Siqian Shen. Department of Industrial and Operations Engineering University of Michigan, Ann Arbor, MI 48109,

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1 Page 1 of 38 Naval Research Logstcs Sngle-commodty Stochastc Network Desgn under Demand and Topologcal Uncertantes wth Insuffcent Data Accepted Artcle Sqan Shen Department of Industral and Operatons Engneerng Unversty of Mchgan, Ann Arbor, MI 48109, sqan@umch.edu; Mngd You Department of Industral and Operatons Engneerng Unversty of Mchgan, Ann Arbor, MI 48109; Ynta Ma Department of Industral Engneerng, Tsnghua Unversty, Bejng, Chna Abstract: Stochastc network desgn s fundamental to transportaton and logstc problems n practce, yet faces new modelng and computatonal challenges resulted from heterogeneous sources of uncertantes and ther unknown dstrbutons gven lmted data. In ths paper, we desgn arcs n a network to optmze the cost of sngle-commodty flows under random demand and arc dsruptons. We mnmze the network desgn cost plus cost assocated wth network performance under uncertanty evaluated by two schemes. The frst scheme restrcts demand and arc capactes n budgeted uncertanty sets and mnmzes the worst-case cost of supply generaton and network flows for any possble realzatons. The second scheme generates a fnte set of samples from statstcal nformaton (e.g., moments) of data and mnmzes the expected cost of supples and flows, for whch we bound the worst-case cost usng budgeted uncertanty sets. We develop cuttng-plane algorthms for solvng the mxed-nteger nonlnear programmng reformulatons of the problem under the two schemes. We compare the computatonal effcacy of dfferent approaches and analyze the results by testng dverse nstances of random and real-world networks. Keywords: Two-stage stochastc optmzaton; robust optmzaton; mxed-nteger lnear programmng (MILP); lnearzaton technques; cuttng-plane algorthms; vald nequaltes 1 Introducton Network desgn problems (NDPs) arse n many applcatons that nvolve servce desgn, constructon, and operatons. They are of vtal mportance for buldng and operatng complex systems of telecommuncaton, energy, transportaton, and logstcs n the modern socety. However, uncertantes assocated wth these systems may degrade ther performance, leadng to potental proft losses and servce-qualty drops. In ths paper, we focus on NDPs under both demand and topologcal uncertantes for operatng flows between supply and demand locatons. Fnte data observatons are known for the two uncertantes, but may not be suffcent for dervng exact dstrbutons. We use general statstcal nformaton of the gven data, ncludng bounds and moments, to construct robust or sem-robust models (explaned below n detal) for desgnng relable networks. We consder varants of the sngle-commodty NDP as follows. A desgner bulds arcs n a network where each arc has fxed constructon cost and capacty. The goal s to mnmze a weghted sum of the arc-constructon cost and a recourse cost ncurred after realzng the two uncertantes. We nvestgate two problem varants that use dfferent schemes for evaluatng the recourse cost, namely, the performance of a desgned network under uncertanty. The frst one apples robust optmzaton to handle the ssue of unknown dstrbutons, and mnmzes the worst-case cost of Ths s the author manuscrpt accepted for publcaton and has undergone full peer revew but has not been through the copyedtng, typesettng, pagnaton and proofreadng process, whch may lead to dfferences between ths verson and the Verson record. Please cte ths artcle as do: /nav Ths artcle s protected by copyrght. All rghts reserved.

2 Naval Research Logstcs Page 2 of 38 supply generatons and arc flows over two ndependent uncertanty sets of the demand and arc capactes. Ths treatment s relatvely conservatve and assumes that the network desgner does not have exact dstrbutonal knowledge but only lmted statstcal nformaton, e.g., bounds on the demand and arc capacty. We refer to the correspondng problem varant as Robust Network Desgn (RND). In the second approach, more data related to the uncertan demands and arc dsruptons can be obtaned by means of smulaton, hstorcal experences, forecastng, etc. From the data we can derve dstrbutons wth emprcal moments and generate a fnte set of samples to formulate a stochastc program, n whch we mnmze the desgn cost and the expected cost of supply generaton and flows. Ths model s also embedded wth a robust constrant that restrcts the worst-case flow cost to be no more than a gven threshold over the two budgeted uncertanty sets. We refer to ths varant as Sem-Robust Network Desgn (S-RND). For both RND and S-RND, we develop cuttng-plane algorthms to teratvely optmze ther mxed-nteger nonlnear programmng reformulatons. The man objectve s to develop effectve means for analyzng stochastc NDPs under nsuffcent data wth more than one uncertanty source. Va extensve computatonal studes based on both randomly generated and real-world networks, we show that S-RND nvolves lttle addtonal computatonal effort to drectly computng a stochastc optmzaton model, but can guarantee the desgn robustness wth hgh confdence. S-RND yelds lower cost of network desgn and operatons as compared to RND, when more nformaton of the uncertantes are avalable for generatng dscrete samples. When only the bounds of the uncertantes are known, RND provdes conservatve and robust solutons to guarantee hgh performance under extreme cases when S-RND and the stochastc optmzaton approach are not applcable. Accepted Artcle 1.1 Lterature Revew NDP studes can be found across a wde range of theoretcal and applcaton areas, due to the common NDP structure n many network plannng and operaton problems n practce. Startng from the study by Magnant and Wong [39], the lterature has tackled NDP wth sngle- and multcommodty flows [see, e.g., 29]. We refer to the thess by Poss [48] for a thorough summary of models and algorthms for varous NDPs. Meanwhle, other broader classes of NDP nclude jont locaton-nventory desgn [38], transportaton-nventory network desgn [54], faclty locaton desgn n supply chans [23], road network desgn [56], and servce network desgn [9]. To consder uncertantes, Lum et al. [37] provde a comprehensve analyss of demand uncertanty n stochastc NDP. Cu et al. [21] optmze faclty locaton desgn under random arc dsruptons, whle Mudchanatongsuk et al. [43] analyze both random transportaton cost and demand uncertanty. Moreover, metrcs other than the expectaton have been used to evaluate the network performance under uncertanty, ncludng network relablty (e.g., probablty of havng unmet demand n supply chans, and probablty of havng traffc losses n a transportaton system) [53, 45, 55], and multple objectves that balance the cost and rsk [18]. Many stochastc NDP studes assume fully known dstrbutons of the uncertanty, and one can formulate the correspondng stochastc programs wth fnte but large-scale samples. A common approach s the L-shaped method (.e., the Benders decomposton method) for dervng vald nequaltes and teratvely optmzng the large-scale samplng-based reformulatons [see, e.g., 47]. Fortz and Poss [24], Botton et al. [16] develop (mproved) Benders decomposton approaches for optmzng dfferent NDP models (wth multple layers and constraned hops, respectvely). Recently, Cranc et al. [19, 20] propose a scenaro decomposton algorthm for stochastc NDP and solve subproblems generated by progressve-hedgng heurstcs and scenaro groupng strateges. Indeed, solvng stochastc NDP models requres suffcently many scenaros to represent the underlyng uncertanty. These scenaros 2 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

3 Page 3 of 38 Naval Research Logstcs can be generated ether from true dstrbutons (e.g., the Sample Average Approxmaton (SAA) method [33]) or statstcal nformaton (e.g., moment-matchng method [31]). When the dstrbuton of the uncertanty s unknown and/or the goal s to plan aganst the worst case, robust optmzaton s the most popular to model NDP. Altn et al. [2], Koster et al. [34] descrbe a varety of formulatons, complextes, vald nequaltes, and computatonal results for RND varants, manly wth uncertan hose demand. Ukkusur et al. [59] present robust optmzaton models that construct arcs or plan arc capactes n transport networks under unknown but bounded demand values. Atamtürk and Zhang [7] propose a two-stage RND wth recourse flows under only uncertan demand. They show that the problem s NP-hard even for bpartte network desgn, and test lot-szng and locaton-transportaton nstances to demonstrate the results as compared to the ones by sngle-level robust optmzaton. Under arc capacty uncertanty Mnoux [42] show that the related RND s NP-hard n general. Álvarez-Mranda et al. [3] dscuss the complexty and heurstc results for sngle-commodty RND varants; Cacchan et al. [17] focus on the dervaton of optmal solutons to the sngle-commodty RND by usng the branch-and-cut algorthm. They consder uncertanty sets of the hose demand modeled as a fnte set of scenaros or as a polytope. Indeed, the RND problem can be modfed n a varety of ways, ncludng consderng dynamc routng decsons for some specfc applcatons such as n Matta [40] and Poss and Raack [49]. The resultng optmzng models for RND and ts varants are often two-stage mxed-nteger lnear programs, and can be optmzed through decomposton, cuttng-plane, and/or column-generaton methods [see, e.g., 36, 14, 8, 60]. The theores of stochastc/robust NDP has been substantally appled to optmze performance of a varety of network classes arsng n the applcatons of telecommuncaton, transportaton, and waster dstrbuton [4, 51, 26, 25]. In such contexts, a network desgner often faces mult-commodty flows rather than sngle-commodty flow optmzaton consdered n ths paper. Later we demonstrate n Remark 1 that our models can be easly modfed to accommodate the mult-commodty settng. However, the extended models requre more complex computatonal approaches. Furthermore, a sgnfcant number of NDP studes are closely related to the survvable network analyss [see 28, 32] as well as reslent network desgn [58]. The lterature has studed multcommodty survvable network optmzaton and focused on advancng soluton methods ncludng cuttng-plane algorthms [22] and polyhedra studes [57]. In partcular, Dahl and Stoer [22] optmze a survvable network desgn under the uncertanty that any one arc n the network may fal and reformulate the problem usng arc nstallaton varables, smlar to the feasblty cuts we ntroduce later n our soluton approaches. Atamtürk [5] and Atamtürk and Rajan [6] nvestgate cut-set and arc-set polyhedron, respectvely, to mprove the computaton of capactated NDP. Accepted Artcle 1.2 Contrbutons We take nto account the dstrbutonal ambguty of two sources of uncertantes, namely, demand and arc-dsrupton uncertantes nvolved n network desgn problems. We frst formulate a RND model, of whch the robust counterpart s a mxed-nteger nonlnear program. We then talor a semrobust stochastc optmzaton model by ntegratng dscrete samples and robust feasblty (.e., S-RND), whch agan has a nonlnear reformulaton. We optmze both mxed-nteger nonlnear programs by usng lnearzaton technques, cuttng-plane algorthms, and vald nequaltes to acheve fast computaton. Through extensve computatonal studes, we show that the soluton gven by S-RND s not as conservatve and costly as the one gven by RND f more nformaton about the uncertantes are gven other than just the lower and upper bounds so that we can derve dscrete samples used n S-RND. Meanwhle, comparng wth a pure stochastc optmzaton model based on the same set of lmted uncertanty samples, the S-RND model provdes a more relable 3 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

4 Naval Research Logstcs Page 4 of 38 desgn by only slghtly ncreasng the computatonal tme. 1.3 Structure of the Paper The remander of the paper s organzed as follows. Secton 2 descrbes a generc model for both RND and S-RND. Secton 3 and Secton 4 specfy the two models and develop cuttng-plane algorthms for each problem, respectvely. Secton 5 derves addtonal vald nequaltes based on specal structures of node degrees and arc capactes. Secton 6 tests the approaches for solvng RND and S-RND on dverse network nstances, descrbes computatonal results, and provdes soluton analyss. Secton 7 concludes the paper and states future research. Accepted Artcle 2 Problem Descrpton and Formulaton Overvew Let G(N, A) be a drected connected graph wth the node set N = N + N = N, where N +, N = and N respectvely represent sets of supply, transmsson, and demand nodes, satsfyng N + N = = N + N = N N = =. The set A N N ncludes all arcs that can be potentally constructed. Assocated wth each arc (, j) A are the constructon cost c j > 0, flow capacty a j > 0, and unt flow cost d j > 0. Let h be the unt cost of supply generaton, S 0 be the generaton capacty at node, N +, and D 0 be the random demand at node, N. Defne bnary varables x j, for all (, j) A, such that x j = 1 f we construct arc (, j), and 0 otherwse. For each (, j) A, defne varable f j 0 as the amount of flow on arc (, j), and for each supply node N +, defne varable g 0 as the amount of supply generated at node N. To model arc dsrupton, we ntroduce a random vector I = [I j, (, j) A] T [0, 1] A, bounded wthn an uncertanty set U I, wth each I j representng the remanng percentage of the capacty at arc (, j), (, j) A after some random dsruptons. The random demand vector D = [D, N ] T s bounded n an uncertanty set U D. We specfy the two uncertanty sets U I and U D as budgeted uncertanty sets [see, e.g., 12] descrbed n Secton 3.1. Denote x as the vector form of varables x j, (, j) A. Let V (x, D, I) be the mnmum flow cost gven arc constructon decson x, demand D, and dsrupton I. V (x, D, I) = mn f,g h g + d j f j (1a) N + (,j) A s.t. f j f j g = 0 N + (1b) j:(,j) A j:(,j) A j:(,j) A f j f j j:(j,) A j:(j,) A j:(j,) A f j = D N (1c) f j = 0 N = (1d) 0 g S N + (1e) 0 f j a j I j x j (, j) A. (1f) In the objectve (1a), N + h g + (,j) A d jf j s the sum of flow cost and supply generaton cost on the remanng network after dsrupton. Constrants (1b) (1d) model the flow balance at nodes n N +, N and N =, respectvely. Constrants (1e) bound varables g from above by S, N +. Constrants (1f) allow flow f j beng postve when x j = 1 and I j > 0, meanng that arc (, j) has been constructed and the remanng arc capacty after dsrupton s postve. 4 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

5 Page 5 of 38 Naval Research Logstcs We consder a two-stage structure for both RND and S-RND problems as follows. At the frst stage, we buld arc capactes,.e., decde the values of x j, (, j) A, before realzng uncertan demand and arc dsruptons. Both decson vectors f and g are recourse decsons at the second stage, and we formulate the second-stage objectve by measurng V (x, D, I) for gven frst-stage decson vector x, and parameters D and I. The goal s to mnmze a weghted sum of cost objectves from both stages. A generc form of the two problem varants reads: Accepted Artcle mn x X ρ (,j) A c j x j + (1 ρ)y, (2) where X {0, 1} A represents a feasble regon, consstng of constrants to ensure that x satsfes supply and demand at all nodes, gven any possble arc dsrupton and uncertan demand. Parameter ρ [0, 1] s a weght related to the desgner s tradeoff preference between the frst-stage constructon cost (,j) A c jx j and the second-stage recourse cost y R +. We calculate y based on two crtera for evaluatng the random cost V (x, D, I). Frst, we follow a robust optmzaton scheme and mnmze the cost of network flows and supply generatons n the worst case (.e., y = max I UI,D U D V (x, D, I)). For a gven x, V (x, D, I) n (1) may be nfeasble for some realzed I or D. Denote R x as the feasble regon of arcs constructed at the frst stage to be robust wth respect to all possble arc dsrupton and demand,.e. gven x R x, Problem (1) s always feasble for any I U I and D U D. We desgn a robust network by solvng RND: ρ c j x j + (1 ρ) max mn x R x (,j) A I U I,D U D V (x, D, I). (3) Alternatvely, a stochastc programmng scheme [cf. 15] consders y = E ξ [V (x, D ξ, I ξ )] as the expected cost of supply generatons and flows, gven random parameters D ξ and I ξ,.e., we can optmze = mn x ρ (,j) A c jx j +(1 ρ)e ξ [V (x, D ξ, I ξ )]. However, when samplng a large number of scenaros, the computaton s generally neffcent. Also, a fully known dstrbuton of ξ may not be avalable due to varous ssues such as the dffculty of data collecton n hghly uncertan envronments. Here we explore an alternatve approach as follows. We generate a fnte set of scenaros, from statstcal nformaton of the uncertantes D ξ and I ξ that one can be derved from gven data. We optmze a two-stage stochastc program formulated by usng the scenaros, n whch we also requre certan level of soluton robustness wth respect to uncertanty sets of demands and arc dsruptons deduced from exstng data. Let (I 1, D 1 ), (I 2, D 2 ),..., (I N, D N ) be an ndependently and dentcally dstrbuted (..d) samples of the random vectors I, D. The correspondng sample average functon s 1 N N ω=1 V (x, Dω, I ω ). As the sample average functon cannot reflect all possble values of I and D, we mpose the followng two types of soluton robustness: Frst, soluton x should be feasble for all realzatons from the uncertanty sets U I and U D. Second, the maxmum possble V (x, D, I) for all I U I, D U D s bounded by an upper bound L. We desgn a sem-robust network by solvng ρ mn x S x (,j) A where feasble regon S x s descrbed as S x = { x R x c j x j + (1 ρ)e ξ [V (x, D ξ, I ξ )], (4) } max V (x, D, I) L I U I,D U D. (5) We consder S-RND as the followng scenaro-based formulaton: S-RND: ρ N c j x j + (1 ρ) V (x, D ω, I ω )/N, (6) mn x S x (,j) A ω=1 5 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

6 Naval Research Logstcs Page 6 of 38 Let V denote the optmal objectve value of Problem (4), and X ɛ denote ɛ-optmal soluton set of Problem (4),.e., f x X ɛ then ρ (,j) A c j x j + (1 ρ)e ξ [V ( x, D ξ, I ξ )] V + ɛ. Smlarly, we defne ˆV N and ˆX N ɛ to be the optmal objectve value of Problem (6) and ɛ-optmal soluton set of Problem (6), respectvely. Accepted Artcle Proposton 1. [33] () As N, ˆV N V wth probablty 1; () for any ɛ 0, the event { ˆX ɛ N X ɛ } happens wth probablty 1 as N, and the probablty of the event ˆX ɛ N X ɛ approaches 1 exponentally fast as N. Proof. The feasble regon of Problem (4) and Problem (6) are fnte because x {0, 1} A. Accordng to the defnton of R x, for any D U D, I U I and x R x, V (x, D, I) has fnte values. Thus, E ξ [V (x, D ξ, I ξ )] < for any x R x. Therefore, the two propertes drectly follow from Proposton 2.1 and Proposton 2.2 n Kleywegt et al. [33]. Proposton 1 guarantees asymptotcal convergence by usng the scenaro-based formulaton to approxmate the true soluton of S-RND. 3 Models and Algorthms of RND For RND, we solve a relaxed master problem at the frst stage: MP1 : mn ρ c j x j + (1 ρ)y (,j) A (7a) s.t. L 1 (x) 0 (7b) L 2 (y, x) 0 (7c) x j {0, 1} (, j) A, y 0, (7d) where y equals to max I UI,D U D V (x, D, I), x R x. Set L 1 (x) 0 conssts of vald feasblty cuts, and set L 2 (y, x) 0 conssts of vald optmalty cuts. We descrbe the methodologcal detals of dervng these two types of cuts n Secton 3.2 and Secton 3.3, respectvely. 3.1 Budgeted Uncertanty Sets For demand uncertanty, let D, N be a random varable wth mean D, lower bound D D, and upper bound D + D,.e., D [ D D, D + D ], N. (Assume that D D 0, N.) We formulate a budget constrant N π D Π, where Π and π, N are fxed parameter. The budgeted demand uncertanty set s U D = D R N + π D Π, D [ D D, D + D ], N. (8) N For arc dsrupton uncertanty, let Γ be a gven parameter that can be vewed as the maxmum sum of percentage of faled arcs allowed n any realzatons. The budgeted uncertanty set of arc dsrupton s U I = I [0, 1] A I j A Γ. (9) (,j) A 6 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

7 Page 7 of 38 Naval Research Logstcs 3.2 A Separaton Problem and Vald Inequaltes for Defnng R x We specfy feasblty cuts n L 1 (x) 0 that provde an exact descrpton of R x. Theorem 1. Gven bnary solutons x j, (, j) A, decson vector x R x f and only f Accepted Artcle N + Ñ S N Ñ D + (,j) A φ + (Ñ ) a j x j I j 0, I U I, D U D, Ñ N. (10) where for any Ñ N, set φ+ (Ñ ) ncludes all the ncomng arcs of the nodes n Ñ. Ths result drectly follows the Gale-Hoffman nequaltes [27, 30], for whch we provde a proof usng Farkas Lemma n Appendx A. The proof does not requre specal types of uncertanty sets U D and U I, and the result can be generalzed to any uncertanty sets rather than the budgeted uncertanty consdered n ths paper. One can change parameters Π of U D and Γ of U I to enlarge or strengthen the feasble regon x R x defned by nequaltes (10). Specfcally, larger Π- and Γ-values wll enlarge sets U D and U I and thus requre the soluton x satsfyng (10) to be more robust. We can also use dfferent U D, U I n (10) and n max I UI,D U D V (x, D, I) by changng ther correspondng Π and Γ. Constrant (10) seeks feasble x wth respect to any possble realzatons of uncertan D and I, and max I UI,D U D V (x, D, I) further examnes the worst-case objectve value under certan uncertanty sets that can be dfferent from the ones used to enforce x R x. One can also vary the upper and lower bounds of D for some N n set U D to reflect dfferent levels of feasblty strctness. Accordng to Theorem 1, gven x j {0, 1}, (, j) A, we formulate a separaton problem to check whether x j belongs to R x,.e., whether x satsfes (10) for any D U D, I U I, and Ñ N. The separaton problem s gven by SP R : R(x) = mn v,w,i,d S v D v + a j x j I j w j (11a) N + N (,j) A s.t. v j v w j (, j) A (11b) v {0, 1} N, w j {0, 1} (, j) A (11c) π D Π (11d) N D D D D + D N (11e) I j A Γ (11f) (,j) A 0 I j 1 (, j) A (11g) where R(x) s the mnmum objectve value of SP R gven any soluton x. In SP R, v = 1 f and only f Ñ N. Due to (11b), w j s forced to be 1 f arc (, j) φ + (Ñ ). Ths s because a jx j I j 0 and we mnmze (11a), then optmal solutons w j to SP R wll dentfy a cut of φ + (Ñ ) and solutons v correspond to a subset Ñ N. Constrants (11d), (11e), (11f), (11g) defne sets U I and U D. Gven x, we optmze SP R by solvng Problem (11). Lettng v = 0, N, w j = 0, (, j) A, we can trvally satsfy all constrants and obtan a feasble objectve value 0. Therefore, R(x) must be 0 or less. If R(x) = 0, (10) holds for any D U D, I U I, and Ñ N, whch verfes x R x. Otherwse f R(x) < 0, then there exst D U D and I U I, such that V (x, D, I ) has no feasble solutons g 0, N + and f j 0, (, j) A. We then generate a feasblty cut nto L 1 (x) 0 to MP1 to exclude such a soluton x. 7 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

8 Naval Research Logstcs Page 8 of 38 We further lnearze D v and I j w j n SP R usng McCormck Inequaltes [41]. Lettng D v b, N and I j w j p j, (, j) A, we reformulate SP R as an equvalent mxed-nteger lnear programmng (MILP) model: Accepted Artcle SP MILP R : mn s.t. N + S v N b + (,j) A a j x j p j (11b), (11c), (11d), (11e), (11f), (11g) (12a) I j + w j 1 p j 0 (, j) A (12b) p j 0 (, j) A (12c) b ( D + D )v 0 N (12d) b D 0 N, (12e) where McCormck nequaltes p j w j, p j I j, and p j 1, for all (, j) A are satsfed automatcally due to the mnmzaton objectve, and thus are omtted n (12). To see that SP MILP R s equvalent to SP R n (11): f w j = 1, snce we mnmze (,j) A a jx j p j and a j x j 0, then p j = I j due to (12b); f v = 0, due to the mnmzaton of N b, we have b = 0 accordng to (12d); f v = 1, then b = D due to (12e). If x / R x, then there exsts an optmal soluton (ˆv, ˆb, ˆp j ) to Problem SP MILP R n (12) such that S ˆv ˆb + a j x j ˆp j < 0. Correspondngly, we generate a feasblty cut to L 1 (x) 0 N + N (,j) A as: S ˆv ˆb + a j ˆp j x j 0. (13) N + N Theorem 2. Cut (13) s vald for all x R x. (,j) A Proof. Note that (ˆv, ˆb, ˆp j ) s a feasble soluton to (12). If x R x, then the correspondng objectve value N + S ˆv N ˆb + (,j) A a j ˆp j x j must be greater or equal to 0. Ths shows the valdty of Cut (13). 3.3 Optmalty Cuts We now derve optmalty cuts L 2 (y, x) 0 by consderng the dual of V (x, D, I), formulated as max D µ + S ν + a j x j I j γ j (14a) N N + (,j) A s.t. µ + ν h N + (14b) µ µ j + γ j d j (, j) A (14c) ν 0 N + (14d) γ j 0 (, j) A (14e) where dual varables µ, ν 0, γ j 0 are assocated wth constrants (1b) (1d), (1e), and (1f), respectvely. Gven that the dual s a maxmzaton problem, we have max V (x, D, I) = max {(14a) : (14b) (14e), (11d) (11g)} (15) I U I,D U D whch s a blnear program. Unlke SP R, all the dual varables n model (15) are not necessarly bnary, and thus we cannot drectly replace the blnear terms D µ and I j γ j wth addtonal varables and lnear constrants. Next, we explore characterstcs of optmal solutons to model (15), and reformulate (15) as an MILP model. 8 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

9 Page 9 of 38 Naval Research Logstcs Proposton 2. When Γ s nteger, all extreme ponts of U I are bnary valued. Proof. For any feasble I U I gven nteger parameter Γ, f t contans a fractonal component, then there are not enough lnearly ndependent constrants tght at ths soluton and thus t cannot correspond to an extreme pont. Accepted Artcle Proposton 3. When Γ s nteger, there exsts an optmal soluton to (15) that has all I j, (, j) A bnary valued. Proof. For a blnear program, a global optmum can be found among pars of extreme ponts from respectve feasble regons [e.g., 44]. Thus, at least one extreme pont of U I, whch has all I j, (, j) A bnary valued accordng to Proposton 2, optmzes (15). Gven the above result, we can generalze all the approaches n ths paper to handle the varant wth complete 0-1 arc closure,.e., I {0, 1} A. For nteger Γ, the lnearzaton steps reman the same due to the fact that the optmal solutons wll always resolve at bnary realzatons of I. Next, we lnearze D µ accordng to the followng propertes of any optmal soluton. Proposton 4. When N π ( D + D ) Π, there exsts an optmal D U D to the mxednteger blnear programmng model (15) satsfyng the followng two propertes: () all D, N ether equals to D D or D + D except for at most one D ; () N π D = Π. Proof. Smlar to Proposton 3, at least one extreme pont of U D optmzes the blnear program. Consderng extreme ponts, or equvalently, the basc feasble solutons of U D, t s easy to see that at least N 1 nequaltes of D D D D + D, N have zero slacks, showng that at least N 1 of D, N ether equal to D D or D + D. To see N π D = Π, we verfy the result by contradcton: Suppose that D, N are optmal to (15) wth N π D < Π, then for any D < D + D, we can ncrease ts value untl ether t reaches the upper bound, or N π D = Π. Such a modfcaton keeps the soluton feasble wthout decreasng the objectve value untl N π D = Π. Ths completes our proof. We now propose an MILP reformulaton of (15) under the assumpton that parameters Γ and Π n the budgeted uncertanty sets are both ntegral. Accordng to Proposton 3 and Proposton 4, model (15) can be solved by computng N subproblems, each of whch sets one D, N n between ts upper bound ( D + D ) and lower bound ( D D ). Therefore, n the k th subproblem, we desgnate D k = (Π ) N \{k} π D /π k wth other D varables ether beng D D or D + D, k. We add superscrpt (k) to all varables n model (15), and present the k th subproblem as ( max D (k) µ (k) + π ) µ (k) π k Π µ (k) k π k + S ν (k) + a j x j I (k) j γ(k) j (16a) k N \{k} N + (,j) A s.t. (14b) (14e),(11f) D (k) { D D, D + D } N \{k} (16b) I (k) j {0, 1} (, j) A, (16c) where (16b) ndcate that D (k) bnary varables z (k) ether equals to D D or D + D, N \{k}. We then defne = 0 f D (k) = D D, and z (k) = 1 f D (k) = D + D, {0, 1}, such that z (k) 9 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

10 Naval Research Logstcs Page 10 of 38 N \{k}. Wth both varables z (k) each k N and obtan Accepted Artcle max s.t. N \{k} α (k) α (k) Π π k µ (k) k and I (k) j (14b) (14e),(11f),(16c) ( D + D ) ( µ (k) α (k) z (k) ( D D ) ( µ (k) beng bnary, we can lnearze subproblem (16) for + S ν (k) + N + + π ) µ (k) π k k ) + π µ (k) π k k (,j) A a j x j σ (k) j ( ) + M 1 1 z (k) N \{k} (17a) (17b) + M 1 z (k) N \{k} (17c) {0, 1} ( N \{k} (17d) ) γ (k) j + M 2 1 I (k) j (, j) A (17e) σ (k) j σ (k) j 0 (, j) A (17f) To see the equvalence of (17) and (16), n (17b) and (17c), M 1 s arbtrarly large and the objectve maxmzes ( N \{k} α(k). Thus, ether α (k) equals to D + D ) ( ) µ (k) + π µ (k) k /π k f z (k) = 1, ndcatng that D (k) = D + D (, or α (k) equals to D D ) ( ) µ (k) + π µ (k) k /π k f z (k) = 0, ndcatng that D (k) = D D. Smlarly, n (17e) and (17f), because a j x j 0, γ (k) j 0 and we maxmze (,j) A a jx j σ (k) j, f I(k) j = 1, then σ (k) j constrants (17e) are relaxed gven an arbtrary large M 2, and σ (k) j equals to γ (k) j ; otherwse f I(k) j = 0, equals to 0 due to (17f). Dervaton of M 1 and M 2 : To tghten the MILP formulaton (17), we derve lower bounds of M 1 and M 2, whch need to guarantee the valdty of ther related constrants n (17). From (17b) and (17c), M 1 must satsfy ( D + D ) ( µ (k) + π ) ( µ (k) π k + M 1 D D ) ( µ (k) + π ) µ (k) k π k, N \{k} (18a) k ( D + D ) ( µ (k) + π ) ( µ (k) π k D D ) ( µ (k) + π ) µ (k) k π k + M 1, N \{k} (18b) k Thus, n the k th subproblem, a lower bound to M 1 s For M 2, t must satsfy M 1 Thus, a lower bound to M 2 s max N \{k} {±2 D ( µ (k) + ( µ (k) k π ) )} /π k. (19) γ (k) j + M 2 0 (, j) A (20) { M 2 max (,j) A γ (k) j } k N. (21) Note that both lower bounds n (19) and (21) nvolve dual solutons to (17) and thus cannot be obtaned a pror to solvng model (17). We descrbe an teratve approach usng (19) and (21) to compute vald M 1 and M 2 as follows. We start wth suffcently large M 1 and M 2 n (17) 10 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

11 Page 11 of 38 Naval Research Logstcs to obtan optmal dual solutons µ (k) and γ (k), usng whch we update the values of M 1 and M 2 based on (19) and (21), respectvely. We re-compute (17) for possbly new optmal solutons of µ (k) and γ (k) usng the new M 1 and M 2. We repeat the process untl the mprovements of M 1 and M 2 are suffcently small. Such an teratve approach for updatng bg-m coeffcents n MILP models has been dscussed and mplemented by, e.g., Qu et al. [50], who teratvely strengthen bg- M coeffcents for MILP models wth 0-1 Knapsack structures, to reformulate chance-constraned lnear programs wth fnte samples. In our later computatonal studes the values of M 1 and M 2 do not sgnfcantly affect the soluton tme and we use M 1 = M 2 = 1000 n all our nstances. We repeatedly solve Model (17) for every k N. Suppose that the ˆk th subproblem yelds the maxmum objectve value among all subproblems, and soluton (α (ˆk) Accepted Artcle the orgnal Problem (15). As a result, y N \{ˆk} α (ˆk) Π πˆk ˆµ (ˆk) ˆk + S ν (ˆk) + N + (,j) A, µ (ˆk) whch we refer to as an optmalty cut to be generated nto set L 2 (y, x) 0. Theorem 3. Cut (22) s vald to MP1, for all x R x., ν (ˆk), σ (ˆk) j ) s optmal to a j σ (ˆk) j x j, (22) Proof. If x R x, then V (x, D, I) s feasble for any I U I, D U D. Recall model (15) n whch varables x j, (, j) A only exst n the objectve functon. The rght-hand sde of (22) provdes a feasble objectve of max I UI,D U D V (x, D, I) for any x R x. Ths completes the proof. 3.4 Cuttng-Plane Algorthms Algorthm 1 demonstrates the detals of a cuttng-plane algorthm for solvng the RND problem n (3). At each teraton, the algorthm solves a relaxed MP1 to obtan a soluton ˆx and a lower bound of the second-stage recourse cost y, and then solves the separaton problem SP MILP R. If R(x) < 0, a feasblty cut (13) s generated nto L 1 (x) 0, and we re-solve MP1. Otherwse, we calculate an upper bound of y by computng max I UI,D U D V (ˆx, D, I). If there exsts a postve gap between the current best upper and lower bounds that s greater than ɛ, we generate an optmalty cut (22) nto set L 2 (y, x) 0, and solve MP1 agan. Recall that n MP1, none of the constrants (7b) (7d) reflect flow balance n the network, ndcatng that solutons gven by MP1 cannot be guaranteed to satsfy flow balance at every node. (We later n Secton 6.3 demonstrate that many feasblty cuts wll be added nto MP1 before very few optmalty cuts are generated.) Here we consder an alternatve master problem formulaton, to mprove the qualty of x computed by MP1 at the frst stage. Alt-MP1 : mn ρ c j x j + (1 ρ)y (23a) s.t. (,j) A (1b) (1f) y N + h g + L 1 (x) 0 L 2 (y, x) 0 (,j) A d j f j (23b) (23c) (23d) x j {0, 1} (, j) A, y 0, (23e) where (1b) (1f) and (23b) ensure that soluton x gven by Alt-MP1 must satsfy flow balance at all the nodes and y equals to the correspondng V (x, D, I) for gven D and I. It follows that for 11 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

12 Naval Research Logstcs Page 12 of 38 Algorthm 1 Cuttng-plane algorthm for solvng the RND problem n (3) Input: An nstance of problem (3). Output: An ɛ-optmal soluton to problem (3), or no feasble soluton exts. Step 0: Set x j = 1, (, j) A. Solve SP MILP R n (12). f R(x) = 0 then go to Step 1. else the nstance s not feasble; ext the algorthm. end f Step 1: Solve MP1 n (7) to obtan the current optmal solutons x and ȳ. Accepted Artcle Step 2: Fx x = x, and solve SP MILP R n (12). f R(x) = 0 then go to Step 3. else generate feasblty cut (13) nto MP1, and go to Step 1. end f Step 3: Solve subproblem (17) for every k N, and choose soluton ŷ that yelds the maxmum objectve value of all subproblems. f (ŷ ȳ) ɛŷ then x s an ɛ-optmal and return the optmal objectve ρ (,j) A c j x j + (1 ρ)ȳ else generate optmalty cut (22) nto MP1, and go to Step 1. end f any I U I, D U D, mn ρ c j x j + (1 ρ)v (x, D, I ) mn ρ c j x j + (1 ρ) max V (x, D, I). (24) x R x x R x I U I,D U D (,j) A (,j) A Together wth cuts n L 1 (x) 0 and L 2 (y, x) 0, Inequalty (24) shows that the optmal objectve value of Alt-MP1 yelds a lower bound to RND. Meanwhle, gven feasble x, we obtan an upper bound to the optmal objectve value. At each teraton, we fx I and D, whch are computed based on subproblems (12) and (17). We develop an mproved Algorthm 1, named Alt-Algorthm 1 by revsng Steps 0, 1, 2, and 3 n Algorthm 1. The key s to always record the optmal values of D and I after solvng subproblem SP MILP R and then re-solve Alt-MP1 n (23) by fxng the recorded values of D and I. Appendx B presents a full descrpton of Alt-Algorthm 1. 4 Models and Algorthms for Optmzng S-RND In ths secton, we descrbe soluton methods for S-RND n (6), whch s a two-stage stochastc program wth an addtonal robust constrant (5). We contnue usng the budgeted uncertanty sets U I and U D n (8) and (9), respectvely. 4.1 Vald Feasblty and Optmalty Cuts We develop a cuttng-plane algorthm for Model (6) that reuses feasblty cuts (13) and also uses generalzed optmalty cuts (22). The followng formulaton MP2 descrbes the master problem of 12 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

13 Page 13 of 38 Naval Research Logstcs S-RND: Accepted Artcle MP2 : mn ρ (,j) A c j x j + (1 ρ) N η ω /N ω=1 (25a) s.t. L 1 (x) 0 (25b) L 2 (η ω, x) 0, ω = 1, 2,..., N x j {0, 1}, (, j) A, η ω 0, ω = 1, 2,..., N (25c) (25d) where η ω provdes a lower bound to V (x, D ω, I ω ), ω = 1, 2,..., N for any x. Sets L 1 (x) 0 and L 2 (η ω, x) 0 respectvely correspond to feasblty and optmalty cuts to be generated from subproblems. To derve feasblty cuts n the set L 1 (x) 0 and enforce x S x, frst note that cut (13) s stll vald to MP2 because S x R x. Also, we requre max I UI,D U D V (x, D, I) L for any x S x. Through the same method of optmzng max I UI,D U D V (x, D, I), we cut off all solutons x R x that wll make max I UI,D U D V (x, D, I) > L. Suppose that (ˆk, ˆα (ˆk) soluton to max I UI,D U D V (x, D, I). We requre L N \{ˆk} ˆα (ˆk) Π πˆk ˆµ (ˆk) ˆk + S ˆν (ˆk) + N + (,j) A, ˆµ (ˆk) ˆk, ˆν(ˆk), ˆσ (ˆk) j ) s an optmal a j ˆσ (ˆk) j x j. (26) The valdty of cut (26) can be verfed gven that the rght-hand sde of (26) s feasble to V (x, D, I) for any x S x, whch s then bounded by L. Cut (26) and cut (13) consttute L 1 (x) 0 n MP2. To derve optmalty cuts n the set L 2 (η ω, x) 0, gven x and parameter D ω, I ω at the ω th subproblem to MP2, we formulate the dual of V (x, D ω, I ω ) as: η ω = max D ω µ ω + S ν ω + a j x j Ijγ ω j ω : (14b) (14e) (27) N N + (,j) A Therefore, the cuts n L 2 (η ω, x) 0, ω = 1, 2,..., N can be obtaned by the standard Benders procedures gven lnear (14b) (14e). Followng the weak dualty, η ω D ω ˆµ ω + S ˆν ω + a j x j Ijˆγ ω j ω (28) N N + (,j) A where ˆµ ω, ˆνω and ˆγ j ω are optmal dual solutons to Model (27). Here we utlze the dervaton of optmalty cuts n RND to buld feasblty cuts n S-RND. Later n our computaton, we also use the optmal objectve value of RND to desgn the threshold value for boundng the worst-case flow cost n S-RND. Although the two models share smlar soluton methods, we pont out that they are desgned for dfferent data avalablty cases S-RND requres knowng fnte samples of the uncertantes to compute the expectaton-based objectve value, whle RND only needs demand upper and lower bounds as well as parameters Π and Γ n sets U I and U D. In data-scarce envronment, t s more approprate to use RND although we computatonally show later that S-RND yelds smlar robust solutons as RND but lower cost, when nformaton about the uncertanty become avalable. 4.2 Cuttng-Plane Algorthm Proposton 1 states that for suffcently large N, the sample average functon N ω=1 V (x, Dω, I ω )/N can provde a reasonable statstcal estmate for E ξ [V (x, D ξ, I ξ )]. Followng smlar setups as the 13 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

14 Naval Research Logstcs Page 14 of 38 SAA method, we solve M..d samples, each wth N realzatons of the random (I, D). Algorthm 2 demonstrates the steps of solvng Model (6) for each sample m, m = 1,..., M. For every sample m, Algorthm 2 Cuttng-plane algorthm for solvng problem (6) n sample m. Input: An nstance of problem (6) wth (I ω,m, D ω,m ), ω = 1, 2,..., N beng the realzatons of (I, D) n sample m. Accepted Artcle Step 0: Set x j = 1, (, j) A. Solve SP MILP R n (12) and subproblem (17) for every k N. f the optmal objectve value of (12) = 0 and the maxmum of the optmal objectve value of each (17) L then go to Step 1. else report that no feasble soluton to ths nstance, and ext. end f Step 1: Solve MP2 n (25) to obtan solutons x and η ω, ω = 1, 2,..., N. Step 2: Fx x = x, and solve SP MILP R n (12). f the optmal objectve value = 0 then go to Step 3. else generate feasblty cut (13) nto set L 1 (x) 0, and go to Step 1. end f Step 3: Solve subproblem (17) for every k N, and choose the soluton wth the maxmum optmal objectve value ŷ. f ŷ L then go to Step 4. else generate feasblty cut (26) nto set L 1 (x) 0, and go to Step 1. end f Step 4: Solve subproblem (27) for ω = 1, 2,..., N. Calculate N ω=1 ηω /N. f N ω=1 ηω /N N ω=1 ηω /N > ɛ N ω=1 ηω /N then generate cut (28) nto set L 2 (η ω, x) 0, ω = 1, 2,..., N, and go to Step 1. else return x m = x and V m = ρ (,j) A c j x j + (1 ρ) N ω=1 ηω /N. end f we obtan an ɛ-optmal soluton x m and ts correspondng optmal objectve value V m. Moreover, V = M m=1 V m /M provdes a statstcal estmate for a lower bound to the optmal objectve of S-RND. To obtan an upper bound, we choose some x m computed by Algorthm 2 for any m = 1,..., M. We generate a reference sample wth realzatons (I ω, D ω ), ω = 1, 2,..., N and N N for post-optmzaton smulaton. The value ˆV = ρ (,j) A c j x m j + (1 ρ) V (x m, D ω, I ω )/N N ω=1 provdes a statstcal estmate for an upper bound to the optmal objectve of S-RND because x m S x. The value ˆV V s the optmalty gap. Remark 1. The proposed approaches for RND and S-RND can be generalzed to tackle NDP wth mult-commodty flows, where the flow varables, supply generatons, and demand parameters are 14 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

15 Page 15 of 38 Naval Research Logstcs all ndexed by commodty n the objectve (1a) and constrants (1b) (1f). As a result, we also ndex the correspondng dual varables and formulate dual constrants wth respect to each commodty to proceed wth the dverse cuttng-plane algorthms. Note that the overall sum of flows of multple commodtes s upper bounded by a j I j x j on each arc (, j) n (1f). Ths prevents us solvng the scenaro-based subproblems n each cuttng-plane scheme va further decomposton by commodty. Thus, the proposed approaches could be neffcent when computng mult-commodty problems under two uncertantes and t s desrable to develop more effectve algorthms by combnng the Benders decomposton (.e., row generaton) wth column generaton [see 11, 1] for handlng the couplng capacty constrant (1f). We leave ths for future research and focus on sngle-commodty NDP n ths paper. Accepted Artcle 5 Vald Inequaltes We propose addtonal vald nequaltes that can be ntegrated n MP1, Alt-MP1 and MP2 to respectvely mprove the performance of Algorthm 1, Alt-Algorthm 1 and Algorthm 2. Compared wth the generc feasblty cuts such as (13) resulted from verfyng (10) for all node subsets Ñ N, of whch the number can be exponental, the vald nequaltes proposed n ths secton are based on graph topologes (e.g., node degrees after buldng arcs) and arc flow capactes requred by any feasble soluton to RND or to S-RND. 5.1 Degree-Based Vald Inequaltes Recall that the set φ + (N ) contans all the ncomng arcs of the nodes n a node set N. We also denote φ (N ) as the set of outgong arcs of the nodes n set N. Any feasble bnary soluton x j, (, j) A satsfes x j 1 N (29a) j:(j,) A (,j) φ (N + ) (j,) φ + (N ) x j Γ + 1 x j Γ + 1 (29b) (29c) x j + x j 1 (, j) A. (29d) Inequalty (29a) ndcates that the number of ncomng arcs to each demand ste s no less than one, so that we can satsfy postve demand at each node N. Inequaltes (29b) (29c) show that Γ + 1 sets a lower bound for the number of arcs needed to be constructed. That s, because the number of arc dsruptons s up to Γ, the number of arcs constructed from set N + to set N needs to be at least Γ + 1 n both RND and S-RND for worst-case flows. Assumng postve arc constructon and flow costs, we buld no more than one arc between any par of nodes and thus (29d) s vald. Smlarly, consder all transmsson nodes n N = wth zero supply/demand. Defne a bnary varable t j, j N = to ndcate whether node j s part of a constructed path from N + to N or s not connected wth any constructed arc n soluton x. For each j N =, we propose x j 1 t j, (, j) A, x j 1 t j, (j, ) A (30a) t j + x j 1, t j + x j 1, (30b) :(,j) A :(j,) A 15 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

16 Naval Research Logstcs Page 16 of 38 where nequaltes (30a) ndcate that we do not construct any ncomng or outgong arcs for node j f t j = 1. In such a case, nequaltes (30b) are satsfed. If t j = 0, because node j s a transmsson node, we have to construct both ncomng and outgong arcs, enforced by nequaltes (30b). The above cuts ensure that we ether do not buld any arcs for a transmsson node or buld both ncomng and outgong arcs at the same tme. Accepted Artcle 5.2 Capacty-Based Vald Inequaltes Consder any feasble bnary soluton x j, (, j) A and vald nequaltes a j x j D j D j j N (31a) :(,j) A (,j) φ (N + ) a j x j N ( D D ). (31b) Inequaltes (31a) guarantee that the total capactes of the ncomng arcs of each demand node are not fewer than the mnmum possble demand. Inequalty (31b) guarantees that the total outgong capactes from all the supply nodes n N + meet the mnmum of the total demand generated from the nodes n N. Moreover, we buld vald nequaltes by usng the max-flow mn-cut theorem [1]. We frst dentfy the mnmum cut of graph G from node set N + to node set N wth all arcs n A beng constructed. Denote A mn A as the mnmum-cut set wth any arc (, j) A mn havng node N + and node j N. For any feasble soluton x j, (, j) A, denote f(x) as the sum of the flow on arcs bult from N + to N. The followng relatonshp holds: ( D j D j ) f(x) a j x j a j x j (32) j N (,j) A: N +, j N (,j) A mn For the frst nequalty from left to rght of (32), the feasblty of soluton x guarantees the total amount of flow from N + to N s no less than the total amount of demand on the left-hand sde. Ths value s bounded by the total capactes of the arcs bult from N + to N by soluton x, and further bounded by the maxmum flow (equvalent to the capacty of the mnmum cut) from N + to N f we construct all arcs n A. We therefore propose the followng nequalty: ( D D ) a j x j (33) N (,j) A mn We develop a smlar type of nequaltes by dervng modfed copes of graph G, denoted by G k for each k N, where we keep the same network structure and topologes, but modfy the demand values such that we only consder a sngleton demand node k for each G k. We then fnd the mnmum cut set from N + to node k, denoted by A k mn, for all k N. The followng set of nequaltes can be derved n lght of the dervaton of (33). D k D k a j x j k N. (34) (,j) A k mn An Example: Fgure 1 llustrates an example where the mnmum cut sets A mn, A k mn, k N are dfferent, and so are the related vald nequaltes (33), (34). Node A s supply mean value s 16, Node B and Node C s demand mean values are 10 and 6, respectvely. Flow capactes are shown along the sde of each arc. 16 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

17 Page 17 of 38 Naval Research Logstcs!"#$ - Accepted Artcle :;<=3>,8=<9?3=@9 9;=9% A464@B@9?8=<9CB;D?3=@9 9;=9% * + / 0 1 / %&"'( #, 0 )&#( (a) Subnetwork G B wth N B = {B}!"#$ :;<=3>,8=<9?3=@9 9;=9) A464@B@9?8=<9CB;D?3=@9 9;=9) * + / 0 1 / %&"'( #, 0 )&#( (b) Subnetwork G C wth N C = {C}!"#$ :;<=3>,8=<9?3=@9 9;=9%A) B464@C@9?8=<9DC;E?3=@9 9;=9%A) * + / 0 1 / %&"'( #, 0 )&#( (c) Network G wth N = {B, C} Fgure 1: An example for demonstratng vald nequaltes (33) and (34). In Fgure 1(a) and Fgure 1(b), we examne graphs G B and G C for nodes B, C N, respectvely. Ther correspondng A B mn and AC mn are {(D, B), (F, B)} and {(E, C), (F, C)}, respectvely. We specfy the correspondng nequaltes (34) as D B D B 4x DB + 6x F B, DC D C 4x EC + 3x F C. For graph G wth N = {B, C}, the correspondng A mn = {(A, E), (D, B), (D, F )} hghlghted n Fgure 1(c). The nequalty (33) s: 6 Computatonal Results ( D B D B ) + ( D C D C ) 9x AE + 4x DB + 3x DF. In ths secton, we demonstrate the computatonal effcacy of vald nequaltes developed n Secton 5 by applyng them to randomly generated network nstances. Then we mplement Algorthm 1, Alt- Algorthm 1 to solve RND and use Algorthm 2 to solve S-RND based on nstances generated based on real-world networks. We compute ten replcatons of each nstance and report the average results unless specfed otherwse. All the experments are performed by usng CPLEX wth C++ language on Workstaton wth Intel(R) Xeon CPU X GHz and 6GB memory. Note that the relaxed master problems MP1, Alt-MP1, and MP2 are MILP models and we use branch-andbound to optmze them n each teraton. When mplementng the cuttng-plane algorthms, we add feasblty/optmalty cuts usng callback functons n CPLEX for both nteger and fractonal temporary solutons. We generate volated cuts at each node n the branch-and-bound tree for solvng the master problem as long as any exsts. 1 The optmalty gap tolerance s 0.01% followng the default settng n CPLEX. We set the threshold for dentfyng volated cuts as 10 4 and use one computatonal thread. 6.1 Expermental Setup and Parameter Desgn We generate test nstances from the followng network structures. We consder three sets of random network nstances, named RG1, RG2, and RG3 wth 5, 6, and 7 nodes, respectvely, n whch 1 Such an mplementaton has been dscussed and shown very effectve for mplementng the Benders decomposton algorthm, e.g., at 17 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.

18 Naval Research Logstcs Page 18 of 38 we randomly select pars of nodes to form the potental arcs n A. We also use real-world networks ABILENE, POLSKA, NOBEL-US from the onlne Survvable Network Desgn Lbrary [46], and the Soux-Falls road network from the Transportaton Test Problems [see 10, 35]. Table 1 provdes the total number of nodes n N and the total number of arcs n A n each type of random or real-world network we test. Accepted Artcle Table 1: Graph sze of each test nstance Network RG1 RG2 RG3 ABILENE POLSKA NOBEL-US Soux-Falls ( N, A ) (5,7) (6,16) (7,20) (12,30) (12,36) (14,42) (24, 76) We follow a Bernoull tral and randomly make each node as a supply, demand, or transmsson node. The average number of supply and demand nodes s about 15%-20% of the total number of nodes n all network nstances. We set the values of constructon cost (c), flow cost (d), generaton cost (h), arc capacty (a), and supply capacty (S) by unformly generatng nteger numbers from the correspondng ntervals gven n Table 2. Table 2: Parameter settngs Parameter c d h a S Interval [8, 12] [1, 3] [3, 5] [20, 60] [20, 60] The parameters for defnng the uncertanty sets U I and U D represent network desgners rsk preference and can be determned va, e.g., cross valdaton results. In ths paper, for the uncertanty set U I n (9), we consder Γ = 2 for RG1 and Γ = 3 for all the other networks; for the uncertanty set U D n (8), we generate the values of D D and D + D from ntervals [1, 5] and [11, 15], respectvely, for each demand node N. We also set π = 1, N. To determne the value of Π, D, N are sampled unformly between D D and D + D, and Π = max s Ω N D s where Ω s the set of all the samples, and D s presents the demand realzaton of node n sample s. Table 3 provdes the Π-values for all the networks. We use M 1 = M 2 = 1000 whch are verfed suffcently large by our results. Table 3: Values of Π for each network Instance RG1 RG2 RG3 ABILENE POLSKA NOBEL-US Soux-Falls Π For RND, we assume that only the parameters for defnng sets U I and U D are gven; we later generate data samples (ncludng realzatons of D and I) based on the desgned demand ntervals, demand dstrbuton type, and parameter Γ for S-RND and the stochastc optmzaton approach. 6.2 Effcacy of Vald Inequaltes Added to Algorthm 1 We frst test Algorthm 1 for solvng RND wth or wthout addng the vald nequaltes proposed n Secton 5. Tables 4 reports the results of testng RND on RG1, RG2, and RG3 nstances. We compute each nstance wth weght parameter ρ = 1, 0.75, 0.25 and 0. Columns t total, t fea and t opt report the average CPU seconds of solvng replcatons of each nstance, of dervng each feasblty cut (13) and of dervng each optmalty cut (22), respectvely. Columns Fea.Cut and Opt.Cut report the number of feasblty cut (13) and optmalty cut (22) beng added nto MP1 when Algorthm 1 termnates. Column Opt.Obj reports the optmal objectve value of MP1. 18 John Wley & Sons Ths artcle s protected by copyrght. All rghts reserved.