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....................................................................................................................... 20 1: - ....................................... 24 1.1 ............................................... 25 1.1.1 ....................... 27 1.1.2 ............................................................ 33 1.1.3 ............... 39 1.2 ............................................... 42 1.2.1 .............................................................................. 48 1.2.2 ............................................................ 50 1.2.3 ................................................... 52 1.3 - ....................................................... 55 1.3.1 ....................................................................................... 55 1.3.2 - .............................................. 57 1.3.3 ..................................................... 61 1.4 ..................................................................................... 64 2: ................................................................................................... 68 2.1 ......................................................... 69 2.1.1 ....................................................................................... 71 2.1.2 ................................................................................. 71 2.1.3 ............................................................................................. 72 2.2 .................................................... 73 2.3 ......................................................................... 79 2.3.1 Tarry ..................................................................................... 81 2.3.2 Manas - Nedoma ................................................................. 85 2.3.3 Simplex ................................................................ 98 2.3.4 ............................................................................ 114 3: ................................................................. 122 3.1 ..................................................................................... 123 3.2 ................................................................................... 132 3.3 ........................................................................................... 144 4: ............................ 146 4.1 ...................................................................................... 147 4.1.1 ................................................................................... 147

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4.1.2 ............................................................................... 151 4.1.3 ..................................................... 152 4.1.4 ........................................................... 154 4.2 ............................. 157 4.3 M UTA .................................................................. 159 4.3.1 .......................................................................................... 159 4.3.2 UTA ............................................................................................... 161 4.3.3 UTASTAR ................................................................................. 165 4.3.4 UTA ................................................ 168 4.4 M MUSA ................................................................................................... 171 4.4.1 .......................................................................................... 171 4.4.2 .................................................................................................. 173 4.4.3 ................................................................ 176 4.4.4 MUSA ................................................................ 179 4.4.5 MUSA......................................................... 182 5: .............. 186 5.1 H UTA .............................................. 187 5.1.1 ................................................... 187 5.1.2 ............................................... 189 5.1.3 ........................................................ 191 5.2 H MUSA ........................................... 195 5.2.1 ....................................... 195 5.2.2 ........................................................ 197 5.2.3 ........................................................................................ 200 5.2.4 MUSA ..................................................................... 209 5.2.5 MUSA ............................ 210 6: MUSA ............................. 212 6.1 ...................................... 213 6.2 .................................................................................... 214 6.3 ......................................................... 217 6.4 ................................................................................................ 221 6.5 ......................................................................................................... 229 6.6 .......................................................................................... 230 7: ............ 238 7.1 ................................................................................................................ 239 7.2 ............................................................................. 241 7.3 ........................................................................... 242 7.4 .................................................................... 244

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7.5 ............................................................................ 249 7.6 ........................................................................................................ 252 8: .......................................................... 254 8.1 E ................................................................................................................ 255 8.2 .......................................................................................................... 255 9: 18 .................................................................... 260 9.1 ................................................................................................................ 261 9.2 ................................ 262 9.2.1 - ...................................................... 262 9.2.2 - ............................................................ 263 9.3 .................................................................................................... 264 9.3.1 - ...................................................... 264 9.3.2 - ............................................................ 265 9.4 ............................. 266 9.4.1 - ...................................................... 266 9.4.2 - ............................................................ 267 9.5 ................................................................................................................. 268 9.5.1 - ...................................................... 268 9.5.2 - ............................................................ 269 9.6 ........................................................................................................... 270 9.6.1 - ...................................................... 270 9.6.2 - ............................................................ 270 9.7 ........................................................................................................... 271 9.7.1 - ...................................................... 271 9.7.2 - ............................................................ 272 9.8 ........ 273 9.8.1 - ...................................................... 273 9.8.2 - ............................................................ 273 9.9 Super Market...................... 274 9.9.1 - ...................................................... 274 9.9.2 - ............................................................ 275 9.10 Super Market .............. 276 9.10.1 - .................................................... 276 9.10.2 - .......................................................... 277 9.11 ................................................................................................................ 278 9.11.1 - .................................................... 278

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9.11.2 - .......................................................... 279 9.12 ............................... 280 9.12.1 - .................................................... 280 9.12.2 - .......................................................... 281 9.13 ................................................... 281 9.13.1 - .................................................... 281 9.13.2 - .......................................................... 282 9.14 ................................................ 283 9.14.1 - .................................................... 283 9.14.2 - .......................................................... 284 9.15 ................................................................... 284 9.15.1 - .................................................... 284 9.15.2 - .......................................................... 285 9.16 ......................................................................................................... 286 9.16.1 - .................................................... 286 9.16.2 - .......................................................... 287 9.17 Logistics.............................. 288 9.17.1 - .................................................... 288 9.17.2 - .......................................................... 288 9.18 .............................................................................................................. 289 9.18.1 - .................................................... 289 9.18.2 - .......................................................... 290 9.19 RAM................................ 291 9.19.1 - .................................................... 291 9.19.2 - .......................................................... 292 ...................................................................................................................... 294 ............................................................................................................... 298 I: Simplex ............................................................................. 318 II: ......................... 324 III: Tarry.................................................. 326 IV: ............................................................. 330 V: .............................................................. 366

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2-1: Manas-Nedoma, Simplex s=0 .......................................... 91 2-2: Manas-Nedoma, Simplex s=1 .......................................... 91 2-3: Manas-Nedoma, Simplex s=2 .......................................... 92 2-4: Manas-Nedoma, Simplex s=3 .......................................... 92 2-5: Manas-Nedoma, Simplex s=5 .......................................... 93 2-6: Manas-Nedoma, Simplex s=6 .......................................... 93 2-7: Manas-Nedoma, Simplex s=6 .......................................... 94 2-8: Manas-Nedoma, Simplex s=7 .......................................... 94 2-9: Manas-Nedoma, Simplex s=8 .......................................... 95 2-10: Manas-Nedoma, Simplex s=9 ........................................ 95 2-11: Manas-Nedoma, Simplex s=10 ...................................... 96 2-12: Manas-Nedoma, Simplex s=11 ...................................... 96 2-13: Manas-Nedoma, Simplex s=12 ...................................... 97 2-14: Simplex p .................................... 99 2-15: Simplex p+1 .............................. 100 2-16: (s=0) Simplex ................... 102 2-17: s Simplex ................... 102 2-18: Simplex ............... 112 2-19: Simplex............................................................................................................ 113 2-20: Simplex........................................................ 118 2-21: ............................................. 121 3-1: Simplex 1.1 .......................... 126 3-2: Simplex s = 0 ............................................................... 127 3-3: s ........ 129 3-4: Simplex ........................... 132 3-5: ........................................... 143 4-1: ...................................................................... 169 4-2: MUSA ....................................................... 174 5-1: UTA ........................................................................................................... 194 5-2: MUSA .............................................................................................................. 199 6-1: ........................................... 230 8-1: ......................................... 256

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8-2: .......................... 257 8-3: .......................... 258 II-1: r ............................................. 324 II-2: r ...................................... 325 V-1: ........................................ 333 V-2: , & ............................................................................................................. 349 V-3: WVi ............................................................... 364

, 1-1: 2 ...................................................................... 46 1-2: 3- simplex.......................................................................... 48 1-3: .................................................. 48 1-4: 2 3 ......................................................... 49 1-5: 2 ...................................... 51 1-6: ........................................................................ 52 1-7: (, 2008) ...................... 58 1-8: ........................................... 60 1-9: ( & , 2000) .................................................................................................... 61 1-10: - (Jacquet-Lagrze & Siskos, 2001) .................................................................... 62 1-11: - (, 1996). 64 2-1: (V,U)........................................................................................ 86 2-2: Manas-Nedoma ................................... 89 2-3: Simplex......................... 107 2-4: Manas-Nedoma 0-1-27-8-5-6-11-4-3-9-10-12.................................................................................... 113 2-5: ............................................ 119 3-1: ............. 130 4-1: Malcolm Baldridge..... 149 4-2: ................ 150 4-3: .................................... 163

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4-4: (Jacquet-Lagrze and Siskos, 2001)............................................................... 166 4-5: ( et al., 1997) ........... 174 4-6: ........................................... 175 4-7: j (Grigoroudis & Siskos, 2002) 176 4-8: zm wik (Grigoroudis & Siskos, 2002) ............................................................................................................... 177 4-9: Y* (Grigoroudis & Siskos, 2002) ............................................................................................................... 180 4-10: (Grigoroudis & Siskos, 2002) .......................................................................... 182 5-1: (Grigoroudis & Siskos, 2002) .......................................................................... 196 5-2: ( & , 2000) ............................................................................................................... 203 6-1: (Sage, 1991).................................................................................................... 216 6-2: .................... 228

1: r .......................................................................... 54 2: r ................................................................. 55

6-1: .......................................................... 231 6-2: ................................................... 232 6-3: ................................................. 232 6-4: MUSA..................................................................... 233 6-5: zoom ............................................. 233 6-6: OPL ASI ............................................................................. 234 6-7: ........................ 234 6-8: ...... 234 6-9: .................................................................................................... 235 6-10: ............................................................ 235 6-11: ......................... 236 6-12: zoom... 236 6-13: ........ 237 6-14: MUSA ........................................................ 237

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7-1: ..................................... 246 7-2: ................................ 247 7-3: .......................................................... 248 7-4: 1 ........................................................... 249 7-5: 2 ........................................................... 250 7-6: 3 ........................................................... 250 7-7: 4 5 ................................................. 251 7-8: 10000 ................................................................... 251 7-9: ......................................... 252

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, . , , - . (2008) , . , , , , . , , - . , , , , . : : ;. (disaggregation models) . , - (aggregation-disaggregation approach) Jacquet-Lagrze & Siskos (1982) . . , (inference paradigm). , ( ), . , , (extrapolation) . UTA Jacquet-Lagrze & Siskos (1982) MUSA

(2000) . . , , . , . , - , ( ), . , UTA Jacquet-Lagrze & Siskos (1982), , . UTA, , , (Siskos et al., 2005). - MUSA (2000), - . , , , . , . (Siskos & Grigoroudis, 2002). (post-optimality analysis) (robustness) . (sensitivity analysis) (Van de Panne, 1975; , 1998) , . , UTA MUSA .

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. , . , . (robustness) 68 (MCDA) 2-3 2008 . , Roy (2008) . , : , . MUSA. . , . / . MUSA / , . 5 6 MUSA ( 5) ( 6). 8 9

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. , 7 8 . 9 . , , .

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1: ( , ) . . , , . . -. .

1: -

1.1 Operational Research Operations Research. , . , , , , , , (, 2008). : , , , , , . , , . ( ). ( ) . , ( ), . (robustness stability) Roy (2007) . , , , ( ) , , . ,

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1: -

(Roy, 2005). , , ( ) . , , . ( ) . , . Dias (2007) , . , (Roy, 1989). . , ( ) , . , (.. ELECTRE), , , , , ... , - ( Roy (2003, 2007)). : (.. , ), (.. ), (.. ) .

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1: -

, .. . (.. ) , , ( ) ( ). , (Beer, 1966: .44). (Roy 2007) (Vincke, 1999a,b). 6 . . ( ) ( ). , .

1.1.1 Roy (2007) 2004, Philippe Vincke , , . , . . , , .

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1: -

. , , . , : 1. ( ) 2. ( ) , , , . Roy (2007) : , , .

-

, ( ) , . , Roy (2005). . i) : ( ), , , ,

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1: -

ii) , , , : , , , iii) ( ) : , , , iv) ( ): , , - , . . . , , (Roy, 1989; Roy, 2005). , , . , . . , . , . : ,

29

1: -

. . Roy (2004, 2005) , , . , ( ). . . . , , . , (Roy, 2005). . , ( ) (, ) . Vincke (1999a,b). , Dias (2007) . . , . , . (Roy, 2002). Rosenhead (2001a), ( ) , . Kouvelis and Yu (1997) (.. max-min),

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1: -

o Aloulou et al. (2005) . Mulvey et al. (1995) , ( ) , , . Sevaux & Sorensen (2004) ( ) . , Hites et al. (2003) . (Roy, 1998; Roy & Bouyssou, 1993), Vincke (1999a) ( ) ( Vincke ). Roy (1998) ( ) - ( ). , (Dias & Climaco, 2002). , ( /). ( ), . . . Dias and Climaco (1999). , .

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1: -

, x 0.7 y 10 () , . , x y x y 0.7 . () , . , x x () . ( , , ). , x y 99% 0.8 95% x . , , ( ...) ( -). , : ( ) . , - , .. . , , ( - /).

32

1: -

1.1.2 . : , . , . : 1) , ( ) , . , . : ( ), ( ). , . , Roy (2007) : Aissi et al. (2005a, 2007), Aloulou, Della Croce (2005), Averbackh, Berman (1997), Averbackh, Lebedev (2004, 2005), Ben-Tal, Nemirovski (1999), Bertsimas, Sim (2003, 2004), Briand et al (2005), Deineko,Woeginger (2006), Guttirrez et al. (1996), Hites (2000), Kala et al. (2005), Kouvelis, Yu (1997), Montemanni et al. (2004), Mulvey et al. (1995), Snyder (2006), Soyster (1973, 1979), Vallin (1999), Yaman et al. (2001),Yu, Yang (1998). : i.

. . : , , , , ,

33

1: -

, , p-median p-centre , ( ). ii. . , . . iii. . . , . , . . . , . iv. . , . , (Kala et al., 2005; Perny & Spanjaard & 2003; Rosenblatt & Lee, 1987; Snyder, 2006; Snyder & Daskin, 2006; Bertsimas & Sim, 2003, 2004; Beuthe & Scannella, 2001; Kala, 2006; Mulvey et al., 1995; Soyster, 1973, 1979). 2) , , , : . . , , (, ) . ( ) . , . -

34

1: -

. - . , , . , : , , , - (Roy, 1998, 2005). , , , , . Aissi (2005c), Aissi et al., (2005a), Aloulou, Portmann (2005) , Carr and al. (2006), Chang, Yeh (2002), Durieux (2003), Espinouse et al. (2005), Gabrel (1994), Kazakci and al. (2006), Rosenblatt, Lee (1987), Roy, Bouyssou (1993), Roy and al. (1986), Sevaux, Srensen (2002, 2004b). . Bertsimas Sim EDF (Roy, 2007) Aloulou (Aloulou & Portmann, 2005). . . , . , Roy (2007) , , ( , Roy, 1985). Roy (2005) . ) ( )

35

1: -

i.

: : , , , ( ). ( ) ( ). , , , . ( ) .

ii. : , . , , : , , , , .. , . . . . , . , , ,

36

1: -

. , , . ) / i. : (, ,) ( ,..). ; .

ii. : , , . n , ( ). . , , , . , . ) (, ,) . i. : , , . .

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1: -

. . ii. : ( ) , . , . iii. , , , , : , . , , , , , . . , , : , / , .

-

, .

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1: -

3) , , , . . , , . . , , (, ) . / , . , . Aloulou, Artigues (2006), Beuthe, Scannella (2001), Billaut, Roubellat (1996), Dias et al. (2002), Elkhyari . (2005), Gupta, Rosenhead (1972), Gutirrez , Kouvelis, (1995), Kouvelis et al. (1992), Malcolm, Zenios (1994), Pierreval, Durieux (2007), Rosenhead (2001a,b), Rosenhead et al. (1972), Sengupta (1991), Sevaux et al. (2005). , .

1.1.3 . , , .

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1: -

, . , , . (.. ). , . : Kouvelis and Yu (1997), Mulvey et al. (1995). ( Hites et al., 2003), . ( ) , (Aloulou et al., 2005, Roy ,2007). , . (Roy and Bouyssou, 1993). , . , x , x y, x y y x . , , . , , . Roy (2002)

40

1: -

. Roy (1998) . , , . , , . , . , , . , . , (.. ). (Jacquet-Lagrze & Siskos, 2001), o . . , / . , ( ) . , . , . , . , .

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1: -

1.2 (optimisation problems). / . . ( ) (.. , , , ) . , , . . . , , (Du et al., 2001). , . ( , ) ( ). , , , . : xj

42

1: -

, , ( ) . : () (Linear Programming). Computing in Science and Engineering Simplex (Dantzig & Thapa, 1997a, 1997b) 10 (Computing in Science and Engineering, volume 2, no. 1, 2000). 52 G. Dantzig L. Kantorowitz 1947, . , - (Decision Support Systems - DSS) (, 1986).

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1: -

. , , , , . , . ( ) , , , (), . . ( ) (, 1996;, 1992). , , , . . - , Simplex ( ), : n m , m , : (n + m) - m = n . , . , , .

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1: -

Simplex (j) . , . cj, bi aij . , , , . , , . (cj), (bi) (aij) . : ( , , ...) ( , , ...) ( , , ...). , , , (, 1991; , 1992). , . , . . , . ( )

45

1: -

, z z* () . . ( ) ( ). . . :

[max ]z = c t x . . 1.1 Ax b x 0 A, x, b c mn, n1, m1 n1 . Rn ( n) ( n-1) . H - . p . n-p (Dantzig, 1997b; Saaty, 1955). n=2 ( ) n-1=1, . p=2 ( n-p=2-2=0). :Ax b x0

1-1: 2

46

1: -

n=3 n-1=2, . p=2 ( n-p=3-2=1). p=3 ( n-p=33=0). . . . (V,U) V () U , Simplex (Manas & Nedoma, 1968). , Simplex () , Simplex (Van de Panne, 1975). . m m-. Simplex -, simplex. , m, () m+1 m- simplex (Dantzig, 1997b). : - simplex - simplex 2- simplex 3- simplex (. 1-2)

47

1: -

4

3

1 1-2: 3- simplex x=Aj m- Simplex 1-3).

2

simplex. (.

1-3: , , ..

1.2.1 Simplex Simplex . (Siskos, 1984).

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1: -

:

[max ]z = c t x . . 1.1 Ax b x 0 A, x, b c mn, n1, m1 n1 . (multiple optimal solutions) , , Simplex . (.I.9) :

j = c j ci y ij = 0 j .i =1

m

, z ( ) , 1992).

Ax b c t x = z* x0

2

3

1-4: 2 3 j , z=z*, (.I.16) k=0

z= z* +

x Br k z=z* . yrk

... Simplex, ,

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1: -

. () 1.1:

1.1

Ax b ct x = z* x0

z=z*(z* ) . 1.1 . , ( ) 1.1 ( ). (Siskos, 1984; , 1991).

1.2.2 . , . , . , (Van de Panne, 1975). , , , . , , . .

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1: -

. . . z* ( ) k. (near optimal solutions). k, , (Siskos, 1984; Van de Panne, 1975; , 1992). - 1.2:

1.2

Ax b c t x z * k k ( ) x0

1.2 1.1 z=z* z z*-k. ( 2 ).

z=z*

z=z*-k

1-5: 2 k=0 z=z* . .

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1: -

z=z*

z=z* z=z*-

z=z*-k

1-6: , . (, 1992). 1.2.. , .

1.2.3 ( 1-5) . .. ( ) . . Matheiss Rubin (1980) . 1952 Charnes (Charnes, 1952; Charnes et al., 1953; Charnes & Cooper, 1961) .. ( ) . 2 . , 1.2., .

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1: -

. Saaty (1955) m + n m+n n m . m n . Klee (1964) , r , m n m+n n-1 m :1 2(n + m ) 2 (2m + n ) m 2m + n r = 1 2 (2m + n 1) 2 m

n (1.1) n

. r : (m=3, n=4), r =14 (m=10, n=6), r =352 (m=10, n=7), r =572 (Klee, 1964). 1.1 Klee (1971) :

n + 1 n + 2 n + m int + n + m int 2 r = 2 m m

(1.2)

(Amani, 1977; Burton et al., 1987). Klee (1.1) m+nn2/4 -1 m n (McMullen, 1970). r . Grunbaum (1967) :

r = ( n 1) m ( n 2 )( n + 1) m n-1 (1.3) Barnette (1971) (1.3) .

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1: -

Berenguer Smith (1986), Armand (1993), Avis Bremmer (1995) , (Cohen & Hickey, 1979; Lasserre, 1983). , , . , Simplex . Schmidt Mattheiss (1975, 1977, 1980) 9,867 4, 7 10. Dunham et al. (1977). (1.2) (1.3) r r . .

180,000 160,000 140,000 120,000 100,000 80,000 3 60,000 5 7 40,000 9 20,000 11 15 m 25 2 3 4 5 6 7 8 9 10 13

n

1: r , r , (.. 25 10 166,257 ),

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1: -

.

160 140 120 100 80 60 3 5 40 7 9 20 11 2 3 4 5 6 7 8 9 10 13 15 m 25 n

2: r

1.3 - 1.3.1 1972, ... , . / . , (, 1986). , , , g (

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, , ) , a, b A: a b g(a)>g(b). . . (Roy & Vincke, 1981). (multicriteria analysis) , (Siskos & Spyridakos, 1999). (ill-structured problems), , , (, 1986, 1998, 2008). (Siskos & Spyridakos, 1999; , 1986): 1. ( ) (multiobjective optimization approach) , . , - , , . . . -. (, 1986). 2. (outranking relation approach), . ,

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1: -

. ELECTRE Roy (1990). 3. (value system approach, multiattribute utility theory), , . . 4. - (aggregation-disaggregation approach) . UTA MUSA 4. Jacquet-Lagrze and Siskos (2001) .

1.3.2 - : : , : , : , V: (Roy, 1985, 1986, Jacquet-Lagrze and Siskos 2001). , . , , . A n- Rn 1-7:

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1: -

x a x x x x

gig(a)

g1 g1(a)

g2(a)

1-7: (, 2008) - : u, A, , , . g1, g2,,gn. . (Roy and Vincke, 1981). (value function) , ( ) , ( ): u(g)=u(g1, g2, ..., gn) (1.4)

P I a b. g(a)=[g1(a), g2(a), ..., gn(a)] a, u: u[g(a)]>u[g(b)]a P b (1.5.) u[g(a)]=u[g(b)]a I b (1.5.) R=PI (Jacquet-Lagreze & Siskos, 1982). u(.), , ( & , 2000): 1. 2.

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, , , . F={g1, g2,,gn} o Keeney and Raiffa (1976) . 1 ( preferential independence): {g1, g2} {g3, g4, ,gn} n 3, g1 g2 . 2 ( mutual preferential independence): {gi, gj} F\{gi, gj} i, j, . (1998) : (gi, gj), i=1, 2, ,n () . :

u[g(a)]=

u (g (a ))i =1 i i

n

(1.6),

a A, u(a) a gi(a) gi. : u[g(a)]=

p g (a )i =1 i i

n

(1.7)

ui(gi(a)) gi pi. , . (Jacquet-Lagrze & Siskos, 1982): u[g(a)]=

p w (g (a )) ,i =1 i i i

n

wi(gi(a))=

1 u i (g i (a )) i. (1.8) pi

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1: - g* g i* , i. i :

n p i =1 i =1 w i ( g i* ) = 0 i * w i ( g i ) =1 i

(1.9)

[0,1] (Jacquet-Lagrze & Siskos, 1982). 1-8 i:

1

g i*

g* i

1-8: - , , ui wi - gi. (1.6), :

n * u i ( g i )=1 i =1 u ( g )= 0 i i i*

(1.10)

pi ( 1.7). . , (trade-offs) . (, 1998)

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1: -

, . . .

1.3.3 , ( ). , : , ; , ; - , . (Jacquet-Lagrze & Siskos, 2001). , a priori , . , .

1-9: ( & , 2000)

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1: -

, , 1-10 (Jacquet-Lagrze & Siskos, 2001).

?

1-10: - (JacquetLagrze & Siskos, 2001)

R. : 1. (AR past actions) 2. , (R A) 3. , (R fictitious actions) / R,

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1: -

. (Jacquet-Lagrze & Siskos, 2001): AR () (weak order relation) AR, , - , -. , , , . (, 1981) ( ), ( ). (extrapolation) AR . ( , 2000) - 1-11, , , .

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1: -

1-11: - (, 1996) - Hammont et al. (1977), UTA (Jacquet-Lagrze & Siskos, 2001), UTASTAR (Siskos & Yannacopoulos, 1985), UTA II (Siskos, 1980) MUSA ( & , 2000). (Despotis et al., 1990; Despotis & Zopounidis, 1995; Jacquet-Lagrze, 1990; Jacquet-Lagrze & Siskos, 2001; Matsatsinis & Siskos, 1999; Siskos, 1980; Siskos et al., 1999; Siskos & Zopounidis, 1987; Stewart, 1987). .

1.4 . (Roy Bouyssou 1993; Roy 1998) . , R , R P . .

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1: -

. . . , ( ) (Roy 1993, Dias & Tsoukias 2004). . , - , . , , . , - . : (.. - ). , , /. (, ) ( .. ) , . .

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1: -

. , () , . , . / . . , , . . MUSA . (. 1.1.2). - (. 1.2.2 1.2.3). , , .

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1: -

67

2: . . . (pivoting) . . - . . Tarry , Simplex C. Van de Panne Manas - Nedoma. , Siskos . . 8 9 - .

2:

2.1 . 1.2.1 1.2.2 () 1.2 n:

1.2

Ax b c t x z * k k k=0 ( x0 )

() 50. Kaibel Pfetsch (2003) 20 . Simplex Dantzig 40 . Simplex . Dantzig (Grunbaum, 1967). , 2- . Simplex . 1947 . , .. 1.1. , r ,

69

2:

r . , , : 1. . 2. , {d1, . . . , dr}, r , , , , r .. Murty (2009) . 1.2.3. , . NPhard, (Khachiyan et al., 2006). , .. , (Provan, 1994). . . . . .

70

2:

(pivoting) . .

2.1.1 , , Simplex Dantzig. . .. 1.1 . , . , -. , perturbation . Simplex 2.3 .

2.1.2 Simplex . Matheiss Rubin (1980)

71

2:

Motzkin, Thompson, Raiffa Thrall 1953. . () , . , . , Duffin (1974) Dantzig Eaves (1973) Fourier-Motzkin. . (Bik & Wijshoff, 1995).

2.1.3 . . : , , , . , . Winkels (1982), . .

72

2:

1984 Siskos . .

2.2 , 50 , . Simplex , , , . Charnes Coopers (1953, 1961) Tarry (1896) Simplex. Remez Shteinberg (1966). . - , , , . . n . - (half-spaces)

73

2:

Gomory (1963). Balinski (1961) - . , . Balinski - . , . , Simplex. Balinski simplex . , Hi. Hi - (faces) 2- , simplex. 2- Hi - Hi , Hj, Hj Hi. (. ), . . Balinski . Murty (1968) . (fixed charge) . v1, v2, , vk-1 c,

74

2:

, vk, . v1, v2, , vk-1 simplex vj j=1,2,,k-1. - vk vk vk-1. . vk+1. , , . : ) , ) ) . 1992 (Murty & Chung, 1992). 1968 Manas Nedoma . . Simplex . , (Hamiltonian Path) (Manas & Nedoma, 1968). . . simplex . . Balinski Pollatschek Avi-Itzhak 1969 ( ) . v1 ctx . v1 v2 ctx ctv2 .

75

2:

, . (pivoting) Balinski . vi ctx ctvi . v1, v2, , vj . , ctx = ctvj (v, vr,) vr ctvr > ctvj v vr ctv < ctvj (Murty, 1971). , . , , (Matheiss and Rubin, 1980). 1971 Silverman Manas Nedoma Hamiltonian G-. G- simplex . G- , . Manas Nedoma (Silverman, 1971). Matheiss and Rubin (1980) Manas Nedoma Dahl Storoy 1973. v1, v2, , v ( x1, x2, , x c) ctx1 ctx2 ctx. Manas Nedoma vi vj d(vi, vj) = ct(xi - xj). Manas Nedoma simplex vj vi. Manas Nedoma. simplex .

76

2:

Mattheiss (1973) n n+1 . n- . , . simplex, . . . , Dyer Proll (1977) Mattheiss , . Mattheiss Rubin (1980) . Burdet (1974) 0 ( ) . ( n) d-1, 0. ( .. ) . . n . , simplex . .. . Dyer Proll (1977) . Van De Panne (1975) . , Simplex,

77

2:

. Van De Panne () simplex Simplex .. () () . simplex . (Siskos, 1984). . . 1977 Amani , Manas Nedoma, Simplex (Amani, 1977). Dyer Proll (1977) (spanning tree) . 1, k- k. 1- . k k- . , k k-1, k+1, k+1 k. Mattheiss Rubin (1977) . simplex - (breadth first) . 1982 Dyer Proll ,

78

2:

Mattheiss Rubin (1977) (Dyer & Proll, 1982). . NP-hard. Avis Fukuda (1992) , Dyer Proll (1977), . - Bland (1977) o Van De Panne (1975). , (network linear programming) (Provan, 1994).

2.3 - . . Tarry . Manas Nedoma ( ) . Simplex Van De Panne simplex . , Siskos

79

2:

. : :

[max ] z = 3 x1 + 4 x2 + 5 x3 + 6 x4 .. x1 + x2 + x3 + x4 18 2 x3 + 3 x 4 6 xi 0, i = 1,2,3,4

( 2.1)

:[max ] z = 3 x + 4 x + 5 x + 6 x + 0 x + 0 x 1 2 3 4 1 2 .. x1 + x2 + x3 + x4 + x1 = 18 2 x3 + 3 x 4 + x 2 = 6 xi 0, i = 1,2,3,4,1,2 x1 , x2 .

Simplex :

cB 0 0

1 1 0

2 1 0

3 1 2

4 1 3

11 0

20 1

xB18 6

1 2

cj

3 3

4 4

5 5

6 6

0 0

0 0 z= 0

j

80

2:

Simplex Simplex :

c B 4 6 2 4

1 1 0

2 1 0

3 1/3 2/3

4 0 1

11 0

2-1/3 1/3

xB16 2

cj

3 -1

4 0

5 -1/3

6 0

0 -4

0 -2/3 z*= 76

j

.1.2

k=20,

, ( ) 20 (76 ), :

2.1

x1+ x2+ x3+ x4 18 2x3+3x4 6 3x1+4x2+5x3+6x4 56 x1, x2, x3, x4 0

Tarry .

2.3.1 Tarry Charnes Cooper . (V,U) V () U ,

81

2:

Simplex. , Simplex . () , , Simplex (Charnes, 1952; Charnes & Cooper, 1961). , , 1.2, .. . . Tarry (1895) , . - - , , , (Charnes, 1952). Tarry ( ) , . , , : , , . (Siskos, 1984) , . , , , , . , . , , .

82

2:

, . (), . . . , , . , , . . ( ) ( ). , , . , , , : , . (Tarry, 1895) 4 6 , ():

83

2:

Tarry :

1

++

+++

2

++

+++

++ +++

........ ( 3-11 )

12 ++

++++

+++ ++

++ +++

++

++ +++

++

++

84

2:

. . , , . (Tarry, 1895) .. ( ) (Van De Panne, 1975; Charnes, 1952). (V,U) Simplex. Tarry Simplex , n 1.1, ( ), r 1.1, nr. 2.1 413=52 Simplex. . (Siskos, 1984; Van de Panne, 1975)

2.3.2 Manas - Nedoma , ( m: n: ) (graph connected) (V,U) V U , Simplex. , Simplex . (V,U). :

V ( ) m ( Simplex) u=(i1, i2, ..., im) 1ijm, j=1,2,...,m.

85

2:

u1=(i1, i2, ..., im) u2=(k1, k2, ..., km) dm d u2 u1. u1 u2 d=1 (. 6.1). (u1, u2) U u1 u2 . O ui N(ui)

u1=(i1, i2, ..., im)

d=1

u2=(k1, k2, ..., km)

2-1: (V,U)

(V,U) . , Simplex, (Manas & Nedoma, 1968). Manas Nedoma 1968 (V,U) : 1) ( ) , 2) Simplex. , , , : - (uo) .

86

2:

(R1, R2, ..., Rs) ( ). (W1, W2, ..., Ws) Rs Simplex.

R W : us ( SIMPLEX) Rs RXs (xB1, xB2, ,xBm+1) us. : Rs= Rs-1 us, RXs= RXs-1 us N(us) ( Rs) Ws. (i1, i2, ..., im) N(us) Ws. : Ws=Ws-1N(us)-Rs Ws us ( d=1). ( Ws) SIMPLEX. dm SIMPLEX. Ws=. Rs RXs.

Manas Nedoma : Ws= Rk=V ( ), (Manas & Nedoma, 1968). 6 .

87

2:

0: uo. 1: ( , ) ( ) 1.2. Simplex uo. 1.2:

ct x Y = z * k Y . 2.3.4.

2: R RX xB. 3: W. 4: W . , -. . () . 5: , unew W Xpress 2. :

88

2:

s=0

Rs= Ws= RXs=

uo ( Simplex)

us s=s+1

Rs+1=Rsus RXs+1

us

Ws+1=Ws+1-us+1

Ws+1=WsN(us)-Rs+1

Ws+1= us+1 Ws+1

NAI

R

: [max]z =+0xk-Mxr

t=1

us+1 Ws+1 d(us,us+1)=t;

t=t+1

2-2: Manas-Nedoma

89

2:

Simplex Manas-Nedoma , Tarry. , r , m n (l=n+m m , n ), Tarry nr Simplex . r mr, . (Siskos, 1984) r (Hamiltonian ) . Hamiltonian (Mattheis & Rubin, 1980). Manas-Nedoma Simplex rm Rs Ws. . . r Simplex, r .. 1.1. (Siskos, 1984). 2.1 Manas-Nedoma . k=20, 3x1+4x2+5x3+6x4 56 3x1+4x2+5x3+6x4 -Y=56 : x1 = x5 , x 2 = x 6 Y = x 7 . , s=0, Rs= Ws=. .., uo={(2,4,7)} z*=76. 2-1 Simplex , , , (j) ( 2.3.4. ).

90

2:

x2 x4 x7

x1 1 0 1

x2 1 0 0

x3 1/3 2/3 1/3

x4 0 1 0

x5 1 0 4

x6 -1/3 1/3 2/3

x7 0 0 1

xB

16 2 20

2-1: Manas-Nedoma, Simplex s=0 R1={(2,4,7)} , . : (uo)={(1,4,7), (2,3,7), (2,4,5), (2,6,7)} W1=Wo(uo)-uo={(1,4,7), (2,3,7), (2,4,5), (2,6,7)}. W1 uo (d=1) . u1=(2,3,7) .

s=1. Simplex : x2 x3 x7 x1 1 0 1 x2 1 0 0 x3 0 1 0 x4 -1/2 3/2 -1/2 x5 1 0 4 x6 -1/2 1/2 1/2 x7 0 0 1xB

15 3 19

2-2: Manas-Nedoma, Simplex s=1 : R2=R1{u1}={(2,4,7), (2,3,7)} (u1)={(1,3,7), (2,4,7), (2,3,5), (2,6,7)} W2=W1(u1)-R2={(1,3,7), (1,4,7), (2,3,5), (2,4,5), (2,6,7)} u2={(2,6,7)}.

91

2:

s=2. Simplex : x2 x6 x7 x1 1 0 1 x2 1 0 0 x3 1 2 -1 x4 1 3 -2 x5 1 0 4 x6 0 1 0 x7 0 0 1xB

18 6 16

2-3: Manas-Nedoma, Simplex s=2

: R3=R2{u2}={(2,4,7), (2,3,7), (2,6,7)} (u2)={(2,6,1), (2,3,7), (2,4,7), (2,6,5)} W3=W2(u2)-R3={(1,3,7), (1,4,7), (2,3,5), (2,4,5), (2,6,1), (2,6,5)} u3={(2,6,1)}.

s=3. Simplex : x2 x6 x1 x1 0 0 1 x2 1 0 0 x3 2 2 -1 x4 3 3 -2 x5 -3 0 4 x6 0 1 0 x7 -1 0 1xB

2 6 16


: R4=R3{u3}={(2,4,7), (2,3,7), (2,6,7), (2,6,1)} (u3)={(3,6,1), (4,6,1), (2,6,5), (2,6,7)} W4=W3(u3)-R4={(1,3,7), (1,4,7), (2,3,5), (2,4,5), (2,6,5), (3,6,1), (4,6,1)} u4={(2,6,5)}.

92

2:

s=4. Simplex : x2 x6 x5 x1 3/4 0 1/4 x2 1 0 0 x3 5/4 2 -1/4 x4 3/2 3 -1/2 x5 0 0 1 x6 0 1 0 x7 -1/4 0 1/4xB

14 6 4


: R5=R4{u4}={(2,4,7), (2,3,7), (2,6,7), (2,6,1), (2,6,5)} N(u4)={(2,6,1), (2,3,5), (2,4,5), (2,6,7)} W5=W4N(u4)-R5={(1,3,7), (1,4,7), (2,3,5), (2,4,5), (3,6,1), (4,6,1)} u5={(2,4,5)}.

s=5. Simplex : x2 x4 x5 x1 3/4 0 1/4 x2 1 0 0 x3 1/4 2/3 1/12 x4 0 1 0 x5 0 0 1 x6 -1/2 1/3 1/6 x7 -1/4 0 1/4xB

11 2 5


: R6=R5{u5}={(2,4,7), (2,3,7), (2,6,7), (2,6,1), (2,6,5), (2,4,5)} N(u5)={(1,4,5), (2,3,5), (2,6,5), (2,4,7)} W6=W5N(u5)-R6={(1,3,7), (1,4,7), (2,3,5), (3,6,1), (4,6,1), (1,4,5)} u6={(2,3,5)}.

93

2:

s=6. Simplex : x2 x3 x5 x1 3/4 0 1/4 x2 1 0 0 x3 0 1 0 x4 -3/8 3/2 -1/8 x5 0 0 1 x6 -5/8 1/2 1/8 x7 -1/4 0 1/4xB

41/4 3 19/4


: R7=R6{u6}={(2,4,7), (2,3,7), (2,6,7), (2,6,1), (2,6,5), (2,4,5), (2,3,5)} N(u6)={(1,3,5), (2,4,5), (2,6,5), (2,3,7)} W7=W6N(u6)-R7={(1,3,7), (1,4,7), (3,6,1), (4,6,1), (1,4,5), (1,3,5)} u7={(1,3,5)}.

s=7. Simplex : x1 x3 x5 x1 1 0 0 x2 4/3 0 -1/3 x3 0 1 0 x4 -1/2 3/2 0 x5 0 0 1 x6 -5/6 1/2 1/3 x7 -1/3 0 1/3xB

41/3 3 4/3


: R8=R7{u7}={(2,4,7), (2,3,7), (2,6,7), (2,6,1), (2,6,5), (2,4,5), (2,3,5), (1,3,5)} N(u7)={(2,3,5), (1,4,5), (1,3,6), (1,3,7)} W8=W7N(u7)-R8={(1,3,7), (1,4,7), (3,6,1), (4,6,1), (1,4,5)} u8={(1,3,7)}.

94

2:

s=8. Simplex : x1 x3 x7 x1 1 0 0 x2 1 0 -1 x3 0 1 0 x4 -1/2 3/2 0 x5 1 0 3 x6 -1/2 1/2 1 x7 0 0 1xB

15 3 4


: R9=R8{u8}={(2,4,7), (2,3,7), (2,6,7), (2,6,1), (2,6,5), (2,4,5), (2,3,5), (1,3,5), (1,3,7)} N(u8)={(2,3,7), (1,4,7), (1,3,5), (1,3,6)} W9=W8N(u8)-R9={(1,4,7), (3,6,1), (4,6,1), (1,4,5)} u9={(1,4,7)}.

s=9. Simplex : x1 x4 x7 x1 1 0 0 x2 1 0 -1 x3 1/3 2/3 0 x4 0 1 0 x5 1 0 3 x6 -1/3 1/3 1 x7 0 0 1xB

16 2 4


: R10=R9{u9}={(2,4,7), (2,3,7), (2,6,7), (2,6,1), (2,6,5), (2,4,5), (2,3,5), (1,3,5), (1,3,7), (1,4,7)} N(u9)={(2,4,7), (1,3,7), (1,4,5), (1,4,6)} W10=W9N(u9)-R10={(3,6,1), (4,6,1), (1,4,5)} u10={(1,4,5)}.

95

2:

s=10. Simplex : x1 x4 x5 x1 1 0 0 x2 4/3 0 -1/3 x3 1/3 2/3 0 x4 0 1 0 x5 0 0 1 x6 -2/3 1/3 1/3 x7 -1/3 0 1/3xB

44/3 2 4/3


: R11=R10{u10}={(2,4,7), (2,3,7), (2,6,7), (2,6,1), (2,6,5), (2,4,5), (2,3,5), (1,3,5), (1,3,7), (1,4,7), (1,4,5)} N(u10)={(2,4,5), (1,3,5), (1,4,6), (1,4,7)} W11=W10N(u10)-R11={(3,6,1), (4,6,1)} u11={(1,4,6)}.

s=11. Simplex : x1 x4 x6 x1 1 0 0 x2 2/3 1/3 -1 x3 1/3 2/3 0 x4 0 1 0 x5 2 -1 3 x6 0 0 1 x7 1/3 -1/3 1xB

52/3 2/3 4


: R12=R11{u11}={(2,4,7), (2,3,7), (2,6,7), (2,6,1), (2,6,5), (2,4,5), (2,3,5), (1,3,5), (1,3,7), (1,4,7), (1,4,5), (1,4,6)} (u11)={(1,2,6), (1,3,6), (1,4,5), (1,4,7)} W12=W11(u11)-R12={(3,6,1)} u12={(1,3,6)}.

96

2:

s=12. Simplex : x1 x3 x6 x1 1 0 0 x2 1/2 1/2 -1 x3 0 1 0 x4 -1/2 3/2 0 x5 5/2 -3/2 3 x6 0 0 1 x7 1/2 -1/2 1xB

17 1 4


: R13=R12{u12}={(2,4,7), (2,3,7), (2,6,7), (2,6,1), (2,6,5), (2,4,5), (2,3,5), (1,3,5), (1,3,7), (1,4,7), (1,4,5), (1,4,6), (1,3,6} (u12)={(1,2,6), (1,4,6), (1,3,5), (1,3,7)} W13=W12(u12)-R12={} . , Hamiltonian . s=12 12 Simplex. , Simplex , s Simplex r s rm.

97

2:

2.3.3 Simplex Simplex (Simplex Inverse) Van de Panne Simplex (Van de Panne, 1975). , Simplex xj xBr, Simplex xBr xj. z z Simplex (Siskos, 1984). Simplex () . 1.2

1.2

Ax b

ct x z* k k=0 ( x0 )

k

k . k (Van de Panne, 1975). Simplex . Simplex . Simplex . k . Simplex Simplex , Simplex .

98

2:

, Simplex , Simplex , (Van de Panne, 1975). Simplex . 2-14 Simplex, p .

cBcB1

1

1 1

k y1k

r 0

m+n Y1(m+n)

xBxB1

cBr

r

0

yrk

1

yr(m+n)

xBr

cBj

jcj

0

yjk

0

yj(m+n)

xBj

c1 -

ck k

cr -

cn n z

j

2-14: Simplex p

p+1 Simplex xk (. k = max j j >0 ).j

xBr :

x x Br = min Bi , y ik > 0 i y y rk ik

(2.1)

2-15 Simplex p+1 :

99

2:

cBcB1

1

1 1

k 0

r -y1k/yrk

m+n y1(m+n)-y1k(yr(m+n)/yrk)

xBxB1- y1k(xBr/yrk)

cBk

k

0

1

1/yrk

yr(m+n)/yrk

xBr/yrk

cBj

jcj

0 c1 -

0 ck -

-yjk/yjk cr -k/yrk

yj(m+n)-yjk(yr(m+n)/yrk) cn n-k(yrn-yrk)

xBj- yjk(xBr/yrk)

j

z+k(xBr/yrk)

2-15: Simplex p+1

k(xBr/yrk) -k/yrk. xr. -k/yrk . z :

z= z + k

x Br k x Br yrk = z yrk yrk yrk

(2.2)

. xB :

x Bi = x Bi x Br

yik y rk

yik y rk x Br 1 y rk y rk

(2.3)

.

100

2:

x Bi yik

x Br * = x Bi yrk

(2.3) :

* x Bi

yik y x + rk Br 0 1 y rk y rk

(2.4)

(2.4) yik >0. (. yrk xk .) xk xr 2-14. Simplex (Van de Panne, 1975). A Simplex . :

z Y = z * k (2.5.), Y . Y A Simplex : =z-(z*-k) 0 (2.6) ks Y s . ks z*-k. Simplex Simplex (2.5). 2.3.4 Simplex , : oj = -j 0. (s=0) Simplex.

101

2:

x1

ao1i ao1j 1

0 0 0

xoB1

xi

aoii

aoij

0

1

0

0

xoBi

xm Y

aomi oi

aomj oj

0 0

0 0

1 0

0 1

xoBm ko(=k)

2-16: (s=0) Simplex

s :

x1 xi xm Y

as1l asil asml

as1j asij asmj

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

xsB1 xsBi xsBm ks

sl

sj

2-17: s Simplex xoBi >0 i oj >0 j z* oj 0. , xk , ok ( (..16) (..20) ) ko . xr , Simplex, : oj =o xo xBr o = min Bi , aij > 0 o i ao arj ij

(2.5)

102

2:

xk . k0j=k-oj(xoBr/aorj ) (2.6) j. ys koj. Y , (k1) :

k1 = max(k oj ,0 ) = k 0 kj

(2.7)

. k1 0, xk , (2.5). k1=0 . (2.5) j aoij 0 i, - xk z* . , k ks=0 , k1=0, . , , Y=k1. k1. ks k k1 , z*, Simplex, Simplex. , Simplex, 2.7) . , k1. Simplex . ( ) ks, . , . , , xk. xj 1j>0. 1jk1

103

2:

. ks k k1, xj 1j>0 , z*- k k1. , (s=1) xj 1j>0, : 1j =

x1 x1 1 Br = min Bi , a ij > 0 i a1 a1 rj ij

(2.8)

y1 xj : k1j=k-1j(x1Br/a1rj ) (2.9)

ks :

k 2 = max k oj , k1 j ,0 k 0 j < k1 , k1 j >0j

(

)

(2.10)

. k0j s=0, k1j s=1. k2=0 . Simplex 6, . 00, j).

104

2:

- xB . Manas-Nedoma Rs RXs.

1: Simplex s. xj sj>0. - asrj ( j sj>0) :s s x Bi x Br s = min s , aij >0 s i a a rj ij

(2.11)

2: Y xj :ksj=ks-sj(xsBr/asrj ) (2.12)

s ksj>0.

3: xj :ks +1 = max koj , k1 j ,..., ksj ,0 kij > 0j

(

)

(2.13)

s kij>0. R RX xB. ks+1=0, 2.

4: xj xr (xj

xr )

Simplex (2.13). 1 :s=s+1.

2 k1j,...,ksj , Simplex z=z*-k.

105

2:

() sj >0, Simplex . , z*-k. z ks Simplex s. :

ks =

sj

s s x Br k s x Br s = s s a rj j a rj

(2.14)

2.15

xj =

x Br a rj

, (2.14) :

x Bj =

ks sj

(2.15)

:s s x Bi = x Bi aij x Bj , i j

(2.16)

Simplex s, Y ( Y=0) xj sj>0. , Simplex .

sj+1 .

:

106

2:

1

s=0 Ks= {0} R s= RXs= oj = -j >=0 Y=k (ko=k) asrj ( j sj>0) Y xj Ks+1=Ks ksj ksj>0 xj :ks +1 = max koj , k1 j ,..., ksj 0 ( k ij > 0j

s=s+1

xj xr Simplex

(

)

Rs+1=Rsus RXs+1 OXI

ks+1=0ss=0 q=0

NAI

2

ss=ss+1

A Y ( Y=0) Simplex ss xj sj>0 Rs+q+1=Rs+qusj RX s+q+1 q=q+1 R

ss=s

2-3: Simplex

107

2:

, .. . Simplex. Simplex . , (Siskos, 1984; Van de Panne, 1975). Simplex 2.1.

1 s=0, Ks={0}, Rs= RXs=. Simplex 00 2-18. k=20, : 3x1+4x2+5x3+6x4 56 3x1+4x2+5x3+6x4 -Y=56, : x1 = x5 , x 2 = x 6 Y = x 7 . 0: (j) . Simplex s=0. 1: sj ( Simplex). : x1, x5, x3, x6. - aorj (2.11): j=1, min{16/1}=16/1, j=5, min{16/1}=16/1, 21=1 25=1

j=3, min 16 , 1 3

2 = 2 32 , 1 3

2

2 3

,

43=2/3

j=3, min

2 = 1 3

46=1/3

108

2:

2: k0j (2.12): k01=20-116/1=4, k05= -44, k03=19, k06=16. K0={k01, k03, k06, 0} 3: (2.13) : k1=max{k01, k03, k06, 0}=k03=19. x3 z z* (z=76-1=75). k03 (2.13). R1={(2,4,7)} 4: x3 x4 Simplex, no1.

s=1 1. 1: : x1, x5, x6. - a1rj (2.11): 121=1 125=1 136=1/2

2: k1j (2.12): k11=4, k15= -41, k16=16. K1={ k01, k06, k11, k16, 0} 3: (2.13) : k2=max{k01, k06, k11, k16, 0}=16 k06 k16. x6 Simplex no0 no1. R2={(2,4,7), (2,3,7)} 4: no0 x6 x4 Simplex, no2.

s=2 1. 1: : x1, x5. - a2rj (2.11):

109

2:

221=1 225=1 2: k2j 2.12): k21= -2, k25= -56. K2={ k01, k11, 0} 3: 2.13) : k3=max{k01, k11, 0}=k01=k11=4 x1 Simplex no0 no1. R3={(2,4,7), (2,3,7), (2,6,7)} 4: no0 x1 x2 Simplex, no3.

s=3 1. 1: : x3, x5, x6. - a3rj (2.11): 315=1 343=2/3 a346=1/3

2: k3j 2.12): k35= -44, k33=4, k36= -2. K3={ k33, 0} 3: (2.13) : k4=max{k33, 0}=k33 =4 x3 Simplex no3. R4={(2,4,7), (2,3,7), (2,6,7), (1,4,7)} 4: no3 x3 x4 Simplex, no4.

s=4 1.

110

2:

1: : x5, x6. - a4rj (2.11): 415=1 436=1/2 2: k4j (2.12): k45= -41, k46= -2. K4={0} 3: (2.13) : k5=max{0}=0 R5={(2,4,7), (2,3,7), (2,6,7), (1,4,7), (1,3,7)} 2.

2 Simplex no 0, 1, 2, 3, 4 1 ( 2-18) (2.15) (2.16) 1. xj Y (x7), (Y=0), 2.15), z=z*-k=56. 2 ( 2-19, no 5-12). no 5 (2.15) (1.16) Simplex no 0 ( k050 : uo xm+n+1 =k. (3.10) u1 xm+n+1 =Y0k. usj xm+n+1 Y0k. Ysk k Y0k , , , Simplex. , (3.10), . , Yok. Simplex . Yok xm+n+1 =Ysj >0. usj xm+n+1 =Ysj >0. usj xm+n+1 =0, xm+n+1 : z=z*-k. 4 usj xm+n+1 =Ysj >0 Simplex xm+n+1 =0. Simplex.

131

3:

3.2 .1.2. . :

[max ] z = 3x + 4 x + 5x + 6 x + 0 x + 0 x 1 2 3 4 1 2 . . x1 + x2 + x3 + x4 + x1 = 18 2 x3 + 3 x 4 + x 2 = 6 xi 0, i = 1, 2, 3, 4,1, 2 z*=76 20 (k=20). z-Y=z*-k Y . x1 = x 5 , x 2 = x 6 Y = x 7 . Simplex :

cB4 6

2 4cj

1 1 0 3 -1

2 1 0 4 0

3 1/3 2/3 5 -1/3

4 0 1 6 0

11 0 0 -4

2-1/3 1/3 0 -2/3

xB16 2

j

z*= 76

3-4: Simplex

j .

0: Simplex 3-4 : z-x7=z*-k x7=z-56 (3.2), (3.3), (3.4), (3.5) (2.24) (2.27) Simplex s=0.

132

3:

x2 x4 x7

x1 1 0 1

x2 1 0 0

x3 1/3 2/3 1/3

x4 0 1 0

x5 1 0 4

x6 -1/3 1/3 2/3

x7 0 0 1

xB

16 2 20

uo=(2,4,7), Ro={(2,4,7)}, No={0} vuo=(16,2,20) RXo.

1: uoj (3.6) (uo): (uo)={(1,4,7), (2,3,7), (2,4,5), (2,6,7)}, : Ko= Wo=(uo)-Ro={(1,4,7), (2,3,7), (2,4,5), (2,6,7)} Wo, 2.

2: u01=(1,4,7). u01. xB1=16/1=16, xB4=2-016=2, xB7=20-116=4 vu01=(16,2,4) Ro=Rou01={(2,4,7), (1,4,7)}, Ko= x7 u01 Y01=4 No={ Y01=4, 0} Wo={(2,3,7), (2,4,5), (2,6,7)} Wo, 2.

2: u03=(2,3,7). u03. xB2=16-1/33=15, xB3=

2 =3, xB7=20-1/33=19 2/3

133

3:

vu03=(15,3,19) Ro=Rou03={(2,4,7), (1,4,7), (2,3,7)}, Ko= x7 u03 Y03=19 No={ Y01=4, Y03=19, 0} Wo={(2,4,5), (2,6,7)} Wo, 2.

2: u05=(2,4,5). u05. xB2=16-5=11, xB4=2-05=2, xB5=20/4=5 vu05=(11,2,5) Ro=Rou05={(2,4,7), (1,4,7), (2,3,7), (2,4,5)}, Ko={(2,4,5)} x7 u05. Wo={(2,6,7)} Wo, 2.

2: u06=(2,6,7). u06. xB2=16+1/36=18, xB6= vu06=(18,6,16) Ro=Rou06={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7)}, Ko={(2,4,5)} x7 u03 Y06=16 No={ Y01=4, Y03=19, Y06=16, 0} Wo=, 3.

2 =6, xB7=20-2/36=16 1/ 3

3: Yqj No. Y03=maxNo No=No-Y03={Y01=4, Y06=16, 0} Y03u03=(2,3,7)

134

3:

4: d(uo, u03)=1 5. 5: Simplex s=0 x3 x4 Simplex. : s=0+1=1. Simplex : x2 x3 x7 x1 1 0 1 x2 1 0 0 x3 0 1 0 x4 -1/2 3/2 -1/2 x5 1 0 4 x6 -1/2 1/2 1/2 x7 0 0 1xB

15 3 19

: u1=u03 R1=Ro={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7)} N1=No={Y01=4, Y06=16, 0} 1.

1: u1j (3.6) (u1): (u1)={(1,3,7), (2,4,7), (2,3,5), (2,6,7)}, : K1= W1={(1,3,7), (2,3,5)} W1, 2.

2: u11=(1,3,7). u11. xB1=15/1=15, xB3=3-015=3, xB7=19-15=4 vu11=(11,2,5) R1=R1u11={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7), (1,3,7)}, K1= x7 u11 Y11=4 N1={Y01=4, Y06=16, Y11=4, 0}

135

3:

W1={(2,3,5)} W1, 2.

2: u15=(2,3,5). u15. xB2=15-119/4=41/9, xB3=3-019/4=3, xB5=19/4 vu15=(41/9,3,19/4) R1=R1u15={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7), (1,3,7), (2,3,5)}, K1={(2,3,5)} x7 u11 W1=, 3.

3: Yqj N1. Y06=maxN1 N1=N1-Y06={Y01=4, Y11=4, 0} Y06u06=(2,6,7)

4: d(u1, u06)=1 5.

5: Simplex s=1 x6 x3 Simplex. : s=1+1=2. Simplex : x2 x6 x7 x1 1 0 1 x2 1 0 0 x3 1 2 -1 x4 1 3 -2 x5 1 0 4 x6 0 1 0 x7 0 0 1xB

18 6 16

136

3:

: u2=u06 R2=R1={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7), (1,3,7), (2,3,5)} N2=N1={Y01=4, Y11=4, 0} 1.

1: u2j (3.6) (u2): (u2)={(2,6,1), (2,3,7), (2,4,7), (2,6,5)}, : K2= W2={(2,6,1), (2,6,5)} W2, 2.

2: u21=(2,6,1). u21. xB2=18-116=2, xB6=6-016=6, xB1=16/1=16 vu21=(2,6,16) R2=R2u21={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7), (1,3,7), (2,3,5), (2,6,1)} K2={(2,6,1)} x7 u21 W2={(2,6,5)} W2, 2.

2: u25=(2,6,5). u25. xB2=18-14=14, xB6=6-04=6, xB1=16/4=4 vu25=(14,6,4) R2=R2u25={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7), (1,3,7), (2,3,5), (2,6,1), (2,6,5)} K2={(2,6,1), (2,6,5)}

137

3:

x7 u25 W2=, 3.

3: Yqj N2. N2 Y01 Y11 u01=(1,4,7) u11=(1,3,7). d(u2,u01)=2 d(u2,u11)=2. Y01 u11=(1,4,7) N2=N2-Y01={Y11=4, 0}

4: d(u1, u01)=2 6.

6: Simplex s=2 x4 x6. Simplex : x2 x4 x7 x1 1 0 1 x2 0 0 0 x3 1/3 2/3 1/3 x4 0 1 0 x5 1 0 4 x6 -1/3 1/3 2/3 x7 0 0 1xB

16 2 20

d(u2,u01)=d(u2,u01)-1=1 4.

4: d(u2, u01)=1 5.

5: Simplex s=2 x1 x2 Simplex.

138

3:

: s=2+1=3. Simplex : x1 x6 x7 x1 1 0 0 x2 1 0 -1 x3 1/3 2/3 0 x4 0 1 0 x5 1 0 3 x6 -1/3 1/3 1 x7 0 0 1xB

16 2 4

: u3=u01 R3=R2={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7), (1,3,7), (2,3,5), (2,6,1), (2,6,5)} N3=N2={Y11=4, 0} 1.

1: u3j (3.6) (u3): (u3)={(2,4,7), (1,3,7), (1,4,5), (1,4,6)}, : K3= W3={(1,4,5), (1,4,6)} W3, 2.

2: u35=(1,4,5). u35. xB1=16-14/3=44/3 xB6=2-04/3=2 xB1=4/3 vu35=(44/3,2,4/3) R3=R3u35={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7), (1,3,7), (2,3,5), (2,6,1), (2,6,5), (1,4,5)}

139

3:

K3={(1,4,5)} x7 u35 W3={(1,4,6)} W3, 2.

2: u36=(1,4,6). u36. xB1=16+1/34=52/3, xB4=2-1/34=2/3, xB6=4/1=4 vu36=(52/3,2/3,4) R3=R3u36={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7), (1,3,7), (2,3,5), (2,6,1), (2,6,5), (1,4,5), (1,4,6)} K3={(1,4,5), (1,4,6)} x7 u36 W3=, 3.

3: Yqj N3. Y11=maxN3 N3=N3-Y11={0} Y11u11=(1,3,7)

4: d(u3, u11)=1 5.

5: Simplex s=3 x3 x4 Simplex. : s=3+1=4. Simplex :

140

3:

x1 x3 x7

x1 1 0 0

x2 1 0 -1

x3 0 1 0

x4 -1/2 3/2 0

x5 1 0 3

x6 -1/2 1/2 1

x7 0 0 1

xB

15 3 4

: u4=u11 R4=R3={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7), (1,3,7), (2,3,5), (2,6,1), (2,6,5), (1,4,5), (1,4,6)} N4=N3={0} 1.

1: u4j (3.6) (u4): (u4)={(2,3,7), (1,4,7), (1,3,5), (1,3,6)}, : K4= W4={(2,6,1), (2,6,5)} W4, 2.

2: u45=(1,3,5). u45. xB1=15-14/3=41/3, xB3=3-04/3=3, xB5=4/3 vu45=(41/3,2,4/3) R4=R4u45={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7), (1,3,7), (2,3,5), (2,6,1), (2,6,5), (1,4,5), (1,4,6), (1,3,5)} K4={(1,3,5)} x7 u45 W4={(1,3,6)}

141

3:

W4, 2.

2: u46=(1,3,6). u46. xB1=15+1/24=17, xB3=3-1/24=1, xB6=4/1=4 vu46=(17,1,4) R4=R4u46={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7), (1,3,7), (2,3,5), (2,6,1), (2,6,5), (1,4,5), (1,4,6), (1,3,5), (1,3,6)} K4={(1,3,5), (1,3,6)} x7 u46 W4=, 3.

3: Yqj N4. maxN3=0, u45=(1,3,5)

4: d(u4, u45)=1 5.

5: Simplex s=4 x5 x7 Simplex. : s=4+1=5. Simplex : x1 x3 x5 x1 1 0 0 x2 4/3 0 -1/3 x3 0 1 0 x4 -1/2 3/2 0 x5 0 0 1 x6 -5/6 1/2 1/3 x7 -1/3 0 1/3xB

41/3 3 4/3

: u5=u45

142

3:

R5=R4={(2,4,7), (1,4,7), (2,3,7), (2,4,5), (2,6,7), (1,3,7), (2,3,5), (2,6,1), (2,6,5), (1,4,5), (1,4,6), (1,3,5), (1,3,6)} N5=N4={0} 1.

1: u5j (3.6) (u5): (u5)={(2,3,5), (1,4,5), (1,3,6), (1,3,7)}, : K5= W5=(u5)-R5= W5= Ns={0}, 7.

7: . usj (Rs) vusj(RXs) zsj

uo =(2,4,7) u01=(1,4,7) u03=(2,3,7) u05=(2,4,5) u06=(2,6,7) u11=(1,3,7) u15=(2,3,5) u21=(2,6,1) u25=(2,6,5) u35=(1,4,5) u36=(1,4,6) u45=(1,3,5) u46=(1,3,6)

vuo=(16,2,20) vu01=(16,2,4) vu03=(15,3,19) vu05=(11,2,5) vu06=(18,6,16) vu11=(15,3,4) vu15=(41/9,3,19/4) vu21=(2,6,16) vu25=(14,6,4) vu35=(44/3,2,4/3) vu36=(52/3,2/3,4) vu45=(41/3,2,4/3) vu46=(17,1,4)

zo=76 z01=60 z03=75 z05=56 z06=72 z11=60 z15=56 z21=56 z25=56 z35=56 z36=56 z45=56 z46=56

3-5:

143

3:

3.3 R5 usj s=0,1,2,3,4 vusj . 6 Simplex, , 12 . ( k=0) . Ks 1 . Simplex. , Manas-Nedoma 12 Simplex, , 12 , , Simplex. Simplex, 2.3.3, 4 Simplex, , 12 4 , , . s, , z . , k. k, , , , . .

144

3:

145

4: . - UTA MUSA. UTA , UTASTAR . , MUSA . MUSA - . , - . MUSA . , , - .

4:

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147

4:

. - conformance to specifications - conformance to requirements . , . (fitness for use), . / (Grnroos, 1984; Parasuraman A. et al., 1985) (Kotler, 1994). . , . , . / . (vretveit 1993; Kotler 1994) . , (conformance). , , , , , . ( ) . , ,

148

4:

. , , : Deming 1951 , Deming, W. Edward Deming, . Deming . Malcolm Baldridge 1980 , , . Baldridge, 4.1 , . : ( ), , , , , . 5. 7. 3. 6.

2.

1.

4.

4-1: Malcolm Baldridge

149

4:

1991 (European Foundation for Quality Management - EFQM) , Baldridge : ( ), , , , , , , . 9% 14% 9%

8%

20%

9%

6%

4-2:

ISO 9000 AQAP (Allied Quality Assurance Publications), BS 5750, 1987 (International Organization for Standardization) ISO 9000 (ISO 9001, ISO 9002, ISO 9003, ISO 9004, ISO 9004-1, ISO 9004-2). ISO . , , , . .

150

15%

10%

4:

4.1.2 . . , , (, 1995). , . . ( & , 2000). , , , , , ... - , (Kotler, 1994). (outcome), (process). . , . . , . , . , ,

151

4:

. , : ) , ) , ) (, 1994). . , (Engel et al., 1978; Engel & Blackwell, 1982; , 1986; , 1990; , 1981; , 1994). O Edosomwan (1993) : (output) (work) (Gale, 1963; Kano et al., 1984; Levitt, 1986; Lowenstein, 1995; Mittal et al., 1998) - . , Must-be Kano, , . - Must-be . , Must-be , Must-be Must-be . (key dissatisfiers).

4.1.3 . (Morgan & Piercy, 1994). .

152

4:

, . , , . , . : ( ) (Multiattribute Attitude Models), Rosenberg, (Basic Multiattribute Model), Fishbein, (Hybrid Models), Fishbein (, 1994). (attributes), . , . (, 1995). (market segmentation) , . (product positioning) , (, 1994). Hauser and Urban (1978), Von Neumann Morgenstern (1947), (descriptive models) . Von Neumann Morgenstern (1947). . () . . (utility function),

153

4:

. , . , , . (utility function) (, 1995). , , .

4.1.4 . ( et al., 1997). (Cadotte et al., 1987; Churchill & Surprenant, 1982; Fornell, 1992; Oliver, 1980, 1997; Oliver & Swan, 1989; Tse & Wilton, 1988; Westbrook, 1987). . , (Oliver, 1980; Oliver & DeSarbo, 1988). (Bolton & Drew, 1991;Fornell, 1992). . , () , (Cronin & Taylor, 1992; Gotleib et al., 1994). ; . , :

154

4:

( ) (Kotler, 1994) , /

. , , . , (Kotler, 1994). , . , , ( , 2000). , . , . . , (Czarnecki, 1999; Dutka, 1995; Gerson, 1993; Morgan & Piercy, 1994). , . , , , .

155

4:

, , ( et al., 1997). , , 1960-1980, . Cardoso, Howard and Sheth Oliver ( , 2000). , . , (brand loyalty) . () . ( et al., 1997). , . , (multiple linear regression analysis) (categorical data analysis), logit analysis loglinear models , . , conjoint analysis, , ( et al., 1997). , Saurais (1997) :

156

4:

1. , a priori , . , , , , . 2. . ( ) ( , ). 3. ( , ) . 4. , (.. , ). (2000). MUSA (Preference Aggregation Disaggregation Models). MUSA , . , . 4.4.

4.2 - 1.3.3 ,

157

4:

. , . , , , . ( ), ( ). . , , (, 1995). 4.3 UTA - . . MUSA (Multicriteria Satisfaction Analysis) , - . - . , , . , MUSA . 4.4. UTA MUSA . (post optimality analysis) .

158

4:

5 UTA MUSA.

4.3 M UTA 4.3.1 UTA1 Jacquet-Lagrze Sisko (1982) ( ) R ( 1.3.3). , AR, . (multiattribute utility theory, MAUT) . , , , , (Jacquet-Lagrze & Siskos, 1982). - (aggregation-disaggregation approach) . UTA , . , , , , , , (Despotis et al., 1990). UTA (ordinal regression) .

1

Utilit Additive

159

4:

, ( ) . , , , , . UTA 1.6 1.10 1.3.2. , , , 1.8 1.9 :

u (g ) = u i (g i )i =1

n

(4.1)

:

u i (g i* ) = 0 i = 1,2,..., n

(4.2) (4.3)

u (g ) = 1i =1 i * i

n

, , (Jacquet-Lagrze & Siskos, 2001). U : (i) , (ii) , (iii) , (iv) (Despotis et al., 1990).

160

4:

4.3.2 UTA UTA (Jacquet-Lagrze & Siskos, 2001) , , . / n . Gi=[ g i* , g* ], i=1,2,...,n i i , G= n Siskos, 1982). Gi g ik , . g i* , g* i i [ g i* , g* ] (i-1) [ g ij , g ik +1 ]. i i ui. g ik :i =1

G (Jacquet-Lagrze & i

g ik = g i * +

j 1 * (g g i * ) j = 1,2,...ai i 1 i

(4.4)

ui( g ik ). aj . , gi(aj) [ g ij ,

g ij+1 ], :ui [g i (a j )] = ui (g ik ) + g i (a ) g ik [ui (g ik +1 ) ui (g ik )] k +1 k g gi (4.5)

Gi , , i . Gi=[5,6,7,8,9,10], i=6 ui(5), ui(6), ..., ui(10). R=(P,I) , P I , , R={a1,a2,,am} G. R a1

161

4:

am . R R. , (aj, aj+1) AR :u[g(a j )] > u[g(b j +1 )] a j b j +1 ( ) u[g(a )] = u[g(b )] a ~ b ( ) j j +1 j j +1

(4.6) (4.7)

UTA Jacquet-Lagrze and Siskos (2001) : 1: u[g(aj)] aj ui(gi) (4.5). , ui(gi) n :ui ( g ik +1 ) ui ( g ik ) si k = 1,2,... i 1, i = 1,2,...n

(4.8)

si0 i. ( si=0). , gi , : ui ( g ik +1 ) = ui ( g ik ) g ik +1g ik . ,

si . , (2.7):

n * u i (g i ) = 1 i =1 u ( g ) = 0 i i i*

(4.9)

2: (4.6), (4.7) (2.3) : u[g(aj)]=

u [g (ai =1 i i

n

j

)] + (aj)

aAR. (4.10)

162

4:

(aj) o ( 4-3) : u[g(aj)]=

u [g (ai =1 i i

n

j

)]

(4.11)

T (aj) u[g(aj)] a .

4-3:

, a1, , am : (aj, aj+1)=u[g(aj)]-u[g(aj+1)] (4.12)

(4.12) m-1, 1. (JacquetLagrze & Siskos, 1982) (4.12) (4.10) (4.11): (aj, aj+1)= u[g(aj)]+(aj)- u[g(aj+1)]-(aj+1) (4.13)

163

4:

3: (aj):m [min]z = (a j ) j =1 : ( j , j +1 ) j j +1 j = 1,2,..., m 1 ( j , j +1 ) = 0 j ~ j +1 k +1 k ui (g i ) ui (g i ) 0 i k n * ui ( g i ) = 1 i =1 u (g ) = 0, u ( g k ) 0, (a ) 0 i , k j i i j i i*

(4.14)

(4.15) (4.16) (4.17)

AR sj, . (4.15) (weak order) , (4.16) (4.17) .

(i =1

n

i

1)

u i (g ik ) |R|=m (aj). , m-1 (4.15)

(i =1

n

i

1) (4.16).

1+n (4.17) .

4: UTA , , , . z* 3 / . (4.15)-(4.17) :

164

4:

z z*+

(4.18)

z* , , z*. 5 . . .

4.3.3 UTASTARTo 1985 Siskos Yannacopoulos UTA, . . UTASTAR (aj) UTA. UTA (aj) u[g(aj)] aj ( 4-3). 4.3. .

165

4:

4-4: (Jacquet-Lagrze and Siskos, 2001)

UTASTAR -(aj) u[g(aj)] aj , +(aj) u[g(aj)] aj ( 4-4). : 1: (4.8) :

w ik = u i (