new cutting plane method for solving the problem. Implementing the proposed method, we have obtained promising computational results. 1. Introduction.
대한산업공학회/한국경영과학회 ’98 춘계공동학술대회 논문집.1998 년 4 월 24-25 일. 경성대학교:Session A 10.3
A Branch and Cut Algorithm for Solving an Intra-Ring Design Problem of Synchronous Optical Networks Youngho Lee, Junghee Han, Seong-in Kim Department of Industrial Engineering, Korea University Abstract In this paper, we deal with a ring network design problem arising from the deployment of synchronous optical network (SONET). This problem can be conceptualized as an edge-capacitated graph partitioning problem with node cardinality constraints. We have formulated the problem as a mixed integer programming model, and developed a new cutting plane method for solving the problem. Implementing the proposed method, we have obtained promising computational results.
1. Introduction In this paper, we consider a network design problem arising from the deployment of synchronous optical network (SONET). Consider a set N of nodes in the network, indexed by i∈N = {1,…,n}, where n ≥ 2. Letting dij be traffic demands (positive integer) between node i ∈ N and node j (i < j) ∈N , we have a set of demand pairs on the edge set E = {(i, j) : i < j , dij > 0}. Also, let M be a set of rings, indexed by k ∈ M = {1,...,m}, where m ≥ 2, and let b be the capacity of the ring. Each node requires an add drop multiplexer (ADM) in order to route the traffic in a ring. And let R be the maximum number of ADMs allowed on a ring. Then, the ring design problem of our concern is to minimize the total number of ADMs, while satisfying all demand pairs on the edge set E without demand splitting. Here, we do not allow traffic among the rings. That is, we do not allow any inter-ring traffic. This problem focuses on logical clustering of demands. Note that the feasible solution of this problem may not form a ring, although all the nodes in a cluster will be connected in a ring. The physical connection (routing) for each cluster is another difficult combinatorial optimization problem (see Lee et al., 1998). For example, consider Figure 1 that illustrates the nature of the problem, where Figure 1(a) shows demand pattern, and Figure 1(b) shows a feasible solution using two rings and eight ADMs. 2
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Figure 1: An example of the problem (b = 10, R = 4). Wu and Burrowes (1990) and Wu (1992) have described in detail the advantages of SONET ring architecture over the traditional hubbing network in terms of cost and survivability. Wasem et al. (1994) have developed SONET design software considering demand growth over the years and two types of network architecture, self-healing ring (SHR) and hubbing network with point-to-point diverse protection (DP) systems, to incorporate SHR into the existing hubbing networks. Cosares and Saniee (1994) have presented a SONET ring design model and heuristic algorithms in the context of minimum traffic load on each ring. However, in this paper, we consider the SONET ring architecture for a single period, and develop a new solution procedure to find the optimal solution. Note that the ring design problem of our study can be conceptualized as an edge-capacitated graph partitioning problem with a node
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대한산업공학회/한국경영과학회 ’98 춘계공동학술대회 논문집.1998 년 4 월 24-25 일. 경성대학교:Session A 10.3
cardinality constraint. Ferreira et al. (1994) have presented polyhedral approach to the node capacitated graph partitioning problem. Laguna (1994) has proposed a mixed integer programming (MIP) model of a logical SONET ring design problem allowing inter-ring traffics, and showed promising computational results by the tabu search algorithm. Since inter-ring traffic is not allowed in our problem, we call our problem an intra-ring design problem. Although there exists an extensive body of literatures on the SOENT ring design problem, this paper is the first effort to deal with the intra-ring SONET design problem. This paper is organized as follows. In the next section, we present a MIP formulation for the problem. In section 3, we describe several classes of valid inequalities that tighten the formulation of the problem. In section 4, we develop a heuristic procedure. We also present computational results of the solution procedures in section 5. Section 6 concludes this paper.
2. Problem Formulation In this section, we develop an optimization formulation for solving the problem. Toward this end, let us define xik = 1 if node i ∈ N is assigned to ring k ∈ M and 0 otherwise, and fijk = 1 if traffic demand between node i ∈ N and j (i < j) ∈ N is assigned to ring k ∈ M and 0 otherwise. Then, the inter-ring design problem, denoted by IR, can be formulated as follows. IR: Minimize ∑ ∑ xik i∈ N k ∈M
Subject to ∑ f ijk = 1,
(i, j ) ∈ E ,
k ∈M
∑ d ij f ijk ≤ b,
(i , j )∈E
∑ xik ≤ R,
i∈N
(1)
k ∈M,
(2)
k ∈M,
(3)
f ijk ≤ xik ,
(i, j ) ∈ E , k ∈ M ,
(4)
f ijk ≤ x jk ,
(i, j ) ∈ E , k ∈ M ,
(5)
f ijk ∈ {0,1},
(i, j ) ∈ E , k ∈ M ,
xik ∈ {0,1},
i ∈ N, k ∈ M.
Constraint (1) indicates that each traffic demand should be assigned to a single ring, and constraint (2) indicates that sum of traffic demands assigned to a ring should be less than or equal to the ring capacity. Also, constraint (3) restricts the number of ADMs allowed on a ring to R, and constraints (4) and (5) force ADMs to be installed to nodes i and j of ring k if traffic demand (i, j) ∈ E is assigned to ring k ∈ M.
Remark 1. Observe that for a given (binary) feasible solution of f, the x variables are automatically binary at optimality, even if treated as continuous. Note also that the number of rings must be greater than or equal to the rounded-up value, denoted by ml, of the total demand divided by the capacity of the ring. Otherwise, even the linear programming (LP) relaxation is infeasible. Hence, ml provides a lower bound on the number of rings, m. That is, ∑ d ij (i , j )∈E ≤ m. ml ≡ b Remark 2. Note that the inherent symmetric nature of the optimal solution makes the problem harder to solve computationally. In order to partially avoid this symmetry, we can set f(ij)1 = 1 for some (i, j) ∈ E. Next, we consider the strength of the formulation IR in terms of the LP relaxation. Let ν(P) be the optimal objective
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