Giáo trình toán kinh tế ts đinh ngọc thanh, ths nguyễn quốc huy, ths trần thị ngọc tuyết

Mot trong nhCfng yeu to trong yeu la doi ngu thay cd, he thong cac giao trinh va trang thiet bi day hoc.Gung vdi cac nganh cac cap trong toan quoc dang day nhanh tien do xay dqng va cung

Nguyen Quoc Huy — Dinh Ngoc Thanh

Tran Thi Ngoc Tuyet

TRUONG DAI HOC DAN LAP CLTU LONG

TRIM DAlkot' DAN LAPC&

S^angky:

Tru'dng Dai hoc Dan lap Ctfu Long muon khdng dinh la trung tam dao tao nguon nhan life cd trinh do cao cho khu vifc dong b^ng song Cufu Long de phuc vu cho edng cuoc edng nghiep hda, hien dai hda dat mid’c thi phai nang cao chat litong dao tao toan dien Mot trong nhCfng yeu to trong yeu la doi ngu thay cd, he thong cac giao trinh va trang thiet bi day hoc.Gung vdi cac nganh cac cap trong toan quoc dang day nhanh tien do xay dqng va cung co vi the trong xu the hoi nhap, trifdng Dai hoc Dan lap Cufu Long dan tiing birdc hoan thien de duPng dau vdi nhilng thach thde do, gdp phan diTa nen giao due dai hoc Viet Nam diing vCfng, ngang tam khu vifc va

An pham nay la giao trinh Toan Kinh Te phan Quy

Hoach Tuyen Tinh cua Khoa Khoa Hoc Co Ban, Tru'dng Dai hoc Dan lap Ctiu Long, do PGS TS Dinh Ngoc Thanh, ThS

Nguyen Qud'c Huy va ThS Tran Thi Ngoc Tuyet bien scan

Lanh dao trctdng Dai hoc Dan lap Cdu Long tran trong nhdng edng hien, dong gdp cua Quy thay cd mdi giang cung nhiT cac thay cd can bd - giang vien co hdu ciia nha tru’dng

ThS Nguyen Cao Dat

Chitorng 1

i = 1,2, ,m , ta tim x°

Ham diTOc goi la ham mac tieu Phan tu’ x e Rn duttc goi

la cac phicang an Phiiong an nao thoa (2) diioc goi la phitong

an chap nhan ditoc Phiiong an chap nhan diiqc nao thoa (1) difoc goi la phitong an tdi itu

Bai toan (1, 2) tong quat neu tren duPc khao sat tren co

so cua bai toan cOc tri co rang buoc va co the giai bang phoong phap nhan td Lagrange trong cac giao trinh Toan cao cap

Rd rang la neu he (2) vd nghiem, nghia la khong co phitong an chap nhan dope, thi bai toan quy hoach titong ilng khong co phitong an tbi itu Neu he (2) cd dung mot nghiem, nghia la cd dung mot phitong an chap nhan ditpc, thi hien nhien phitong an do chinh la phitong an tbi itu Neu he (2) cd nhieu hon mot nghiem, nghia la cd nhieu hon mot phitong an

sb f : 1R 1 -> R va gj : R

g Rn sao cho f(x°) dat gia tri nhd nhat (hay Idn nhat) trong sb cac gia tri f(x), vdi

x g Rn thoa dieu kien gj (x) = 0, i = 1,2, , m Khi do, ta ditpc

bai toan quy hoach :

Tim x g Rn sao cho

tinh va qua do giup sinh vien cd the tiep can de dang phirong phap don hinh cho cac bai toan quy hoach tuyen tinh sau nay.

gi (x1,x2, ,xn

cj’

■n) = c l x l+c 2x2 +- + c n x :

bi b2

vdi moi i = 1,2, trong do

j = 1,2, , n , va c0 la cac hang so

Khi do, ta nhan dupe bai toan quy hoach tuyen tinh :

+ c 0

allxl + a12x2 + + alnxn

a21xl + a22x2 + ••• + a2nxn

a 1J5

bm

n^n

mn x n

) “ a ilX l + ai2 x2 + + ain x n “ bj ,

+ c0 -> min (hay max),

chap nhan dupe, thi bay gid ta can tim phuPng an tdi Ou trong

sb cac phoong an chap nhan dope (cac nghiem cua he 2) bbng each tim gia tri nhd nhat (hay Ion nhat) cua ham muc tieu tren tap cac phitong an chap nhan dope

Triidng hop dac biet cd nhieu Png dung xay ra khi cac ham so f va gj, i = 1,2, ,m, la cac ham bac nhat theo cac bien, nghia la

f (X) = f (x1,x2, ,x

bj, i= l,2, ,m,

Tim x e Rn sao cho

f(x) = cixi +c2x2 + + cnxn

bi aii ai2

+

ta difc/c bang cac he sb

536

X2 a12

X 12x}

he sb cua cac an tifcmg dng trong phiicmg trinh do,

SHTDbi

X1 an

amnbm

Xn aln

ta cd

5 -71

SHTD536

SHTD5-71

x2

-112

x3

2-3 -3

manh, ta them cot

ACS

X1

x2

-112

X3

2

1 1

X1

100

X1

121

x3

2

-3 -3

^2

-1 -1

Dung cac phep bien doi (2) := (2) - 2(1) va (3) := (3) - (1),

Chu y r^ng trong he phifcmg trinh nay, hai phiTcfng trinh cudi chi con lai hai an va neu ta giai diipc he hai phifcmg trinh nay, ta tfnh dupe x2, x3 va the vao phiTcfng trinh dau tien, ta nhan diipc xT An xx chi xuat hien trong phitong trinh dau va

cd he so tiiong ilng bang 1 va ta goi la an ca sd De nhan

an co so (ACS) vao bang cac he so'

1

0 0Tiep tuc dung lan loot cac phep bien doi (1) := (1) + (2) va (3) := (3) - 2(2), ta cd bang

x 2

SHTD-2-715

SHTD123

x3

-1-33

X1

1

0 0

x3

0

0 1

ACS

X1 x2

X1

1

0 0

x2 010

x3Nhir vay, ta da giai diroc he (4) Can chu y them r^ng nghiem nay cd the nhan diTtfc td bang cac he sb bang each cho

cac an d cot an co so (ACS) bang vdi cac gia tri toong ring d cot

sb hang to do (SHTD)

Vi du nay minh hoa cho phiiong phap Gauss-Jordan de

giai he phifOng trinh tuyen tinh bang bang cac he so Tong quat, ta co

1.1 Giai thuat Gauss-Jordan tren bang cac hb sb.

Sau khi thanh lap bang cac he sb cho mot he phuOng trinh tuyen tinh gom cac cot : Cot an co sb (ACS), cot cac sb hang tu1 do (SHTD) va cac cot chtfa he sb cac an (cot Xj,

i = l,2, ,n), trong do cot ACS chu’a ten cac an co sb (an chi xuat hien trong mot dong vbi he sb tOOng ling la 1),

xi

^2

0

10

va dupe them an co sb x2 Tiep tuc dung (3):=^(3);(1) := (1) + (3) va (2) := (2) + 3(3), ta co

ACS

X1 x2 x3

va nhan dope he phdong trinh toong doong

123

ta duyet qua cac dong chifa co an co sb Tren cac dong nay, chon mot an cd he sb titong Ong khac 0 de chuyen an nay thanh an co sb Vi tri cua he sb khac 0 do ditoc goi la phdn til

true xoay va de chuyen an difoc chon thanh an co sb, ta chu y

tbi cot chda phan th true xoay, chang han vbi dong thu" i va an

co sb la x2 (co he sb titong ling ai2 0 ),

(!)(1)- aiz (i); (2) := (2) - a12 (i); (m) := (m) - a12 (i), tadiiqc bang cac he sb co x2 lam an co sb (b hang thuf i),

am2 ••• amn

thanh an co sb bang each dung lan loot cac phep

ain ai2

I?! - a

ain m2 a

Ta co cac kha nang sau :

Kha ndng 1 : Moi dong deu co an co sd, nghia la tren moi dong chifa cd an co so deu cd it nhat mot he so khac 0 Khi do, cac an con lai trd thanh an tif do va dope phep lay gia tri tuy y

va ta cd the tinh dope gia tri cua cac an co so theo cac an tp

do Ta cd hai triTdng hop :

Tritang hop 1 : Khdng cd an tp do He phifong trinh cd

duy nhat mot nghiem

Trildng hop 2 : Cd it nhat mot an

cd vd sb nghiem

Khd ndng 2 : Ton tai mot sb dong khdng cd an co so, nghia la

ton tai mot sb dong ma moi he sb tren dong do deu la sb 0 Ta

cd hai truPng hop :

Truong hop 1 : Cd mot dong (khdng cd an co so) ma sb hang tp do toong Png b * 0 Khi do, dong nay cho phoong trinh dang

0x1 + 0x2+ + 0xn = b PhuPng trinh nay vd nghiem nen he vd nghiem

Truong hop 2 : Moi dong (khdng cd an co sb) deu cd sb hang tp do toong Png bang 0 Cac dong nay cho phuOng trinh dang

bm"3!

X1

all“a12i7

xnaln-a12^

SHTD

_5_

13 7_

132

X1

1

0 0

x2

-31313

X4

-122

x i4X1

2X1

x4

-1-20

x 4

"13 _2_

13

0

X3

2 -5

-5

x42X4

x3

11 13 5_

13

0

x3

2 3-1

true Xoay) lam ACS

3x2 + x2 +

7x2

-0X1 + 0X2 + + 0Xn = 0 Cac phitcmg trinh nay nhan moi gia tri cua an lam

nghiem nen ta co the bo cac dong do ma khong lam mat nghiem cua he Chuyen ve kha nang 1

Vi du 1 : Giai he phiTcmg trinh tuyen tfnh

2Xs X4 = 23X3 - 2X4 = 1

Tren dong 2, chon x2 (tefong ting vdi he so 13 lam phan til true Xoay) lam ACS Lan liipt bien doi (2):=^(2); (1) :=(!) +3(2); (3) := (3) -13(2), ta dtioc

Lap bang cac he so

2 1-1

Tren dong 1, chon X1 (tdong tfng vdi he sb 1 lam phan td

Lan lirpt bien doi (2) := (2) - 4(1);

SHTD5-11-8-29-32

x212246

X4

2133-2

x214378

x i2xx

xi

3X!

2x:

x30-10-3-4

X4

25592

3x3

4x3

Tren dong 2, chon x2

true xoay) lam ACS Lan lurot bien doi (2) := |(2); (1) := (1)

Dong 3 khong cd an co so vi moi he sb deu la sb 0 (Kha nang 2) Dong do cd SHTD toong dng la 2 (TrOdng hop 1) toong ilng vdi philong trinh

OX} + 0x2 + 0x3 + 0x4 = 2

Phoong trinh nay vd nghiem nen he vd nghiem

Vi du 2 : Giai he phitong trinh tuyen tinh

-i

o

-3-4(tiiong ting vdi he sb 2 lam phan tuf

(2);

24

1

2132

Tren dong 1, chon (tifong dng vdi he sb 1 lam phan to true xoay) lam ACS Lan loot bien doi (2) := (2) - 2(1); (3) ;= (3) - (1); (4) := (4) - 3(1); (5) := (5) - 2(1), ta dope

+ x2+ 4x2

+ 3x2

+ 7x2

+ 8x2Lap bang cac he sb

ACS

00

10

SHTD

29 3

-2

n3 3

0

SHTD

21 2

n

2

3-71

(3) := (3) - 2(2); (4) := (4) - 4(2); (5) := (5) - 6(2), ta ditoc

ACS

x41

2 3 2

23-3

x2

01000

x2

01000

x4 3 2

1

2

21-5

X1

10000

X1

1

x30

X4

0

X1

x2 x3

-4

3 -4 4

Tren dong 4, chon x4 (tiTOng dng vdi he sb 3 lam phan th true xoay) lam ACS Lan iPpt bien doi (4):=1(4);

(2):=(2)-f(4); (3) := (3) - 2(4);

(5) := (5) 4- 3(4), ta dilpc

ACS

1 -2

29

3

-2

17 3

4

SHTD51-9-10

51-9-10

x3-2-131

X11

x2 0

X44

2 -6

-2

xi

3x1 xi

12xt

x 30X1

Dong 5 khong co an co sd vi moi he so deu la so 0 (Kha nang 2) Dong do co SHTD tifOng Ong la 0 (TrOdng hop 2) tirong ilng vdi phtfong trinh

Ox! + 0x2 + 0x3 + 0x4 = 0 Phoong trinh nay nhan moi gia tri ciia an lam nghiem nen ta cd the bo di ma khong lam mat nghiem

ACS

x 4Chuyen ve Kha nang 1 Moi an deu la an co sd (Triidng hop 1) nen he cd duy nhat nghiem va nghiem nay nhan dope bdng each cho cac an co so lay gia tri cua so hang tp do toong Ong, X1 =^; x2 =-2; x3 =^; x4 =-|

Vi du 3 : Giai he phoong trinh tuyen tinh

Tren dong 1, chon

true xoay) lam ACS

(3) := (3) - (1); (4) := (4) - 12(1), ta dnpc

ACS

X1 x2

-2-7

X1

1

0

0 0

x2-12210

SHTD5-14-14-70

x3-25525

x4

4 -10 -10 -50

X3 1

2 5

00

Tren dong 2, chon x2

true xoay) lam ACS Lan lifpt bien doi (2) := |(2); (1) := (l) + (2);

hop 2) tiTOng ilng vdi phirong trinh

0 (Kha nang 2) Cac dong nay cd SHTD toong ilng la 0 (TrOdng

m(-ta nhan thay rang nghiem cua

duttc tif ba nghiem dac biet Nghiem thtf nhat

ci

+ Ym+ i

ajiYj -aiiYi

++

a mi Xi CjXi

-amYi

-biYi

a b

a jnYj hjYj

co cac an co so la ym+1, ym+i, ■■■, ym+n> g, ya cac an tu do

la y15 yj, ym

Hai he nay dooc goi la doi ngau nhau difa vao cac yeu to sau :

Noi kliac di, vdi phitong phap Gauss-Jordan bang bang cac he sb, neu he cd vo sb nghiem thi nghiem tong quat nhan dope bang each lay tong cua nghiem co ban voi mot to hop

tuyen tinh bat ky cac nghiem rieng.

1.2 Giai thuat Gauss-Jordan vdri he phtfefng trinh dbi

tif do cua he nay

He so cac an tn do trong pht/ong trinh thtf cua he (I) va

he so cua an ty do thu1 j cua he (II) (he so cua yj trong cac phirong trinh cua he (II)) thi doi nhau

So hang tp do cua he (I) va he so' cac an ty do cua phu’ong

trinh cudi he (II) thi doi nhau.

Sb hang ty do cua he (II) va he sb cac an ty do cua phu’ong trinh cubi he (I) thi bting nhau

Sb hang ty do phu’ong trinh cubi trong hai he thi doi

nhau.

Xn

y m+n

Khi do, ta co sy tiiong quan gitta cac an

vdi cac an co so cua he kia

theo nghia la he sb cua cac an ty do he (I) trong phu’ong trinh thd j (vdi an co sd xn+j) va cac he sb cua an ty do thuf j (yj) trong cac phirong trinh he (II) thi dbi nhau,

He sb ciia cac an ty do he (II) trong phirong trinh thir i (vdi an co sd ym+i) va cac he sb cua an ty do thtr i (Xj) trong cac phirong trinh he (I) thi doh nhau

Nhac lai rhng, sb hang ty do thd j ciia he (I) (vdi an co sd

n+j) va he sb cua an ty do thd (yj) cua phirong trinh cubi he (II) thi doi nhau

Sd hang ty do thd i ciia he (II) (vdi an co sb yin + i) va he

sb ciia an ty do thu- i ( Xj) ciia phu’ong trinh cubi he (I) thi bhng nhau

Bay gib, neu ta dbi vai trb cac an trong he (I) : Xj trd

j trd thanh an ty do (dieu kien a^ 0)

Trong he thu’ nhat, ta doi x2 thanh an cn so, x6 thanh

an ter do va titong ring trong he thd nhi, ta doi y3 (tufefng dng vdi x6) thanh an co so, y5 (tirong hng vdi x2) thanh an to do

b^ng each bien doi :

Trong he thd nhat, dung (1) := (1) + (3); (2) := (2) + (3); (4) := (4) - 2(3) va trong he thd nhi, dimg (2):=-(2);

(1) := (1) - (2); (3) := (3) - (2); (4) := (4) + 6 (2), ta diioc hai he mdi

+ f

de nhan dooc he (F) Titong Ong, trong he (II), ta doi yj (dng vdi xn+j) thanh an co so, ym+i (ufng vdi xj thanh an td do de

nhan duoc he (IF) Ta diioc

Hai he (F) va (IF) cung doi ngau nhau.

Phan chdng minh ditoc trinh bay trong phan phu luc Sau

day ta minh hoa bang vf du cu the :

Xet hai he phiiong trinh

2x2x3 X3 3xs

-Yi - 2y2 - y3

Yi + Y2 - Ys

-2yi - y2 - y3

-5yi - 3y2 - 6y3

Ta cd sd tifong quan gida cac an

3y2

yz2y2

9y2

+

++

+++

211

20-73

1-154

-1-2110

119

6 -10

2x6

3x3

2x3 x3 x3

5 x4

22x4

2 x4 7x4

2x33x3

Giai cac he phitong trinh tuyen tlnh

phap Gauss-Jordan bang bang cac he sb

3x2

x 2 7x2

Ta thay hai he nhan difac nay cung ddi ngau ven nhau Nhan xet nay se giup ich nhieu trong phan sau khi ta khao sat cac bai toan ddi ngau bang thuat giai don hinh ddi ngau.

+ g

+

0

+5)

1

+

2+

X4

2x 4

7x 4 + 8x4

Chut&ng 2

Trong phan md dau, ta da de cap tdi bai toan quy hoach tuyen tmh khi xet bai toan cite tri co rang buoc trong do ca ham muc tieu lan cac ham rang buoc deu la ham bac nhat theo cac bien Trong chitong nay, chung ta lan liTOt khao sat mot sb bai toan cu the trong kinh te di/pc difa ve bai toan quy hoach tuyen tinh cung y tifdng hinh hoc de giai bai toan nay Sau do,

ta phan loai cac bai toan quy hoach tuyen tinh va gidi thieu mot cong cu manh, phiTOng phap don hinh, de giai cac dang cua

bai toan quy hoach tuyen tinh nhan dope

Noi dung trong chitong nay la noi dung chu yeu cua giao trinh nen trong tifng muc chinh, nhieu vi du minh hoa difpc gidi thieu de sinh vien ndm rd y tubng chinh Cudi moi muc la phan bai tap cua muc do de sinh vien thifc tap cac giai thuat

va phan cudi chifong la bai tap on cho ca chuong

Trade het, ta xet mot so bai toan trong kinh te dope mo hinh hda bang bai toan quy hoach tuyen tinh

1 VI DU CHO BAI TOAN QUY HOACH TUYEN TINH1.1 Bai toan van tai

Mot nha may gom ba phan xirdng ?!, P2 , P3 cung san xuat mot loai hang hda vdi san lapng hang thang la mp m2, m3 Hang hda se dope trpe tiep chuyen chd th cac phan xifdng

den hai dai ly Dj, D2 vdi nhu cau hang thang la , n2

Gia str code phi chuyen chd hang hda tuf phan xirdng den dai ly thi ti le thuan vdi so liipng hang hda van chuyen va chieu dai quang difdng Bai toan van tai dat ra la lap ke hoach van chuyen hang hda tif cac phan xubng den cac dai ly sao cho i-aVn- rhi nhf van chuven la than nhat

bang nhau

(1.1)

(1.2)

Kha nang cung m2

Goi Xy la so laong hang hoa van chuyen til phan xudng

Pi den dai ly Dj va ky la chi phi van chuyen tren mot don vi hang hoa til P, den Dj, vdi i = 1,2,3 va j = 1,2

Chang han, x12 la so luong hang hoa van chuyen til ?!

den D2 vdi chi phi van chuyen tren m6t don vi hang hoa la k,, Khi do, chi phi van chuyen sb lupng hang hoa nay la

(1.7)dirc/c gpi la mot phitong an.

Tu1 do, ta nhan difpc bai loan quy hoqch tuyen tinh tim ke

hoach tdi u'u (tot nhat) cho bai toan van tai : Tim nghiem (x11,x12,x21,x22,x31,x32) thoa (1.2), (1.3) va (1.4) sao cho gia tri f cua (1.1) la nhd nhat, f min

Ta viet lai he thong tren di/di dang

x 22

+ X31

+ x32

Xjj > 0, vdi i = 1,3 , j = 1,2

Mot bo gia tri cac an x^

Phitong an nao thoa (1.6) difpc goi la plulang an chap nhan

di/qc Mbi mot phifOng an chap nhan difpc cho ta mot chi phi van chuyen f ma ta con gpi la ham mac tieu Phitong an chap

nhan difpc nao cho gia tri ham muc tieu nhb nhat chinh la nghiem cua bai toan dang khao sat va difpc gpi la phitong an

tdi itu.

Vi du 1 Mot cong ty bach hoa co 4 cda hang B!, B2, B3,

co nhu cau ve mot loai hang tifpng dng la 40, 75, 60, 70 (tan) Cong ty da dat mua loai hang do d 3 xi nghiep Aj , A2,

+ x12 + x22 + x 32

hang tren co sd yeu cau cua

828080

+ 79x14 ++ 79x

+ 82x

737577

748177

797982

B3 b4 b2

A3 vdi khoi krang Wang ling la 45, 90, 110 (tan) Gia cade van chuyen hang (ngan dong/tan) tir mot xi nghiep den m&t cda hang cho trong bang sau

Vi cd ng ty bach ho a dat mua

cac eda hang va

Tong phat = 45 + 90 + 110 = 245 (tan)

Tong thu = 40 + 75 + 60 + 70 = 245 (tan)

Do do, co sir can bang gitfa tong sb Itrong hang dat mua d

cac xi nghiep (tong phat) vdi tdng so' Wang hang ma cac eda hang yeu cau (tong thu)

Goi x^ la sb tan hang van chuyen tir A; den Bj, vdi

= 45: xn + x12 + x13

+-> min

:xn

+ x 33 + X31

+ x24

+ X 21 + x 32

1.2 Bai toan vat tit

Mot xi nghiep san xuat hai mat hang , M2 th ba loai

vat th chu yeu V1, V2, V3

Goi a^ la sb don vi vat tn V; dung de san xuat mot don

vi san pham M3 va c, la loi nhuan thu dope tren mot don vi mat hang Mj, vdi i = 1,2,3 , j = 1,2

(1-8)-> max

Vi du 2 Mot xi nghiep san xuat giay hien co so liftfng hot

go va chat ho keo tiiong ting la 5.580m3 va 90 tan Cac yeu to

san xuat khac cd so Mong Idn Xi nghiep co the san xuat ra ba

loai giay A, B, C Biet so lieu cac loai nguyen lieu de san xuat

ra 1 tan giay thanh pham dupe cho trong bang sau

Ngoai ra, Sb don vi vat tif Vj dung de

done qua sb loqng vat to ton kho bi, nghla la

<

<

<

allXl + a12X 2 ' a21 X l + a22x 2

nghiep dat loi nhuan cao nhat

Goi x15 x2 la sb don vi san pham MT va M2

xuat Hien nhien, ta co

>0,

1.624

l.Sxj + 1.8x2 + 1.6x3

20xt + 30x2 + 24x3

Ngoai ra, gia sO rang san pham san xuat ra deu co the

tieu thu dupe het vdi Ipi nhuan khi san xuat 1 tan giay A, B, C

tuong dng la 2.7; 3.6; 3 (trieu dong) Yeu cau lap ke hoach san xuat tdi Uu

Tong Ipi nhuan thu dupe la

Tong ket : Xuat phat tif hai bai toan tren, ta co

Xet mot ham f (dang (1.5), (1.8)) va mot he phaong trinh hay bat phu’ong trinh tuyen tlnh (dang (1.6), (1.9) hay hon hop

ca hai)

Tim nghiem khong am (> 0) cua he phiiong trinh hay bat phiiong trinh sao cho f nho nhat (min) hay ficin nhat (max)

- Ham f dope goi la ham muc tieu.

- Cac phtfOng trinh hay bat phitong trinh goi la he rang buoc cua bai toan.

- Cac dieu kien (1.7) hay (1.10) ddoc goi la cac rang buoc

dan.

Cac bai toan co dang vda neu dope goi la cac bai toan quy

hoqch hay cac bai toan toi itu Mot bp gia tri cua cac an dope

goi la mot phicong an PhuPng an nao thoa tat ca cac rang buoc

dope goi la phitang an chap nhqn diloc PhOOng an chap nhan

dope nao cho gia tri ham muc tieu nho nhat (hay Idn nhat)

dope goi la phuang an toi i£u (hay con goi la nghiem toi itu).

Trong cac bai toan ma ta khao sat, bieu thefe cua ham muc tieu cung nho cua cac phOOng trinh va bat phoong trinh deu la bac nhat theo cac bien nen con dope goi la bai toan quy hoqch tuyen tinh va mon hoc khao sat no dope goi la quy hoach tuyen tinh

Qua bai toan van tai va vat to neu tren, ta co mot so nhan xet cho giai thuat thanh lap bai toan quy hoach tuyen tinh nho sau :

Bilac 1 : Xdc dinh phuang an.

Ch&ng han phoong an cua bai toan van tai la bp cac con

so Xy chi sb lopng h&ng hoa can van chuyen tO phan xodng P, den dai ly Dj (phoong an van chuyen hay lenh dieu xe)

Phuong an trong bai toan vat tit la bo cac con sb Xj chi sb don vi san pham M; can san suat (phtfong an san xuat hay don dat hang).

Bildc 2 : Xdc dinh ham muc tieu.

Chang han ham muc tieu trong bai toan van tai la tong chi phi van chuyen (can it nhat) va trong bai toan vat to la tong Ipi nhuan thu dupe (can nhieu nhat)

Bude 3 : Xdc dinh cdc rang bu.oc.

Cac rang buoc d day co the hieu la cac dieu kien de mot phoong an trb thanh chap nhan dope (phirong an co the thpc hien dope) Chang han, trong bai toan van tai cac dieu kien nay la : sb hang lay or mot phan xobng bang (hay khong the vOpt qua) sb lopng co the san xuat cua phan xOOng do va sb

hang chuyen chb den mot dai ly bang (hay khong the it hon) nhu cau tieu thu ciia dai ly do

Trong bai toan vat to, cac dieu kien rang buoc la : tong sb vat to de san xuat cho mot phoong an san xuat khong dope vOpt qua sb lopng vat to nay dang ton kho

Ngoai ra, thong thodng cac bien trong mot phoong an kinh te thodng khong am nen ta thobng co them cac rang buoc

ve dau

1.3 Bai tap thiTc hanh

1 Nhan dip tet trung thu, xi nghiep san xuat banh "Trang"

muon san xuat 3 loai banh : dau xanh, thap cam va banh deo

nhan dau xanh De san xuat 3 loai banh nay, xi nghiep can dodng, dau, bot, trOng, mOt, lap xOOng, Gia sb sb dodng co the chuan bi dope la 500kg, dau la 300kg, cac nguyen lieu khac muon bao nhieu cuong cP Lopng doPng, dau can thiet va Ipi nhuan thu dope tren mot cai banh moi loai cho trong bang sau

Banh deoBanh

—> max

3

c

BA

200100100

Banh dau xanh

60

80 2000

Banh thap cam

40

0 1700

Nguyen lieu _

Dufrng (g) _

Dau (g) _ _

Leri nhuan (dong)

Can lap ke hoach san xuat

100100200

dau xanh, banh thap cam va banh deo can san xuat Ta co bai toan quy hoach

Tim x = (x1,x2,x3) sao cho

f = 200X! + 1700x2 + 1800x3

J0.06X! + 0.04x2 +0.07x3

[0.08x2 + 0.04x:

Biet Ipi nhuan thu difpc khi san xuat mot met vai cac loai

A, B, C tuong ring la 350, 480, 250 (dong) San pham san xuat

Xj >0j = l,3

2 Mot xi nghiep det hien cd 3 loai spi : Cotton, Kate, Polyester vdi khoi lifpng tiiong ring la 3; 2.5; 4.2 (tan) Cac yeu to san xuat khac cd so lifpng Idn Xi nghiep cd the san xuat ra 3 loai vai A, B, C (vdi khd be rpng nhat dinh) vdi mdc tieu hao cac loai spi de san xuat ra mot met vai cac loai cho trong bang sau

Loai vai

1621557

7020800

Tim so bo moi loai can nuoi sao cho tong tien ldi la Idn nhat Biet rang so bo sera khong qua 18 con

Huong d&n Goi X1, x2 va x3 lan lupt la so la so bo stfa,

bo cay va bo thit can nuoi Ta co bai toan quy hoach

Tim x = (x1,x2,x3) sao cho

f(x) = 59X1 +49x2 +57x3

3 -> max

ra deu co the tieu thu dupe het vdi sb luqng khong han che, nhung ty le ve sb met vai ciia B va C phai la 1 : 2

Hay xay dung bai toan tim ke hoach san xuat tbi

Huong d&n Goi xx, x2 va x3 lan luqt la so met vai cac

loai A, B va C can san xuat Ta co bai toan quy hoach

Tim x = (x1,x2,x3) sao cho

f(x) = 350X! +480x2 + 250x

200X1 +200x2 +100x3100x] + 200x2 +100x3' 100X! + 100x2 + 200x3

2x2 - x3

Xj > 0, j = 1,3

3 Mot trai chan nuoi dinh nuoi 3 loai bo : bo sUa, bo cay va bo thit Sb lieu dieu tra dupe cho trong bang sau, veri don vi tinh la ngan dong / con

Loai bo

Tai nguyen

3523

5063

3122

381828

310177

104128

376

de dvio’c

Ca chua 55

Khoai tay 26

Dit truf 1892

an phan phoi dat trdng cac loai rau

Xj > 0, j = 1,5.

5 De san xuat 3 loai san pham I, II, HI, ngPdi ta can diing 4

loai nguyen lieu Nj, N2 , N3, N4, vdi cac so lieu duoc cho trong bang sau

ldi nhieu nhat

Hildng d&n Goi X1, x2, x3, x4 va x5 lan lupt dien tich (sao) can phan phoi de trong cac loai bap cai, ca chua, dau, khoai tay, hanh Ta cd bai toan quy hoach

Tim x = (x1,x2,x3,x4,x5) sao cho

4 Mot dpi san xuat dit dinh dung 31 sao dat de trong bap cai,

ca chua, dau, khoai tay, hanh Cac so lieu cho trong bdng sau

Bap cai 79

162118

San pham_ II31035

San pham III

1

0 _

3

4 6

Dif true (kg) San pham

mua hat’giong la 3200$ So ngay cong cham sdc cho cac loai cay A, B, C tren mot mau toong cfng la 1,2, 1 So ngay cong toi

da cd the cd la 160 Neu loi nhuan tren mot mau cua moi loai cay cho bdi : A la 100$, B la 300$, C la 200$, thi phai trong moi loai cay bao nhieu mau de thu Iqi nhuan tdi da

Hitting d&n Goi xT, x2 va x3 Jan loot la so mau dat dq dmh trong 3 loai cay A, B, C Ta cd bai toan quy hoach

J

21.505

II

432_00.036

III1.521.5

3 0.078

So litong yen can

A >12

B = 8

C<6 D>7

I3

0 _ 1

_20.01

them d thi trildng Cac so lieu cho trong bang

Tim x = (x1,x2,x3) sao cho

Ta co bai toan quy hoach

Tim x = (x1,x2,x3,x4) sao cho

f(x) = 7X! + 6x2 + 8x3 + 5x4 min

21 12 X]1 + x 12

+ x22

Xildng A

Xitong B

Dai ly I6$

4$

Dai ly II5$

8$

<

>

12867

xj > 0, j = 1,4

8 Mot hang san xuat may

X 21 + X 21

X 1

2X-L

+ x22

x 12 X11

Xij > 0 , i = 1,2 , j - 1,2

’ ' vi tinh cd hai phan xirdng lap rap A,

B va hai dai ly phan phd'i I, II Xadng A cd the rap tdi da 700 may/thang va xPdng B rap tdi da 900 may/thang Dai ly I tieu thu it nhat 500 may/thang va dai ly II tieu thu it nhat 1000 may/thang Cade phi van chuyen mot may tit cac xildng den cac dai ly cho trong bang sau

Sx! + 4x2 + 1.5x3

3x2 + 2x3 + 2x4 + 2x2 + 1.5x3 + 2x4

Tim x = (x11,x12,x21,x22) sao cho

f(x) = OX}! + 5xiq + 4x91 + 8x

A

-> min

x

1012

1420

3017

Cung cap

_JL

II

10020075125100

X 21 + X22 + X 23 + X21

Muon chuyen chd khoai tay vdi tong ciidc phi nhd nhat Lap mo hinh bai toan

Htfdng dan Goi xi;

sb TP do xet y nghia hinh hoc cung nhif phiiPng phap giai bai toan quy hoach tuyen tfnh hai an bang do thi

9 Cd 2 ntfi cung cap khoai tay I va II theo khdi laong lan lP0t

la 100 tan va 200 tan Co 3 noi tieu thu khoai tay: A, B, C vdi yeu cau Wng ting la 75 tan, 125 tan va 100 tan Citfc phi van chuyen (ngan/tan) van chuyen tP cac noi cung cap den noi tieu

thu dope cho trong bang sau

Tieu thu

, i = 1,2 , j = 1,2,3, la sb tan khoai tay

dupe cho tif noi cung cap I, II den noi tieu thu A, B, C Ta co

bai toan quy hoach

Tim X = ( x 11,X12> x 13> x 21’X22’X23)

f(x) = 6xn + 5x12 + 4x21 + 8x22

X11 + x 12 + X13

(1.11)min (max),

(1.12)

(1.13)

ajXj + a2x2 = m DtTdng thdng nay chia mat ph^ng ra

gom cac diem x = (x1,x2)gK2 sao cho

lam hai mien : miena^x-j + a2x2 > m va

X = (x1,x2) e R2

chinh la diidng thdng

2 Y NGHIA HINH HOC CUA BAI TOAN QUY HOACH TUYEN TINH

Xet mot bai toan quy hoach tuyen tmh theo hai an :

Tim x = (x1,x.2) g R2 sao cho

De cd the xac dinh mien “phdcmg an chap nhan ddpc” cung nhd “gia tri cua ham muc tieu” tai cac diem phdcmg an chap nhan ddge”, ta khao sat ham bac nhat

f(x) = f(x1,x2) = a1x1 +a2x2

Lfng vdi moi gia tri m g R , tap cac diem

sao cho f (x) = f (x1, x2) = a1x1 + a2x2 = m

trong mat phdng Ox1x2 cd phdcmg trinh