Branch-And-Price-And-Cut Approach to the Robust Network Design Problem Without Flow Bifurcations

Submitted to Operations Research manuscript (Please, provide the mansucript number!)

Branch-and-Price-and-Cut Approach to the Robust Network Design Problem without Flow Bifurcations

Chungmok Lee Department of Industrial and Systems Engineering, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea, lcm@kaist.ac.kr Kyungsik Lee Department of Industrial & Management Engineering, Hankuk University of Foreign Studies, 89 Wangsan-ri, Mohyeon-myon, Yongin-si, Gyeonggi-do 449-791, Republic of Korea, [email protected] Kyungchul Park College of Business Administration, Myongji University, 50-3, Namgaja-dong, Seodaemun-gu, Seoul, 120-728, Republic of Korea, [email protected] Sungsoo Park Department of Industrial and Systems Engineering, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea, [email protected]

This paper presents a robust optimization approach to the network design problem under traﬃc demand

uncertainty. Here, we consider the speciﬁc case of the network design problem in which there are several

alternatives in edge capacity installations and the traﬃc cannot be split over several paths. A new decom-

position approach is proposed which yields a strong LP relaxation and enables traﬃc demand uncertainty

to be addressed eﬃciently through localization of the uncertainty to each edge of the underlying network. A

branch-and-price-and-cut algorithm is subsequently developed and tested on a set of benchmark instances.

Key words : network design problem, robust optimization, integer programming, branch-and-price-and-cut,

robust knapsack problem

Subject classiﬁcations : Networks/graphs: network design. Programming, integer:

branch-and-price-and-cut. Programming, stochastic: robust optimization for solving hard IP with data

uncertainties.

1. Introduction

Demand uncertainty is an inherent property in many network design problems, and the ability to properly address this uncertainty is a crucial factor in successful network design. The use of average demand to estimate demand uncertainty may lead to severe service quality degradation, whereas the use of extreme demands may result in an overly conservative network design which translates into unnecessary capital expenses.

1 Lee et al.: Robust Network Design Problem under Demand Uncertainty 2 Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!)

There are two main approaches to addressing demand uncertainty in network design: the stochastic programming approach and the more recent robust optimization approach (Ben-Tal and

Nemirovski 2008). While the stochastic programming approach assumes that the probability dis- tribution of the demand is known in advance, the robust optimization approach only requires that the uncertain demand be a member of a pre-speciﬁed set, denoted the uncertainty set. As such, when only a limited amount of information is available on the future demand, a methodology based on robust optimization may be the preferred approach.

In this paper, we present a robust optimization approach to the network design problem with traffic demand uncertainty as modeled by Bertsimas and Sim (2003). We consider the specific network design problem in which there are several alternatives in edge capacity installations and the traffic cannot be split (non-bifurcated) over several paths. One potential application of the problem appears in telecommunication network design using asynchronous transfer mode (ATM) (Stallings

1998, Atamt¨urk and Rajan 2002) or multi-protocol label switching (MPLS) traﬃc engineering

(Xiao et al. 2000). In these cases, exactly one path (ATM virtual circuit or MPLS label switched path) should be speciﬁed for each traﬃc demand.

The uncertainty model proposed by Bertsimas and Sim (2003), which we call here the Γ-model, assumes that each uncertain traffic demand lies in an interval specified by a nominal value and the maximum possible deviation from this value. Since it is highly improbable that all of the traffic demands reach their maximum values simultaneously, these authors introduced a parameter (the uncertainty budget) which limits the number of uncertain demands that can simultaneously be their maximum values. The Γ-model can be useful for solving many telecommunication network design problems when the precise demand cannot be predicted but the possible range of each traffic demand can be estimated (e.g. based on historical experience). In addition, the ability to control the conservativeness of the solution using the uncertainty budget is highly desirable for comparing and evaluating the resulting network design alternatives. Several studies, such as those of Altın et al. (2007), Klopfenstein and Nace (2009), and Koster et al. (2010), have considered the Γ-model in the context of network design. An alternative uncertainty model that is frequently Lee et al.: Robust Network Design Problem under Demand Uncertainty Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!) 3 adopted in network design problems (especially in virtual private network design) is the so-called hose model (Altın et al. 2010). This model assumes that for each node, only the upper bound for the sum of incoming and outgoing traffic is known in advance. While this latter approach is useful when very limited information on the traffic demand is available, the result may be an overly conservative network design, as noted in Altın et al. (2010). Hence, when at least some degree of basic information is available on the point-to-point traffic demands, the Γ-model may be the preferred approach to avoid overly conservative network design. This is especially true for large telecommunication operators since they have accumulated sufficient experience in estimating traffic demand.

Our model assumes that both the capacity of each edge and the path for each traﬃc demand are to be determined simultaneously, so that the ﬂow costs can be considered explicitly. Similar models in the network design context have been presented by Altın et al. (2007, 2010) and Koster et al. (2010), while Mudchanatongsuk et al. (2008) and Mattia (2010) consider models in which the design decision is made prior to the routing decision (the recourse or two-stage model). In general, the recourse model gives a less conservative network design, but its application to the network design problem with demand uncertainty generally leads to two challenging problems.

First, it is generally much more difficult to solve (see Ben-Tal et al. 2004), and secondly, it may be hard to explicitly consider the flow costs. Thus even when the recourse model is seemingly more appropriate, we believe that—in terms of practical applications—our model may be considered the more suitable choice since it can be solved relatively efficiently.

The major difficulty in the robust network design lies in the fact that the resulting optimization model generally does not admit an efficient solution procedure. To overcome this difficulty, we propose a branch-and-price-and-cut approach using a Dantzig–Wolfe decomposition formulation, which provides a tight linear programming (LP) relaxation bound. One of the notable aspects of the proposed approach is that the uncertainty in the traffic demand can be localized to each edge—that is, the robustness of the solution can be guaranteed by considering the robust knapsack problem, which is defined for each edge. This approach enables the user to address the uncertainty Lee et al.: Robust Network Design Problem under Demand Uncertainty 4 Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!) very efficiently, since it is sufficient to consider the demand uncertainty issues only when solving the much smaller subproblems. In addition, our simulation tests demonstrate that a sufficient level of robustness of the solution can be guaranteed, with a minimal conservativeness and penalty in the cost that is acceptable in practical settings. We therefore expect that the model and algorithm proposed here will be of practical benefit in the design of telecommunication network infrastructures with a built-in flexibility for uncertainty in future traffic demand.

2. Network Design Problem under Demand Uncertainty

In this section, we first present the standard flow-based formulation of the robust network design problem. We then present a stronger formulation and theoretical results which will be used to develop an efficient solution procedure for the problem.

2.1. The Flow-based Formulation of the Robust Network Design Problem

For a given undirected graph G =(V, E), let A := {(i, j), (j, i)|{i, j}∈ E} and {(sk,tk) | k ∈ K} be

k a set of ordered node pairs (commodities). For each k ∈ K and e ∈ E, let ce be the unit ﬂow cost of commodity k on edge e. For each edge e ∈ E, a set of possible facilities Fe is given where each

e e facility f ∈ F has capacity hf and installation cost bf .

The demand uncertainty is modeled as follows. For each commodity k ∈ K, let rk and dk ∈

R+ denote the nominal demand and the maximum possible deviation from the nominal demand, respectively; i.e., demand quantity for k ∈ K is an unknown value between rk − dk and rk + dk

(rk ≥ dk is assumed). Let Γ ∈ Z+ be the degree of robustness in the sense of Bertsimas and Sim

(2004), i.e., the edge capacity constraints should be protected against simultaneous perturbations

e of Γ commodities from their nominal values. Let Mf be the maximum number of installations of facility f ∈ Fe on edge e ∈ E, which can be obtained readily from the total sum of demand quantities. Then each capacity constraint for edge e is given as:

k k k k e ¯ e e rk(xij + xji) + max dk(xij + xji)+ hf λf ≤ hf Mf , ∀e ∈ E, (1) S⊂K,|S|≤Γ k∈K k∈S f∈F f∈F e e Lee et al.: Robust Network Design Problem under Demand Uncertainty Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!) 5

k ¯e e e e where xij is the binary ﬂow variable for k ∈ K and (i, j) ∈ A, λf := Mf − λf and λf are the nonnegative integer design variables for e ∈ E and f ∈ Fe. Note that the design variables (λ) are complemented, so that the above constraint has a knapsack-like form.

The robust network design problem (RNDP) can now be stated as:

e k k e e ¯e (RNDP) max − rkck(xij + xji) − bf (Mf − λf ), (2) k∈K i,j e E e∈E f∈F { }= ∈ e −1, if i = sk; subject to xk − xk = 1, if i = t ; ∀k ∈ K, i ∈ V, (3) ji ij  k {j|(j,i)∈A} {j|(i,j)∈A} 0, otherwise.  (1),  k ¯e xij ∈ {0, 1}, ∀ (i, j) ∈ A, k ∈ K, λf ∈ Z+, ∀ f ∈ Fe,e ∈ E. (4)

The objective is to minimize the sum of total ﬂow and installation cost. Inequalities (3) are the well-known ﬂow conservation constraints.

As shown in Bertsimas and Sim (2004), by applying LP duality to constraints (1), (RNDP) can be reformulated as a mixed integer programming (MIP) problem (see Section B.5 in the Electronic

Companion), which we will denote as (RCNDP) from this point onwards. However, the resulting

MIP formulation is of a substantially larger size and seems to give weak LP relaxation bounds, which makes it diﬃcult to use for solving practical problems (see Section 3.2).

2.2. The Edge-Pattern-based Reformulation: Dantzig–Wolfe Decomposition

Using the Dantzig–Wolfe decomposition approach, we can replace each capacity constraint of (1) with an extended constraint that has exponentially many variables. Park et al. (2003) have proposed a similar decomposition approach for the (non-robust) bandwidth packing problem in which the capacity of each edge is ﬁxed.

F low Design |K| |Fe| For each edge e = {i, j} ∈ E, we deﬁne a set Ge := {g = (g ,g ) ∈ B × Z+ |

r gF low d gF low he gDesign he M e k∈K k k + maxS⊂K,|S|≤Γ k∈S k k + f∈Fe f f ≤ f∈Fe f f }, which is called the set of feasible (robust) edge patterns. An edge pattern speciﬁes a set of commodities (gF low), together with a facility conﬁguration (gDesign), which can be admitted on the edge concerning the uncertainty of the demands. Lee et al.: Robust Network Design Problem under Demand Uncertainty 6 Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!)

Using the set of feasible edge patterns Ge and the Dantzig–Wolfe decomposition approach, we introduce the extended formulation (RNDPe):

e k k e e (RNDPe) max − rkck(xij + xji)+ cgzg , k∈K i,j e E e∈E g∈G { }= ∈ e subject to (3),

e zg ≤ 1, ∀ e ∈ E, (5) g∈G e k k F low e xij + xji ≤ gk zg , ∀k ∈ K, {i, j} = e ∈ E, (6) g∈G e k e xij ∈ B, ∀ (i, j) ∈ A, k ∈ K, zg ∈ B, ∀ g ∈ Ge,e ∈ E, (7)

ce be gDesign M e ze g G where g = f∈Fe f ( f − f ). Variable g has a value of 1 if pattern ∈ e is selected for edge e ∈ E, 0 otherwise. The following proposition addresses the strength of the LP relaxation of

(RNDPe). The proof of the proposition can be found in A.1 in the Electronic Companion.

Proposition 1. The LP relaxation of (RNDPe) has the same optimal value as that of the LP relaxation of (RNDP) augmented with valid inequalities which completely describe the convex hull of feasible solutions satisfying each inequality of (1).

2.3. The Column Generation Subproblem: Robust Knapsack Problem

k e Let πi , βe, and γk denote the dual variables associated with constraints (3), (5), and (6), respectively. The column generation subproblem for edge e ∈ E can then be stated as:

e F low e Design e e ξe(π,β,γ) = max γkgk + bf gf − bf Mf − βe, (8) k∈K f∈F f∈F e e F low F low e Design e e subject to rkgk + max dkgk + hf gf ≤ hf Mf , (9) S⊂K,|S|≤Γ k∈K k∈S f∈Fe f∈Fe Design e 0 ≤ gf ≤ Mf , ∀f ∈ Fe, (10)

F low Design |K| |Fe| (g ,g ) ∈ {0, 1} × Z+ . (11)

The above problem is a robust bounded integer knapsack problem (Bertsimas and Sim 2003, 2004,

Bertsimas and Weismantel 2005). Lee et al.: Robust Network Design Problem under Demand Uncertainty Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!) 7

It is well known that a bounded integer knapsack problem can be transformed to an equivalent

0–1 knapsack problem, as shown in §3 of Martello and Toth (1990). By transforming the integer

Design variables gf into binary variables, the above problem can be transformed to a robust 0–1 knapsack problem, which is an ordinary binary knapsack problem but with uncertain coeﬃcients.

Bertsimas and Sim (2003, 2004) reformulated the robust 0–1 knapsack problem as an MIP. A similar problem was considered by Klopfenstein and Nace (2009), who investigated polyhedral properties of the robust knapsack problem and applied the results to solving the robust bandwidth packing problem.

To solve the edge pattern generation problem, we need an eﬃcient algorithm for the robust 0–1 knapsack problem. Rather than using the MIP reformulation provided in Bertsimas and Sim (2003,

2004), we will show that the problem can be solved by solving the ordinary 0–1 knapsack problems at most n − Γ+1 times, where n is the number of items in the 0–1 knapsack problem and Γ is the degree of robustness that is given as a parameter, where 1 ≤ Γ ≤ n.

Robust 0–1 Knapsack Problem Let us consider the feasible set of a robust 0–1 knapsack problem deﬁned as:

n S = x ∈ B | aj xj + max dj xj ≤ b T ⊆N,|T |=Γ j∈N j∈T n = x ∈ B | aj xj + dj xj ≤ b, for all U ⊆ N with |U| = Γ , (12) j∈N j∈U where N = {1, 2,...,n}. For j ∈ N, aj and dj are nonnegative integers. We assume that the indices are ordered such that d1 ≥ d2 ≥···≥ dn and deﬁne dn+1 =0. Let us deﬁne L = {Γ, Γ+1,...,n −

n + 1,n +1}, and Sl = x ∈ B | aj xj + (dj − dl)xj ≤ b − Γdl , where Nl = {j ∈ N | j ≤ l} j∈N j∈Nl and N + = N ∪ {n +1}. The following proposition shows that the column generation subproblem

ξe(π,β,γ) can be solved by solving at most |K|− Γ+1 ordinary 0–1 knapsack problems (see A.2 in the Electronic Companion for the proof.)

Proposition 2. ∗ The robust knapsack problem Z = max j∈N cjxj | x ∈ S , can be solved by

∗ solving at most n − Γ+1 ordinary 0–1 knapsack problems Zl = max j∈N cjxj | x ∈ Sl , for all l ∈ L. Lee et al.: Robust Network Design Problem under Demand Uncertainty 8 Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!)

3. Solution Method and Computational Results

3.1. Solution Method

Since (RNDPe) has exponentially many variables (the robust edge patterns), we use the branch- and-price algorithm (Barnhart et al. 2000). In addition to performing the column generation procedure at each branching node, we added a simpliﬁed version of the cut-set inequalities proposed by Raack et al. (2011). Various techniques for improving the performance of the column generation algorithm were also incorporated: (i) early termination by inspection of the reduced costs; (ii)

k e ﬁxing original variables (xij and λf ) from the optimal dual solution of the extended formulation;

(iii) eﬃcient branching rule based on the divergence of ﬂow paths. Details on the computational aspects of the algorithm are presented in Appendix B of the Electronic Companion.

3.2. Computational Results

All computational tests presented here were performed on an AMD X2 2.9GHz PC with 4GB RAM.

We implemented our own branch-and-price-and-cut algorithm by C++ and CPLEX 10.1 for the linear programming solver. For solving (RCNDP), we used the MIP solver of CPLEX 10.1. We used two sets of problem instances: randomly generated problem instances (see C.1 in the Electronic

Companion for details of the problem generation), and problem instances from SNDlib (Orlowski et al. 2010). Among the SNDlib instances, the results for pdh instance are reported in this paper.

Results for the other SNDlib instances can be found in Section C.3 of the Electronic Companion.

For each problem instance, we used two parameters, namely, degree of robustness Γ and maximum deviation ratio dev to deﬁne the uncertainty set. For example, given dev =15% andr ˆk =100, we have rk =100 and dk = 15.

Table 1(a) summarizes the results of the proposed algorithm with varying network sizes. To compare the tightness of the LP relaxation bounds of diﬀerent formulations, we calculated the gap percentages from the best solution value among the formulations for the given test problem. Let

1 2 zlp, zroot, zroot, and zbest denote the LP relaxation value of (RCNDP), the optimal LP value at the root node of the branch-and-price-and-cut tree for (RNDPe), the optimal value of (RCNDP) with Lee et al.: Robust Network Design Problem under Demand Uncertainty Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!) 9

CPLEX’s default cutting planes at the root node, and the best known objective value for the problem,

1 respectively. We define GAP1 as zroot−zbest × 100 (GAP2 is defined similarly). In a similar way, lpGAP zbest z z is defined as lp− best × 100. In general, lpGAP reflects the difficulty of the original problem, and zbest GAP1 (GAP2) describes the LP relaxation strength of the algorithms’ base formulations—(RNDPe) for the proposed algorithm and (RCNDP) with CPLEX’s default cutting planes. The asterisk(*) indicates that the given time limit (one hour) was reached. For comparison purposes, we report the results obtained by solving (RCNDP) directly, which are shown under the heading CPLEX.

[Table 1 about here.]

As shown in Table 1(a), the proposed algorithm outperformed CPLEX with (RCNDP) formulation.

The high values of lpGAP show that (RCNDP) has a very poor LP relaxation bound. It should be noted that CPLEX signiﬁcantly reduced the gaps by its default cutting plane procedures. However, as shown in the table, our proposed algorithm provided gaps tighter than those provided by CPLEX; in particular, the numbers of branching nodes are much smaller in the proposed algorithm. The results also show that, for the proposed algorithm, only a small portion of computational time was spent in solving the subproblem, which justiﬁes one of the key objectives of using our approach: localization of the uncertainty to relatively small subproblems. Based on these results, the number of commodities would appear to have the greatest impact on the performance of the algorithm.

Table 1(b) lists the results when the values of parameters (Γ and dev) are varied, clearly showing that the proposed algorithm performed nearly independently of the values of Γ and dev. It should be recalled that, in the proposed algorithm, the only diﬀerence between the deterministic case and the robust case is the number of times needed to solve the knapsack problems. Since the knapsack problem can be solved fairly eﬃciently by a specialized algorithm, the incorporation of robustness has little impact on the overall performance.

3.3. Evaluation of Robustness of Solutions by Simulation Tests

To evaluate the robustness of the solution, we conducted two simulation tests designed to reﬂect real-life situations. The detailed description of these simulation tests is presented in Section C.4 Lee et al.: Robust Network Design Problem under Demand Uncertainty 10 Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!) of the Electronic Companion. Figure 1 plots the number of infeasible scenarios with variations in the simulation parameter (∆) for the solutions given in Table 1(b). The simulation parameter ∆ determines the variance of ﬂuctuations in the demand values. The results clearly demonstrate that the robustness of the solution is greatly improved by the robust optimization approach even at the smallest protection level, such as Γ=1. A comparison of the two results when Γ is zero and one, shows that the penalties in the objective values are only 4.51% and 6.09%, respectively (refer to

Table 1(b)), while the gains in robustness are rather large in all cases.

[Figure 1 about here.] 4. Conclusion

We have presented a practically applicable branch-and-price-and-cut algorithm for the design of telecommunication networks under demand uncertainty. A novel Dantzig–Wolfe decomposition approach was proposed to localize the uncertainty, which makes it possible to address network-wide uncertainty very eﬃciently.

For our approach to be implemented in real-life applications, two parameters, Γ and dk, need be speciﬁed to determine the uncertainty set. The deviation of demand (dk) can be estimated from historical data, physical limitations of the devices, and/or business contracts. A well-designed simulation test may be helpful to determine the appropriate value of Γ.

The proposed algorithm is readily applicable to the robust bandwidth packing problem. More- over, the edge based decomposition approach can be used in the path-based formulation. In this case, it is possible to introduce a number of path-related constraints, such as the hop-limit and path delay constraint. It should be recalled that the feasible set of the robust knapsack problem is a union of the feasible sets of the ordinary knapsack problems. It can be easily shown that this result still holds even when there are additional constraints on the compatibility among commodities, such as commodity-wise interference constraint. 5. Electronic Companion

An electronic companion to this paper is available as part of the online version that can be found at http://or.journal.informs.org/. Lee et al.: Robust Network Design Problem under Demand Uncertainty Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!) 11

Acknowledgments

The authors would like to thank several anonymous referees for their numerous and very helpful comments

and suggestions. This work was supported by National Research Foundation of Korea Grant funded by the

Korean Government (KRF-2008-2-D01197).

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Figure 1 The numbers of infeasible scenarios in the simulation for problems |N| = 20/|M| = 30/|K| =20 ((a) and

(b)) and pdh ((c) and (d)). (a) Dependency on Γ, dev = 20%. (b) Dependency on dev, Γ = 3. 1,000 1,000 Γ = 0 d = 0% Γ = 1 d = 10% Γ = 2 d = 20% 800 Γ = 3 800 d = 30% Γ = 4 d = 40% Γ = 5 d = 50% 600 600 35.5% 35.5% 38.2% 38.2% 40.1% 40.1% 400 42.4% 400 42.4% #infeasible #infeasible 43.5% 43.5% 44.4% 44.4% 41.4% 41.4% 200 35.8% 200 35.8% 26.1% 26.1%

5.5% 5.5% 0 0 10 20 30 40 50 10 20 30 40 50 ∆(%) ∆(%)

(c) Dependency on Γ, dev = 20%, pdh. (d) Dependency on dev, Γ = 3, pdh. 1,000 1,000 Γ = 0 d = 0% Γ = 1 d = 10% Γ = 2 d = 20% Γ = 3 800 800 d = 30% Γ = 4 d = 40% Γ = 5 d = 50% 600 600 58.7% 58.7% 66.3% 66.3% 72.2% 72.2% 75.5% 75.5% 76.2% 76.2% 400 75.3% 400 75.3% #infeasible 72.2% #infeasible 72.2% 67.5% 67.5% 59.6% 59.6% 49.7% 49.7% 200 200

0 0 10 20 30 40 50 10 20 30 40 50 ∆(%) ∆(%) Lee et al.: Robust Network Design Problem under Demand Uncertainty 14 Article submitted to Operations Research; manuscript no. (Please, provide the mansucript number!)

Table 1 Performance of the proposed algorithm. (a) Randomly generated networks when Γ = 2 and dev = 30%

Proposed Algorithm CPLEX

|N| |E| |K| Time for lpGAP # of Time for Total Opt. # of Total # of master BnB subprob. GAP1 time (Best) BnB GAP2 time gen. col. prob. nodes (sec.) (sec.) sol. nodes (sec.) (sec.)

10 46 263 0.55 0.59 9.12 2.03 3204 1543 16.68 6.05 63.30 20 20 139 945 34.08 10.30 9.54 55.60 5059 3193 12.39 23.93 38.85 30 1211 2203 1138.57 266.47 6.70 1833.89 6747 48479 13.98 510.76 27.76 10 10 69 342 1.80 0.92 8.84 4.35 3601 9036 16.13 39.73 59.82 30 20 1443 1090 603.28 144.54 11.53 956.42 5645 25195 15.67 243.30 33.92 30 2923 2354 1990.24 659.68 11.09 3600* 7189 154501 17.89 3600* 30.17

10 63 544 4.07 2.37 9.90 8.74 6320 5341 15.59 42.11 61.35 30 20 1245 1917 966.57 186.73 12.85 1500.35 9758 240907 17.85 3600.00 38.31 30 940 4315 2446.07 339.62 11.21 3600* 13097 132801 15.84 3600* 28.24

10 355 530 11.64 6.42 8.75 29.36 4931 46405 16.10 458.28 71.20 20 40 20 583 1449 331.47 62.06 12.11 526.27 7742 159501 18.10 3600* 54.12 30 211 3417 1726.63 98.71 4.99 2125.30 10046 109114 9.35 3600* 33.37

10 143 503 6.16 4.24 4.97 16.29 6002 30040 12.00 241.58 59.52 50 20 1327 1925 1386.98 195.20 8.98 2049.15 8909 148076 16.42 3600* 42.25 30 551 4824 2687.74 265.33 9.87 3600* 11361 65459 14.58 3600* 34.13

10 269 764 39.63 13.12 11.68 68.03 7280 353355 16.02 3600.00 60.80 40 20 563 2091 1350.17 107.79 11.17 1781.98 10765 142901 15.51 3600* 48.03 30 520 4273 2671.48 227.53 15.56 3600* 14072 78501 17.85 3600* 37.85

10 57 817 12.72 5.32 11.78 21.93 6165 21906 16.38 348.08 66.94 30 50 20 4679 2483 2054.62 494.33 18.46 3600* 10794 62223 23.42 3600* 55.84 30 903 5541 2519.75 335.79 18.93 3600* 13979 47701 22.88 3600* 41.77

10 139 1186 68.85 17.79 10.56 103.94 6854 141986 15.20 2874.93 57.51 60 20 330 3017 3113.49 117.22 14.41 3600* 10858 55081 20.37 3600* 40.77 30 374 6006 2851.16 272.80 19.30 3600* 14298 29001 23.97 3600* 43.04

(b) Performance dependency on the values of Γ.

Proposed Algorithm CPLEX

Prob. Γ dev Time for lpGAP # of Time for Total # of Total # of master Opt. (Best) Inc. obj. BnB subprob. GAP1 time BnB GAP2 time gen. col. prob. sol. val. (%) nodes (sec.) (sec.) nodes (sec.) (sec.)

0 0 2146 1827 1634.36 69.37 13.05 2171.27 9024 0.00 1119422 16.60 3600* 44.94

1 3287 1933 1791.04 461.46 14.38 3028.83 9431 4.51 405837 18.01 3600* = 20 42.34 | 2 2278 1915 1563.55 335.07 13.89 2486.61 9496 5.23 276388 18.65 3600* 40.03 K | 320 1707 1891 1318.58 237.70 13.04 2018.98 9729 7.81 211201 19.04 / 3600* 39.90 4 1569 1946 1274.53 208.26 12.24 1919.85 9735 7.88 202439 17.59 3600* 39.05

= 30 5 1412 1913 1378.58 198.05 11.83 2009.22 9735 7.88 202359 16.60 3600* 38.57 | M | 10 3111 1875 2152.65 298.63 13.98 3193.98 9431 4.51 253501 18.44 3600* 42.23 / 20 1707 1891 1318.58 237.70 13.04 2018.98 9729 7.81 211201 19.04 3600* 39.90 3

= 20 30 1573 1937 1448.51 235.99 12.60 2144.53 9933 10.07 210301 18.16 3600* 37.07 | 40 761 1847 836.05 135.29 12.05 1250.09 10353 14.73 191516 18.08 3600* 35.93 N | 50 2730 1967 2291.86 406.00 13.61 3516.79 10811 19.80 211543 19.29 3600* 35.09

0 0 13440 1283 2541.06 165.50 12.83 3600* 12327076 0.00 217077 21.45 2931.90 62.74

1 8990 1292 2095.77 636.74 14.40 3600* 13077409 6.09 46434 23.53 3600* 57.80 2 2089 1192 1549.79 315.38 12.55 2306.93 13077409 6.09 53080 21.93 3600* 57.80 20 3 5313 1242 1490.91 362.62 11.31 2413.01 13077409 6.09 58305 20.59 3600* 57.80 4 4073 1233 1061.70 237.58 10.66 1689.98 13077409 6.09 25290 19.35 1034.59 57.80 5 4094 1143 927.30 202.94 10.01 1483.42 13077409 6.09 50068 18.70 2505.50 57.80

from SNDlib 10 9953 1289 2090.45 620.50 13.98 3600* 13077409 6.09 41112 23.34 3600* 61.29 20 5313 1242 1490.91 362.62 11.31 2413.01 13077409 6.09 58305 20.59 3600* 57.80 pdh 3 30 13867 1191 1887.92 733.55 18.33 3600* 14748959 19.65 75415 26.27 3600* 59.26 40 4659 1293 2160.72 621.84 17.49 3600* 15078138 22.32 96819 25.19 3600* 57.16 50 9981 1279 1901.33 727.54 20.12 3600* 16702854 35.50 95805 27.38 3600* 56.78