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BQ GIAO DUG VA DAO TAO TRUONG DAI HOC DAN LAP CLTU LONG Nguyen Quoc Huy — Dinh Ngoc Thanh Tran Thi Ngoc Tuyet Giao Trinh TOAN KINH TE Phan QUY HOACH TUYEN TINH TRIM DAlkot' DAN LAPC& thUvien S^angky: Vinh Long - 2007 LCfl DE TUA Tru'dng Dai hoc Dan lap Ctfu Long muon khdng dinh la trung tam dao tao nguon nhan life cd trinh 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 nhCfng yeu to yeu la doi ngu thay cd, he thong cac giao trinh va trang thiet bi day hoc Gung vdi cac nganh cac cap toan quoc dang day nhanh tien xay dqng va cung co vi the 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 the gidi Ke hoach xay difng ve mot Bo giao trinh stf dung cho cac giang vien lam tai lieu co ban de giang day, cho sinh vien nghien cufu tham khao hoc tap la nhu cau cap bach cua nha tru’dng An pham 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, 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 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 Hy vong r^ng, mot khong xa Trifdng Dai hoc Dan lap Cufu Long se co day du mot Bo giao trinh hoan chinh dung nha trPdng Tran gidi thieu giao trinh den Quy thay co va cac em sinh vien triidng Dai hoc Dan lap CuP Long Vinh Long, thang 09 nam 2007 Q HIEU TRLfdNG ThS Nguyen Cao Dat Chitorng PHUONG PHAP GAUSS - JORDAN Nhieu bai toan hng dung Khoa hoc Ky thuat noi chung va Kinh te noi rieng diroc quy ve viec tim gia tri Idn nhat hay nhd nhat cua mot ham so' vdi mot sb rang buoc Chang han, vdi cac ham sb f : 1R lRn1 -> R va gj : R lRn R, i = 1,2, ,m , ta tim x° g Rn cho f(x°) dat gia tri nhd nhat (hay Idn nhat) 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 cho f(x) (hay max), (1) gi (x) = 0, i = l,2, ,m (2) thoa cac rang buoc Ham f 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 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 chinh la phitong an tbi itu Neu he (2) cd nhieu hon mot nghiem, nghia la cd nhieu hon mot phitong an chap nhan dupe, thi bay gid ta can tim phuPng an tdi Ou 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 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.■n) = clxl+c2x2+- + cnx: n^n + c0 va Si(x) vdi moi gi (x1,x2, ,xn ) i = 1,2, “ ailXl + ai2x2 + cj’ a1J5 + ainxn “ bj , bj, i = l,2, ,m, j = 1,2, , n , va c0 la cac hang so Khi do, ta nhan dupe bai toan quy hoach tuyen tinh : Tim x e Rn cho f(x) = cixi +c2x2 + + cnxn + c0 -> (hay max), (3) thda cdc rang buoc + alnxn bi a21xl + a22x2 + ••• + a2nxn b2 amixi +am2x2 + + a mn xn bm allxl + a12x2 + (4) Trifdc het, ta can xac dinh cac phiTcfng an chap nhan ditoc cua bai toan quy hoach tuyen tmh, nghia la xac dinh tap nghiem ciia he phifcfng trinh tuyen tinh (4) Viec giai he (4) da ditoc trinh bay cac giao trinh Toan cao cap Tuy nhien, phan md dau nay, ta nhac lai chi tiet cung nhii trinh bay lai phitong phap Gauss-Jordan de giai he phuPng trinh tuyen tinh va qua giup sinh vien cd the tiep can de dang phirong phap don hinh cho cac bai toan quy hoach tuyen tinh sau PhiTOng phap Gauss noi chung va phitong phap GaussJordan noi rieng diTa tren nhan xet rang : Neu ta nhan hai ve cua mot phuong trinh he (4) cho mot hbng sb khac hay thay mot phiTOng trinh he (4) bang phifong trinh cong vcfi mot phuWng trinh khac he nhan cho mot hbng sb thi ta nhan diioc mot he phirong trinh tuyen tinh mdi titong diTOng vdi he ban dau, nghia la tap nghiem cua chung khong dbi Do do, ta bieu dien he phiiong trinh tuyen tinh (4) bbng bang cac he sb, moi mot dong titong hng vdi mot phiTOng trinh cua he, cot dau tien danh cho cac sb hang tii (SHTD) va cac cot lai danh cho cac he sb cua cac an tifcmg dng phiicmg trinh do, SHTD X1 X2 Xn bi an a12 aln bi aii ai2 ain bm aml am2 amn Ta co the chuyen he (4) he tifcfng diicmg bbng each thiic hien cac phep bien dbi tren cac dong cua bang cac he so : Nhan dong thil i cho mot so a 0, ky hieu (i) := a(i), Thay dong thd i bbng dong cong cho mot hdng so p nhan cho dong thd j , ky hieu (i) := (i) + p(j) Chang han, vdi he phiicmg trinh + 2x3 X1 x2 2x} x2 x3 xi x2 x3 ta difc/c bang cac he sb SHTD X1 ^2 X3 -1 -1 1 Dung cac phep bien doi (2) := (2) - 2(1) va (3) := (3) - (1), ta cd SHTD X1 x2 x3 -7 0 -1 2 -3 -3 va nhan ditpc he phifcfng trinh tuyen tlnh titong du’Ong Xj - x22 X + t- 2x X2 - x3; 3x 2x2 — x3 3x -7 Chu y r^ng he phifcmg trinh nay, hai phiTcfng trinh cudi chi 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 phitong trinh dau va cd he so tiiong ilng bang va ta goi la an ca sd De nhan manh, ta them cot an co so (ACS) vao bang cac he so' ACS SHTD X1 -7 1 0 x2 x3 -1 2 -3 -3 Tiep tuc dung lan loot cac phep bien doi (1) := (1) + (2) va (3) := (3) - 2(2), ta cd bang ACS SHTD X1 ^2 x3 X1 -2 -7 0 -1 -3 15 0 x2 va dupe them an co sb x2 Tiep tuc dung (3):=^(3); (1) := (1) + (3) va (2) := (2) + 3(3), ta co ACS SHTD X1 X1 x2 x3 x3 x2 0 va nhan dope he phdong trinh toong doong xi x2 x3 Nhir vay, ta da giai diroc he (4) Can chu y them r^ng nghiem 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 (SHTD) Vi du 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 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 (SHTD) va cac cot chtfa he sb cac an (cot Xj, i = l,2, ,n), cot ACS chu’a ten cac an co sb (an chi xuat hien mot dong vbi he sb tOOng ling la 1), 142 (3,3) khong tao vong vdi cac chon khac Giup ta tlnh du cac UjVavj Khong nSm vong chuyen sang phuong an sau : Kiem tra tinh tdi tfu A12 A13 = -5bj xn + m £ 0 0 aJi i aji _ anii aji aji a+— L aji xn + j Cj ”^7 Ta bien dbi he sau mot each tirong ilng : Chuyen an til yj an co sd Dieu thhc hien dope -a ji va do, an co sd ym+i trd an til Bien ddi (i) := (n):=(n) a jn aii (i); (n + !) := (n + 1) aii (0; aj> v ' 152 ACS y m+1 -an iZi + ajiaii ajj ! ajla mi + aji -aml- + ajnali ain b aji bjaii Cl ym +1 anc- c aii aii ajnaini amn aii ym + n £ 0 0 aii i aii i ajnc> cn aii bj^rni bm ym + i aji anii aii y m+n s ym yj g yi _ o a ji aii Khi hai he mdi cung dbi ngbu Chang han, 1) An co so X; cua he trbdc thong dng vdi an th ym+i cda he sau, an th xn+j cua he trhdc thong hng vdi an co so yj cua he sau, va he sb cua xn+j bieu thdc (dong) chha Xi la , he so cua ym+i bieu thdc (dong) chda yj la - a • 2) An th t cua he trhdc thong dng vcri an co so g cua he sau, an co so x, cua he trhdc thong dng vdi an th ym+i cua bj he sau, va he sb cua t bieu thdc (dong) chda X; la he sb cua ym+i bieu thdc (dong) chda g la , 3) An til xn+j ciia he tritac tiiang dng vdi an co so y3 cua he sau, an co so f cua he tnldc tilong dng vdi an tu s cua he sau, vd he so cua xn+j bieu thdc (dong) chda f la he so cua s bieu thilc (dong) cluia yji Zd dji 4) An th t cua he tritfc thong hng vdi an co sb g cua he cua he sau, an co sb f cua he trdcic thong dng vdi an th s sau, va he sb cua t bieu thdc (dong) chda f la i- he sb cua s bieu thtfc (dong) chila g bjc> la -“"V a + c,bj a ’ aji 153 TAI LIEU TEAM KHAO [1] Nguyen Thanh Ca, Toi ilu hoa, phdn Quy hoach tuyen tinh, NXB Thong Ke, 2005 [2] A C Chiang, Fundamental Methods of Mathematical Economics, Me GrawHill, Inc., 3rd edition, 1984 [3] Dang Han, Quy hoach tuyen tinh, Trufrng Dai Hoc Kinh Te Tp Ho Chi Minh, 1995 154 155 MUC LUC Chirang : PHUONG PHAP GAUSS - JORDAN 1.1 Giai thuat Gauss-Jordan tren bang cac he sb 1.2 Giai thuat Gauss-Jordan vdi he phu’Ong trinh ddi ngau 18 Bai tap 21 Chirong : BAI TOAN QUY HOACH TUYEN TINH 23 Vi du cho bai toan quy hoach tuyen tfnh 23 Y nghia hinh hoc cua bai toan quy hoach tuyen tfnh 39 Cac dang cua bai toan quy hoach tuyen tfnh 48 Phitong phap dan hinh 55 Bai tap 81 Chtrong : BAI TOAN DOI NGAU 99 Thanh lap bai toan ddi ngau 100 Y nghia kinh te cua bai toan ddi ngbu 102 Cap bai toan ddi ngau tong quat 104 Phirong phap don hinh ddi nghu 114 Bai tap 119 156 Chircmg : BAI TOAN VAN TAI 125 Thanh lap bai loan van tai 125 Cac tinh chat cua bai toan van tai 128 Cac phifcfng phap tim philcfng an xuat phat 130 Thuat toan the vi giai bai toan van tai 135 Trifbng hop khong can bang thu phat 143 Bai tap 146 PHU LUC 149 TAI LIEU THAM KHAO 153 MUC LUC 155