Global optimization of capacity expansion and flow assignment in multicommodity networks

Global optimization of capacity expansion and flow assignment in multicommodity networks

Ricardo Poley Martins Ferreira^I,*; Henrique Pacca Loureiro Luna^II; Philippe Mahey^III; Mauricio Cardoso de Souza^IV

^IDepartamento de Engenharia Mecânica, Universidade Federal de Minas Gerais - UFMG, Brazil. Phone: [phone]. E-mail: [e-mail]

^IIInstituto de Computação, Universidade Federal de Alagoas - UFAL, Maceió, Brazil. E-mail: [e-mail]

^IIILIMOS UMR 6158 CNRS, Université Blaise Pascal, Clermont-Ferrand, France. E-mail: [e-mail]

^IVDepartamento de Engenharia de Produção, Universidade Federal de Minas Gerais - UFMG, Brazil. E-mail: [e-mail]

ABSTRAC

This paper describes an exact algorithm to solve a nonlinear mixed-integer programming model due to capacity expansion and flow assignment in multicommodity networks. The model combines continuous multicommodity flow variables associated with nonlinear congestion costs and discrete decision variables associated with the arc expansion costs. After establishing precise correspondences between a mixed-integer model and a continuous but nonconvex model, an implicit enumeration approach is derived based on the convexification of the continuous objective function. Numerical experiments on medium size instances considering one level of expansion are presented. The results reported on the performance of the proposed algorithm show that the approach is efficient, as commercial solvers were not able to tackle the instances considered.

Keywords: capacity expansion, flow assignment, global optimization, implicit enumeration, multicommodity flow problems.

1 INTRODUCTION

We consider a model for the joint problem of capacity expansion and flow assignment in multicommodity flow networks which takes into account congestion effects. The resulting optimization problem relies on the combination of two conflicting criteria: the expansion and the congestion cost functions. Capacity expansion cost functions are discrete as only a finite number of capacity sizes are available. On the other hand, congestion cost functions are nonlinear convex increasing functions as they try to capture queueing effects on the network [3, 4, 18, 23]. For instance, Ishfaq & Sox [18], in the context of an intermodal hub network, deal with shipment delays due to limited resources at logistics hubs for measuring service performance. The authors conducted a study of strategies to integrate hub operation queuing model and the hub locationallocation model on a 25-city road-rail intermodal logistics network. Our aim is to study the general multicommodity flow problem under such assumptions, with no particular applicationin mind.

Typically, the two decision levels of capacity expansion and flow assignment are decoupled to treat the corresponding difficulties separately, and few works, at least those considering queueing effects, have been done on the joint optimization problem. Gerla & Kleinrock [14] stated the general Capacity and Flow Assignment problem (CFA), decomposed it into simpler subproblems, and then suggested a heuristic procedure that alternates a capacity assignment phase with a flow assignment phase. Similar approaches have been proposed in successive papers by Gavish and co-authors [11, 12] and Gerla et al. [15]. Most of that literature concerns heuristic procedures, but the paper by Mahey et al. [20] is an exception for this rule, showing that a generalized Benders decomposition method can find exact solutions for the CFA problem.

The capacity expansion problem is a special case of CFA where initial capacities are already installed on each arc of the network. Given a traffic requirement matrix between origin-destination pairs, the problem consists in jointly deciding which arc capacities, if any, should be expanded and the flow assignment leading to a feasible routing that minimizes expansion and congestion costs. Thus, the problem results in finding a trade-off between investment and routing costs.

Luna & Mahey [19] modeled the capacity expansion and flow assignment as a piecewise convex multicommodity flow problem. Congestion is modeled by a convex increasing function for a given capacity and, at given breakpoints, which represents the maximum tolerable congestion for the users (thus strictly lower than the available capacity), expansion to a higher capacity is decided, decreasing the marginal congestion cost in a discontinuous way. The combinatorial nature of the problem, related to arc expansion decisions, is therefore embedded in a continuous objective function that encompasses congestion and investment costs. The resulting objective function is continuous, but it is nonconvex and nonsmooth. Mahey and Souza [22] derived local optimality conditions for the model proposed in [19]. By exploiting complete optimality conditions for local minima, Souza et al. [25] give the convergence analysis of the negative-cost cycle canceling method. Remark that these former works only consider the case of simple expansions from one installed capacity to a new one. The case of the general expansion problem where several capacity expansion values are available for each arc is analyzed by Ferreira & Luna [7], where a method to find solutions with performance guarantee is introduced.

The contribution of this paper is to provide an algorithm to assure the global optimality of the problem, also including computational experiments to show the efficiency of the method to cope with the case of simple expansions. After showing the correspondences between the continuous model of Luna & Mahey and a mixed-integer model, we analyze the numerical behavior of an implicit enumeration algorithm. The discrete model is used to assign capacities to a subset of arcs and hence to define what is called a partial solution. The continuous model is used to compute, given a partial solution, a lower bound on the value of the best solution that can be obtained with the assigned capacities. The resulting procedure is tested on different types of networks and shown to be quite efficient to solve the mixed-integer nonlinear model.

2 RELATIONS BETWEEN THE CONTINUOUS AND MIXED-INTEGER MODELS

This section presents the network expansion model. The basic component of the model is a digraph G = (V, E) with n nodes and m arcs. Any kind of traffic between a given pair of nodes is treated as a separate commodity k. Let T be a (n ×n) traffic requirement matrix such that t_ijis the traffic between origin i and destination j. We will consider the problem of deciding which arcs should be expanded from a given installed capacity c0 to a greater capacity c_lwhile minimizing the total congestion and expansion costs.

Given a commodity k, we consider the set of directed paths P_kjoining the corresponding origin and destination. Let x_kpbe the amount of flow of commodity k through the path p ∈ P_kand a_kpits arc-path incidence vector defined by

The vector x is composed by the component xe which denotes the total flow on arc e, and also by the component x_kpwhich denotes the flow of commodity k routed through path p. These two components are related by

The set of multicommodity flow vectors, denoted by M(T ) can be described by the arc-path formulation, i.e., for each commodity k flowing between nodes i and j, the active paths must satisfy

That implicit formulation (as the paths are not known in advance) is generally preferred to thenode-arc formulation where x_e = and each x^k is a flow vector on G satisfying flow constraints for commodity k.

We assume now that for each arc e in the topology is assigned a positive capacity c_0e that is expandable to a larger capacity chosen among a given set of capacities c_le < ... < c_Neat given fixed costs π_le, l = 1,..., N . Let δ_le =c_le -c_0e, l = 1,..., N , be the increment of capacity to the l-th capacity value. The capacity expansion model will minimize the total congestion cost plus the expansion fixed costs. Let Φ(c_e, x_e) be the arc congestion function for a given capacity c_e. It is assumed that Φ is expressed in terms of monetary values and that it is convex smooth and increasing up to infinity on the interval [0, c_e]. A common choice is the Kleinrock's average delay function valid for M/M/1 queues which is proportional to, see [3, 14].

A mixed integer model makes use of a binary variable y_le, l = 1,..., N, e ∈E, that assumes 1 if capacity of arc e is to be expanded from c_0eto c_leand 0 otherwise. We can now define a mixedinteger nonlinear model for the capacity expansion problem (DCE):

We will now study the relationship between (DCE) and a continuous model which does not makeuse of any boolean decision variables y (CCE):

where we assume π_0e= 0, ∀e ∈ E, and f_e(x_e) = min{Φ(c_le, x_e) + π_le, l = 0,..., N }.

Remarks.

1. Thanks to the feasibility assumption above and the fact that Φ(c_Ne, x_e) → +∞ whenever x_e↑ c_Ne, we do not need any capacity constraint in the continuous model.

2. As shown on Figure 1, where the nonconvex resulting arc cost function of (CCE) is represented by a bold line, we denote by γ_(l-1)e, l = 1,..., N the breakpoint at which expansion occurs from c_(l-1)e to c_le. The breakpoint can thus be interpreted as the capacity where congestion is such that the network manager is willing to pay for a new expansion. Thus, π_le - π(l-1)_e = Φ(c_(l-1)e, γ_(l-1)e) - Φ(c_le, γ_(l-1)e) is the new expansion cost converted in congestion cost units.

3. The arc cost function in (CCE) is continuous but nonconvex and nonsmooth at the break-points γ_le. It is shown in [19] how one can easily compute a lower bound on the optimal value of (CCE) by taking the convex envelope of each arc cost function.

Proposition 1. If (x, y) is feasible for (DCE), x is feasible for (CCE); If x is feasible for (CCE), then there exists y such that (x, y) is feasible for (DCE); If one of both problems is infeasible, so is the other one.

The proof is straightforward and it is omitted here for sake of simplicity, as the correspondence between (DCE) and (CCE) works with feasibility the same way it works with optimality, as developed below.

The following lemma is a direct consequence of the cost structure of (DCE).

Lemma 1. Let (x*, y*) be an optimal solution of (DCE); then, we have the correspondences:

Moreover, if there exists an arc e with = γ_(l-1)e, then either y*_(l-1)e = 1 and = 0 or y*_(l-1)e = 0 and = 1, so the optimal solution is not unique.

The two cases where is not a breakpoint are straightforward. If = γ(_l-1)e, we have:

Φ(c_(l-1)e, γ_(l-1)e) + π_(l-1)e = Φ(c_le, γ_(l-1)e) + π_le

which shows that the value of the arc cost function does not change whenever y*_(l-1)e = 1 and = 0 or, conversely, y*_(l-1)e = 0 and = 1. The correspondence between optimal solutions of (DCE) and (CCE) follows immediately.

Proposition 2. i) If (x* , y* ) is an optimal solution of (DCE), then x* is optimal for (CCE) and the cost values are equal.

ii) If x* is an optimal solution of (CCE), then (x* , y* ) is optimal for (DCE) with:

where l = 1,..., N, and the cost values are equal.

Finally, we would like to point out that the tight relationship between the optimal solutions of both models does not mean that they are equivalent. In general, the continuous model is not able to take into account additional constraints on the topology which, unlike, can be generally done by the y-variables. Nevertheless, we will mention a few common situations where it is possible to convert such constraints from (DCE) to (CCE):

a. Many models of network design require the same capacity on arcs (

i, j

) and (

j, i

) between two adjacent nodes

i

and

j

in G. The orientation of arcs (

i, j

) and (

j, i

) being mainly to model the flow that pass from

i

to

j

and from

j

to

i

in a common physical link. As these two flows actually share a common link, it is required in such models symmetry between capacities of arcs (

i, j

) and (

j, i

). This is modelled in (DCE) by the constraint

y_ij = y_ji

for some arc

e

= (

i, j

). To obtain the same effect, we must add the following constraints for each

l

in (CCE):

(x_ij- γ_lij )(x_ji- γ_lji ) > 0

b. Cutset constraints: Let A be a subset of nodes of V and C_Athe corresponding cutset. If subset A contains somehow crucial nodes for the network, the arcs in the cutset, i.e., those with one extremity in A and the other in V \ A, may be considered bottleneck arcs as they are the only to carry flow between nodes in A and nodes in V \ A. Thus, it may be of interest forcing the subset A to be connected to the other nodes by at least one expanded arc. This is modelled in (DCE) by the constraint Σ e ∈ C_Ay_le> 1, which is equivalent in(CCE) to:

Observe that both constraints derived in a. and b. define polyhedral nonconvex regions of ^m .Such a situation could not be treated in the following approach, that requires convexity to assureglobal optimality.

3 GLOBAL OPTIMIZATION STRATEGY

In this section, we follow the exposition provided by Geoffrion [13] to describe the application of an implicit enumeration algorithm to solve the capacity expansion and flow assignment problem. Geoffrion [13] presents, including his own developments, an unified approach of the works of Balas [1] and Glover [16] on the implicit enumeration scheme. See also Balas [2]. We consider that each arc e is expandable from an installed capacity c0e to a capacity c_1e, although the procedure can be generalized to deal with more than one possibility of expansion.

The implicit enumeration algorithm combines information from both discrete and continuous models. A partial solution S defines the capacities of a subset of arcs. Here, the discrete model is used to assign capacities to the arcs of . According to the notational convention introducedin [13], for each discrete variable associated to an arc in , the symbol e (resp. -e) denotes y_e= 1 (resp. y_e= 0). The discrete variables associated to arcs not in are called free. As an example, suppose a small network with five arcs and a partial solution S^t={1, 3, -5}. In this example, y₁ = 1, y₃ = 1, y₅ = 0, and y₂ and y₄ are free. A completion of a partial solution S is defined as a solution that is determined by Stogether with a binary specification of the values ofthe free variables. It is said to be a feasible completion if the assignment of values to the binary variables leads to a feasible solution. The four possible completions for Sin the above example are {1, 3, -5, 2, 4} where the free variables assume y₂ = 1 and y₄ = 1; {1, 3, -5, -2, 4} where y₂ = 0 and y₄ = 1; {1, 3, -5, 2, -4} where y₂ = 1 and y₄ = 0; and {1, 3, -5, -2, -4} where y₂ = 0 and y₄ = 0.

A key feature of implicit enumeration is the ability to generate information that can be used to exclude all the completions of a partial solution S from further consideration. Here, the continuous model is used either to provide a lower bound on the value of the best feasible completion of S, i.e., a feasible completion that minimizes the objective function among all feasible completions of S, or to show that S has no feasible completion. To do this, we solve a convex multicommodity flow problem PS:

where _e, e ∈ , is fixed at partial solution S and conv( f_e(x_e)) is the convex envelope of function f_e(x_e), c.f., Figure 1. Any efficient algorithm designed for convex multicommodity flow problems (see [24] for instance) can be employed to solve P_S. The value of the optimum solution of P_Sis a lower bound on the value of the best completion of S.

An upper bound is always possible to be derived if P_Shas an optimum solution. Let be an optimum solution of P_S, then

is an upper bound. Remark that for feasible problems, an upper bound is derived even if S = Ø. We store the best upper bound found during the search as the incumbent. If a partial solution S leads to an upper bound that improves upon the incumbent, then it replaces the latter as the new incumbent. If the lower bound is greater than or equal to or P_Sis infeasible, then S is fathomed. And in this case all completions of S have been implicited enumerated as they can be excluded from further consideration.

A partial solution is said to be nonredundant if it cannot generate a completion equal to one generated with a previous solution that was fathomed. Geoffrion [13] gives a procedure for generating nonredundant partial solutions that terminates fathoming all feasible solutions. By starting with = Ø, i.e., S⁰ has no capacity previously assigned to any arc, P_S0 gives a lower bound ₀ to the optimum value and a first upper bound _S0 taken as the incumbent . Then, the procedure augment the partial solution by assigning a capacity to an arc at a time. Now suppose a partial solution S^t is fathomed. Nonredundancy is achieved by having at least one element of subsequent partial solutions complementary to S^t. The last element that was added to S^t-1 to generate S^tis then underlined and changed to its complement. In the above example, the sequence would be S¹ ={1}, S² ={1, 3}, S³ ={1, 3, -5}. If S³ could be fathomed, then the next partial solution in the sequence would be S⁴ ={1, 3, 5}, which means change capacity assigned to arc 5 from c_0,5 in S³(y₅ = 0) to c_1,5 in S⁴(y₅ = 1). If S⁴ could also be fathomed, then S⁵ would be S⁵ ={1, -3}. Otherwise, S⁵ would be generated by assigning a value to a free variable and by adding it to S⁴, for instance S⁵ ={1, 3, 5, -2}. After fathom a solution S^t, the procedure locates the rightmost element of S^tthat is not underlined. If there is none, then end with the stored incumbent as being the optimal solution. Otherwise, replace this element by its complement underlined e →-e and delete all elements that are on the right. In the example, if S⁵ ={1, 3, 5, -2} and also S⁶ ={1, 3, 5, 2} could be both fathomed, the next solution generated would be S⁷ ={1, -3}.

We now discuss some strategies to choose an arc e ∈ \ E with which augment a partial solution S^t that cannot be fathomed. Note we chose an arc among those such that conv( f_e(_e)) < f_e(_e).Such strategies rely on the flow distribution of an optimum solution of P_St. Various strategies have been tested:

• assign c_1eto the arc e with the highest value f^'+_e (_e) of the right partial derivative of arc e cost function f_ewith respect to _e.

• assign c_0eto the arc e with the lowest value f^'-_e(_e) of the left partial derivative of arc cost e function fe with respect to _e.

• the arc inducing the highest value of when P_Sis defined for ∪{e} assigning c_0e to eif _e < γ_e and c_1eotherwise;

• the arc with the highest flow value;

The last rule was found to be the most efficient, and was adopted.

We employ algorithms proposed in [7], which have guaranteed performance, to find a good initial upper bound ₀. Those algorithms find a feasible solution and then gradually reduce the objective value of the obtained solution until no better solution can be found. They are based in two phases: a common first phase where a multicommodity flow problem taking the convex envelope of arc cost functions is solved and a lower bound is found; and a second phase where the obtained routing is used as starting point for switching methods between capacity assignment and the application of a local search algorithm until no more improvement occurs.

The pseudo-code of the implicit enumeration algorithm is as follows:

Implicit enumeration procedure

Step 1 - Initialize with S⁰ =Ø. Set t = 0 and = ₀. Step 2 -

Solve P_St to compute the lower bound z*_St to the best completion of S^t. If P_St is feasible, then obtain the respective upper bound _St and check to update .

Step 3 - If it is possible to fathom S^t, i.e., P_St is infeasible or z*_St > , then go to Step 5. Else go to Step 4. Step 4 - Augment S^t by adding a free variable ye with e or -e to obtain S^t+1. Set t =t + 1 and return to Step 2. Step 5 -

Locate the rightmost element of S^t not underlined. If none exists, then stop. Elsechange such element by its complement underlined e →-e and delete all elements to the right. Set t =t + 1 and return to Step 2.

4 NUMERICAL TESTS

Numerical tests were performed to analyze the numerical behavior of the proposed algorithm and the influence of the parameters on its performance. Three different network topologies were used for the computational tests: the C-NET introduced in [24]; the RING introduced in [11]; and the NTS100 generated using a special program driver [6, 7]. For the sake of illustration, Figure 2 and 3 present the topologies of networks C-NET and NTS100, respectively. Table 1 gives the main characteristics of the three networks used in our numerical experiments. Given a root node r, the hop-depth of a node i ∈ V \{r} is the number of arcs in the path between r and i that has minimum length. The hop-based diameter of the graph is the largest hop-depth among the nodes of the network. C-NET and RING are full duplex networks, i.e., flow can be sent between two nodes in both directions and simultaneously.

Two sets of tests were made with these topologies:

- The first set concerns the C-NET network [24]. The aim is to verify the influence of thetraffic throughput increase and the capacity expansion factor (c_1e/c_0e) on the number of iterations performed by the implicit enumeration algorithm. Results are compared with the ones obtained solving the mixed integer nonlinear formulation of the problem (DCE) using two commercial solvers: BARON and LINDOGlobal [5, 17, 27] and modeling the problem with GAMS [10]. The problems were solved with help of the NEOS Server [5, 17];

- The second test set was performed on the other two topologies RING and NTS100. The aim is to assess the effectiveness of the proposed implicit enumeration algorithm to solve the capacity expansion problem with different scenarios of larger networks and heterogeneous traffic requirement demands.

The computation of the lower bound was performed by a specialized Flow Deviation algorithm for the convex multicommodity flow problem. That algorithm has been shown to be efficient since the early work of Fratta et al. [9], but it is generally known to become very slow when the algorithm approaches the optimal solution (see [3] for instance). To accelerate its convergence, a parallel tangent procedure (PARTAN, see [8]) was introduced in the direction finding step. The algorithm was coded in C.

The following notations were used:

• Φ* is the global optimal value;

•

optimal value of the convexified problem;

• Φ_pg is the solution obtained by the performance guaranteed algorithms [7];

• Φ_Baron is the solution obtained with BARON [27];

• Φ_LG is the solution obtained with LINDOGlobal [5, 17];

• N_ggo is the number of partial solutions enumerated to guarantee the global optimality;

• N_ois the number of partial solutions enumerated until the optimal solution is found be not necessarily proven to be optimal;

• N_Baronis the number of visited nodes made by the Baron solver;

• N_LGis the number of visited nodes made by the LINDOGlobal solver;

• α =

;

• α_pg = .

4.1 First test set

In these experiments, the congestion cost is derived from the Kleinrock's average delay function for M/M/1 queueing networks and defined by ρ . We fix ρ = 1, see [11]. All arcs have the same initial capacity. And the expansion cost π_1e, for each arc e ∈ E, is given, for each value of the parameter γ = equal to 0.7 and 0.9, by

Tables 2 to 5 present the results for the C-NET network varying traffic requirement demands, parameter γ and the available capacity for expansion.

The results show that the algorithm is rather sensitive to the capacity expansion factor. When c_1e = 4c_0e the number of iterations increase significantly, as observed for N_ggoand N_oin Tables 4 and 5 in contrast with N_ggoand N_oin Tables 2 and 3. Increasing γ affects the efficiency of the algorithm. Both effects can be illustrated by the values of α and α_pg.

As the efficiency of the algorithm depends on the quality of the lower bound to exclude completions of a partial solution from further consideration, the weakest is the convexification bound, the biggest is the number of solutions enumerated. Observe that the global optimal solution was reached in many instances, some of them well before N_ggo» N_o.

Table 6 shows the results obtained solving the mixed integer nonlinear formulation of the problem (DCE) using two commercial solvers: BARON and LINDOGlobal [5, 17] and the resultsof the proposed algorithm. The Baron and the LINDOGlobal solvers were not able to guarantee the optimality of the results within the adopted time limits and or the number of iterations limits. They were not able to obtain the guaranteed global optimum of the C-NET problem. The LINDOGlobal solver uses linear approximations of the original problem which explains the large number of iterations within the limits of runtime.

4.2 Second set of tests

The second set of tests concern the RING and the NTS100 topologies with heterogeneous trafficrequirement demands. The congestion costs, as in the precedent set of experiments, is given by ρ

. The following scheme was adopted for all these experiments:

- An initial feasible solution is computed using algorithms proposed in [7].

- The initial demand between each OD-pair was fixed to 10 for the RING network and to 3 for the NTS100 network. The installed capacities are given in Table 7.

• In a first set of instances, the demand is uniformly multiplied by a throughput factor of 20%, 40%, 60%, 80% and 100% until capacity expansion becomes economically interesting.

- In the second set, the demand increase is no more uniform: half of the demands receive the same throughput factor, but 25% receive 50% more, and 25% receive 25% less. Theresults are exposed in Tables 8 to 11.

- An initial distribution of the increase of demand between the OD-pairs was defined randomly and fixed for each instance.

- The parameter ρ used to compute congestion costs is set to 500, 1000, 5000, and 10000.

A small number of iterations was needed to solve the proposed instances. Considering the complexity of the studied problem (CNET = 2³⁴ = 17,179,869,184 solutions, RING = 2⁶⁰ solutions, NTS100 = 2¹⁸⁷ solutions), one can evaluate the effectiveness of the proposed method. The second set of test problems was easier to be solved. This can be explained by noting that for these instances, the quality of the lower bound obtained by solving the problem with the convex envelope of the integrated function of congestion and expansion costs, c.f., Figure 1, is very good.

At each iteration of the implicit enumeration algorithm a convex multicommodity flow problem is solved. A convex multicommodity flow problem can be difficult, especially if feasibility problems appear when an arc flow is close to the arc capacity. The decision to apply the Partan was primarily due to the gain in speed in solving problems is adequate for a precision of about 1% or better. Each multicommodity flow subproblem spent from a second fraction to several minutes.

All test problems have many local optima with objective function value close or equal. The implicit enumeration depth search approach has modest memory requirements. It needs to store only the path from the root node to the leaf node in the search tree. As the problem has many solutions, the depth search has a good chance of finding an optimal solution after exploring a small portion of the entire search space. The proposed algorithm can be parallelized and we believe that a good scalability can be get.

The proposed algorithm can be further specialized if more information about the problems are considered. For example, in real scenarios budget and operational constraints may restrict the set of candidate arcs for expansion. The proposed procedure can also be used for capacity reduction problems.

5 CONCLUSION

We have applied successfully an adapted implicit enumeration scheme to a mixed-integer nonlinear model for the capacity expansion and flow assignment in multicommodity networks. The key fact to explain the good performance of the method is the use of a reliable bounding procedure at each node of the enumeration tree, which is based on the convex hull of the equivalent continuous objective function.

The literature has scarce approaches to find exact solution of the discrete capacity expansion and flow assignment problem. Basically two lines of approaches have been adopted: Benders decomposition [20] and methods based on difference of convex functions - DC programming [21]. Recently a conic quadratic formulation was proposed [26]. The results show that the implicit enumeration algorithm proposed here can be considered as an alternative available to solve to global optimality such large scale problems.

Future research is to be done on the extension of the proposed approach to deal with multiple choices of available capacities for expansion on each arc. It remains to show that lower bounds obtained with the convex envelope of the integrated function of expansion and congestion costs remain sharp in the presence of multiple expansions.

ACKNOWLEDGMENTS

This work has been supported by CNPQ (305164/2010-4) and (473763/2011-7).

Received December 9, 2011

Accepted January 22, 2013

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Corresponding author

Publication Dates

• Publication in this collection
  16 July 2013
• Date of issue
  Aug 2013

History

• Received
  09 Dec 2011
• Accepted
  22 Jan 2013