Online Supplement for The Robust Network Loading Problem under Hose Demand Uncertainty: Formulation, Polyhedral Analysis, and Computations


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1 Onine Suppement for The Robust Network Loading Probem under Hose Demand Uncertaint: Formuation, Pohedra Anasis, and Computations Aşegü Atın Department of Industria Engineering, TOBB Universit of Economics and Technoog, Sögütözü Ankara, Turke Hande Yaman, Mustafa Ç. Pınar Department of Industria Engineering, Bikent Universit, Ankara, Turke For i W and e E, et u e i be a a unit vector of dimension W E where the entr corresponding to e and i is 1 and other entries are zero. For e E and L, et ve be a unit vector of dimension E L where the entr corresponding to e and is one and other entries are zero. Let M be a ver arge integer. Proof of Proposition 3.1 The dimension of P and P is at most the number of variabes, i.e., ( W + L ) E. Suppose that a points in P (resp. P ) satisf αλ + β = γ. Let e E, L and (λ, ) P (resp. (λ, ) P ). Observe that the vector (λ, + v e) is aso in P (resp. in P ). Hence βe = 0. Let i W and e E. Consider the vectors (λ, ) where λ = i W e E ue i ue i and = L e E Mv and (λ + e ue i, ). Since these two vectors are in P (resp. in P ), we have αi e = 0. As there is no equait that is satisfied b a points in P (resp. in P ), pohedron P (resp. P ) is fudimensiona, i.e., its dimension is ( W + L ) E. Next we give two emmas which are often used in the seque. Lemma 1. Let F (resp. F ) be a projection of F (resp. of F ) into the subspace of a subset of variabes and et αλ + β γ be a facet defining inequait for P (resp. for P ). If the coefficients of the variabes that are projected out are zero in the inequait αλ + β γ, then the inequait aso defines a facet of conv(f ) (resp. of conv(f )). Proof. Foows from Coroar 3.6 in Baas and Oosten (1998). Lemma 2. Suppose that a points in P (resp. in P ) such that αλ + β = γ aso satisf αλ + β = γ. Let e E and L be such that βe = 0. Then β e = 0. 1
2 Proof. Let (λ, ) P (resp. in P ) such that αλ + β = γ and = + v e. The vector (λ, ) is aso in P (resp. in P ) and satisfies αλ + β = γ since β e = 0. Hence β e = 0. Proof of Proposition 3.2 B Lemma 1, if inequait σλ σ 0 is facet defining for P (resp. for P ) then it is facet defining for F λ (resp. for F λ ). Suppose that inequait σλ σ 0 is facet defining for F λ (resp. for F λ ) and a points in P (resp. in P ) such that σλ = σ 0 aso satisf αλ + β = γ. Then b Lemma 2, β = 0. As σλ σ 0 is facet defining for F λ (resp. for F λ ) and F λ = P roj λ (F ) (resp. F λ = P roj λ(f )), (α, γ) is a mutipe of (σ, σ 0 ). Thus inequait σλ σ 0 is facet defining for P (resp. for P ). Proof of Theorem 3.1 Let e E be such that δ(s) \ {e} for ever S V such that there exists q Q with s(q) S and t(q) V \ S. It foows from Lemma 1 that if inequait αλ e + β e γ is facet defining for P (resp. for P ) then it defines a facet of P e. Suppose that inequait αλ e + β e γ is facet defining for P e (resp. for P e). Suppose aso that a points in P (resp. in P ) such that αλ e + β e = γ aso satisf αλ + β = γ. B Lemma 2, β e = 0 for a e E \ {e} and L. Let i W and e E \ {e}. Let (λ e, e ) be a vector in P e (resp. in P e) such that αλ e + β e = γ. Consider (λ, ) where λ = i W λe i u e i + i W e E\{e} ue i u e and i = L e E\{e} Mv + e L eve. Since δ(s) \ {e} = for ever S V such that q Q with s(q) S and t(q) V \ S, this vector is in P (resp. in P ) and satisfies αλ e + β e = γ. Consider aso (ˆλ, ŷ) where ˆλ = λ + u e i and ŷ =. This vector is aso in P (resp. in P ) and satisfies αˆλ e + βŷ e = γ. Hence α e i Now we can concude that αλ + β = γ is a mutipe of αλ e + β e = γ as F e = P roj (λ e, e )(F ) (resp. F e = P roj (λ e, e )(F )). Proof of Proposition 3.3 Suppose that S V such that the subgraphs induced b S and V \ S are both connected and B(S) > 0. Let δ(s) F (S) and assume that B(S) = b(s). The proof for the other case is simiar. Define (ˆλ, ŷ) as foows: ˆλ e i = 1 for e E \ δ(s) and i W, ˆλ e i = 0 for e δ(s) and i W \ S, ŷ = L e δ(s) eve + L e E\δ(S) Mv e. Notice that since the subgraphs induced b S and V \ S are both connected, for S V 2
3 such that S S, S V \ S, and δ(s ) \ δ(s). Hence, if (ˆλ, ŷ) satisfies ˆλ e i = 1 i S W (1) e δ(s) ˆλe i L C ŷ e e δ(s) (2) ˆλ e i 0 i S W, e δ(s) (3) then (ˆλ, ŷ) is in F and F. B Farkas Lemma, this sstem is feasibe if and on if α i + C ŷeβ e 0 (4) for a (α, β) such that e δ(s) L α i + β e 0 i S W, e δ(s) β e 0 e δ(s). Since L C ŷe 0 for a e δ(s), we can imit ourseves to (α, β) such that β e = ( ) + α max i for a e δ(s). Now, we can rewrite the eft hand side of (4) as α i + ( ) + C ŷe αi max. (5) e δ(s) L ( ) + α If max i = 0, then αi 0 for a i S W, and (5) is nonnegative. Suppose now that max ( α i ) + = α i Now as e δ(s) > 0 for some i S W. Then (5) is α i α i L C ŷ e b(s) and α i α i α i b(s) = e δ(s) L C ŷe. (6) > 0, (6) is greater than or equa to (α i α i ). As α i α i for a i S W, b (α i α i b b i) 0. So the sstem (1)(3) awas has a i i soution and F (S) P roj δ(s) (F ) and F (S) P roj δ(s) (F ). As L e δ(s) C e B(S) is a vaid inequait for F and F, P roj δ(s) (F ) F (S) and P roj δ(s) (F ) F (S). Proof of Theorem 3.2 Let S V be such that the subgraphs induced b S and V \ S are 3
4 both connected and suppose that inequait L e δ(s) β ee β 0 is facet defining for P (S) and for each e δ(s) there exists a vector δ(s) F (S) such that L e δ(s) β ee = β 0 and L C > B(S). Assume that B(S) = b(s). The proof for the other case is simiar. e Suppose that a points in P such that L e δ(s) β e e = β 0 aso satisf αλ + β = γ. B Lemma 2, β e = 0 for a e E \ δ(s) and L. Define λ δ(s) S W to be the restriction of the vector λ to its entries with edges in δ(s) and vertices in S W. Let δ(s) F (S) be such that L e δ(s) β ee = β 0 and λδ(s) S W be a soution to the sstem (1)(3) with δ(s). Let i W and e E \ δ(s). Let λ = i W e E\δ(S) ue i u e + i e δ(s) λe i u e i and = e δ(s) eve + e E\δ(S) Mv e. The vector (λ, ) is in P and satisfies L e δ(s) β e e = β 0. Consider aso (ˆλ, ) where ˆλ = λ + u e. This atter vector is aso in P and satisfies i L e δ(s) β e e = β 0. Hence α e = 0. i Let i W, e δ(s) and δ(s) F (S) be such that L e δ(s) β ee = β 0 and L C > b(s). Let λs W e δ(s) be a soution to the sstem (1)(3) with δ(s). Consider the vector (λ, ) where λ = i W e E\δ(S) ui e + e δ(s) λi eu i e and = e δ(s) eve + e E\δ(S) Mv e. This vector is in P and satisfies L e δ(s) β e e = β 0. Now et ɛ > 0 be ver sma and consider aso the vector (ˆλ, ) where ˆλ = λ + ɛu e. This vector is aso in P i since L C > b(s) and satisfies e L e δ(s) β e e = β 0. Thus α e = 0. i As F (S) = P roj δ(s) (F ) and as L e δ(s) β ee β 0 is facet defining for P (S), αλ + β = γ is a mutipe of L e δ(s) β ee = β 0. Proof of Proposition 3.5 Suppose that S V such that the subgraphs induced b S and V \ S are both connected, L such that R (S) > 0 and C 1 = 1. The cutset inequait (23) is facet defining for P (S) (Yaman 2007). To show that it aso defines a facet of P, we need to prove, for each e δ(s), the existence of a vector δ(s) F (S) which satisfies the inequait (23) with equait and L C e > B(S). For edge e δ(s), consider the vector δ(s) = ve. This vector satisfies (23) with equait, and we have L C e > B(S) B(S) C since R (S) > 0. Proof of Proposition 3.6 Inequait i S λ e i L C e is a vaid inequait for P. We substitute λ e i = 1 λ e i for i S and obtain b(s) L C e + i S λ e i. Let L and divide the ast inequait with C. This ieds b(s) C L C C e + i S C λe i. Let 4
5 L u = { L : g(, ) > r (S)} and L d = L \ L u. The ast inequait impies b(s) C C C e + g(, ) C e + C C e + λ e C i. L d L d L u i S Here C L d C e + C L u C e is integer and g(, ) L d C e + i S C λe i is a nonnegative rea. Hence the MIR inequait g(, ) C e + ( b(s) ) λ e C i r (S) Ld C e Lu C C C C C e L d i S is vaid for P. Mutiping both sides with C and organizing the terms, we obtain ( ) C g(, ) + r (S) C e + C r (S) C e + λ e b(s) i r (S) C L u i S L d which is the same as ( C r (S) + min { g(, ), r (S) }) C e + i S L λ e b(s) i r (S) C. Substituting λ e i = 1 λ e i for i S ieds inequait (24). Experimenta Resuts Beow, we provide detaied resuts of our experimenta tests. The entries in the tabes contain the foowing information: the instance characteristics, i.e., the name of the instance as we as the numbers of nodes V, inks E, and terminas W (the number of commodities is W ( W 1)), faciit capacities, i.e., C for the singe faciit case, C 1 and C 2 for the twofaciit case, the best upper bound z cp, the gap at termination g cp, the soution time t cp in CPU seconds, and the number of B&C nodes # cp for Cpex, the best upper bound z B&C, the gap at termination g B&C, the soution time t B&C in CPU seconds, and the number of B&C nodes # B&C for our B&C agorithm, where gap at termination is the gap between the best upper and ower bounds at termination defined as (upper ower)/ower 100 for each method. Given the difficut of the probem, it is not surprising that some of the instances are not soved to optimait within two hours 5
6 time imit. We indicate such cases with a in the corresponding z coumn. Moreover, the gaps at termination (g) for them are provided under the corresponding time (t) coumns in parenthesis. Besides, we mark those cases for which soving the NLP hose GD or NLP hose modes did not ied even a feasibe soution within the time imit with NoI whereas MA indicates a termination due to insufficient memor aocation. The gap at termination for such cases are noted as INF in a resut tabes. Instance ( V, E, W,C) z cp t cp (g cp ) # cp z B&C t B&C (g B&C ) # B&C metro (11,42,5,24) nsf1b (14,21,10,24) 97, , atcep1 (15,22,6,24) 51, , pacbe (15,21,7,24) 11, , bhv6c (27,39,15,24) 840, , 251 (1.22%) 37,702 bhvdc (29,36,13,24) 1.1e e pdh (11,34,6,480) 1.1e e poska (12,18,12,155) 44, 253 (1.16%) 10,793 44, 287 (0.42%) 46,133 poska (12,18,12,100) ,719 dfn (11,47,11,155) 52, 380 (6.34%) , 416 (3.74%) 6624 nework (16,49,16,1000) 1.499e + 6 (56.74%) e + 6 (45.4%) 21 france (25,45,14,2500) 17, 400 (3.61%) , 400 (5.45%) 304 ncep2 (16,49,9,24) (3.8%) (2.52%) 7998 atanta (15,22,15,1000) 4.7e + 8 (0.2%) e + 8 (0.54%) 19,135 tai (24,51,19,504k) 1.6e + 12 (99.7%) e + 7 (19.94%) 13 janos (26,42,26,64) 1.29e + 9 (99.7%) e nobeeu (28,41,28,20) 1.47e + 10 (99.7%) e + 6 (2.01%) 8 sun (27,51,24,40) 6.3e + 10 (99.7%) (20.5%) 6 Tabe 1: Resuts for the singe faciit probem. In Tabe 1, we provide the resuts for the singe faciit case. In Tabe 2, for each of the four settings, F, F &D, F &R, and a, we report the percentage gap between the optima vaue and the ower bound at the root node under the gap coumns as we as the corresponding soution times under the t coumns. Fina, in Tabe 3, we provide the detaied resuts for the twofaciit case. References Baas, E., Oosten, M On the dimension of the projected pohedra. Discrete App. Math
7 Instance t F gap F t F +D gap F +D t F +R gap F +R t a gap a metro % % % % nsf1b % % % % atcep % % % % pacbe % % % % bhvdc % % % % pdh % % % % Tabe 2: Resuts with different cuts for the singe faciit case. Instance ( V, E, W, C 1, C 2 ) z cp t cp # cp z B&C t B&C # B&C metro (11, 42, 5, 1, 24) nsf1b (14, 21, 10, 1, 24) 96, , atcep1 (15, 22, 6, 1, 24) 50, , pacbe (15, 21, 7, 1, 24) 11, , bhv6c (27, 39, 15, 1, 24) 826, ,548* (0.5%) 36,108 bhvdc (29, 36, 13, 1, 24) 1.09e e pdh (11, 34, 6, 30, 480) 8.18e e poska (12, 18, 12, 155, 622) 34,006* (2.18%) ,348* (3.11%) 50,403 dfn (11, 47, 11, 155, 622) 44,352* (21.37%) ,316* (12.43%) 21,736 nework (16, 49, 16, 1k, 4k) NoI MAINF 2 1.6e+6* (70.4%) 30 atanta (15, 22, 15, 1k, 4k) 1.376e+8* (1.15%) e+8* (1.44%) 14,598 tai (24, 51, 19, 504k, 1008k) NoI INF e+7* (10.48%) 8702 nobeeu (28, 41, 28, 20, 40) NoI INF e+6* (1.6%) 690 pioro (40, 89, 40, 155, 622) NoI INF e+6 (2.47%) 2 norwa (27, 51, 27, 1k, 4k) NoI INF 2 NoI INF 1 cost266 (37, 57, 37, 7560, 30240) NoI INF e+7* (29.75%) 2 gui39 (39, 86, 39, 160, 320) NoI INF 2 NoI INF 1 Tabe 3: Resuts for the two faciit probem. 7
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