Department of Industrial Systems and Information Engineering, Korea University, Seoul, Korea. Keywords: Optical Internet, WDM systems, Branch-and-Cut, Tabu ...
A Location-Routing Problem in Designing the Optical Internet with WDM Systems Youngho Lee, Seungjun Lee, Seong-in Kim Department of Industrial Systems and Information Engineering, Korea University, Seoul, Korea. Keywords: Optical Internet, WDM systems, Branch-and-Cut, Tabu search, Location-Routing, Multi-commodity flow. 1. Introduction In this talk, we consider a location-routing problem in designing the optical Internet with WDM systems. This problem arises from the design of broadband local access networks that deliver high-speed access service to residential subscribers. In this problem, we consider fiber routing and gateway location. For routing problem, we determine the optimal dimensioning of link capacity, and seek to find the fiber routing of traffic between demand node and gateway node. In the optical Internet architecture of this talk, we consider WDM systems as a viable means to increase capacity. Couple with fiber routing problem, we address the optimal location problem of the gateway for minimizing the total cost of routing and WDM systems. This problem would be solved as the subproblem of the multi-period network planning problem faced by network planners at telecommunications company. To solve the large-scale problems, network planners estimate the cost of building new networks under a variety of conditions involving different demand patterns and WDM cost. Such studies may require solving iteratively a series of related network design problems. And solving this problem many times without consumption of too much computation time is very important. Furthermore, given the significant cost of the WDM equipments, it is also crucial to have the optimality guarantee of solutions for solving real-world problems. To provide the optimality guarantee, we develop a strong formulation that generates a tighter lower bound for the problem. Then, we get some useful constructs for the heuristic, by exploiting the special structure of the proposed strong formulation. The location-routing problem is similar to the well known bandwidth packing problem (BWP). The BWP is to find a solution of maximizing the total revenue from the routed calls when each call has a given revenue and requires a nonnegative bandwidth in a capacitated arc graph. Anderson, et al [1], and Laguna et al. [2] developed a tabu search heuristic algorithm to solve the BWP, and Laguna et al. [2] suggested an integer programming formulation and solved some instances by using commercial package. In the BWP, the location of the gateway is fixed, while the location of the gateway in our problem is determined by considering the total cost of fiber routing and WDM systems. 2. Solution Procedures We develop two alternate formulations for the problem: a path model and a multi-commodity flow model. The path model has been explored extensively for the last ten years in the BWP. It has been known that the path model is more effective than the multi-commodity flow model because the path model provides strong lower bounds, which in turn reduce the effort of the branch-and-bound. However, in this talk, we demonstrate that the augmented multi-commodity flow model with strong valid inequalities provides competitive lower bounds. With the tight lower bound of the LP relaxation of the model, we develop a branch-and-cut algorithm. Then, by using the inherent special structure of the problem, we propose an effective tabu search heuristic algorithm. Recently, in solving the classical location-routing problem of trucking company, Tuzun et al. [3] proposed a sequential two-phase tabu search algorithm that first determines the location of depot and then finds a routing of vehicle for a given location of the depot. Since the overall cost of routing depends on the location of the gateway, the quality of the proposed sequential method could be extremely poor in some
problem instances. However, in this paper, we develop an effective tabu search algorithm with various long term memory strategies for simultaneously solving location and routing problems. First, we devise a simple greedy algorithm for an initial solution. In this algorithm, we find a minimum cost path for each demand to the candidate gateway. Then, a set of gateways are selected in a greedy fashion. Then, we rearrange the path for each demand by iteratively swapping gateways. For the description of our tabu search algorithm, let a WDM link be an edge that deploys a WDM equipment and a WDM forest be a set of the WDM links which are connected. Then, we define two moves of our tabu search algorithm: WDM link move and gateway swapping move. The WDM link move changes the member of a WDM forest by deleting a WDM link of low utilization and by adding one that is promising and not prohibited by tabu memory. Since WDM equipments are expensive, this move improves the quality of the solution. Note that the number of edges in a WDM forest does not change after the move. Next, we allow an infeasible solution in the swapping move of gateways, which hopefully improves the chance of recovering better feasible solutions in the later phase of the tabu search. The frequency that each location is selected as a candidate location and the frequency of the edge used as WDM link are monitored during the normal phase. Then, we use the frequency information for intensification and diversification phases of the proposed tabu search. In the intensification phase, we select some of the locations which are frequently selected as candidate locations. Then, with these locations, we examine various routing solutions intensively. Note, however, that the WDM link information is used as the tabu memory during this phase. But, in the diversification phase, we include the least used location as the candidate location and it enables the exploration of unvisited solution space. In our tabu search algorithm, we apply a longer tabu tenure for the normal edges than for the WDM link. 3. Computational Results We have tested the proposed algorithm for various problem instances. For a real network topology of 50 nodes and 133 edges, problem instance are generated with different parameters of the cost for WDM and routing. From our computation results, we observed that our tabu search algorithm provides optimal or near optimal solutions in many problem instances. References [1] Anderson A., K. Fraughnaugh, M. Parker, J. Ryan, “Path assignment for call routing: An application of tabu search,” Annals of Operation Research, Vol. 41, 1993. [2] Laguna, M., F. Glover, “Bandwidth packing: a tabu search approach,” Management Science, Vol. 39, No. 1, 1993. [3] Tuzun D., L. I. Burke, “A two-phase tabu search approach to the location routing problem,” European Journal of Operational Research, Vol. 116, pp.87–99, 1999.
