We consider the network loading problem (NLP) under a polyhedral uncertainty description of traffic

Transcription

1 INFORMS Journa on omputing Vo. 23, No. 1, Winter 211, pp issn eissn informs doi /ijoc INFORMS The Robust Network Loading Probem Under Hose Demand Uncertainty: Formuation, Poyhedra Anaysis, and omputations Ayşegü Atın Department of Industria Engineering, TOBB University of Economics and Technoogy, 656 Sögütözü, Ankara, Turkey, Hande Yaman, Mustafa Ç. Pınar Department of Industria Engineering, Bikent University, 68 Bikent, Ankara, Turkey We consider the network oading probem (NLP) under a poyhedra uncertainty description of traffic demands. After giving a compact muticommodity fow formuation of the probem, we state a decomposition property obtained from projecting out the fow variabes. This property consideraby simpifies the resuting poyhedra anaysis and computations by doing away with metric inequaities. Then we focus on a specific choice of the uncertainty description, caed the hose mode, which specifies aggregate traffic upper bounds for seected endpoints of the network. We study the poyhedra aspects of the NLP under hose demand uncertainty and use the resuts as the basis of an efficient branch-and-cut agorithm. The resuts of extensive computationa experiments on we-known network design instances are reported. Key words: network oading probem; poyhedra demand uncertainty; hose mode; robust optimization; poyhedra anaysis; branch and cut History: Accepted by S. Raghavan, Area Editor for Teecommunications and Eectronic ommerce; received February 28; revised November 28, May 29, October 29; accepted December 29. Pubished onine in Artices in Advance March 23, Introduction onsider the probem of deciding the optima (i.e., resuting in the east tota instaation cost) number of devices of unit capacity to be instaed on the inks of the simpe network in Figure 1(a) to support the communication demands between the nodes. The number on each edge gives the instaation cost of a unit capacity device on that edge. Each pairwise demand is cited with its source and destination; i.e., AB is the demand from A to B, whereas BA is the demand in the reverse direction. Suppose that a communication demands except AD, DA, AE, and EA are forecasted to be one unit of traffic fow. The aforementioned four pairs are not expected to exchange any traffic, and hence these demands are zero. Suppose that we seek a design where ink capacities are sufficient to accommodate the tota fow on each ink in both directions and we aow mutipath routing. Then, an optima capacity instaation is given in Figure 1(b) with a tota cost of 13. Now suppose that the communication demands are reaized to be different than expected, namey, AD, AE, BD, and BE are one unit more than forecasted, whereas AB, BA, DE, and ED are one unit fewer than forecasted. As a resut, the current capacity of ink D woud not be sufficient to route a traffic requests simutaneousy. In teecommunications networks, such a deficiency causes a deay whose consequences become more severe as the deviation from expectations and the strategic vaue of the data traffic increase. In this paper, we discuss the design of networks that can support changing communication patterns in the east costy manner. More precisey, we study the robust network oading probem (NLP) under a poyhedra uncertainty definition of possibe traffic demands. The traditiona NLP assumes that pairwise demands are known. The purpose is to determine the east costy aocation of discrete units of capacitated faciities on the inks of the given network. In this work, we do not assume that demands are known a priori, but we consider a poyhedra definition of feasibe demands. Our motivation for this study is to design networks robust to fuctuations in demand estimates, which are amost sure to happen in reaife appications. Hence, we want our east-cost design to remain operationa for any feasibe reaization in a prescribed poyhedra set. It is we accepted that data are aways subject to some uncertainty in rea-ife probems. On some occasions researchers competey ignore uncertainty and 75

2 76 INFORMS Journa on omputing 23(1), pp , 211 INFORMS (a) Initia network (b) Minimum cost design for the deterministic demand A A (.5) 4 (.5) 8 (2) (1.5) (.5) 8 B (.5) 6 E D E D Figure 1 Exampe of Network apacity Loading use nomina vaues to represent the expected average behavior of the system. On the other hand, stochastic programming (SP) has been widey used to dea with uncertainty. SP yieds decisions that might become infeasibe with some probabiity, but in some cases, such a toerance is not favorabe, and robust optimization (RO) is more usefu because it aims to make the best decision that remains operationa for any reaization of data within a prescribed uncertainty set. An overview of some topics in the RO domain is given by Ben-Ta and Nemirovski (28). In RO, one decides on an uncertainty set, which defines a ikey data reaizations for which one is wiing to be prepared, without making any assumption on the stochastic mode of the data. Then, a robust design is the one whose worst performance over is the best. There are various ways of defining the uncertainty set : a set of finite/infinite number of scenarios, finite intervas, or a poyhedra or an eipsoida set (see, e.g., Atamtürk 26; Atamtürk and Zhang 27; Ben-Ta and Nemirovski 1998, 1999, 28; Ben-Ta et a. 24; Bertsimas and Sim 23, 24; Mudchanatongsuk et a. 28; Ordoñez and Zhao 27; Yaman et a. 27). An important uncertain component in network design probems is the traffic matrix, i.e., the demand between origin destination pairs. In practice, it is not ikey for network designers to have a precise estimate of the traffic matrix, and ignoring this uncertainty may ead to a faiure to meet service-eve agreements. To overcome this obstace, Duffied et a. (1999) and Fingerhut et a. (1997) independenty proposed a fexibe mode (hose mode) that specifies aggregate traffic upper bounds for seected endpoints of the network. Since then, the hose mode has gained significant popuarity because of its ease of specification (Fingerhut et a. 1997) as we as the resource-sharing fexibiity and mutipexing gains it provides (Duffied et a. 1999). The hose mode is initiay used to design virtua private networks (VPNs). Among these efforts, Gupta et a. (21), Itaiano et a. (22), Grandoni et a. (28), and Goya et a. (28) address the computationa compexity of the resuting combinatoria optimization probems; Goya et a. (28) prove that B the VPN design probem with fractiona ink capacities and singe-path routing of symmetric traffic matrices can be soved in poynomia time. Simiary, Gupta et a. (23), Kumar et a. (21), and Swamy and Kumar (22) deveop approximation agorithms for the probem with different hose definitions. In the same vein, Ben-Ameur and Kerivin (25) discuss the poyhedra mode, where the feasibe demand reaizations are defined by an arbitrary poyhedron. They deveop an iterative agorithm based on enumerating the vertices of the demand poyhedron so as to determine robust minimum-cost spittabe routing and edge capacity configurations. Later, Atın et a. (27) propose a compact mixed-integer programming mode for VPN design with continuous capacity expansion under unspittabe routing aong with a branch-andprice-and-cut agorithm. Their mode considers a traffic matrices simutaneousy. On the other hand, the growth in the size and appication types in IP networks has inspired severa works in this domain as we (Beotti and Pınar 28, Atın et a. 21). The number of different faciity types avaiabe for instaation, the use of different cost functions with fow costs, and technica restrictions on the routing of demands give rise to variants of the deterministic NLP (Atamtürk and Rajan 22; Avea et a. 27; Berger et a. 2; Bienstock and Günük 1996; Bienstock et a. 1998; Günük 1999; Brockmüer et a. 24; Magnanti and Mirchandani 1993; Magnanti et a. 1993, 1995; Mirchandani 2; Rardin and Wosey 1993; van Hoese et a. 22). The capacity expansion probem (EP), where the decision is to determine a capacity expansion pan for a given network, is aso cosey reated with NLP (Atamtürk and Günük 27, Atamtürk and Rajan 22, Berger et a. 2, Bienstock and Günük 1996, Günük 1999). Because NLP is strongy NP-hard, there have been various efforts for soving it as efficienty as possibe through the use of aternative formuations and heuristics, and by a thorough poyhedra anaysis (Magnanti and Mirchandani 1993, Magnanti et a. 1993, van Hoese et a. 22, Atamtürk and Günük 27). The most common approach in the iterature to hande NLP efficienty is to define some strong vaid inequaities to strengthen the inear programming reaxations. Projection of the feasibe set onto the space of discrete design variabes has aso been a common point of interest (Atamtürk and Rajan 22; Avea et a. 27; Bienstock et a. 1998; Bienstock and Günük 1996; Magnanti and Mirchandani 1993; Magnanti et a. 1993, 1995; Mirchandani 2; Rardin and Wosey 1993). Because the demand between each origin destination pair can be considered as a singe commodity, NLP is of a muticommodity fow nature. Athough

3 INFORMS Journa on omputing 23(1), pp , 211 INFORMS 77 the probem for singe-commodity fow with two faciity types is very we studied, and the poyhedra of feasibe fows is fuy characterized (Mirchandani 2), the muticommodity fow version remains hard, and metric inequaities are used to define the projection of the corresponding poyhedron on the space of discrete design variabes (Onaga and Kakusho 1971). Against this background, the main contribution of this paper to the existing body of iterature on singe-stage robust NLP is to reax the assumption of known traffic demands prior to designing the network. Whereas NLP with known (deterministic) demands is we studied, the iterature on robust NLP is rather imited. For the singe-stage robust NLP under poyhedra uncertainty, we are not aware of any other attempt with the exception of an earier reference by Karaşan et a. (25), where uncertainty was incorporated into the design of fiber optic networks with an emphasis on modeing rather than on a detaied poyhedra anaysis and branch and cut. On the other hand, Atamtürk and Zhang (27) study the two-stage robust NLP, where the capacity is reserved on network inks before observing the demands and the routing decision is made afterwards in the second stage. Furthermore, Mudchanatongsuk et a. (28) study an approximation to the robust EP with recourse, where the routing of demands (recourse variabes) is imited to a inear function of demand uncertainty. Our formuation for NLP with poyhedra uncertainty is interesting because we avoid using metric inequaities because of a decomposition property obtained from a projection on the design components. A simiar projection is used in Mirchandani (2) for deterministic singe- and muticommodity NLP, where a extreme rays of the reated projection cone for the singe-commodity case were characterized. However, ony necessary conditions were obtained for the deterministic muticommodity variant. The atter probem is difficut because the couping bunde constraints prevent the decomposition of the probem into singe-commodity subprobems. However, we bypass that difficuty by observing that we can decompose the projection probem into many smaer singe-commodity probems for which the resuts of Mirchandani (2) remain vaid. This observation consideraby simpifies the formuations, but the probem sti remains difficut and requires intensive efforts for deveoping an efficient soution agorithm. onsequenty, it opens the way to a thorough poyhedra anaysis based on which we deveop a branch-and-cut agorithm aong with a simpe but effective heuristic, and we use it to sove severa we-known network design instances. Studies on the poyhedra properties of deterministic NLP are mosty imited to the case of at most three faciity types where the capacity of a faciity is an integer mutipe of the capacity of the smaer faciity. Atamtürk (22) gives vaid inequaities for the deterministic probem with genera capacity moduarities and an arbitrary number of faciities. More recenty, Raack et a. (21) derive a genera definition of fowcutset inequaities as mixed-integer rounding inequaities for deterministic NLP with directed, bidirected, and undirected networks. They aso consider arbitrary capacity structures for mutipe faciities, where they study the facia structure of the cutset poyhedra and its reation to the deterministic NLP. The second main contribution of this paper is that we present vaid inequaities for robust NLP with an arbitrary number of faciities and arbitrary capacity structures. The rest of this paper is organized as foows. In 2 we describe our probem and give a compact mixedinteger programming formuation and its projection onto the space of design variabes. We move on to the hose mode in 2.2 and carry out a thorough poyhedra anaysis for NLP under hose uncertainty in 3. Then we continue with separation agorithms for various vaid inequaities and heuristics, a incorporated into a branch-and-cut agorithm in 4. We give a summary of our computationa resuts in 5 and concude in 6 with some directions for future work. 2. Probem Definition The deterministic NLP is defined as foows. Let G = V E be an undirected graph where V is the set of nodes and E is the set of edges. Let Q denote the set of commodities, i.e., the set of origin destination pairs with traffic demand. The origin of commodity q Q is s q and its destination is t q. A set of faciity aternatives with different capacities and costs can be used to carry fow through the network. The probem is to determine the number of faciities instaed on the edges such that a demand can be routed and the instaation cost is minimized. Then NLP can be modeed as min s.t. h k E p hk y hk (1) 1 h = s q f q hk f q kh = 1 h = t q k h k E otherwise, h V q Q (2) f q hk + f q kh d q y hk h k E (3) q Q y hk and integer h k E L (4) f q hk fq kh h k E q Q (5) where d q is the forecasted demand for commodity q Q, L is the set of faciity aternatives, p hk is the cost

4 78 INFORMS Journa on omputing 23(1), pp , 211 INFORMS of instaing one faciity of type L on edge h k E, and is the transmission capacity of type L faciity. Variabes of the mode are y hk for the number of type L faciities oaded on the edge h k E and f q hk for the fraction of d q routed on the edge h k E in the direction from h to k. onstraints (2) are the usua fow conservation constraints for each demand pair at each node. Finay, the constraints (3) are the edge capacity constraints, which ensure that the tota capacity instaed on each edge is enough to support the tota fow on it in both directions Robust Network Loading Probem with Poyhedra Demands Demand forecasts may not be precise and the reaized demand is very ikey to be different from what is expected. Our aim is to design a network that is viabe for any demand reaization in the poyhedra set D = d Q Ad d, where A m Q and m. We assume that D is bounded and nonempty. This eads to the foowing poyhedra NLP mode NLP POL : min h k E p hk y hk s.t f q hk + f q kh d q y hk h k E (6) max d D q Q Unike the deterministic case, NLP POL is a semiinfinite optimization mode as a resut of the infinite number of inequaities we need to consider over the demand poyhedron for each edge h k E. However, foowing the method commony used in robust optimization (see, e.g., Atın et a. 27, Ben-Ta and Nemirovski 1999, Bertsimas and Sim 23), we can give a compact inear mixed-integer programming (MIP) formuation for NLP POL.InNLP POL, for a given fow vector f and an edge h k E, the worst-case capacity requirement can be found by soving max f q hk + f q kh d q (7) q Q s.t. a q z d q z z = 1 m (8) q Q d q q Q (9) Notice that (7) (9) is a inear programming mode and its dua is m min z hk z (1) s.t. z=1 m z=1 a q z hk z f q hk + f q kh q Q (11) hk z z = 1 m (12) where hk z is the dua variabe corresponding to (8). Since (7) (9) is feasibe and bounded, we can use a duaity transformation simiar to the one of Soyster (1973). Hence for each edge h k E, we can repace (6) with { m y hk min z hk z } 11 and 12 z=1 Then, we can omit the min since we try to minimize the sum of the design variabes y hk with nonnegative weights. Hence, assuming that demand is subject to poyhedra uncertainty, NLP POL can be reformuated as the foowing inear MIP mode NLP GD : min h k E p hk y hk (13) s.t m z hk z y hk h k E (14) z=1 m f q hk + f q kh a q z hk z q Q h k E (15) z=1 hk z z = 1 m h k E (16) As there is no fow cost in our mode, we can obtain a formuation of our probem in the space of m E and design variabes y L E. Mirchandani (2) characterized a extreme rays of the projection cone reated to the singe-commodity NLP. However, ony necessary conditions for the muticommodity variant are given. In this case, the resuting projection inequaities are the we-known metric inequaities. Athough we do not provide the compete machinery of the projection process, we note here a particuar decomposition property for NLP GD. Observe that after the duaity transformation we have used above, there are no constraint bunding fow variabes associated with different commodities in NLP GD. Hence, the existence of a muticommodity fow f can be certified by checking the existence of Q singe-commodity fows; i.e., the projection cone for the muticommodity probem can be decomposed into Q cones with one cone for each commodity q Q. Based on this observation and using the extreme rays mentioned in Mirchandani (2) for the singe-commodity probem, we obtain the foowing mathematica mode NLP PRO in the space of and y variabes: min p e y e e E s.t m a q z e z e E q Q (17) z=1

5 INFORMS Journa on omputing 23(1), pp , 211 INFORMS 79 e S z=1 m a q z e z 1 q Q S V s q S t q V \S (18) where (17) and (18) are the reated projection inequaities. We denote an edge h k as e when there is no need to specify its endpoints. For S V, S denotes the set of edges with ony one endpoint in S. To concude this section, we remark that mode NLP GD has an interesting property. onsider the case where D = d Q Id= d and I is an identity matrix of size Q. Note that this corresponds to the deterministic case where d q = q for each q Q. For this particuar definition of D, constraints (14) (16) in the mode NLP GD become q Q d q hk q y hk h k E f q hk + f q kh hk q q Q h k E Here, the variabe d q hk q can be interpreted as the capacity on edge h k E aocated to commodity q Q. Rardin and Wosey (1993) use simiar variabes to express the fow requirements using cut constraints and obtain an extended formuation for the uncapacitated fixed-charge network fow probem. Then they project out these variabes and obtain the so-caed dicut coection inequaities. Labbé and Yaman (24) do a simiar anaysis on the fow formuations for the uncapacitated hub ocation probem and show that the famiy of dicut coection inequaities contains the metric inequaities. Notice that for a genera demand poyhedron D, in our mode, the variabes hk z are not additiona variabes that are used to get an extended formuation; rather, they come out of the duaity transformation that is used to convert the semi-infinite optimization mode NLP POL to a mixed-integer programming mode NLP GD. Sti, the same duaity transformation resuts in a system where fow variabes reated to different commodities are not bunded together any more and permits the use of cut inequaities to mode the fow requirements as we did in NLP PRO The Hose Demand Uncertainty ase Duffied et a. (1999) proposed the hose mode to carry out fexibe resource management in VPN. Independenty, Fingerhut et a. (1997) discuss the same fexibe specification of nonsimutaneous traffic requirements for a more effective design of broadband networks. Since then, the hose mode has become popuar in the teecommunications community. Rather than the point-to-point demand estimations, it uses the traffic bandwidth of some specia nodes caed VPN terminas to characterize the feasibe demand matrix reaizations. The difficuty of the VPN design probem (with continuous ink capacities) depends on the bandwidth definition (symmetric, asymmetric, and sumsymmetric) and the technica constraints on the routing scheme (singe-path, mutipath, tree, and termina tree routing). An intriguing question is the compexity of the symmetric case with singe-path routing. Hurkens et a. (27) prove that it can be soved in poynomia time if the backbone network of the VPN is a circuit. However, NLP with symmetric demands remains a chaenging probem as our test resuts in 5 show. In the rest of this paper, we consider the foowing symmetric hose mode of demand uncertainty: { D hose = d Q d q b i i W q Q s q =i or t q =i } d q q Q (19) where W V is the set of VPN terminas; i.e., W = i V q Q with s q = i or t q = i and b i is the bandwidth capacity of the termina node i W. In the cassica symmetric mode; demand is undirected; i.e., the demand from s to t is equa to the demand from t to s. However, in (19), we aow directed demand as ong as the cumuative bounds are respected. The importance of the hose mode can be demonstrated by returning to the simpe exampe in Figure 1, where we consider a singe-faciity type with unit capacity. Reca that the optima capacity aocation woud be as shown in Figure 1(b) with a tota cost of 13 when the demands are assumed to be known. Now consider the corresponding hose mode where the bandwidth of nodes from A to E are 4, 8, 8, 6, and 6 units, respectivey. Then, the optima design for the hose poyhedron is as shown in Figure 2(a) with a tota cost of 15. Notice that even though the tota design cost has increased sighty, the poyhedra design is more robust to fuctuations in demand. Reca the scenario we discussed in 1 where some of the pairwise demands have deviated by one unit from their expectations. Athough the deterministic design fais in that case, the robust one in Figure 2(a) remains operationa. Next, consider the demand uncertainty definition that we ca the BS mode, deveoped by Bertsimas and (a) Minimum cost design for the hose mode A 4 8 E Figure 2 6 D (b) Minimum cost design for the BS mode A B E Minimum ost Robust Designs 1 D B

6 8 INFORMS Journa on omputing 23(1), pp , 211 INFORMS Sim (23), where each demand d q takes a vaue in the range d q d ˆ q d q + d ˆ q such that at most commodities woud attain their maximum vaues. For the exampe above, we et the mean demand estimations d q and deviations d ˆ q be one unit so that both the expected and reaized demand matrices beong to the demand poyhedron. Then, even for the not-so-conservative case with = 2, the optima design is as in Figure 2(b) with a tota cost of 22. Athough this design aso remains operationa for the aforementioned scenario, it eads to a significant increase in the design cost. An increase in the tota design cost is a natura consequence of having a robust design. We provide some experimenta resuts on this issue ater in 5. However, this exampe shows that the hose mode can be more advantageous than some other uncertainty definitions. The hose mode enabes the transfer of unused capacity for a pairwise demand to another demand that goes beyond its estimation. Hence, capacities of edges for the hose mode can be ess than required by the point-to-point pipes as a resut of statistica mutipexing. The next proposition gives a formuation of NLP under hose demand uncertainty. Proposition 2.1. The projection of NLP GD onto the space of y variabes for the hose mode NLP hose is as foows: min p e y e e E s t b i e i y e e E (2) i W e s q + e t q 1 e S y e q Q S V s q S t q V \S (21) and integer e i i W e E e E L 3. Poyhedra Anaysis In this section we present resuts on the facets of the poyhedron associated with the network oading probem under hose uncertainty NLP hose. In the seque, we assume that is a positive integer for L and that the set L is ordered such that for 1 and 2 in L such that 1 < 2 we have 1 < 2. Let F = y W E + E L + : (2) and (21) and P = conv F. Observe that adding constraints e i 1 i W e E (22) does not change the vaidity of the mode when the costs are nonnegative (see Karaşan et a. 25). Let F W E = F y + E L + : (22) and P = conv F. First, we investigate the dimension of the poyhedra P and P. The proofs of a the resuts presented in this section as we as two emmas are given in the Onine Suppement at Proposition 3.1. The dimension of P and P is W + L E. Proof. See the Onine Suppement Projection onto the Subspace of Let F W E = Proj F = + : (21) and F = Proj F W E = F + : (22). Now, we reate facet defining inequaities of F and F with those of P and P. Proposition 3.2. Inequaity is facet defining for P (respectivey, for P ) if and ony if it is facet defining for F (respectivey, for F ). Proof. See the Onine Suppement Projection into the Subspace of e y e For e E, define F e = e W y e + L + : (2), P e = conv F e, F e = F e e W, y e + L + : (22), and P e = conv F e. Observe that if S \ e for every S V such that there exists q Q with s q S and t q V \S, then F e = Proj F and F e y e e = Proj F e y e.in the foowing theorem, we investigate how the facetdefining inequaities of P e and P e are reated to those of P and P. Theorem 3.1. Let e E be such that S \ e for every S V such that there exists q Q with s q S and t q V \S. Inequaity e + y e is facet defining for P e (respectivey, for P e) if and ony if it is facet defining for P (respectivey, for P ). Proof. See the Onine Suppement Projection into the Subspace of Design Variabes Associated with the Edges of a ut For S V, define b S = i S W b i and B S = min b S b V \S. Notice that in the worst case a terminas in S V woud want to use a of their bandwidths to exchange traffic with the nodes in V \S. As a resut, the worst-case traffic on the cut S woud be the minimum of these requirements, i.e., B S (see Gupta et a. 21, Karaşan et a. 25). Let S V be such that the subgraphs induced by S and V \S are both connected. Let y S be the restriction of the vector y to edges e S, F S = y S S L + e S ye B S, and P S = conv F S. Proposition 3.3. Let S V be such that the subgraphs induced by S and V \S are both connected and B S >. F S = Proj y S F = Proj y S F. Proof. See the Onine Suppement.

7 INFORMS Journa on omputing 23(1), pp , 211 INFORMS 81 Now, we can reate facet-defining inequaities of P S to those of P. Theorem 3.2. Let S V be such that the subgraphs induced by S and V \S are both connected and B S >. If inequaity e S e y e is facet defining for P S, and for each e S there exists a vector y S F S such that e S e y e = and ye >B S, then the inequaity is facet defining for P. Proof. See the Onine Suppement utset and Residua apacity Inequaities Now, we modify two we-known famiies of vaid inequaities for NLP to render them vaid for our probem. These inequaities are the cutset inequaities and arc residua capacity inequaities (see, e.g., Magnanti et a. 1993). Both inequaities can be generated as mixed-integer rounding (MIR) inequaities. Let X = x 1 x 2 + x 1 + x 2. The MIR inequaity x 1 x 2 is vaid for X (see, e.g., Wosey 1998, ornuéjos 28). The specia cases of the cutset and residua capacity inequaities for the network oading probem under hose uncertainty with a singe-faciity type are presented and used in Karaşan et a. (25) to strengthen the inear programming (LP) reaxation bound. The set F S is an integer knapsack cover set. Its convex hu is a specia case of the singe-commodity mutifaciity cutset poyhedron studied in Atamtürk (22). Yaman (27) gives a famiy of vaid inequaities caed the ifted rounding inequaities for the integer knapsack cover set. These inequaities generaize the cutset inequaities and are specia cases of the mutifaciity cutset inequaities of Atamtürk (22). As they are vaid for P S, they are aso vaid for P and P. For S V and L, et Y S = y e and e S b S r S = b S B S R S = B S For 1 and 2 in L, et g 1 2 = 1 1 <B S 2 2 Proposition 3.4. For S V and L such that R S >, the cutset inequaity ( ) R S +min g R S Y S + B S is vaid for P and P. B S B S R S Y S R S (23) Inequaity (23) is obtained from the inequaity Y S B S / using sequence-independent ifting in Yaman (27). The same inequaity can be obtained as MIR inequaity. Yaman (27) proves that if 1 = 1, then the cutset inequaity (23) for L such that R S > is facet defining for P S. Using Theorem 3.2, we can state the foowing proposition. Proposition 3.5. Let S V be such that the subgraphs induced by S and V \S are both connected, and ast L be such that R S >. If 1 = 1, then the cutset inequaity (23) is facet defining for P. Proof. See the Onine Suppement. Notice that if 2 L are divisibe by 1, then we can scae the b s vaues and the vaues by dividing with 1 so that 1 = 1. Moreover, if L =1 and R 1 S >, then the cutset inequaity (23) is facet defining for P for S V such that the subgraphs induced by S and V \S are both connected. Next, we generate residua capacity inequaities as MIR inequaities. Proposition 3.6. Let e E, L, and S W be such that r S >. The residua capacity inequaity ( r S is vaid for P. ) + min g r S y e + b S b i 1 e i r S i S Proof. See the Onine Suppement. (24) If L =1, the residua capacity inequaity becomes r 1 S y 1 e + b S b i 1 e i r 1 S (25) i S 1 Magnanti et a. (1993) prove the foowing: if b S / 1 2, then this inequaity defines a facet of P e.if b S / 1 =1, then the inequaity defines a facet of P e if S =1. Using Theorem 3.1, we can prove the foowing. oroary 3.1. Let e E be such that S \ e for every S V such that there exists q Q with s q S and t q V \S. Suppose that L =1 and et S W be such that r 1 S >. The residua capacity inequaity (25) defines a facet of P if b S / 1 2 or if b S / 1 =1 and S =1.

8 82 INFORMS Journa on omputing 23(1), pp , 211 INFORMS 4. Branch-and-ut Agorithm Because we have an exponentia number of constraints (21) in NLP hose, we use a branch-and-cut (B&) agorithm, which starts with a arger feasibe W E set y + E L + 2 and adds the vioated inequaities iterativey. In this section, we first expain our separation agorithms for the feasibiity cuts (21), as we as the demand cutset (23) and residua capacity (24) inequaities. Then, we briefy describe our upper bounding procedure Separation of Feasibiity uts Inequaities (21) can be separated by soving minimum cut probems. Given a pair ȳ, we construct an auxiiary graph Ḡ q = V E for each commodity q Q such that the capacity of each edge e E is set to be e s q + e t q. If the capacity of the minimum cut q separating s q and t q is ess than one, then we have a vioated inequaity (21) for commodity q. Otherwise, no inequaity (21) is vioated for q by the pair ȳ. Hence, we add at most Q feasibiity cuts at each iteration Separation of Demand utset Inequaities We have a heuristic separation agorithm for (23). For each commodity q Q, we use the cut q for which a feasibiity cut (21) is vioated. If the pair ȳ aso vioates a demand cutset inequaity for q and the faciity type L, then we add the corresponding cut to the probem. Thus, we add at most Q L such inequaities at each iteration Separation of Residua apacity Inequaities We do not know any poynomia-time agorithm to separate inequaities (24), but we can separate a reaxed version of these inequaities in poynomia time. Let e E, L, and S W. Define the reaxed residua capacity inequaity as ( ) r S y e + b S b i 1 e i r S (26) i S which is vaid for P as it is impied by inequaity (24). Moreover, it is a MIR inequaity. For a given edge e E, a faciity type L, and a pair e ȳ e, finding a vioated reaxed residua capacity inequaity or showing that there is no such inequaity is equivaent to soving the probem e = min S W { b i 1 e i r S i S ( b S )} ȳ e If / ȳe + e, then e ȳ e satisfies a (26) for e E and L. Otherwise, we have a vioated reaxed residua capacity inequaity defined by a minimizing set S. Since (26) is a MIR inequaity, if / ȳe b S / or / ȳe b S / 1, it cannot be vioated. This is because it woud be dominated by i S b i 1 e i and ye + i S 1 e i b i b S otherwise. Then, using the arguments in Atamtürk and Rajan (22), we can show that the reaxed residua capacity inequaities can be separated in the foowing way. For each e E and L, we construct the minimizing set S e = and et { i W e i > S e = i S e } ȳ e ȳ e b i 1 e i r S e ( b S e ) ȳ e Note that S e incudes nodes with negative objective function coefficients in the separation probem (Atamtürk and Rajan 22). onsequenty, (26) for edge e E, faciity type L, and the set S e is vioated if / ȳe < b S e / < / ȳe and / ȳe + S e <, where the former condition ensures that S e characterizes a feasibe soution to the separation probem. Otherwise, no inequaity (26) for this e E and L is vioated. Hence, for a given edge e E and faciity type L, the separation of the reaxed residua capacity inequaities can be done in O W time. This means that the compexity of the overa agorithm is O W E L. We use Agorithm 1 to separate the reaxed residua capacity inequaities. Note that we sove the separation probem for the reaxed inequaities but add the stronger ones in case of a vioation. Another aternative is to use a hybrid separation method, where for each edge e and faciity type, we check if any strong residua capacity inequaity is vioated for the set S e. We have impemented both methods and observed that the former method is as efficient as the atter one. Hence, we use the former method dispayed in Agorithm 1 for the reaxed inequaities. Agorithm 1 (Residua capacity inequaity separation) for a edge e E do for a faciity type L do Y e = ȳ e S e = i W e i > Y e Y e

9 INFORMS Journa on omputing 23(1), pp , 211 INFORMS 83 S e = b i 1 e i r S e i S e ( b S e ) ȳ e if Y e < b S e < Y e and ) ( ȳ e + S e < then i S e Add the vioated residua capacity inequaity ( r S e ) + min g r S e y e + b S e b i 1 e i r S e Heuristics Given the difficuty of the probem, we expect it to be usefu to incorporate approximation heuristics into our B& agorithm. These agorithms yied easy-tocompute upper bounds, usefu especiay for the arge instances that are reativey more difficut to sove. We appy a simpe rounding heuristic to get upper bounds on the optima soution. Thus, at each node of the B& tree, if we cannot find any vioated inequaity, then we have a feasibe soution for the LP reaxation of the NLP hose probem. Let ȳ be the current fractiona soution. We simpy generate a feasibe soution ŷ such that ŷe = ȳ e for a e E and L. Bienstock et a. (1998) aso use a simiar method and mention that it is efficient. We have aso adapted the approximation agorithm of Gupta et a. (21) for designing VPNs with continuous capacity reservation to our probem. However, based on some preiminary tests we chose to use the rounding heuristic. 5. Experimenta Resuts In this section we report the resuts of a computationa study for NLP hose with a singe faciity and with two faciities. Let NLP hose GD be the NLP GD mode for the hose uncertainty definition, which we sove using ILOG PLEX. Then, we compare our B& agorithm with PLEX on instances from the network design iterature. The instances poska, dfn, newyork, france, janos, atanta, tai, nobe-eu, pioro, gui39, cost266, norway, and sun are from the SND website (Zuse-Institute Berin), whereas the remaining seven instances are the ones used in Atın et a. (27) for a VPN design probem. For the SND instances the average pairwise demand estimates d q are avaiabe. Hence, to generate an initia hose poyhedron, we et the bandwidth of each termina node be the tota demand incident to it; i.e., b i = q Q s q =i or t q =i d q for a i W. Naturay, this is an assumption we make to construct an initia hose poyhedron. The choice of most effective bandwidth vaues is beyond the scope of the current study. However, we discuss the sensitivity of the routing performance to the choice of bandwidth vaues in 5.3. Moreover, we compare the hose mode and the BS mode in 5.1. For the atter mode, we consider the interva d q / d q for each commodity q Q. We have used AMPL to mode NLP hose GD as we as PLEX 9.1 MIP sover to sove it. The B& agorithm is impemented in using MINTO (Mixed INTeger Optimizer; see Nemhauser et a. 1994) and PLEX 9.1 as LP sover. We have set a two-hour time imit both for AMPL and MINTO. The branching rue for the B& agorithm is to choose the integer variabe with fractiona part cosest to.5. Node seection is done using best-bound search. We discuss our resuts for singe- and two-faciity cases in 5.1 and 5.2, respectivey. See aso the Onine Suppement for detaied test resuts Singe-Faciity NLP hose Here, we assume that there is ony one type of faciity avaiabe with a capacity of units. Then the demand cutset inequaities (23) reduce to B S Y 1 S S V (27) which ensure that the tota capacity across a cut is sufficient to support the tota demand between a termina pairs whose endpoints are on different shores of the cut. Moreover, the residua capacity inequaities are i S W b i 1 e i ( )( ) b S b S b S y e S V e E (28) Notice that the inequaities (24) and (26) are identica for the singe-faciity case. Thus, we impement an exact separation agorithm for the residua capacity inequaities (24). First, we compare our B& agorithm with soving using PLEX. We use the demand cutset inequaities (27) and the arc residua capacity inequaities (28) together with the feasibiity cuts (21) in our B& agorithm. We coud sove 7 out of 18 instances to optimaity in two hours using both PLEX and B&. Figure 3(a) shows the change in soution time as a resut of using our B& agorithm rather than PLEX to sove these seven instances. We see that B& yieds significanty shorter soution times in a these instances, which grows as arge as 99.7% for bhvdc. Moreover, we provide a comparison of termination gaps with PLEX and our B& agorithm for the remaining 11 instances in Figure 3(b). the singe faciity NLP hose GD

10 84 INFORMS Journa on omputing 23(1), pp , 211 INFORMS (a) Reduction in soution times if we use B& rather than PLEX for the seven instances which we coud sove to optimaity within two hours with both methods poska1 metro pacbe at-cep1 nsf1b pdh bhvdc (b) Gaps at temination for PLEX and the B& agorithm 1 9 PLEX B& Figure 3 france bhv6c atanta poska155 ny-cep2 dfn newyork sun tai nobe-eu janos omparison of Soution Times and Termination Gaps for the Singe-Faciity ase Even though PLEX gives better upper bounds than B& in dfn, ny-cep2, and atanta, the gaps at termination are better for the B& agorithm in the first two of these instances. On the other hand, B& is ceary superior for newyork, tai, janos, nobe-eu, and sun. The most important observation here is the significant degradation in the performance of PLEX reative to the B& agorithm as the network size increases. The instances tai, janos, nobe-eu, and sun are very good exampes of this behavior. Except tai, a of the nodes are demand nodes in these instances, and we observe that among such cases, ony in dfn and atanta has PLEX performed sighty better than B&. The upper bound of PLEX is just 7% and 2% tighter than the one of B& in dfn and atanta, respectivey. On the other hand, the upper bounds we obtain with B& are 1% better than the bounds with PLEX in tai, janos, nobe-eu, and sun. Finay, a comparison of the gaps at termination shows that the B& agorithm is ceary superior in 8 of the 11 instances with much ower gaps for tai, nobe-eu, and sun, in addition to the zero gap for janos. In sum, B& is superior in terms of soution times or termination gaps in 15 of the 18 instances. We have aso investigated the individua and joint infuence of the two types of cuts on the root reaxation soution quaities and the tota soution times. We consider the four cases F, F&D, F&R, and a, where each capita etter shows which of the feasibiity cuts (F ), demand cutset inequaities (D), and residua capacity inequaities (R) are used throughout the B& agorithm. We have considered six instances that were soved to optimaity in reativey shorter times. In Figure 4, we dispay the percentage of improvement for soution times and the reative change for root gaps when we use each setting rather than F, e.g., the change in soution time for F&D is time F &D time F / time F 1 and the change in root gap is gap F &D gap F. Figure 4 shows that the impact of demand cutset inequaities both on root gaps and soution times is significant. The residua capacity inequaities aso yied reasonabe improvements in root gaps. Athough adding residua capacity and demand cutset inequaities together does not improve the root gaps, it improves the soution times. Average improvements in root gaps and soution times are 86% and 84.72%, respectivey, for the setting a. Next, we compare the design cost for the hose mode with the BS mode for = 1 Q, = 15 Q, and = 25 Q. We show the percentage increase in design costs for the BS mode, which is measured as cost BS cost hose /cost hose 1, in Figure 5. We see that the BS mode eads to more-costy designs with respect to the hose mode as, i.e., the eve of conservatism, increases. Average differences are 36%, 4 83%, and 1 1%, respectivey. Finay, we consider the set of instances for which we coud sove both the deterministic and robust probems in ess than two hours, and we show the hange in soution times metro nsf1b at-cep1 pacbe bhvdc pdh hange in root gaps metro nsf1b at-cep1 pacbe bhvdc pdh F&D F&R a Figure 4 Impact of Different uts

11 INFORMS Journa on omputing 23(1), pp , 211 INFORMS For 1% For 15% For 25% then sove (1) (5). For the above instances, the cost of this worst-case deterministic mode is 6 to 25 times arger than the cost with the hose mode. We show the magnitudes of increase in terms of the ratio of the worst-case cost to the hose design cost in Figure 6(b) Figure 5 metro at-cep1 nsf1b bhv6c bhvdc Increase in ost if We Use the BS Mode with Different Rather than the Hose Mode change in the optima capacity instaation costs in Figure 6(a). The average increase in the tota reservation cost as we shift to the robust counterpart from the deterministic NLP is 17.62%. Athough we have to pay for the additiona fexibiity that the hose mode provides, we avoid overconservative designs by expoiting the hose mode. Suppose that we have the bandwidth capacities for a nodes and we ook for a design that can support the worst case that can happen based on the given information. eary, the safest approach woud be to fix the demand to its worstcase vaue as d q = min b s q b t q for each q Q and (b) Ratio of the worst-case design cost to the hose design cost Figure 6 (a) Increase in design cost due to robustness metro nsf1b at-cep1 pacbe bhv6c bhvdc pdh metro nsf1b at-cep1 pacbe bhv6c bhvdc Hose Design ost vs. the Deterministic and the Worst-ase Design osts 5.2. Two-Faciity NLP hose In the two-faciity case, we consider two types of faciities, namey, ow-capacity (LF) and high-capacity (HF) faciities with transmission capacities of 1 and 2 units, respectivey. Naturay, the cost of instaing each faciity is different and economies of scae prevai; i.e., the cost of 2 / 1 LFs is more than the cost of one HF. For S V, the demand cutset inequaities (23) reduce to the foowing inequaities: The LF case, i.e., = 1, where the resuting inequaities can be as foows: R 1 S Y 1 S + ( R 1 S 2 1 +min g 2 1 R 1 S B S Y S R 1 S if 1 2 <B S 1 Y 1 S +Y 2 S 1 if 1 2 B S B S B S Y 1 S + Y 2 S 1 if 1 <B S and 2 B S The HF case, i.e., = 2 where we can have min 1 R 2 S Y 1 S + R 2 S Y 2 S B S R 2 S if 1 2 <B S 2 Y 1 S + Y 2 S Y 1 S + B S Y 2 S B S ) if 1 2 B S if 1 <B S and 2 B S The two types of residua capacity inequaities (24) for each edge e E and set S V are ) 2 r 1 S ye (r S + min g 2 1 r 1 S 1 b S b i e i r 1 S b S for = 1 i S W 1 min 1 r 2 S ye 1 + r 2 S ye 2 b i e i b S r 2 S b S 2 i S W y 2 e for = 2 The number of residua capacity and demand cutset inequaities are doubed as we move from the singefaciity case to the two-faciity case. As a resut, the

12 86 INFORMS Journa on omputing 23(1), pp , 211 INFORMS LP modes we sove at each iteration of the B& agorithm can rapidy get arge. Therefore, we have tried the foowing five different schemes for adding vioated cuts: HA: add ony HF-type inequaities in a nodes of the B& tree; HR: add ony HF-type inequaities ony at the root node; GHA: add HF-type inequaities graduay i.e., add a vioated HF residua capacity inequaity ony if no HF demand cutset inequaity is vioated, in a nodes of the B& tree; GHR: graduay add HF-type inequaities i.e., add a vioated HF residua capacity inequaity ony if no HF demand cutset inequaity is vioated, ony at the root node of the B& tree; and GAR: graduay add a vaid inequaities i.e., add vioated LF and HF residua inequaities ony if no LF or HF demand cutset inequaity is vioated, at the root node. We compared the performances of the five settings in terms of the gaps at termination as shown in Figure 7. The instances for which the B& agorithm coud not find a feasibe soution within the two-hour time imit are assigned a 15% gap. Furthermore, we eave the bhv6c instance out of this anaysis because a schemes stopped with the same gap. onsequenty, we see that the average gaps at termination for these 11 instances are 32.6%, 38.5%, 31.1%, 31.2%, and 56.9% for HA, HR, GHA, GHR, and GAR, respectivey. The average number of nodes in the B& tree for these five settings are 13,968, 11,769, 7,869, 8,93, and 8,629. An important point to note here is that the number of nodes is one for those instances terminated with no feasibe soution. Thus, even though the highest number of such cases are observed for GAR, the size of the B& tree is smaer for GHA on average. In what foows, we provide the resuts with GHA. Initiay, we consider the six instances, which we coud sove to optimaity both with PLEX and the B& agorithm. Figure 8(a) shows the change in soution times defined as time B& time PLEX / time PLEX 1. We see that B& is faster than PLEX for a of these instances. PLEX was faster in ony pacbe, which we do not show in Figure 8(a) in order not to bur the figure. Athough the percentage change seems quite significant for this instance, the difference is actuay in seconds, and we coud sove it in ess than one minute in both cases. Next, we provide the test resuts for the remaining 11 instances in Figure 8(b). The termination gap for the instances, for which we coud not sove the LP reaxation in two hours, is taken to be 15%. We see that our B& agorithm is superior to PLEX, especiay for the arge instances where a nodes are demand nodes just ike the singe-faciity case. This is quite obvious especiay for tai, nobe-eu, pioro, and cost266 because the MIP sover coud not find even a feasibe soution in two hours, whereas the B& agorithm successfuy produced some upper bounds. Specificay, the upper bounds for nobe-eu and pioro are quite promising. Moreover, the NLPGD hose probem coud not be soved for newyork because of insufficient memory. In two cases, i.e., norway and gui, we coud not find any upper bound with either of the methods. On the other hand, the B& agorithm is better in six of the remaining nine instances with much ower gaps for dfn, tai, nobe-eu, pioro, and cost266. A fina anaysis in Figure 9 is about the price of robustness measured in terms of the percent change in the fina design cost for the two-faciity case. The average increase in the optima reservation costs of 12 HA HR GHA GHR GAR poska dfn newyork atanta tai nobe-eu pioro norway cost266 gui39 Figure 7 Percent Gaps at Termination for Each Scheme

13 INFORMS Journa on omputing 23(1), pp , 211 INFORMS 87 (a) hange in soution times if we use B& rather than PLEX metro nsf1b at-cep1 bhvdc pdh (b) Gaps at termination for PLEX and the B& agorithm bhv6c PLEX poska B& dfn newyork atanta tai nobe-eu pioro norway cost266 gui39 Average traffic routing Design cost , (a) Routing performance for different bandwidth definitions K =.25 K =.33 K =.5 K = Tau Tau (b) hange in tota design cost as a function of tau Figure 8 omparison of Soution Times and Termination Gaps for the Two-Faciity ase Figure 1 Impications of the Bandwidth Definition on Routing Performance and Design ost Figure 9 metro nsf1b at-cep1 pacbe bhv6c bhvdc Increase in Design ost as a Resut of Robustness the six instances for which gaps coud be cacuated is 18.86% Parametric Hose ase In this section, we consider the metro instance and anayze the sensitivity of the robust design to the choice of bandwidth capacities. First, we generate 2 demand matrices d 1 d 2, where the demand d j q for each q Q is normay distributed with mean d q and standard deviation Kd q for K 1. Next, for R ++, we et b i = q Q s q =i or t q =i d q for a i V and sove the corresponding NLP GD to get the optima capacity configuration y. Then, for each j = 1 2, we determine the maximum tota fow F j we can route given demand matrix d j and ink capacities y by soving a inear programming probem. We cacuate the fraction of demand routed as F j / q Q d j q and take the average over 2 demand matrices to evauate the performance of the optima hose design y for a given. We have performed severa tests with K and eight different vaues of Figure 1(a) shows the average percentage of traffic we coud route under different hose definitions and K vaues. As expected, Figure 1(a) shows that independent of what K is, the traffic routing rate improves as we consider a arger hose poyhedron, i.e., as grows. This is natura because a arger hose poyhedron impies a more conservative design. On the other hand, for a given, the demand satisfaction rate is negativey affected by demand deviations. However, higher protection comes at a cost, and Figure 1(b) shows how the tota cost changes with. The proper choice of is reated to the accuracy of the demand information as we as the trade-off between the design cost and the service eve. We study the hose poyhedron for = 1 for our tests in 5.1 and 5.2. The resuts above show that the average routing rates of the corresponding robust design for K are 98 62%, 97 92%, 96 1%, and 92 81%, respectivey. 6. oncusion In this paper we studied the network oading probem where the pairwise traffic demands are not assumed to be known in advance. We used a poyhedra definition of traffic demands and sought to design a network that is capabe of supporting infinitey many