Introducción a la programación lineal.pdf

1. Álgebra superior. Curso completo Cálculo integral de varias variables Curso intermedio de probabilidad Geometría analítica Introducción a la geometría avanzada Introducción a la probabilidad Introducción a la teoría del riesgo María del Carmen Hernández María del Carmen Hernández Ayuso Introducción a la programación lineal PANTONE 3262 U PANTONE 655 U PANTONE 621 U L aprogramaciónlinealesunaramadelainvestigacióndeoperacionesqueestudia problemas de optimización en los cuales se desea maximizar o minimizar una función lineal restringida mediante ecuaciones o desigualdades lineales. Esta disciplina debe su fama principalmente a dos puntos; el primero es que gran variedad de problemas, en diversos campos, pueden ser formulados como modelos lineales y el segundo es que se han desarrollado algoritmos sencillos para resolver este tipo de problemas. En 1974, George B. Dantzig publicó su algoritmo simplex para reslover modelos de programación lineal. En los siguientes años se hicieron modificaciones que mejoraron su eficiencia y, aunque cualquier variante es no polinomial, actualmente este algoritmo es el más utilizado en la práctica debido principalmente a su buen desempeño y a las herramientas matemáticas relativamente sencillas que usa. Este libro constituye un apoyo para cursos introductorios de investigación de operaciones o programación lineal de las carreras de actuaría, matemáticas, ingeniería y administración. Tiene un nivel básico y el único requisito para su comprensión es contar con conocimientos de álgebra lineal. Se presentan ejemplos de problemas clásicos de programación lineal, las propiedades del modelo, el algoritmo simplex y especializaciones de éste para los problemas de transporte y de ordenamiento. En cada capítulo se han incluido ejemplosyseplanteanejerciciosparaponerenprácticalosconocimientosadquiridos. Estudió la carrera de matemáticas en la Fa- cultad de Ciencias de la unam y la Maestría en Investigación de Operaciones en la División de Estudios de Posgrado de la Facultad de Inge- niería de la unam. Ha impartido cursos a nivel licenciatura en el área de investigación de operaciones, en las facultades de Ciencias y de Ingeniería de la unam; en la División de Ciencias Básicas de la uam-Azcapotzalco y en la Escuela de Actuaría de la Universidad Anáhuac del Sur. A lo largo de su actividad académica, ha publicado diversas notas para cursos de esta rama de las matemáticas aplicadas, así como el libro Introducción a la teoría de redes editado por la Sociedad Matemática Mexicana. Otros títulos de esta colección: Carmen Gómez Laveaga Javier Páez Cárdenas Luis Rincón Ana Irene Ramírez-Galarza Ana Irene Ramírez-Galarza José Seade Kuri Luis Rincón Luis Rincón 9 786073 001472 ISBN: 978-607-30-0147-2

2. María del Carmen Hernández Ayuso NTRODUCCIÓN A LA PROGRAMACIÓN LINEAL FACULTAD DE CIENCIAS, UNAM

3. Introducción a la programación lineal 1º edición, 2007 Universidad Nacional Autónoma de México. Impreso y hecho en México. 2º edición, 2010 1º reimpresión, 2013 3º edición, 10 de diciembre de 2017 Diseño de portada: Laura Uribe Hernández ISBN: 978-607-30-0147-2 © D.R. 2017. Facultad de Ciencias. Ciudad Universitaria. Delegación Coyoacán, C. P. 04510, México, Distrito Federal. editoriales@ciencias.unam.mx Prohibida la reproducción parcial o total de la obra por cualquier medio, sin la autorización por escrito del titular de los derechos patrimoniales. Hernández Ayuso, María del Carmen Introducción a la programación lineal / Ma. del Carmen Hernández Ayuso. -- 3 edición. -- México : UNAM, Facultad de Ciencias, 2017. Reimpresión. xv, 233 páginas : ilustraciones ; 22 cm. - (Temas de matemáticas) Bibliografía: páginas 231-233 ISBN: 978-607-30-0147-2 1. Programación lineal. 2. Programación lineal Problemas, ejerci- cios, etc. 3. Investigación operativa. 4. Optimización matemática. I. Universidad Nacional Autónoma de México. Facultad de Ciencias. II. título. III. Serie. 519.72-scdd21 Biblioteca Nacional de México a 1º reimpresión, 2019

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335. 9@ 7CBIBHC 89 GC@I7=CB9G :57H=6@9G 89 9G 897=F

336. 8 BHCB79G

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338. ,95 7 8 9 ,9 DFC65FX EI9 7

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344. ,8/0(+ -+%-)!9* (*#( (/1564$,@0 ,95

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349. 8 8 3 !

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351. 8 8 Puntos que satisfacen 3 x1 + 2 x2= 18 Puntos que satisfacen 3 x1 + 2 x2 18 Puntos que satisfacen 3 x1 + 2 x218 x2 x1 )'52!

352. ,8/0(+ -+%-)!9* (*#( F5OEI9ACG 9B 9@ D@5BC @5 F97H5

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354. 8 8 M CHFC EI9 BC %5 F9;=B :57H=6@9 89@ DFC6@9A5 9GHX HCH5@A9BH9 7CBH9B=85 9B 9@ DF=A9FC )5F5 89H9FA=B5F 9B 7IX@ 89 9@@CG G9 9B7I9BHF5 65GH5 DFC65F G= IB DIBHC 7I5@EI=9F5 G5H=G:579 C BC @5 89G=;I5@858 )CF 99AD@C 9@ CF=;9B @5 G5H=G:579 9BHCB79G @5 F9;=B :57H=6@9 9GHX 7CBH9B=85 9B 9@ G9A=9GD57=C EI9 7CBH=9B9 5 9GH9 DIBHC B @5 O;IF5 G9 5 GCA6F958C 9GH9 G9A=9GD57=C Puntos que satisfacen x1 + x2= 5 Puntos que satisfacen x1 + x2 5 Puntos que satisfacen x1 + x2 5 x2 x1 )'52!

355. 7/++ %-64!+ +9D=H5ACG @CG D5GCG M 7CB @5 F9GHF=77=B 8 8 @ G9A=9GD57=C 89OB=8C DCF 9GH5 F9GHF=77=B G9 AI9GHF5 9B @5 O;IF5 %5 F9;=B :57H=6@9 9GHX 7CAD@9H5A9BH9 7CBH9B=85 9B @5 =BH9FG977=B 89 9GHCG 8CG G9A=9GD57=CG ACGHF585 9B @5 O;IF5 Puntos que satisfacen 3 x1 + 2 x2  18 y x1 + x2  5 x2 x1 )'52!

356. +9D=H5ACG @CG D5GCG M 7CB @5G F9GHF=77=CB9G 8 8 8 %5 =BH9FG977=B 89 HC8CG @CG G9A=9GD57=CG 89OB=8CG 9G @5 F9;=B :57H=6@9 89@ DFC6@9A5 GH5 G9 AI9GHF5 9B @5 O;IF5

357. ,8/0(+ -+%-)!9* (*#( Región factible x2 x1 )'52! .B5 J9N EI9 9ACG ;F5O758C @5 F9;=B :57H=6@9 DFC7989F9ACG 5 89H9F A=B5F 9@ DIBHC EI9 A5L=A=N5 @5 :IB7=B : CB 9GH9 C69HC ;F5O75F9ACG F97H5G 7CFF9GDCB8=9BH9G 5 8=:9F9BH9G J5@CF9G 89 : M C6G9FJ5F9ACG 9B EIY 8=F977=B 7F979 )CF 99AD@C @5G F97H5G D5F5@9@5G : :

358. : G9 AI9GHF5B DIBH9585G 9B @5 O;IF5 B 9GH5 O;IF5 G9 5 8=6I58C H5A6=YB 9@ ;F58=9B H9 89 @5 :IB7=B : EI9 =B8=75 @5 8=F977=B 89 7F97=A=9BHC 89 YGH5 'CH9ACG

359. 7/++ %-64!+ EI9 7585 F97H5 :

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361. ,8/0(+ -+%-)!9* (*#( Punto óptimo 3 x1 + 6 x2 = 54 Figura 2.6 x1 x2 B 7CB7@IG=B @5 GC@I7=B DH=A5 89@ DFC6@9A5 9G 8 M 9@ J5@CF DH=AC 89@ DFC6@9A5 9G : : -5A6=YB 8969ACG C6G9FJ5F EI9 9@ DH=AC G9 8= 9B IB JYFH=79 DIBHC 9LHF9AC 89 @5 F9;=B :57H=6@9 GHC 9G JX@=8C 9B ;9B9F5@ G= 9G EI9 9L=GH9 H5@ DH=AC 7CAC 89ACGHF5ACG 9B @5 G977=B 5BH9F=CF $515 (52(,$.(5 B 9GH5 D5FH9 89@ 75DZHI@C J9F9ACG A98=5BH9 99AD@CG @CG 75GCG EI9 DI989B DF9G9BH5FG9 7I5B8C F9GC@J9ACG IB DFC6@9A5 @=B95@

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363. 8 8 3 ! 8 8 8

364. 8 8 8

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367. DFC8I7=9B8C IB AZB=AC J5@CF 89 : =;I5@ 5 : :8

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369. CBG=89F9ACG 5CF5 9@ DFC6@9A5 ). : 8 8 3 ! 8 8 8

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371. ,8/0(+ -+%-)!9* (*#( B @5 O;IF5 DC89ACG C6G9FJ5F EI9 @5 :IB7=B C69H=JC 9G D5F5@9@5 5 @5 F97H5 89OB=85 DCF @5 DF=A9F5 F9GHF=77=B 'CH9ACG EI9 @5 8=F977=B 89 7F97=A=9BHC 89 : =AD@=75 EI9 9@ DH=AC M5 EI9 9G IB DFC6@9A5 89 A=B= A=N57=B G9 5@75BN5 9B IB G9;A9BHC 89 8=75 F97H5 9B 9GH9 75GC H9B9ACG 9BHCB79G IB5 =BOB=858 89 DIBHCG DH=ACG z = 4 z = 0 Soluciones óptimas 5 5 Dirección de crecimiento de z x2 x1 )'52! G =ADCFH5BH9 BCH5F EI9 8CG 89 9GHCG DH=ACG GCB DIBHCG 9LHF9ACG 8 M ) 8

373. @CG 89AXG DH=ACG GCB DIBHCG BC 9LHF9ACG %5 75F57H9F=N57=B 89 HC8CG 9@@CG 9G

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375. 8 3 ! 8 8 8

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378. .B5 A5B9F5 89 75F57H9F=N5F HC8CG @CG DH=ACG 9B 9GH9 75GC 9G 8

379. 8 8

380. @ J5@CF DH=AC 89@ DFC6@9A5 9G : :8

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382. 8 8 3 ! 8 8 8

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385. 8 8 + 9L=GH9 CHF5 F97H5

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387. + EI9 =BH9FG97H5 @5 F9;=B :57H=6@9 GHC =AD@=75 EI9 : BC 9GHX 57CH585 GID9F=CFA9BH9 9B @5 F9;=B :57H=6@9 DCF @C EI9 BC 9L=GH9 GC@I7=B DH=A5 D5F5 9@ DFC6@9A5 9GH9 H=DC 89 DFC6@9A5 @C @@5A5F9ACG 4630)1% 23 %'38%(3 Dirección de crecimiento de z z = 3 z = - 3 z = 6 5 5 )'52!

388. ,8/0(+ -+%-)!9* (*#( -(/2.1 ,95 9@ DFC6@9A5 ). : 8 8 3 ! 8 8 8 8 8 8

389. 8 8 %5 =BH9FG977=B 89 @CG G9A=9GD57=CG 89OB=8CG DCF 7585 IB5 89 @5G F9G HF=77=CB9G 9G 9B 9GH9 DFC6@9A5 J57Z5 7CAC DC89ACG J9F 9B @5 O;IF5 B 9GH9 DFC6@9A5 BC 9L=GH9 B=B;IB5 GC@I7=B EI9 G5H=G:5;5 HC85G @5G F9GHF=7 7=CB9G 5 @5 J9N DCF @C EI9 @C @@5A5F9ACG 4630)1% 23 *%'8-0) x2 x1 )'52! -(/2.1 7CBH=BI57=B G9 F9GC@J9FX CHFC 99AD@C 9B 9@ 7I5@ @5

390. 7/++ %-64!+ GC@I7=B 9G ]B=75 ,95 9@ DFC6@9A5 !8 7 8 8 3 ! 8 8 8 8 8

391. 8 8 -5 5 10 5 -5 x1 x2 2x1 + x2 = 8 x1 + 2 x2= 7 x2 = 3 )'52!

392. ,=;5ACG 9@ DFC798=A=9BHC 89G7F=HC 5@ DF=B7=D=C 89 @5 G977=B D5F5 89 H9FA=B5F @5 F9;=B :57H=6@9 89@ DFC6@9A5 B @5 O;IF5 G9 AI9GHF5 GCA6F9585 YGH5

393. ,8/0(+ -+%-)!9* (*#( @ ;F58=9BH9 89 @5 :IB7=B : 9G 9@ J97HCF )I9GHC EI9 YGH9 =B8=75 @5 8=F977=B 89 7F97=A=9BHC 89 @5 :IB7=B C69H=JC M 9B 9@ DFC6@9A5 G9 89G95 A5L=A=N5F 7CB7@I=ACG EI9 9@ DH=AC G9 85 9B 9@ DIBHC 9LHF9AC 8

394. DFC8I7=9B8C IB J5@CF 89 : =;I5@ 5 : :8 'CH9ACG EI9 @5 ]B=75 GC@I7=B DH=A5 9G 8 $4$6(4,;$,@0 '( 270615 (964(/15 CBG=8YF9G9 9@ G=GH9A5 89 97I57=CB9G @=B95@9G 8 8CB89 9G IB5 A5HF=N - . M 8 GCB J97HCF9G 7C@IAB5 ,95 IB5 GI6A5HF=N 89 89 CF89B -- BC G=B;I@5F 9G 897=F IB5 65G9 89@ 9GD57=C 7C@IAB5 89 M GIDB;5G9 EI9 @5G . - 7CADCB9BH9G 89 8 5GC7=585G 7CB @5G 7C@IAB5G 89 EI9 BC 9GHXB 9B GCB =;I5@9G 5 79FC 9BHCB79G @5 GC@I7=B 89@ 7CBIBHC 89 97I57=CB9G F9GI@H5BH9 G9 89BCA=B5 51.7,@0 %=5,$ 7CB F9GD97HC 5 @5 65G9 GZ A=GAC @5G 7CADCB9BH9G 89 8 5GC7=585G 5 @5G 7C@IAB5G 89 G9 @@5A5B J5F=56@9G 6XG=75G ,= 5@;IB5 89 YGH5G H=9B9 J5@CF 79FC G9 8=79 EI9 @5 GC@I7=B 9G '(*(0(4$'$ (6GYFJ9G9 EI9 9@ G=GH9A5 89 97I57=CB9G @=B95@9G A9B7=CB58C DC8FZ5 G9F 9@ 7CBIBHC 89 F9GHF=77=CB9G 89 IB DFC6@9A5 89 DFC;F5A57=B @=B95@ DCF @C EI9 DC89ACG F9:9F=FBCG 5 @5G GC@I7=CB9G 6XG=75G 89@ DFC6@9A5 'H9G9 H5A6=YB EI9 9@ B]A9FC 89 9@@5G 9G OB=HC 9B 9:97HC DI989 G9F 5 @C AXG =;I5@ 5 @5G 7CA6=B57=CB9G 89 . 9B - B CHF5G D5@56F5G 8 9G 51.7,@0 %=5,$ 89 IB DFC6@9A5 @=B95@ 9B :CFA5 9GHXB85F G= @5G 7C@IAB5G 89 @5 A5HF=N 89 F9GHF=77=CB9G 5GC7=585G 5 @5G 7CADCB9BH9G 8=:9F9BH9G 89 79FC GCB @=B95@A9BH9 =B89D9B8=9BH9G G 897=F G= G9 D5FH=7=CB5 9B M 8CB89 9G @5 A5HF=N :CFA585 DCF @5G 7C@IAB5G 89 EI9 BC D9FH9B979B 5 9@ G=GH9A5 89 97I57=CB9G 8 DI989 9G7F=6=FG9 8 8

395. 8 8

396. -!/#-3!9* # ,0*/+. #1/-#)+. 8CB89 8 M 8 GCB @CG J97HCF9G :CFA58CG DCF @5G J5F=56@9G 5GC7=585G 5 @5G 7C@IAB5G 89 M F9GD97H=J5A9BH9 9GD95B8C 8 89 @5 9LDF9G=B 5BH9F=CF G9 C6H=9B9 8 8 ,= 5G=;B5ACG J5@CF 5 @5G 7CADCB9BH9G 89 8 9BHCB79G 8 M 9@ J97HCF 8 8 8

397. 9G IB5 51.7,@0 %=5,$ %5G 7CADCB9BH9G 89 8 G9 @@5A5B J5F=56@9G 6XG=75G %5G 89 8 GCB BC 6XG=75G -(/2.1 CBG=89F9ACG 7CAC 99AD@C 9@ G=GH9A5

398. 8 8 8

399. 8 8 8 8 8 ,95 @5 A5HF=N

400. # M G95B 8 89 7CADCB9BH9G 8 ) M

401. 9@ @58C 89F97C 89@ G=GH9A5

402. # 8 $ $ $ $ 8 8 8

403. 8 8 % % % % # #

404. ,8/0(+ -+%-)!9* (*#( @ G=GH9A5 DI989 9G7F=6=FG9 8

405. # $ $ $ $ 8 8 8

406. 8 8 % % % % # # 9BCH9ACG 7CB ! ) 5 @5G 7C@IAB5G 89 ,95 ! ! !

407. # BHCB79G ! ! # 8 8 8 8

408. # 8 8 8

409. %5 A5HF=N 9G BC G=B;I@5F M H=9B9 =BJ9FG5 # @ G=GH9A5 8 DI989 F99G7F=6=FG9 8 8

410. 8 8 9G 897=F

411. # 8 8 8

412. # # 8 8

413. # BHCB79G 8 8 8 8

414. # 8

415. -!/#-3!9* # ,0*/+. #1/-#)+. ,= 8 8 8

416. 89 @5 9LDF9G=B 5BH9F=CF 8 # # # 8 8 8

417. # DCF @C H5BHC ) 8 9G IB5 GC@I7=B 6XG=75 (6GYFJ9G9 EI9 9@ G=GH9A5 7CBG=89F58C 9G 9@ 7CBIBHC 89 F9GHF=77=CB9G 89@ AC89@C 9LDF9G58C 9B :CFA5 9GHXB85F F9GI9@HC ;FXO75A9BH9 9B 9@ 99AD@C .BC 89 @CG DIBHCG 9LHF9ACG 89 @5 F9;=B :57H=6@9 89 9GH9 DFC6@9A5 9G

418. 9@ 7I5@ 9G @5 =BH9FG977=B 89 @5 G9;IB85 M H9F79F5 F9GHF=77=CB9G 89@ DFC6@9A5 G 897=F 9G @5 GC@I7=B 89 8 8 8 %5 GC@I7=B 6XG=75 ) 8 7CBH=9B9 HC85 9GH5 =B:CFA57=B DI9G GIG DF=A9F5G 8CG 7CADCB9BH9G GCB 9@ J5@CF 89 @5G J5F=56@9G 8 M 8 M @5G F9GH5BH9G =B8=75B @5 8=GH5B7=5 5@ @58C 89F97C M5 EI9 GCB @5G J5F=56@9G 89 C@;IF5 9 9GH9 AC8C 8 8 =AD@=75 EI9 9@ DIBHC 9GHX GC6F9 @5G F9GHF=77=CB9G G9;IB85 M H9F79F5 M 8

419. 9G @5 8=GH5B7=5 5 @5 DF=A9F5 F9GHF=77=B 7CBH=BI57=B G9 C6H9B8FXB HC85G @5G GC@I7=CB9G 6XG=75G 89 9GH9 G=GH9A5 ,95 ! ! !

420. #

421. # BHCB79G 8

422. # #

423. # 8 8 8 # DCF H5BHC * 8

424. 9G IB5 GC@I7=B 6XG=75

425. ,8/0(+ -+%-)!9* (*#( GH5 GC@I7=B 7CFF9GDCB89 5@ DIBHC 9LHF9AC 89OB=8C DCF @5 =BH9FG977=B 89 @5G F9GHF=77=CB9G DF=A9F5 M H9F79F5 ,95

426. ! ! !

427. #

428. # BHCB79G 8

429. # #

430. # 8 8 8 # DCF H5BHC 8

431. 9G IB5 GC@I7=B 6XG=75 GH5 GC@I7=B 7CFF9GDCB89 5@ DIBHC 9LHF9AC 89OB=8C DCF @5 =BH9FG977=B 89 @5G F9GHF=77=CB9G DF=A9F5 M G9;IB85 'H9G9 EI9 G9 9B7I9BHF5 :I9F5 89 @5 F9;=B :57H=6@9 ,95 ! !

432. !

433. #

434. # BHCB79G 8

435. # # # 8 8

436. 8 # DCF H5BHC 8 9G IB5 GC@I7=B 6XG=75 GH5 GC@I7=B 7CFF9GDCB89 5@ DIBHC 9LHF9AC 89OB=8C DCF @5 =BH9FG97 7=B 89 @5G F9GHF=77=CB9G H9F79F5 M 89 BC B9;5H=J=858 89 8 'H9G9 EI9 G9 9B7I9BHF5 :I9F5 89 @5 F9;=B :57H=6@9 ,95 ! !

437. !

438. #

439. # BHCB79G 8

440. # #

441. # 8 8

442. 8 #

443. -!/#-3!9* # ,0*/+. #1/-#)+. DCF H5BHC 8

444. 9G IB5 GC@I7=B 6XG=75 GH5 GC@I7=B 7CFF9GDCB89 5@ DIBHC 9LHF9AC 89OB=8C DCF @5 =BH9FG977=B 89 @5G F9GHF=77=CB9G G9;IB85 M 89 BC B9;5H=J=858 89 8 'H9G9 EI9 G9 9B7I9BHF5 :I9F5 89 @5 F9;=B :57H=6@9 ,95 ! ! !

445. #

447. # BHCB79G 8

449. # # # 8 8 8 # DCF H5BHC ) ) 8 9G IB5 GC@I7=B 6XG=75 GH5 GC@I7=B 7CFF9GDCB89 5@ DIBHC 9LHF9AC 89OB=8C DCF @5 =BH9FG977=B 89 @5G F9GHF=77=CB9G DF=A9F5 M 89 BC B9;5H=J=858 89 8 'H9G9 EI9 G9 9B7I9BHF5 :I9F5 89 @5 F9;=B :57H=6@9 ,95 ! !

450. ! # 9GH5 A5HF=N 9G G=B;I@5F DCF @C EI9 BC 7CBGH=HIM9 IB5 65G9 89@ 9GD57=C 7C@IAB5 89 M DCF H5BHC BC 89OB9 IB5 GC@I7=B 6XG=75 %5G F9GHF=77=CB9G H9F79F5 M 89 BC B9;5H=J=858 89 8 BC G9 =BH9FG97H5B ,95 ! !

451. ! #

452. # BHCB79G 8 # # # 8 8

453. 8 # DCF H5BHC * * 8 9G IB5 GC@I7=B 6XG=75

454. ,8/0(+ -+%-)!9* (*#( GH5 GC@I7=B 7CFF9GDCB89 5@ DIBHC 9LHF9AC 89OB=8C DCF @5 =BH9FG977=B 89 @5G F9GHF=77=CB9G G9;IB85 M 89 BC B9;5H=J=858 89 8 ,95 ! ! ! #

455. # BHCB79G 8

456. # # # 8 8 8 # DCF H5BHC 8 9G IB5 GC@I7=B 6XG=75 GH5 GC@I7=B 7CFF9GDCB89 5@ DIBHC 9LHF9AC 89OB=8C DCF @5 =BH9FG977=B 89 @5G F9GHF=77=CB9G DF=A9F5 M 89 BC B9;5H=J=858 89 8 ,95 !

457. ! ! #

458. # BHCB79G 8 # # # 8

459. 8 8 # DCF H5BHC 8 9G IB5 GC@I7=B 6XG=75 GH5 GC@I7=B 7CFF9GDCB89 5@ DIBHC 9LHF9AC 89OB=8C DCF @5 =BH9FG977=B 89 @5G F9GHF=77=CB9G 89 BC B9;5H=J=858 89 8 M 89 8 EI9 9G 9@ CF=;9B 'H9G9 EI9 BC 9G :57H=6@9 CAC C6G9FJ57=CB9G 58=7=CB5@9G G9 H=9B9 EI9 9@ B]A9FC 89 GI6A5HF=79G

460. 9B 9GH9 99AD@C 9G

462. 89 @5G 7I5@9G 7CBGH=HIM9B 65G9G 89@ 9GD57=C 7C@IAB5 89 M DCF H5BHC 9@ B]A9FC 89 GC@I7=CB9G 6XG=75G 9G 89 @5G 7I5@9G :I9FCB :57H=6@9G

463. -!/#-3!9* # ,0*/+. #1/-#)+. ,=B 9A65F;C D5F5 DFC6@9A5G 89 8=A9BG=B A5MCF 9G B979G5F=5 @5 75F57 H9F=N57=B 5@;96F5=75 89 IB DIBHC 9LHF9AC H5@ 7CAC G9 A9B7=CB 5@ DF=B7= D=C GH5 75F57H9F=N57=B G9 DFCDCF7=CB5 9B @CG G=;I=9BH9G H9CF9A5G !(14(/$ CBG=8YF9G9 9@ DFC6@9A5 89 DFC;F5A57=B @=B95@ 9B :CFA5 9GHXB85F ! ! ! ! !8 : #8 3 ! 8 8 ,95

464. 9@ 7CBIBHC 89 GC@I7=CB9G :57H=6@9G 89 ,= 9L=GH9 IB 7CBIBHC 89 + + - 7C@IAB5G 89 @=B95@A9BH9 =B89 D9B8=9BH9G ! ! ! H5@9G EI9 8 ! 8! 8! 7CB 8 ) + 9BHCB79G 8 8 8 8 9G IB DIBHC 9LHF9AC 89

465. (/1564$,@0 (6GYFJ9G9 DF=A9F5A9BH9 EI9 @5 =DH9G=G 9GH56@979 EI9 8 9G GC@I7=B 6XG=75 89 ,95B 9 7

466. H5@9G EI9 8 9 7 D5F5 5@;IB5 5GH5 DFC65F EI9 9 7 8 D5F5 7CB7@I=F EI9 8 9G DIBHC 9LHF9AC 89

467. %5 = YG=A5 7CADCB9BH9 89 8 9G 8 9 7 ) . DCF @C EI9 9 7 ) + . CAC 9 7

468. 9BHCB79G 9 7 ) . 89AXG @I9;C 9 7 89 8CB89 9 7 D5F5 ) + . )5F5 J9F=O75F @5 J5@=89N 89 9GH5 =;I5@858 D5F5 ) + 7CBG=8YF9G9 @C G=;I=9BH9

469. ,8/0(+ -+%-)!9* (*#( CAC 9 7

470. 9BHCB79G 9 ! 9! 9! 9 ! 9! 9! 7 ! 7! 7! 7 ! 7! 7! 89AXG DCF =DH9G=G 8 ! 8! 8! +9GH5B8C 89 M 89 G9 C6H=9B9 8 9 ! 8 9 ! 8 9 ! 9 7 ! 9 7 ! 9 7 ! CAC ! ! ! GCB @=B95@A9BH9 =B89D9B8=9BH9G 8 9 9 7 ) + 89 8CB89 8 9 7 ) + )CF H5BHC 8 9 7 M 89 5EIZ G9 7CB7@IM9 @5 5OFA57=B 89@ H9CF9A5 @ F97ZDFC7C 89 9GH9 H9CF9A5 9G JX@=8C M 5 7CBH=BI57=B G9 89AI9GHF5 !(14(/$ ,95 IB DFC6@9A5 9B :CFA5 9GHXB85F M G95

471. 9@ 7CBIBHC 89 GC@I7=CB9G :57H=6@9G 89 ,95 8 8 8 8 IB DIBHC 9LHF9AC 89

472. BHCB79G @5G 7C@IAB5G 89 5GC7=585G 5 @5G 7CADCB9BH9G DCG=H=J5G 89 8 GCB @=B95@A9BH9 =B89D9B8=9BH9G (/1564$,@0 ,IDB;5G9 G=B DYF8=85 89 ;9B9F5@=858 EI9 @5G DF=A9F5G + 7CADCB9BH9G 89 8 GCB DCG=H=J5G CAC 8

473. 9BHCB79G 8 ! 8! 8!

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