University of Arkansas, Fayetteville ScholarWorks@UARK Theses and Dissertations 5-2014 Robust Network Interdiction with Invisible Interdiction Assets Nail Orkun Baycik University of Arkansas, Fayetteville Follow this and additional works at: http://scholarworks.uark.edu/etd Part of the Operational Research Commons Recommended Citation Baycik, Nail Orkun, "Robust Network Interdiction with Invisible Interdiction Assets" (2014). Theses and Dissertations. 2273. http://scholarworks.uark.edu/etd/2273 This Thesis is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of ScholarWorks@UARK. For more information, please contact scholar@uark.edu, ccmiddle@uark.edu.
25 60 R =0 73 R =4 45 R =6 38 R =13 28 R R =1 417 R =5 212 R =15 198 R =66 171
n
c d c
x y
z y x G =(N,A) N = {1,...n} A (i, j) c ij c ij + d ij d ij 0 x ij 1 (i, j) A x =ˆx (i, j) c ij +d ij ˆx ij z ij 1 (i, j) A 0 y ij 0 y ij d ij ˆx ij (i, j) c ij + y ij
y =ŷ 1 n c ij +ŷ ij (ŷ) ŷ)] : z [c ij +ŷ ij ] z ij, (i,j) A j (i) j (i) 1 i =1 z ij z ji = 0 i =2,...,n 1 1 i = n z ij 0, (i, j) A. z u 1 u n, u i u j c ij +ŷ ij, (i, j) A, u i,i N i
d ij x ij x =ˆx (ˆx) (ˆx y,z [c ij + d ij ˆx ij ] z ij, (i,j) A y ij R, (i,j) A 0 y ij d ij ˆx ij (i, j) A, z y. R (i, j) d ij x ij =1 x ij =0 z y x =ˆx
(ˆx u,y,z (i,j) A (i,j) A [c ij + d ij ˆx ij ] z ij, y ij R, 0 y ij d ij ˆx ij, (i, j) A, j (i) z ij j (i) 1 i =1 z ji = 0 i =2,...,n 1 1 i = n u i u j c ij + y ij, (i, j) A, [c ij + y ij ] z ij = u 1 u n, (i,j) A z ij {0, 1}, (i, j) A. y ij z ij z w ij (i, j) A y ij z ij w ij y ij w ij d ij ˆx ij w ij 0 w ij y ij d ij ˆx ij (1 z ij ) (i, j) A x =ˆx
ˆx) u,w,y,z [c ij + d ij ˆx ij ] z ij, (i,j) A y ij R, (i,j) A 0 y ij d ij ˆx ij, (i, j) A, 1 i =1 z ij z ji = 0 i =2,...,n 1 j (i) j (i) 1 i = n u i u j c ij + y ij, (i, j) A, [c ij z ij + w ij ]=u 1 u n, (i,j) A w ij y ij, (i, j) A, w ij d ij ˆx ij, (i, j) A, w ij y ij d ij ˆx ij (1 z ij ), (i, j) A, w ij 0, (i, j) A, z ij {0, 1}, (i, j) A. (i,j) A (c ijz ij +y ij z ij ) u 1 u n w ij y ij z ij, (i, j) A z ij {0, 1} (i,j) A (c ijz ij + w ij ) (i,j) A (c ijz ij + y ij z ij ) u 1 u n w ij = y ij z ij
(i, j) w x ij f(x) x X { f(x) (x) X = x : } (i,j) A x ij B; x ij {0, 1} ; (i, j) A z 1 n 1 n I = {1,..., m} 1 n λ i 1 i I 0 i P i γ(p i ) (k,l) A (c kl + d klˆx kl ) i
λ,y γ(p i )λ i, i I λ i =1, i I [c kl + y kl ] M(1 λ i ) (k,l) P i (k,l) P i [c kl + y kl ] 0, i, i I,i i (k,l) A y kl R, 0 y kl d klˆx kl, (k, l) A λ i {0, 1}, i I. R y y kl 0 i I λ i =1
x x X f(x) f(x) x X x x x d
(c ij v ij +(c ij + d ij )w ij ), (i,j) A 1 i =0 (v ij + w ij ) (v ij + w ij )= 0 i =1,...,n 1 (i,j) (i) (i,j) (i) 1 i = n u i u j d ij x ij c ij, (i, j) A, u n u 0 + ( cij v ij +( c ij + d ) ij )w ij =0, (i,j) A v ij + x ij 1, (i, j) A, w ij x ij 0, (i, j) A, v ij,w ij 0, (i, j) A, x X. c ij d ij c ij d ij x ij (i, j) A u i i N v ij w ij c = c d = d c d R
x x x T T α x x n f x x f x >f x x x e Δf(x)/T Δf(x) =f(x ) f(x 25 60 0 24 R B =3 R =0 73
x x x x x t =1,.., m i =1,.., n x f(x ) f(x ) f(x ) f(x ) x x U(0, 1) e (f(x ) f(x ))/T x x x x f(x ) f(x ) x x T := α T (c ij,d ij : y ij ) (0, 9), (9, 10) (10, 19) 0 9 10 19 24 R =0 R =4 45 (0, 9), (2, 3) (10, 19) 0 1 2 3 4 21 24
(0, 9) (10, 19) R =0 R =6 38 (2, 3), (9, 10) (10, 19) 0 9 10 3 4 21 24 (2, 3) (10, 19) R =13 28 (0, 1), (0, 9) (0, 11) 0 9 10 19 24 (0, 1) (0, 11) R R R
5 (1, 15) 6 7 8 (2, 9) (3, 27) (4, 5) 20 (3, 26) (2, 16) (3, 19) (2, 30) (2, 31) (3, 8) (3, 27) (2, 14) (3, 25) (3, 17) 1 (4, 17) 2 3 4 (1, 32) (1, 9) (3, 24) 21 (2, 28) (2, 29) (4, 24) (2, 13) (2, 33) (3, 7) (3, 25) (1, 6) (2, 5) (2, 27) 0 (1, 19) 9 10 19 (2, 26) (3, 19) (3, 9) 24 (2, 31) (2, 21) (1, 24) (3, 32) (1, 27) (2, 20) (2, 10) (4, 33) (4, 30) (2, 12) 11 (1, 9) 12 13 14 (3, 16) (3, 8) (4, 21) 22 (3, 14) (1, 7) (3, 9) (3, 8) (2, 22) (4, 25) (1, 13) (1, 21) (3, 25) (1, 25) 15 (2, 25) 16 17 18 (4, 10) (3, 19) (3, 23) 23 25 60 (i, j) (c ij,d ij )
5 (1, 15) 6 7 8 (2, 9) (3, 27) (4, 5) 20 (3, 26) (2, 16) (3, 19) (2, 30) (2, 31) (3, 8) (3, 27) (2, 14) (3, 25) (3, 17) 1 (4, 17) 2 3 4 (1, 32) (1, 9) (3, 24) 21 (2, 28) (2, 29) (4, 24) (2, 13) (2, 33) (3, 7) (3, 25) (1, 6) (2, 5) (2, 27) 0 (1, 19) 9 10 19 (2, 26) (3, 19) (3, 9) 24 (2, 31) (2, 21) (1, 24) (3, 32) (1, 27) (2, 20) (2, 10) (4, 33) (4, 30) (2, 12) 11 (1, 9) 12 13 14 (3, 16) (3, 8) (4, 21) 22 (3, 14) (1, 7) (3, 9) (3, 8) (2, 22) (4, 25) (1, 13) (1, 21) (3, 25) (1, 25) 15 (2, 25) 16 17 18 (4, 10) (3, 19) (3, 23) 23 R =0 73 (i, j) (c ij,d ij )
5 (1, 15) 6 7 8 (2, 9) (3, 27) (4, 5) 20 (3, 26) (2, 16) (3, 19) (2, 30) (2, 31) (3, 8) (3, 27) (2, 14) (3, 25) (3, 17) 1 (4, 17) 2 3 4 (1, 32) (1, 9) (3, 24) 21 (2, 28) (2, 29) (4, 24) (2, 13) (2, 33) (3, 7) (3, 25) (1, 6) (2, 5) (2, 27) 0 (1, 19 : 3) 9 10 19 (2, 26) (3, 19 : 1) (3, 9) 24 (2, 31) (2, 21) (1, 24) (3, 32) (1, 27) (2, 20) (2, 10) (4, 33) (4, 30) (2, 12) 11 (1, 9) 12 13 14 (3, 16) (3, 8) (4, 21) 22 (3, 14) (1, 7) (3, 9) (3, 8) (2, 22) (4, 25) (1, 13) (1, 21) (3, 25) (1, 25) 15 (2, 25) 16 17 18 (4, 10) (3, 19) (3, 23) 23 R =4 45 (i, j) (c ij,d ij )
5 (1, 15) 6 7 8 (2, 9) (3, 27) (4, 5) 20 (3, 26) (2, 16) (3, 19) (2, 30) (2, 31) (3, 8) (3, 27) (2, 14) (3, 25) (3, 17) 1 (4, 17) 2 3 4 (1, 32 : 3) (1, 9) (3, 24) 21 (2, 28) (2, 29) (4, 24) (2, 13) (2, 33) (3, 7) (3, 25) (1, 6) (2, 5) (2, 27) 0 (1, 19) 9 10 19 (2, 26) (3, 19 : 3) (3, 9) 24 (2, 31) (2, 21) (1, 24) (3, 32) (1, 27) (2, 20) (2, 10) (4, 33) (4, 30) (2, 12) 11 (1, 9) 12 13 14 (3, 16) (3, 8) (4, 21) 22 (3, 14) (1, 7) (3, 9) (3, 8) (2, 22) (4, 25) (1, 13) (1, 21) (3, 25) (1, 25) 15 (2, 25) 16 17 18 (4, 10) (3, 19) (3, 23) 23 R =6 38 (i, j) (c ij,d ij )
5 (1, 15) 6 7 8 (2, 9) (3, 27) (4, 5) 20 (3, 26) (2, 16) (3, 19) (2, 30) (2, 31) (3, 8) (3, 27) (2, 14) (3, 25) (3, 17) 1 (4, 17) 2 3 4 (1, 32) (1, 9) (3, 24) 21 (2, 29 : 6) (2, 28) (4, 24) (2, 13) (2, 33) (3, 7) (3, 25) (1, 6) (2, 5) (2, 27) 0 9 10 19 24 (1, 19) (2, 26) (3, 19) (3, 9) (2, 21) (1, 24) (3, 32) (2, 31 : 7) (1, 27) (2, 20) (2, 10) (4, 33) (4, 30) (2, 12) 11 (1, 9) 12 13 14 (3, 16) (3, 8) (4, 21) 22 (3, 14) (1, 7) (3, 9) (3, 8) (2, 22) (4, 25) (1, 13) (1, 21) (3, 25) (1, 25) 15 (2, 25) 16 17 18 (4, 10) (3, 19) (3, 23) 23 R =13 28 (i, j) (c ij,d ij )
Δ
B R c d B R c d U (1, 25) U (1, 25) U (1, 20) U (0, 50) U (1, 15) U (1, 50) U (1, 10) U (1, 50) Δ Δ Δ m T α Δ m m n
1000 Δ m 50 100 100 = 0 m =50 50 > 0 α Δ 0.10 R 2 0.23
2.92 10 7 R 2 m R 2 α 7.9 10 16 T 9.3 10 20 1.5 10 7 R 2 m R 2 α 3.6 10 3 T 16 10 3 8.4 10 6 R 2 m R 2 α T 2 10 3 2.4 10 6 R 2 m R 2 α 6.9 10 3 T 9.1 10 4 0.88 Δ
m α T 1 0.01 m (0.01/0.01) 100 = 100 2 m (0.05/37) 100=0.14 0 1, 2, 3 4 m n α T
α
α
α
9 2, 3 4 α T m m T m n 1000/m n m n 1000 α α
α 1 92 90 2 379.6 375.4 3 361.4 334.8 4 216.4 207.2 ( )
0.074 361 1362 c 1 10 d 1 50 20 R 5 R R
R R 171 R =66 R R R R
R =1 417 R =5 212
R =15 198 R =66 171
361 1362 R
n k n