MOLP MOLP MOLP Corresponding author,

Transcription

1 h://jnrm.rbau.ac.r پژوهشهای نوین در ریاضی دانشگاه آزاد اسالمی واحد علوم و تحقیقات * Correondng auhor, Emal: j.val@abrzu.ac.r. *

2 تخمین

3 x x x, c x e d.. A x B b.. A x B b, A d,, e n2 cx, n A 2 mn2 m2n2 B B 2 m n m2n x, c x d.. A x B b e.. A x B b ( x, ) x A x B b.

4 Y X E { x X x X; Cx Cx, Cx Cx} Y { Cx x X E }. Cx Cx Cx xx X WE xx x mn cx mn cx 2 mn cx.. Ax b x 0, Y W b m x n A mn c, c2,, c n XWE { xx xx; Cx Cx} Y { Cx xx }. W WE. Y YW XEXWE n X{ x Axb, x0} c x C Y{ Cxx X} xx Y xx.. Y 0, Cx Cx, Cx Cx Cx X E x

5 .. Y,2,, 2 I mn.. Y Y.. 0, Y I,, 2,, Y I mn.. Y I ( I, I,, I 2 ),2,, (,,, 2 )

6 M, c.. Y M.. Y 0 Y (, ) z.. c Y z c M 2,.. Y.. Y 0 c.. Y Y Y 0

7 c.. Y W Y Y YW YW ( *, 0) * Y * * M z ( *, 0) * c. 2 c M (),.. Y.. Y 0 ( *, 0) z z z 2 2 ( * *, 0) z2z ( *, *) * 0 * Y * * (, ) * 0 M z ( * *, ) ( * 0, *), z 2 2 * * (, ) e c ( ) ce Y z2z * ( *, 0) * * * ( *, ) ( * c 0 ). z z z z 2 2 z2z

8 v ( v C w A) x 0 ( v ) 0 ( Ax) 0 w b x 0, 0, w 0 x,, x, x, M.. Cx Ax b x 0.. Cx Ax b x 0, 0 K 0,,( ) 0 v C w A 0 x 0 v C w A x v, 0, ( v ) 0 w 0 Ax b w b Ax,, ( ) 0 0 v C w A, 0 x ( ), K K n {0,}, j,2,, n j 0 v q, 0 ( q), K K q {0,},,2,, 0 w K, 0 b Ax K( m ), {0,},,2,, m KKT 0 M x,, x,.. Cx Ax b x 0 Cx Ax b v C w A 0

9 x x2 x3 x4 x5 x6 x7 x, x2, x3, x4, x5, x6, x70 -x2 x3 2x49x5 9x69x7 2 -x x3 9x4 2x5 9x69x x x2 9x49x5 2x62x7 3 x x2 x3 x4 x5 x6x7 0 v 2v 3 w K 0v v3 w K 2 0v v 2 w K 3 02v 9v 29v 3 w K 4 09v 2v 29v 3 w K 5 09v 9v 22v 3 w K 6 09v 9v 22v 3 w K 7 v, v 2, v 3 0 x K ( ) 0 x2k ( 2) 0 x3 K ( 3) 0 x4k ( 4) 0x5 K ( 5) 0 x6k ( 6) 0 x7k ( 7), 2, 3, 4, 5, 6, 7{0,}. Mn -x 2 -x3-2x 4-9x5-9x6 + 9x 7 Mn -x -x 3-9x 4-2x5-9x6 +9x7 Mn x x x x 2x x x x 2 + x3 + x 4 + x5 + x6 + x7 = x, x 2, x3, x 4, x5, x6, x7 0 (0, 0, 0). Max -M( ).. = -x2-x3-2x4-9x5-9x6+ 9x7 2 = -x -x3-9x4-2x5-9x6+9x7 3 = -x-x2-9x4-9x5-2x6-2x = x x x x x x x x, x2, x3, x4, x5, x6, x70 Max = -x2 -x3-2x4-9x5-9x6+ 9x7 2 2= -x -x3-9x4-2x5-9x6 +9x7 3 3= -x -x2-9x4-9x5-2x6-2x7 x + x2 + x3 + x4 + x5 + x6 + x7 = x, x2, x3, x4, x5, x6, x7 0 K Max.. -x2x32x49x59x69x7 2 -x x39x42x59x69x7 3 -x x2 9x49x52x62x7

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11 [8] Deou, M. I., Gha, M., Dav, W. J., (986). Emae of he mnmum nondomnaed creron value n mulle-crera decon-mang. Engneerng Co and Producon Economc, 0, [9] Der, J.S., Fhburn, P. C., Seuer, R. E., Wallenu, J., Zon, S., (992). Mulle creron decon mang, mul arbue ul heor: The nex en ear. Managemen Scence, 38, [0] Ecer, J. G., Hegner,. S., Kouada, I. A., (980). Generang all mal effcen face for mulle objecve lnear rogram. Journal of Omzaon Theor and Alcaon, 30, [] Ehrgo, M., (2005). Mulcrera omzaon. Srnger, Second Edhon, Berln, German. [2] Ehrgo, M., Tenfelde-Podehl, D., (2003). Comuaon of deal and nadr value and mlcaon for her ue n MCDM mehod. Euroean Journal of Oeraonal Reearch, 5,9-39. [3] Hor, R., Thoa,. V., Yamamoo, Y., Zene, D., (2007). On omzaon over he effcen e n lnear mul crera rogrammng. Journal of Omzaon Theor and Alcaon, 34, [4] Iermann, H., (977). The enumeraon of he e of all effcen oluon for a lnear mulle objecve rogram. Oeraonal Reearch Quarerl, 28, [5] Iermann, H., Seuer, R. E., (987). Comuaonal exerence concernng aoff able and mnmum creron [] Armand, P., Malver, C., (99). Deermnaon of he effcen e n mulobjecve lnear rogrammng. Journal of Omzaon Theor and Alcaon, 70, [2] Benon, H. P., (984). Omzaon over he effcen e. Journal of Mahemacal Anal and Alcaon, 98, [3] Benon, H. P., Lee, D., McClure, J. P., (998). Global omzaon n racce: An alcaon o neracvemulle objecve lnear rogrammng. Journal of Global Omzaon, 2, [4] Benon, H. P., Sun, E., (2002). A wegh e decomoon algorhm for fndng all effcen exreme on n he oucome e of a mulle objecve lnear rogram. Euroean Journal of Oeraonal Reearch, 39, [5] Calvee, H. I., Gale, C., Maeo, P. M., (2008). A new aroach for olvng lnear blevel roblem un ggenec algorhm. Euroean Journal of Oeraonal Reearch, 88,4-28. [6] Deb, K., Chaudhur, S., Menen, K., (2006). Toward emang nadr objecve vecor ung evoluonar aroache. In: Kejzer, M. e al. (Ed.), 2006 Genec and Evoluonar Comuaon Conference (GECCO2006), Seale,Wahngon, USA,, [7] Deme, S., (2002). Foundaon of Blevel Programmng. Kluwer Academc Publher.

12 Euroean Journal of Oeraonal Reearch, 36, [23] Sh, C., Lu, J., Zhang, G., (2005). An exended Kuhn-Tucer aroach for lnear blevel rogrammng. Aled Mahemac and Comuaon, 62, [24] Sh, C., Lu, J., Zhang, G., Zhou, H., (2006). An exended branch and bound algorhm for lnear blevel rogrammng. Aled Mahemac and Comuaon, 80, [25] Sh, C., Zhou, H., Lu, J., Zhang, G., Zhang, Z., (2007). The Kh-be aroach for lnear blevel mul follower rogrammng wh aral hared varable among follower. Aled Mahemac and Comuaon, 88, [26] Tu, T. V., (2000). Omzaon over he effcen e of a aramerc mulle objecve lnear rogrammng roblem. Euroean Journal of Oeraonal Reearch, 22, [27] Yamamoo, Y., (2002). Omzaon over he effcen e: Overvew. Journal of Global Omzaon, 22, [28] Yan, H., We, J., (2005). Conrucng effcen oluon rucure of mulobjecve lnear rogrammng. Journal of Mahemacal Anal and Alcaon, 307, value over he effcen e. Euroean Journal of Oeraonal Reearch, 33, [6] Jorge, J. M., (2005). A blnear algorhm for omzng a lnear funcon over he effcen e of a mulle objecve lnear rogrammng roblem. Journal of Global Omzaon, 3, -6. [7] Korhonen, P., Salo, S., Seuer, R. E., (997). A heurc mehod for emang nadr creron value n mulle objecve lnear rogrammng. Oeraon Reearch, 45, [8] Lechne, T. M., Wallenu, H., Verdn, W. A. (992). Ineracve mulobjecve anal and amlave caac baed ocean doal decon. Euroean Journal of Oeraonal Reearch, 56, [9] Lv, Y., Hu, T., Wang, G., Wan, Z., (2007). A enal funcon mehod baed on Kuhn-Tucer condon for olvng lnear blevel rogrammng. Aled Mahemac and Comuaon, 88, [20] Meev, B., Valev, V., (2003). A Mehod for adr Pon Emaon n Problem. Cbernecand Informaon Technologe, 3, [2] Pourarm, L., Zareheh, M., (2007). A dual-baed algorhm for olvng lexcograhc mulle objecve rogram. Euroean Journal of Oeraonal Reearch, 76, [22] Reeve, G. R., Red, R. C., (988). Mnmum value over he effcen e n mulle objecve decon mang.