a tabu search approach towards congestion and total flow minimization in optical networks

ISSN 1004-3756/04/1302/180 JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING CN11-2983/N ©JSSSE 2004 Vol. 13, No. 2, pp180-201, June, 2004

A TABU SEARCH APPROACH TOWARDS CONGESTION AND

TOTAL FLOW MINIMIZATION IN OPTICAL NETWORKS* Valter BOLJUNČIĆ Darko SKORIN-KAPOV Jadranka SKORIN-KAPOV

Faculty of Economics and Tourism “Dr.Mijo Mirković”

Pula University of Rijeka, Preradovićeva 1/1, 52100 Pula, Croatia School of Business

Adelphi University, Garden City, NY, 11530, USA W.A.Harriman School for Management and Policy

State University of New York at Stony Brook, Stony Brook, NY 11794-3775, USA [email protected] [email protected] [email protected]

Abstract

This paper considers rearrangeable multihop lightwave networks whereby each network node is equipped with a number p of transmitters and receivers, and a spectrum of wavelengths is accessible by, and shared among, all nodes by using the Wavelength Division Multiplexing (WDM). Depending on input traffic flow, nodal transmitters and receivers can be re-tuned to create virtual connectivity best suited with respect to a given optimization criterion. We present an efficient heuristic algorithm that combines two criteria for optimization: throughput maximization, as well as total flow minimization. Throughput maximization criterion is equivalent to congestion minimization, while minimizing total flow under the assumption of having links with equal lengths implies minimization of the average number of hops. Taking into account lengths of the links (i.e. link costs proportional with distances), the total flow minimization becomes equivalent to the total delay minimization. Tabu search is implemented as a two-phase strategy dealing with diversification as well as intensification of search. Computational experiments include consecutive runs with different sets of weights associated with the two criteria. Results for a benchmark set of problems are presented.

Keywords: Heuristic solvability, tabu search, multihop, rearrangeable optical networks, minimal total flow, maximal throughput

* This work was supported partly by the National Science Foundation Grant ANI 9814014, by the project 036033-Architectural Elements for Regional Information Infrastructure, funded jointly by the Ministry of Science, Education and Sports of the Republic of Croatia and the Istrian County, and by the project 067010-Models and Methods of Operational Research funded by the Ministry of Science, Education and Sports of the Republic of Croatia.

1. Introduction Optical networks present a fast and reliable

medium for transferring information, especially high volume data. Due to their structure, optical networks have possibilities for reconfiguration

BOLJUNČIĆ, SKORIN-KAPOV and SKORIN-KAPOV

JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING / Vol. 13, No. 2, June, 2004 181

by re-tuning node transmitters and receivers to different wavelengths. Namely, we assume that each network node is equipped with a number p of transmitters and receivers, and that a spectrum of wavelengths is accessible by, and shared among, all nodes by using the Wavelength Division Multiplexing (WDM). Tuning a transmitter of node i and a receiver of node j to the same wavelength establishes a logical link (i,j) through which traffic can be sent. Re-tuning transceivers results in new connectivity diagrams, making logical connectivity independent of physical architecture. This provides a possibility to design the so-called virtual topology which is, then, imposed on the existing physical topology consisting of fiber links. Depending on the input traffic flow, optical networks can be optimized by taking into account different criteria. This, in turn, contributes to their cost effectiveness. The advantage of an optical network is the ability to retune optical transmitters/lasers and/or receivers directly on the nodes of the network in order to change their connectivity (assuming a broadcast physical architecture). In a general non-broadcast physical network one would rely on a sub-layer of electronic or optical cross-connects to change the connectivity among service nodes.

Relevant previous research reported in literature considered congestion minimization or, alternatively, maximization of throughput, as well as delay minimization. A discussion of different design objectives is presented in (Labourdette 1998). Furthermore, previous research distinguished between arbitrary and regular network topologies. This work deals with arbitrary networks since they are more

general and exist for every network size. In our application the input flow related to a

source-destination pair of nodes can be split and travel via different routes, and can traverse a number of intermediate nodes. In literature, these types of networks are called multihop networks. When optimizing only the throughput through the network, the flow will tend to pass through many hops, in turn increasing the total network flow. Increased network flow might not be desirable from the managerial point of view since it will require bigger resources, and will make it more difficult for network owners to devise a fair cost allocation scheme among users.

This paper presents an efficient tabu search algorithm that combines both criteria for optimization: throughput maximization, as well as total flow minimization. Throughput maximization criterion is equivalent to congestion minimization and we want to minimize the maximal flow on a link. The total flow is the sum of flows on all links. Minimizing the total flow under the assumption of having links with equal lengths implies minimization of the average number of hops. Virtual topologies with a small number of hops are desirable from the economical point since that leads to the reduction of the cost of the station equipment (Stern and Bala 2000). Taking into account lengths of the links (i.e. link costs proportional with distances), the total flow minimization becomes equivalent to the total delay minimization. This might be desirable in a nation-wide network in which the total cost is related to the total bandwidth-miles. In the context of broadcast optical networks, (Labourdette 1998) presents a survey of

A Tabu Search Approach towards Congestion and Total Flow Minimization in Optical Networks

182 JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING / Vol. 13, No. 2, June, 2004

different minimization objectives. For wavelength-routed networks, minimization of network congestion, as well as minimization of average hop distance is surveyed in a recent book (Jia et al. 2002).

Tabu search has been successfully used in numerous applications and its guiding principles are presented in, e.g. (Glover and Laguna 1997). For example, a tabu search algorithm for topological optimization of telecommunication networks, whereby the number and the locations of the multiplexing centers have to be decided, is presented in (Costamagna et al. 1998.) An adaptation of tabu search to the problem of establishing the system of virtual paths in an ATM network is described in (Shyur and Wen 2001).

Combining meta-heuristics with multiobjective programming proved to be a fertile area of research. A recent overview of multi-objective meta-heuristics (including tabu search, genetic algorithms and simulated annealing) is presented in (Jones et al. 2002). An annotated bibliography of multiobjective combinatorial optimization is given in (Ehrgott and Gandibleux 2000). Some representative papers include the following: in (Viana and deSousa 2000) a resource constrained project scheduling is solved via multiobjective tabu search and simulated annealing; (Alves and Climaco 2000) provides an interactive method for solving general 0-1 multiobjective problems using simulated annealing and tabu search; multicriteria tabu search applied to 0-1 knapsack problems is presented in (Gandibleux and Freville 2000). Benefits of using multicriteria models in the context of telecommunication networks are presented in (Antunes et al. 1998).

Simultaneous considerations of multiple criteria make trade-offs between sometimes conflicting objectives easier to grasp and evaluate.

The heuristic approach presented in this paper adapts the tabu search strategy proposed in (Skorin-Kapov and Labourdette 1995) for throughput maximization, to the multicriteria problem of simultaneously minimizing congestion and total flow. In addition, strategic algorithmic enhancements are twofold: 1) a more elaborate diversification of search implemented via two different long term memory functions, and 2) an improved intensification of search implemented via the so-called Phase 2 of search. The proposed method improves the best known solutions for the 24-dimensional benchmark problems from (Skorin-Kapov and Labourdette 1995). The tabu search strategy is incorporated in an iterative process of changing weights associated with two different optimization criteria, in the process utilizing the information from past search, and identifying solutions performing well over a range of weight-pairs.

Following a strategy from (Skorin-Kapov and Labourdette 1995), we decompose the problem into two subproblems: the connectivity and the routing problems. The connectivity problem is a linear assignment problem that seeks a 0/1 solution maximizing the one hop path traffic. This criterion, originally used in the context of minimizing congestion, is even more suitable when minimizing the total flow. The routing problem is a multicommodity flow problem with the objective function that is a weighted sum of both criteria. A local search based on branch exchange and tabu strategy is used to explore different virtual topologies.



Computational results are obtained for benchmark sets of networks of size 14 and 24. In this paper we develop a strategy to obtain robust solutions. In particular, our task is to find connections among nodes that support both the maximal (or very good) throughput, yet do not increase excessively the total network flow.

The paper is organized as follows. In Section 2, the formulation of our problem as a mixed integer problem is presented. An outline of our tabu search algorithm is given in Section 3. Computational results are presented in Section 4. Conclusions and future research are indicated in Section 5.

2. Problem Formulation In this paper we modify the formulation used

in (Skorin-Kapov and Labourdette 1995). The minimal congestion problem (entitled The Flow and Wavelength Assignment (FWA) problem) was previously considered in a number of studies including (Labourdette and Acampora 1991), (Yener and Boult 1994), (Bienstock and Gunluk 1995). We formulate the problem as a mixed integer program as follows. The input N×N traffic matrix Tst presents the traffic flow from source s to destination t. The number of transmitters and receivers is set to p for any node, and the capacity of any link equals C. (For simplicity the capacities of all channels are assumed to be equal and large enough, so that the feasible solution exists.) Let zij be a {0,1} variable indicating whether or not the virtual link (i,j) is used in the network, and let fkij be a continuous variable indicating the amount of flow originating at source k, sent through the link (i,j). We would like to minimize congestion

(denoted as F), as well as the total flow sent through the network and obtained by summarizing flows on all virtual links (denoted as T). The above criteria might be conflicting for the best-obtained solutions since congestion minimization implies sending flow over longer paths (i.e. making more hops), in turn contributing to larger total flow. We will include both criteria with respective weights in the objective function. The formulation is then:

min

s.t.F TF Tω ω+

kijk i j

f T, ,

≤∑ (1)

for allkijk

f F i j i j≤ , , ≠∑ (2)

for allkij ijk

f Cz i j i j≤ , , ≠∑ (3)

for allkij kji kji j i j

f f T k j k j≠ ≠

− = , , ≠∑ ∑ (4)

for allijj i

z p i≠

= ,∑ (5)

for alljij i

z p i≠

= ,∑ (6)

0 {0 1} for allkij ijf z k i j i j≤ , ∈ , , , , , ≠

Constraint (1) assures that the total network flow is being minimized with the weight ωT, (2) assures that with weight ωF the maximum flow on any link is minimized, (3) enforces the capacity constraints on links, (4) are conservation of flow constraints, and (5, 6) are assignment type constraints assuring p transmitters and receivers on any node.

The above formulation does not take into account lengths of links, hence the total flow minimization is equivalent to minimizing the number of hops. This criterion might be



economically justified when distance is not a constraining factor, and the increased number of hops (i.e. re-tuning of transmitters/receivers) adds to the complexity of the network. However, without loss of generality, constraint (1) can be restated as ∑k,i,j di,j fkij ≤ T where dij represents the length of link (i,j). Minimizing the total flow then becomes equivalent to minimizing the total delay. This minimizes, for example, the total bandwidth-miles of the network, and the criterion might be useful for finding the capacities to assign to links.

As in (Skorin-Kapov and Labourdette 1995), the problem is first decomposed into two subproblems: the connectivity and the routing problems. The connectivity problem (CP) is a linear assignment problem maximizing the one hop path traffic:

max ij ij ijT z∑

s.t.

for allijj i

z p i≠

= ,∑ (7)

for alljij i

z p i≠

= ,∑ (8)

0 1 for allijz i j i j≤ ≤ , , ≠

The solution to the connectivity problem,

ijz , is used in the formulation of the routing

multi-commodity flow problem (RP):

min F TF Tω ω+

s.t.

kijk i j

f T, ,

≤∑ (9)

for allkij ijk

f z F i j i j≤ , , ≠∑ (10)

for allkij kji kji j i j

f f T k j k j≠ ≠

− = , , ≠∑ ∑ (11)

0 for allkijf k i j i j≤ , , , , ≠

Since the complete problem is very difficult to solve, even with today’s state of the art mixed integer programming solver, we resort to its heuristic solvability. The tabu search strategy presented in (Skorin-Kapov and Labourdette 1995) is modified in order to improve diversification as well as intensification of search. Furthermore, the search takes into account both criteria for optimization, hence the modification involves a different rule for neighborhood evaluation. Our main tasks in this work are:

(1) to systematically run the algorithm with different values of weights associated with our two criteria (congestion minimization and total flow minimization) in order to observe the behavior of the proposed solutions;

(2) to assess the robustness level of proposed solutions; and

(3) to propose guidelines for decision makers regarding good values for simultaneous weight allocation for both criteria.

3. Tabu Search Strategy The tabu search algorithm of (Skorin-Kapov

and Labourdette 1995) was developed for the FWA problem whereby the sole criterion for optimization was congestion minimization. In their work, the neighborhood of a feasible solution consisting of the connectivity matrix, the flow routing matrix, and the maximal flow an any link, was defined as the set of feasible solutions obtained after performing a branch-exchange (BE) operation and resolving



the routing sub-problem. The BE operation replaces two existing links (i,j) and (k,l) by the two new links (i,l) and (k,j). Due to the computational complexity of evaluating the complete neighborhood, the neighborhood was evaluated only partially by employing the so-called candidate list strategy. Based on the intuitive fact that least utilized links do not contribute to solution quality, the BE operation was performed on K least utilized links which did not disconnect the network. Diversification strategy was implemented via a frequency-based long term memory function used in generating new starting solutions.

In this paper we modify the strategy of (Skorin-Kapov and Labourdette 1995) by taking into account the two criteria: congestion minimization and total flow minimization. We perform tabu search runs with different sets of weights associated with the two criteria and try to utilize the search information from consecutive runs. Our task is to find a solution that performs well on both criteria simultaneously. Furthermore, we enhance the tabu search strategy of (Skorin-Kapov and Labourdette 1995) by adding an additional diversification function, and by adding an intensification phase when “reasonably" good solutions have been obtained.

Following the notation of (Skorin-Kapov and Labourdette 1995), let us denote our solution as (con, rout, F, T), where con denotes the connectivity matrix, rout is the corresponding matrix of flow values, F is the maximal flow value, and T is the total network flow. For a given solution, its neighbor is denoted as (conn, routn, Fn, Tn ) and presents the feasible solution obtained by performing a branch-exchange (BE)

operation, and by re-solving the routing problem. For a network with N nodes and p transceivers, under the assumption that the available spectrum of wavelengths is accessible by all nodes, there will be Np available links for transferring the flow. This implies that the evaluation of complete neighborhood results in O(N2p2) routing problems, each having O(N3) variables and O(N2) constraints. Except for small problems (e.g. with less than 10 nodes), the evaluation of complete neighborhood is computationally intractable. Hence, we decided to evaluate a subset of “promising" solutions, referred to as the candidate list (i.e. the list of candidate solutions for limiting the search). First, we ordered all the links in decreasing order of their utilization. Preliminary strategy showed that when bigger weight was given to congestion minimization, better results were obtained by considering the K least utilized links. When the weight for total flow minimization was bigger, the strategy of using K most utilized links was preferable. In the spirit of simultaneous optimization on both criteria, we decided to perform branch exchanges by considering pairs of most and least utilized links. Hence, given K most utilized and K least utilized links, the K- neighborhood of a given solution consists of all feasible solutions obtained by branch exchanges involving those links.

An iteration of the search involves evaluation of the K-neighborhood of a current solution. Namely, all K-neighbors of a current solution are evaluated and the admissible neighboring solution with the smallest objective value (i.e. the weighted sum of objectives) is selected as the next current solution. The reversal of this move is temporarily forbidden



by recording it in a tabu list. The tabu list is implemented as an N×N matrix with (i,j) entry denoting the iteration at which link (i,j) is not tabu. We start with the matrix of 0’s, meaning that all links can be exchanged at the first iteration. During the course of the algorithm, the tabu list is modified as follows: if at iteration I the old links (i,j) and (k,l) are replaced by the new links (i,l) and (k,j), then in order to forbid the reversal of this move for the tabu_size number of iterations, we update the matrix by recording tabu_list(i,j)=I+tabu_size and tabu_list(k,l)=I+tabu_size. The tabu status of a link becomes inactive only if the move including this link leads to a solution better than the best found previously. In order to avoid possible cycling, tabu_size is randomly perturbed using a specified interval before starting with a new initial solution.

Our tabu search strategy is designed for consideration and comparison of different weights given to the two criteria for optimization. We start with 100% weight given to congestion minimization and 0% weight given to total flow minimization. In consecutive runs the weights are modified for a prescribed percentage until we reach the opposite end: 0% weight for congestion minimization and 100% weight for total flow minimization. In each consecutive change of weights, the best-obtained connectivity matrix from the previous set of weights is selected as the initial connectivity for the next set of weights. The best connectivity matrix obtained for each pair is saved and used in re-evaluation of best routings for all weight pairs. Our strategy was to (possibly) identify a small number of connectivity matrices that would provide good throughput and relatively

small total flow for a range of different weights associated with congestion and total flow. In a sense, we wanted to generate a good and robust solution.

Tabu strategy is further enhanced by providing a two-phase search. The first phase, termed the initial tabu search, is designed for more aggressive diversification. It employs two different diversification strategies implemented via the corresponding N×N matrices. The first long term memory strategy, implemented via the LTM1 matrix, constructs new initial solutions by modifying the original “flow" entries as follows. When the old links (i,j) and (k,l) are replaced by the new links (i,l) and (k,j), we modify LTM1(i,l)=LTM1(i,l)+Til and LTM1(k,j)= LTM1(k,j)+Tk,j. Given such modifications, when the connectivity problem is resolved in order to generate the new initial solution, the search will be geared towards selecting promising links. Hence, this diversification strategy has elements of intensification as well. Similar diversification strategy was proposed in (Skorin-Kapov and Labourdette 1995). In this work we enhance the search by proposing an additional diversification strategy, implemented via the LTM2 matrix. It has a role somehow opposite to LTM1: for the non-zero entries in the best obtained connectivity matrix we set the corresponding entries of LTM2 to zero, and the other entries of LTM2 are set to the initial traffic values. Hence, for the links comprising the best virtual topology, we set the corresponding initial traffic values to zero and resolve the connectivity subproblem. The obtained solution will exclude the “best” links, providing a diversified re-start.

After completing Phase 1, further intensification around the best-obtained solution



is provided in Phase 2. For the first set of weights we start with the best solution from Phase 1. For consecutive sets of weights, we employ two starting solutions: (i) the best connectivity matrix from the previous set of pairs, and (ii) the best solution for the current set of pairs from Phase 1. At the end, as in Phase 1, we further try to improve solutions by re-running the routing subproblem for all pairs on all best connectivity matrices associated with pairs of weights. By using the best solutions from Phase 1 as starting points, this phase intensifies the search in a promising area of feasible space.

The pseudocode of our tabu search strategy follows. Input data: number of nodes N, number of

transceivers p, traffic matrix T. Initialization: Set the input parameters: basic

tabu_size, random perturbation interval δ, number of diversification restarts LTM1_restart and LTM2_restart, number of iterations per restart max_iter, size of neighborhood K, rate of change in pairs of weights δw, percent increase in K for Phase 2 (δK), percent decrease in LTM1_restart for Phase 2 (δLTM1). Initialize tabu_list to the matrix of zeros and LTM1 to the initial traffic matrix. Start the counters LTM1_count=0 and LTM2_count=0. Set ωF=1, ωT=0.

1. Phase 1 – Initial Tabu Search 1a. Phase 1 – Connectivity

Solve the connectivity problem (CP) with the traffic matrix LTM1 to obtain con.

1b. Phase 1 – Routing Solve the routing problem (RP) to obtain

rout, F, T. 1c. Phase 1 – Tabu Search Improvements

iter = 0. While iter ≤ max_iter do

For each admissible BE evaluate the exchange by solving the routing problem. Perform the best exchange if it improves the incumbent, and record it as the new incumbent (con*,rout*, F*,T*). Otherwise, perform the best non-tabu exchange. Update the current solution (con, rout ,F, T). Update the tabu_list. Update the LTM1 matrix. Set iter=iter + 1

LTM1_count = LTM1_count + 1. 1d. Phase 1 – Diversification 1

While LTM1_count ≤ LTM1_restart do Reset the short term tabu_list memory to the zero matrix. Set tabu_size = tabu_size + random perturbation Go to Step 1a to apply the long term memory traffic matrix, LTM1, to generate a new starting solution.

LTM2_count = LTM2_count+1. 1e. Phase 1 – Diversification 2

While LTM2_count ≤ LTM2_restart do Reset the short term tabu_list memory to the zero matrix. Set tabu_size = tabu_size + random perturbation. Generate the LTM2 matrix as follows: for zero entries in the best connectivity matrix set the



corresponding LTM2 values to initial traffic values, otherwise set the LTM2 values to 0. Go to Step 1a.

1f. Phase 1 – Output The best solution (con*,rout*,F*,T*), the iter, the LTM1_count, and the LTM2_count when the solution was obtained.

1g. Phase 1 – Change of Weight Pairs While ωF ≥ 0 and ωT ≤ 1do

ωF=ωF – δw and ωT=ωT+δw Go to Step 1b using the best connectivity matrix from the previous set of weight pairs.

1h. Phase 1 – Re-Evaluation of Best Solutions for All Weight Pairs For every pair of weights (ωF,ωT) = (1,0)…(0,1) (in increments of δw) solve the routing problem with all best connectivity matrices obtained. For each pair of weights record the best solution as the incumbent (con*,rout*,F*,T*).

2. Phase 2 – Intensification Set K=K∗(1+δK). Set LTM1_restart = LTM1_restart ∗ (1-δLTM1). Set LTM2_restart = 0. Set ωF=1, ωT=0.

2a. Phase 2 – Start Perform Steps 1c – 1d starting with the best solution from Phase 1. Record the best solution as the incumbent (con*,rout*,F*,T*).

2b. Phase 2 – Change of Weight Pairs While ωF ≥ 0 and ωT ≤ 1 do ωF=ωF – δw ; ωT=ωT+δw

i) Perform Steps 1b–1d starting with the best connectivity matrix from

the previous set of weight pairs ii) Perform Steps 1c–1d starting with the best solution from Phase 1. Compare the outputs from i) and ii) and record the better one as the incumbent.

2c. Phase 2 – Re-Evaluation of Best Solutions for All Weight Pairs For every pair of weights (ωF,ωT) = (1,0)…(0,1) (in increments of δw) solve the routing problem with all best connectivity matrices obtained. For each pair of weights record the best solution as the incumbent (con*,rout*, F*,T*).

3.Calculate % Improvement in Objective

Value For every pair of weights (ωF,ωT) = (0,1)…(1,0) (in increments of δw) Calculate 100(objective value_Phase 1 – objective value_Phase 2)/objective value_Phase 1.

4. Computational Results The computational experiments were

performed on a Dell Dimension XPS B8000 PC with Pentium III processor at 800 Mhz. The algorithm was coded in C, and the routing subproblems were solved using Cplex 7.0 callable library from Cplex Optimization, Inc. It should be noted that the complete formulation (1)–(6) with eight nodes and four transceivers (the quasiunif1 data set from (Skorin-Kapov and Labourdette 1995) with the sole objective of minimizing congestion was solved optimally via the exact algorithm provided by the standard CPLEX implementation. However, it took about



thirty six hours. The 14-dimensional program was running for four days, after which time it was killed. Hence, it seems that heuristic approaches are still the only available option for relatively larger problems.

Data sets used in our computational study are benchmark data as follows:

1. 14-dimensional traffic matrix from (Mukherjee et al., 1996) which consists of bytes/sec and is an actual measurement of the traffic on the T1 NSFNET backbone for a 15-minute period (11:45pm to midnight) on January 12, 1993;

2. Three 24-dimensional quasiuniform matrices with different variability as presented in (Skorin-Kapov, Labourdette, 1995). The parameters include: ρ determining the number of non-zero entries as well as their initial value; g determining the scaling of non-zero entries; and ∆ determining the spread around the scaled value of non

zero-entries. The data instances have different variability as follows: for 24-1 instance, ∆=10; for 24-2 instance, ∆=50; and for 24-3 instance, ∆=250.

As an example of a physical network, Figure 1 presents a 14-node NSFNET network.

Our algorithm was tested for networks with p=2,3,4 transceivers. The current topologies allow for larger sizes of p, however, the increased size of p would result with less constrained problem and, hence, with the problem that would be easier to solve. For comparison purposes, and without loss of generality, we decided to use the same p sizes as used in (Skorin-Kapov and Labourdette 1995).

We started with the 100% weight given to congestion minimization and 0% weight given to minimization of total flow. In order to deal with different order of magnitude of congestion and total flow appearing in the objective function, we applied range equalization factors as proposed in (Steuer 1986) for multiple criteria

Figure 1 CNSFNET T1 Network



optimization. To obtain suitable ranges for congestion and total flow, we first run the algorithm with the sole objective of minimizing congestion. Let the obtained congestion value be F1 , and the corresponding total flow T1. We then run minimization of total flow only, obtaining T2 and its respective congestion F2.

The range equalization factors were calculated as

1

2 1 2 1 1 2

1 1 1Fp

F F F F T T

−⎛ ⎞

= × +⎜ ⎟− − −⎝ ⎠

and 1

1 2 2 1 1 2

1 1 1Tp

T T F F T T

−⎛ ⎞

= × +⎜ ⎟− − −⎝ ⎠.

The objective function used in routing subproblems was then min ωF(pFF)+ωT(pTT).

Our tabu search strategy is split in two phases: Phase 1 initializes the search and performs aggressive diversification implemented through two different long term memory functions, while Phase 2 tries to improve the incumbent solution via intensification in the promising region of the feasible space. Regarding the tabu search strategy, our tasks in running the computational experiments were as follows:

(1) Assess the merit of diversifications in Phase 1 in obtaining good quality solutions;

(2) Assess the merit of Phase 2 by specifying what percentage of solution improvement can be obtained through application of Phase 2 with respect to Phase 1.

Regarding the consideration of multiple objectives, we wanted to evaluate the efficient

solutions for each pair of weights, with respect to the linear combination of the objectives, by systematically changing weights associated with congestion and total flow. The consecutive weights were modified in decrements (resp., increments) of 10% for congestion (resp., total flow). For the initial pair of weights, the starting solution (con, rout, F, T) was obtained by first solving the connectivity subproblem, followed by the routing subproblem. For subsequent weight-pairs it was observed that, in many instances, better final solutions were obtained when starting from the best connectivity matrix from the immediately preceding weight-pair. Moreover, since the initial solution directs the trajectory of search and evaluation of a subspace of the feasible set, ’building up’ upon the previous search resulted with final connectivity matrix similar to the final connectivity matrix from the preceding weight pair. Hence, it was decided that in each consecutive step the best connectivity matrix from the previous step is to be used as a starting solution. In many instances, our results indicated few connectivity matrices that support best routings over a large range of possible pairs of weights. This suggests that some solutions are robust and could be preferable in a dynamic environment necessitating change of weights.

Results related to the 14-dimensional NSFNET data from (Mukherjee et al. 1996) are presented in Table 1. The weight for congestion minimization is denoted by ωF , while ωT denotes the weight for total flow minimization. Parameters for Phase 1 of tabu search were set to K=5, tabu_size=4, δ=[1,4], max_iter=10, LTM1_restart =3, and LTM2_restart=2. For Phase 2 we used K=9, max_iter=20, and



LTM1_restart =1. For each weight pair we list the congestion

and total flow obtained at the end of Phase 1. We also list percentage of improvement in the combined objective function attributed to Phase 2 search. The overall improvement sometimes leads to deterioration in one of the two criteria. These percentages are displayed in the last two columns of Table 1.

For a given number of transceivers p , Phase 1 tabu search for all eleven pairs of weights took less then two hours. Diversification was used more aggressively for initial pairs of weights. However, since for each pair of weights the initial starting solution is set to the best solution from the immediate predecessor pair, the impact of previous diversification is carried further. When at the end of Phase 1 in Step 1h we re-evaluated best connectivity matrices for all weight pairs, it appeared that some connectivity patterns were best for a range of weight pairs. For p=2, the connectivity matrix obtained for the pair (ωF,ωT)=(0.2,0.8) performed best for six different weight pairs; for p=3, the connectivity matrix obtained for (ωF,ωT)=(0.8,0.2) was best in ten out of eleven cases. For p=4 three different connectivity matrices proved best in three or four other cases as well. These results point to some robust connectivity patterns that support a significant range of weight changes. The results also justify a systematic consecutive evaluation of pairs of weights across their range: it seems that search “intelligence" gets accumulated in consecutive runs leading to better and more robust solutions.

Phase 2 for the 14-dimensional NSFNET data set run for about seven or eight hours regardless of the number of transceivers used.

(The time includes all eleven runs for different weight pairs.) In the description of the algorithm we indicated that we restart Phase 2 twice: first with the best connectivity matrix from the previous set of weights and, second, with the best solution for the current set of weights from Phase 1. The results reveal that this strategy has merit since in some cases (i.e. for some weight pairs) the best solution is attained from the first initial solution, and in other case from the second initial solution. The improvements in the objective function overall were about few percent, sometimes resulting with deterioration of either congestion or total flow. Of course, the most desirable overall improvement is the one that simultaneously improves both criteria. In any case, the fact that the improvement phase did not produce more significant results indicates that Phase 1 already gives very good results. (This is further evidenced in runs for 24-dimensional data sets from (Skorin-Kapov and Labourdette 1995) whereby our solutions improve on the best published results.)

It seems that congestion is more volatile and prone to sudden increases or drops when weights change. The total flow seems to change more uniformly. The intuition is that the objective of minimizing congestion allows for less flexibility in attaining it and, hence, forces solutions to change more significantly when its weight is changed. The flexibility in routing should be related to the number of transceivers used: higher number of transceivers allows for more freedom in routing, decreasing the congestion, as well as total flow. To that end, Table 1 displays the ratio

Total flowCongestion



Table 1 Results for the 14-dimensional NSFNET data set from (Mukherjee et al. 1996)

No. Weight pairs Phase 1 – basic Phase 2 – % improvement transceivers ωF ωT congestion total flow Total flow

Congestion objective congestion total flow

p=2 1 0 743.98 17828.09 23.96 1.02 1.04 -0.25

0.9 0.1 795.10 16637.11 20.92 2.64 5.30 0.93

0.8 0.2 799.73 16595.09 20.75 2.10 3.58 1.67

0.7 0.3 819.58 16495.84 20.13 1.80 5.92 1.08

0.6 0.4 819.58 16495.84 20.13 2.06 -4.77 2.82

0.5 0.5 978.76 16217.47 16.57 2.53 -4.02 3.12

0.4 0.6 978.76 16217.47 16.57 2.71 -4.02 3.12

0.3 0.7 1077.40 16118.83 14.96 2.65 5.50 2.53

0.2 0.8 1077.40 16118.83 14.96 2.60 5.50 2.53

0.1 0.9 1077.40 16118.83 14.96 2.56 5.50 2.53

0 1 1077.40 16118.83 14.96 2.53 5.50 2.53

p=3 1 0 399.92 14185.42 35.47 4.09 4.11 1.25

0.9 0.1 404.47 13373.52 33.06 2.22 4.95 -3.81

0.8 0.2 409.07 13141.53 32.13 0.55 1.74 -0.66

0.7 0.3 414.82 13001.63 31.34 0.20 1.93 -0.85

0.6 0.4 414.82 13001.63 31.34 0.34 -2.62 1.50

0.5 0.5 422.25 12924.67 30.61 0.64 -1.70 1.26

0.4 0.6 427.81 12883.47 30.11 0.87 -4.26 1.79

0.3 0.7 427.81 12883.47 30.11 1.16 -4.26 1.79

0.2 0.8 427.81 12883.47 30.11 1.41 -4.37 1.80

0.1 0.9 538.91 12687.19 23.54 1.49 -16.56 2.19

0 1 538.91 12687.19 23.54 2.18 -16.56 2.19

p=4 1 0 272.75 12007.66 44.02 2.18 2.17 4.03

0.9 0.1 272.88 11989.83 43.94 2.70 2.22 3.89

0.8 0.2 273.23 11967.02 43.80 2.99 2.35 3.71

0.7 0.3 285.65 11436.59 40.04 2.29 6.45 -0.65

0.6 0.4 287.22 11402.21 39.70 1.54 6.67 -0.81

0.5 0.5 290.93 11346.60 39.00 0.91 7.33 -1.09

0.4 0.6 290.93 11346.60 39.00 0.44 5.28 -0.56

0.3 0.7 305.43 11236.00 36.79 0.19 -4.98 0.92

0.2 0.8 316.43 11179.81 35.33 0.29 -1.85 0.47

0.1 0.9 316.43 11179.81 35.33 0.39 -1.85 0.47

0 1 384.00 11112.24 28.94 0.57 -9.65 0.57



which can be viewed as an index combining both criteria. Higher ratio indicates a more uniform usage of a network, while lower ratios indicate higher congestion relative to a total flow. Whenever a sudden drop in this ratio appears, it is an indication of a significant congestion increase relative to a decrease in total flow. For the 14-dimensional NSFNET data set Figure 2 displays the ratio

Total flowCongestion

for all weight pairs for different values of p. The graph reveals that the increase in the

number of transceivers results with smaller weight for congestion where the significant drops in the ratio connecting total flow and congestion occur. This stems from the topology of the network allowing for more diversified paths associated with more transceivers. When there are only two transceivers, congestion increases significantly when its weight is changed from 0.6 to 0.5. For three transceivers, the significant jump in congestion occurs when

its weight is changed from 0.2 to 0.1; and for p=4, the biggest jump in congestion happens when its weight changes from 0.1 to 0.

The comparison of change in congestion and total flow for the endpoints of the range of weights was interesting. Namely, the comparison of the minimal congestion obtained when it was the sole criterion (i.e. ωF=1) and when the complete weight was given to minimization of total flow (i.e. ωF=0), reveals that congestion increase is much more significant than the corresponding drop in total flow. Table 2 displays percentage of increase in congestion for change of ωF=1 to ωF=0, and percent decrease of total flow for ωT changing from 0 to 1.

We calculated the percentage change for best solutions from Phase 1 and Phase 2. The results reveal that, on average, congestion increase of 46.7% resulted with 9.2% decrease in total flow. Hence, almost 50% higher congestion was attained for decreasing the total flow for about 9%. The network manager should decide how much congestion needs to be sacrificed in order

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

1.0 - 0.0 0.9 - 0.1 0.8 - 0.2 0.7 - 0.3 0.6 - 0.4 0.5 - 0.5 0.4 - 0.6 0.3 - 0.7 0.2 - 0.8 0.1 - 0.9 0.0 - 1.0

Weight pairs

Tota

l flow

/Con

gesti

on

p=2 p=3 p=4 Figure 2 Display of Total flow

Congestion for different number of transceivers



Table 2 Percent increase in congestion and decrease in total flow over the complete range of weight changes for the 14-dimensional NSFNET data set

No. transceivers %increase congestion %decrease total flow

p=2 Phase 1 44.81 9.59

Phase 2 38.27 12.09

p=3 Phase 1 34.75 10.56

Phase 2 63.8 11.41

p=4 Phase 1 40.79 7.46

Phase 2 57.8 4.12

to bring down the total network flow. Our analysis can identify points where more drastic changes take place, in turn helping a manager to make an economically justified decision.

We remarked earlier that, without loss of generality, the model can accommodate link costs, for example proportional with link lengths. In such case minimizing total flow becomes equivalent to minimizing total delay. For the 14-dimensional NSFNET data instance we calculated distances (in miles) between nodes and resolved the problem with distances taken into account. The results showed similar behavior, revealing that a small number of connectivity matrices proved best over the large range of weight pairs, and that the congestion is more volatile than total delay. When running through all sets of weight pairs, from (ωF=1,ωT=0) to (ωF=0,ωT=1), the percent increase in congestion was drastically bigger than percent decrease in total delay. For example, for p=2, 110% increase in congestion resulted with 39% decrease in total delay; for p=3, 141% increase in congestion resulted with 26% decrease in total delay, and for p=4, 144% increase in congestion resulted with 10% decrease in total delay.

In order to test our approach further, we used the 24-dimensional instances from (Skorin-Kapov and Labourdette 1995). These are randomly generated data, hence we run our algorithm with uniform link lengths. The results are presented in Tables 3, 4 and 5. Parameters of tabu search were set as follows: for Phase 1, K=3, tabu_size=2, δ=[1,4], max_iter=5, LTM1_restart =1, LTM2_restart =1 ; for Phase 2, K=5, max_iter =7, LTM1_restart =1.

The analysis of obtained results follows similarly as for the 14-dimensional NSFNET data set. Due to increased problem size, the computational times were increased resulting in about 3 hours spent in Phase 1 and 11 hours in Phase 2. (This is cumulative time over all eleven pairs of weights.) Conclusions obtained for the 14-dimensional data set are confirmed with the three 24-dimensional instances.

The effectiveness of Phase 1 can be assessed by comparing relevant solutions with the published results: for exclusive congestion minimization we were able to improve the relevant results from (Skorin-Kapov and Labourdette 1995) for eight out of nine data instances. This is presented in Table 6. (The 14- dimensional data set used in (Skorin-Kapov



Table 3 Results for the quasiuniform 24-1 data matrix from (Skorin-Kapov and Labourdette 1995)



p=2 1 0 717.31 31712.98 44.21 3.64 3.63 3.44

0.9 0.1 717.31 31712.98 44.21 3.57 3.63 3.44

0.8 0.2 717.58 31692.54 44.17 3.53 3.66 3.38

0.7 0.3 717.58 31692.33 44.17 3.49 3.66 3.35

0.6 0.4 721.31 31604.00 43.81 3.42 4.16 3.11

0.5 0.5 747.80 31214.80 41.74 3.20 7.37 1.97

0.4 0.6 757.00 31132.00 41.13 2.84 8.39 1.73

0.3 0.7 768.50 31040.00 40.39 2.41 9.43 1.49

0.2 0.8 812.50 30864.00 37.99 1.95 13.13 1.04

0.1 0.9 839.00 30811.00 36.72 1.46 11.92 1.07

0 1 839.00 30811.00 36.72 1.43 -4.65 1.43

p=3 1 0 356.27 23596.00 66.23 0.98 0.98 0.56

0.9 0.1 356.27 23596.00 66.23 0.95 0.98 0.57

0.8 0.2 356.29 23591.71 66.21 0.91 0.99 0.57

0.7 0.3 356.29 23591.71 66.21 0.87 0.99 0.57

0.6 0.4 356.34 23584.20 66.18 0.83 0.99 0.56

0.5 0.5 356.92 23534.40 65.94 0.78 1.08 0.44

0.4 0.6 357.30 23513.70 65.81 0.72 1.10 0.43

0.3 0.7 358.07 23481.58 65.58 0.64 1.24 0.34

0.2 0.8 359.53 23442.78 65.20 0.59 0.43 0.64

0.1 0.9 361.17 23419.83 64.84 0.65 0.37 0.69

0 1 401.00 23380.00 58.30 1.11 -10.97 1.12

p=4 1 0 231.33 20469.55 88.49 0.35 0.35 2.24

0.9 0.1 231.38 20436.02 88.32 0.75 0.37 2.11

0.8 0.2 232.18 20299.81 87.43 1.00 0.70 1.48

0.7 0.3 233.12 20194.98 86.63 1.06 0.77 1.33

0.6 0.4 234.05 20134.17 86.03 1.09 1.06 1.10

0.5 0.5 235.64 20061.75 85.143 1.04 1.43 0.88

0.4 0.6 236.52 20034.14 84.70 0.99 1.54 0.84

0.3 0.7 246.67 19879.45 80.59 0.96 5.03 0.20

0.2 0.8 249.75 19842.75 79.45 0.64 6.09 0.04

0.1 0.9 255.58 19817.00 77.54 0.33 7.68 -0.04

0 1 255.58 19817.00 77.54 0.33 -2.90 0.33






p=2 1 0 734.91 32414.50 44.11 0.76 0.77 0.99

0.9 0.1 734.91 32414.50 44.11 0.80 0.77 0.99

0.8 0.2 734.91 32414.50 44.11 0.82 0.77 0.99

0.7 0.3 735.06 32402.00 44.08 0.85 0.77 0.99

0.6 0.4 735.70 32364.60 43.99 0.87 0.75 1.00

0.5 0.5 737.76 32277.88 43.75 0.86 0.99 0.76

0.4 0.6 744.31 32114.19 43.15 1.09 -0.66 1.94

0.3 0.7 749.57 32029.36 42.73 1.31 -0.26 1.80

0.2 0.8 754.69 31974.38 42.37 1.53 -1.54 2.09

0.1 0.9 755.75 31967.38 42.30 1.82 -1.75 2.11

0 1 809.00 31910.00 39.44 2.25 -3.83 2.25

p=3 1 0 370.94 24456.08 65.93 0.68 0.67 1.55

0.9 0.1 371.54 24254.51 65.28 0.83 0.75 1.05

0.8 0.2 372.48 24162.62 64.87 0.89 0.85 0.93

0.7 0.3 372.85 24142.10 64.75 0.96 0.56 1.22

0.6 0.4 374.11 24091.89 64.40 1.00 0.45 1.25

0.5 0.5 375.78 24053.04 64.01 1.05 0.67 1.17

0.4 0.6 378.06 24013.47 63.52 1.07 0.54 1.17

0.3 0.7 378.45 24008.85 63.44 1.12 0.18 1.24

0.2 0.8 383.33 23974.67 62.54 1.30 -5.78 1.84

0.1 0.9 396.50 23922.00 60.33 1.54 -3.86 1.73

0 1 426.00 23863.00 56.02 1.65 -8.45 1.65

p=4 1 0 234.43 20093.15 85.71 1.82 1.82 0.36

0.9 0.1 234.43 20093.15 85.71 1.62 1.77 0.79

0.8 0.2 234.58 20043.01 85.44 1.45 1.74 0.82

0.7 0.3 234.65 20031.86 85.37 1.35 1.70 0.88

0.6 0.4 235.34 19960.06 84.81 1.21 1.94 0.58

0.5 0.5 236.12 19920.41 84.37 1.08 2.10 0.49

0.4 0.6 237.25 19877.94 83.78 0.93 2.16 0.45

0.3 0.7 238.92 19831.62 83.01 0.77 1.99 0.46

0.2 0.8 246.33 19728.67 80.09 0.64 4.17 0.09

0.1 0.9 247.00 19722.00 79.85 0.34 4.37 0.07

0 1 271.00 19698.00 72.69 0.14 0.00 0.14






p=2 1 0 1200.00 51408.00 42.84 3.40 3.39 3.89

0.9 0.1 1200.00 51408.00 42.84 3.48 3.39 3.89

0.8 0.2 1200.00 51408.00 42.84 3.55 3.39 3.89

0.7 0.3 1200.00 51408.00 42.84 3.63 3.31 4.05

0.6 0.4 1200.00 51408.00 42.84 3.71 3.31 4.05

0.5 0.5 1200.00 51408.00 42.84 3.79 4.11 4.08

0.4 0.6 1201.33 51377.33 42.77 3.98 1.87 4.77

0.3 0.7 1210.13 51260.87 42.36 4.20 2.22 4.68

0.2 0.8 1345.50 50244.50 37.34 4.12 10.37 3.11

0.1 0.9 1365.00 50186.00 36.77 3.90 2.53 4.00

0 1 1801.00 49317.00 27.38 2.57 19.60 2.56

p=3 1 0 582.45 36878.29 63.32 0.75 0.75 1.32

0.9 0.1 584.22 36689.82 62.80 1.40 0.54 2.38

0.8 0.2 587.29 36535.96 62.21 1.70 0.71 2.21

0.7 0.3 593.33 36381.07 61.32 1.83 0.83 2.14

0.6 0.4 602.14 36256.14 60.21 1.97 -0.33 2.43

0.5 0.5 608.56 36195.22 59.48 2.14 -0.89 2.55

0.4 0.6 632.20 36007.20 56.96 2.14 -0.34 2.37

0.3 0.7 642.00 35968.00 56.02 2.22 -3.50 2.57

0.2 0.8 702.20 35804.00 50.99 2.25 5.37 2.13

0.1 0.9 702.20 35804.00 50.99 2.28 0.81 2.30

0 1 702.20 35804.00 50.99 2.30 0.81 2.30

p=4 1 0 370.20 31280.54 84.50 2.46 2.46 1.58

0.9 0.1 371.87 30888.14 83.06 1.88 2.41 1.26

0.8 0.2 376.15 30610.26 81.38 1.51 3.31 0.56

0.7 0.3 384.88 30313.28 78.76 1.34 0.43 1.63

0.6 0.4 391.00 30187.83 77.21 1.46 0.77 1.60

0.5 0.5 392.32 30170.66 76.90 1.57 -0.71 1.89

0.4 0.6 393.80 30158.00 76.58 1.73 -6.76 2.52

0.3 0.7 414.22 30054.78 72.56 1.97 -1.50 2.19

0.2 0.8 414.22 30054.78 72.56 2.05 -1.50 2.19

0.1 0.9 414.22 30054.78 72.56 2.13 -1.50 2.19

0 1 490.00 30048.00 61.32 2.18 13.47 2.18



Table 6 Comparison of results for exclusive congestion minimization for 24 dimensional data instances from (Skorin-Kapov and Labourdette 1995)

Data instances best published current best % improvement

from (Skorin-Kapov, Labourdette, 1995) congestion congestion current over published

24-1 p=2 733.520 691.29 5.76

p=3 366.275 352.77 3.69

p=4 234.965 230.52 1.89

24-2 p=2 777.062 729.28 6.15

p=3 374.925 368.45 1.73

p=4 235.810 230.16 2.40

24-3 p=2 1239.935 1159.28 6.50

p=3 575.354 578.10 -0.48

p=4 379.043 361.09 4.96

and Labourdette 1995) was taken from (Mukherjee et al. 1994), and it differs from the one from (Mukherjee et al. 1996) used here. When we solved the 14-dimensional data set from (Mukherjee et al. 1994) with the strategy proposed in this paper, the exclusive congestion minimization obtained the same results as in (Skorin-Kapov and Labourdette 1995).

Regarding the benefit of Phase 2, it seems that it is best visible for data with larger variability among its entries. Table 5, presenting the 24-3 data instance for which the variability among entries is the largest, shows the biggest percentage in overall improvement attributed to Phase 2. The reason might be in the feasible set containing solutions with more diverse values of the objective function. A single pairwise exchange as performed in our tabu search algorithm can then substantially change the objective function value, i.e the intensification of search can “hit" substantially better solution more

easily. Moreover, the improvement more often happens on both criteria: congestion as well as total flow minimization. Actually, for the total flow we always achieved a few percent improvement. For congestion minimization, sometimes the overall improvement resulted with some deterioration.

5. Conclusions and Future Research This paper presents our approach towards

designing optical networks with desirable properties regarding congestion minimization as well as total flow minimization. Depending on input data, virtual topologies are identified that perform well under different weighting schemes. Tabu search algorithm is proposed as a two phase strategy. The first phase performs more aggressive diversification of search, while Phase 2 attempts to intensify search around the best obtained solutions. The results reveal that Phase 2 provides a few percent increase in the overall objective which



sometimes translates to improvement in one criterion, while degradation in the other. A decision maker has to decide how beneficial it is to perform the second phase. In any case, the first phase already delivers good solutions. For example, when only congestion is concerned (i.e. ωF=1, ωT=0), we can compare our results to the best published ones. In this case, our strategy improves the best published results for 24-dimensional instances from (Skorin-Kapov, Labourdette, 1995) in eight out of nine instances with an average improvement of 3.62%.

Our future work will address strategies to obtain approximations of efficient frontiers of solutions by employing different weighting schemes. Further, we intend to evaluate regular topologies under multiple criteria for optimization.

References [1] Alves, A.J. and J. Climaco, “An iterative

method for 0-1 multiobjective problems using simulated annealing and tabu search”, Journal of Heuristics, Vol. 6, pp385-403, 2000.

[2] Antunes, C.H. and J.F. Craveirinha, J. N. Climaco, “Planning the evolution to broadband access networks: A multicriteria approach”, European Journal of Operational Research, Vol. 109, pp530-540, 1998.

[3] Bienstock, D. and O. Günlük, “Computational experience with a difficult mixed-integer multicommodity flow problem”, Mathematical Programming, Vol. 68, pp213-237, 1995.

[4] Costamagna, E., A. Fanni and G. Giacinto,

“A tabu search algorithm for the optimization of telecommunication networks”, European Journal of Operational Research, Vol. 106, pp357-372, 1998.

[5] “CPLEX 7.0” ILOG, 2000. [6] Ehrgott, M. and X. Gandibleux, “An

annotated bibliography of multiobjective combinatorial optimization”, working paper, Universitat Kaiserslautern, No.62, 2000.

[7] Gandibleux, X. and A. Freville, “Tabu search based procedure for solving the 0-1 multiobjective knapsack problem: The two objectives case”, Journal of Heuristics, Vol. 6, pp361-383, 2000.

[8] Glover, F., M. Laguna, Tabu Search, Kluwer Academic Publishing, Hingham, MA, 1997.

[9] Jia, X., X.-D. Hu and D.-Z. Du, Multiwavelength Optical Networks, Kluwer Academic Publishers, 2002.

[10] Jones, D.F., S.K. Mirrazavi and M. Tamiz, “Multi-objective meta-heuristics: An overview of the current state-of-art”, European Journal of Operational Research, Vol. 137, pp1-9, 2002.

[11] Labourdette, J-F., “Traffic optimization and reconfiguration management of multieavelength broadcast lightwave networks”, Computer Networks and ISDN Systems, Vol. 30, pp981-998, 1998.

[12] Labourdette, J-F. and A. Acampora, “Logically rearrangeable multihop lightwave networks”, IEEE Transactions on Communication, Vol. 39, pp1223-1230, 1991.

[13] Mukherjee, B., S. Rammamurthy, D.



Banerjee and A. Mukherje, “Some principles for designing a wide-area optical network”, Proceedings Infocom ’94, Toronto, Canada, 1994.

[14] Mukherjee, B., S. Rammamurthy, D. Banerjee and A. Mukherje, “Some principles for designing a wide-area WDM optical network”, IEEE/ACM Transactions on Networking, Vol. 4, No. 5, pp684-696, 1996.

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Valter Boljunčić received his Ph.D., M.Sc. and B.Sc. in Mathematics from the Faculty of

Mathematical and Natural Sciences of the University of Zagreb in Croatia. His research and publication record includes efficiency evaluation, especially sensitivity analysis of different DEA models, and applications of discrete optimization to telecommunications. He is the recipient of a research grant from the Ministry of Science, Education and Sports of Republic of Croatia, and is a former Fulbright scholar at State University of New York at Stony Brook. He is an Assistant Professor of Mathematics at the Faculty of Economics and Tourism “Dr. Mijo Mirković” in Pula, University of Rijeka. He is a member of Croatian Mathematical Society and Croatian Operational Research Society. Darko Skorin-Kapov received his Ph.D. in Management Science from the University of British Columbia, B.C., Canada, and his M.Sc. and B.Sc. degrees in Information Science and Mathematics, respectively, from the University of Zagreb, Croatia. His research interest is in the areas of optimization and cost allocation in telecommunication, transportation and financial networks. He has published numerous articles in those areas and was also a recipient of a number of grants, including National Science Foundation grants and a Fulbright Award. Dr. Skorin-Kapov is currently a Full Professor of Operations and Information Technology Management at the School of Business, Adelphi University, Garden City, New York. He is a member of INFORMS and the Croatian Operational Research Society. Jadranka Skorin-Kapov received her Ph.D. in Operations Research from the Faculty of



Commerce and Business Administration, the University of British Columbia in Canada, and her M.Sc. and B.Sc. degrees in Mathematics from the Faculty of Natural and Mathematical Sciences of the University of Zagreb in Croatia. Her research and extensive publication record includes applications of discrete optimization to telecommunications, network design, manufacturing design, scheduling, and location and layout. She develops algorithms (heuristic search and learning, and polynomial

algorithms for special cases) for difficult decision problems arising in engineering and business areas. Dr. Skorin-Kapov is a recipient of a number of grants, including six National Science Foundation grants. She is a Full Professor at the W.A. Harriman School for Management and Policy, College of Business, State University of New York at Stony Brook. She is a member of INFORMS and the Croatian Operational Research Society.

a tabu search approach towards congestion and total flow minimization in optical networks

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