn (ca hng) Bj, j=1,n vi nhu cu

hng ho tng ng b1, b2,,bnCn xc nh phng n vn ti sao cho tng chi ph

vn chuyn l cc tiu, bit cc ph n v hng vnti cij trn cc on ng t ni pht i n ni nhn jGi xijl s lng hng cn vn chuyn t ni pht i

n ni nhn j. Khi bi ton vn ti c m ttrn bng 1-3

Bng 1 3


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Bng 1-3

Ni nhn

Ni pht

B1 B2 Bn Dunglng

A1 c11

x11

c12

x12

c1n

x1n

a1

A2 c21

x21

c22

x22

c2n

x2n

a2

..Am cm1

xm1

cm2

xm2

cmn

xmn

an

Dung lng b1 b2 bn 1 1

m n

i j

i ja b


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Bi ton vn ti c m t di dng ton hc nh sau:Xc nh xij sao cho

ij ij

1 1min

m n

i j

c x

ij

1

ij

1

ij

; 1,

; 1,

0; 1, ; 1,

n

i

j

m

j

i

x a i m

x b j n

x i m j n

Vi gi thit tng dung lng hng pht i bng tng dunglng hng ni nhn, ta c

1 1

m n

i j

i j

a b


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1

axn

j j

j

c x m

Vi cc iu kin:1

, 1,

0, 1,

n

ij j j

j

j

a x b i m

x j n

c) Bi ton ci ti

Ni dung: Mt ngi du lch mun em theo mtci ti nng khng qu b Kg trong khi c n

loi vt cn mang theo.Mi vt loi j ckhi lng aj(kg) v c gi tr cj. Ngi du lchmun cht vo ti cc vt sao cho tng gi trln nht.

Gi xjl s vt loi j. Ta c m hnh ton hcnh sau:

; xj nguyn

y l bi ton quy hoch nguyn

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