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Linear programming is considered to be, beyond any doubt, one of the most popular tools of operational research and management science in general. In many decision problems, which are modeled by using lineal systems, there is a necessity of finding new solutions (post-optimal), not so far from the optimal one. The subject of the present thesis is the study of several post-optimal analysis methods (robustness analysis) in linear programming problems and the proposition of a new approach. In order to compare and evaluate the post-optimal analysis methods with each other and with the new approach, these methods were implemented in software (using C# as the programming language). Following, these software modules were intergraded in a decision support system which also includes the MUSA (Mulitctiteria Satisfaction Analysis) method, developed by Grigoroudis and Siskos for measuring customers’ satisfaction. For the implementation of the relevant evaluation experiments regarding these methods, data from real-world surveys were used as well as synthetic data sets which were produced by a data set generator, also developed under this thesis and intergraded in the aforementioned decision support system. This DSS supports the analysts, who use MUSA method, in selecting the most appropriate post-optimal analysis method in each case and also in selecting the levels of the two basic method parameters.
Citation preview
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2009
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, , , , , , , & , , . , , , , & ,
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. 1995 . , . , 3. 18 9. , , . .
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: . , , , , . , , , . , , . () (near optimal solutions) , . (post optimal analysis - robustness analysis) . ( C#) MUSA . . MUSA .
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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
22
. , 7 8 . 9 . , , .
23
1: ( , ) . . , , . . -. .
1: -
1.1 Operational Research Operations Research. , . , , , , , , (, 2008). : , , , , , . , , . ( ). ( ) . , ( ), . (robustness stability) Roy (2007) . , , , ( ) , , . ,
25
1: -
(Roy, 2005). , , ( ) . , , . ( ) . , . Dias (2007) , . , (Roy, 1989). . , ( ) , . , (.. ELECTRE), , , , , ... , - ( Roy (2003, 2007)). : (.. , ), (.. ), (.. ) .
26
1: -
, .. . (.. ) , , ( ) ( ). , (Beer, 1966: .44). (Roy 2007) (Vincke, 1999a,b). 6 . . ( ) ( ). , .
1.1.1 Roy (2007) 2004, Philippe Vincke , , . , . . , , .
27
1: -
. , , . , : 1. ( ) 2. ( ) , , , . Roy (2007) : , , .
-
, ( ) , . , Roy (2005). . i) : ( ), , , ,
28
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),
30
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). , .
31
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. : , , . .
37
1: -
. . ii. : ( ) , . , . iii. , , , , : , . , , , , , . . , , : , / , .
-
, .
38
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 . , , .
39
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 . . , / . , ( ) . , . , . , . , .
41
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).
43
1: -
. , , , , . , . ( ) , , , (), . . ( ) (, 1996;, 1992). , , , . . - , Simplex ( ), : n m , m , : (n + m) - m = n . , . , , .
44
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).
48
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, ,
49
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). , , , . , , . .
50
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* . .
51
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., .
52
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) .
53
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 ),
54
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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1: -
, , ) , 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), . ,
56
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:
57
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.
58
1: -
, , , . 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
59
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)
60
1: -
, . . .
1.3.3 , ( ). , : , ; , ; - , . (Jacquet-Lagrze & Siskos, 2001). , a priori , . , .
1-9: ( & , 2000)
61
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,
62
1: -
. (Jacquet-Lagrze & Siskos, 2001): AR () (weak order relation) AR, , - , -. , , , . (, 1981) ( ), ( ). (extrapolation) AR . ( , 2000) - 1-11, , , .
63
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 . .
64
1: -
. . . , ( ) (Roy 1993, Dias & Tsoukias 2004). . , - , . , , . , - . : (.. - ). , , /. (, ) ( .. ) , . .
65
1: -
. , () , . , . / . . , , . . MUSA . (. 1.1.2). - (. 1.2.2 1.2.3). , , .
66
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
2-4: Manas-Nedoma, Simplex s=3
: 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
2-5: Manas-Nedoma, Simplex s=5
: 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
2-6: Manas-Nedoma, Simplex s=6
: 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
2-7: Manas-Nedoma, Simplex s=6
: 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
2-8: Manas-Nedoma, Simplex s=7
: 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
2-9: Manas-Nedoma, Simplex s=8
: 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
2-10: Manas-Nedoma, Simplex s=9
: 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
2-11: Manas-Nedoma, Simplex s=10
: 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
2-12: Manas-Nedoma, Simplex s=11
: 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
2-13: Manas-Nedoma, Simplex s=12
: 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:
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4: . - UTA MUSA. UTA , UTASTAR . , MUSA . MUSA - . , - . MUSA . , , - .
4:
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147
4:
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148
4:
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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:
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4.1.3 . (Morgan & Piercy, 1994). .
152
4:
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153
4:
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154
4:
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155
4:
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156
4:
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4.2 - 1.3.3 ,
157
4:
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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 (