Kyungchul Park

An extended formulation approach to the edge-weighted maximal clique problem

We consider an extended formulation approach to the edge-weighted maximal clique problem. The problem is formulated by using additional variables for the set of nodes with the natural variables for the set of edges. We show that the proposed formulation is superior to the natural formulation both theoretically and practically. By using the projection technique, we can also derive new classes of facet-defining inequalities for the lower-dimensional polytope of the natural variables. Computational results are reported.

Modeling and solving the spatial block scheduling problem in a shipbuilding company

Abstract We consider the block scheduling problem arising in a shipyard. The problem is complicated since we are considering both scheduling and spatial allocation of each block simultaneously. Moreover, varying conditions in several work spaces should be reflected. We develop a scheduling algorithm using partial enumeration and decompostion. An efficient heuristic search procedure for the spatial allocation of blocks is also devised.

Lifting cover inequalities for the precedence-constrained knapsack problem

This paper considers the polyhedral structure of the precedence-constrained knapsack problem, which is a knapsack problem with precedence constraints imposed on the set of variables. The problem itself appears in many applications. Moreover, since the precedence constraints appear in many important integer programming problems, the polyhedral results can be used to develop cutting-plane algorithms for more general applications. In this paper, we propose a modification of the cover inequality, which explicitly considers the precedence constraints. A combinatorial, easily implementable lifting procedure of the modified cover inequality is given. The procedure can generate strong cuts very easily. We also propose an additional lifting procedure, which is a generalization of the lifting procedure for cover inequalities. Some properties of the lifted inequality are analyzed and the problem of finding an optimal order of lifting is also addressed.

A heuristic for an assembly line balancing problem with incompatibility, range, and partial precedence constraints

This paper considers a new type of an assembly line balancing problem which occurs in a company manufacturing electronic home appliances. The problem has some special characteristics. The main difficulty in this problem is that the precedence diagram is not enough to describe the precedence relationships between the tasks. It is shown that the usual line balancing model is a special case of the problem considered in this paper. After considering two subproblems, a heuristic algorithm for the problem is proposed. The algorithm is based on some network theories. Computational results based on some real world problems show the proposed algorithm works well in practice. A full system which incorporates graphic user interface is developed and currently in use.

Design of capacitated networks with tree configurations

This paper considers the problem of designing a capacitated network with a tree configuration (CTP). For a given set of nodes with their capacities,k types of link facilities with various characteristics, and installation cost for connecting each pair of nodes using each type of link facility, the problem is to find a tree network which satisfies the given traffic requirements between all pairs of nodes and minimizes total installation cost. We formulate (CTP) as an integer programming problem using path variables. To solve the linear programming relaxation which has exponentially many variables, we develop a polynomial-time column generation procedure. Moreover, to tighten the formulation, an efficient preprocessing procedure is devised and some classes of valid inequalities are found. Using the results, we develop a branch- and-cut algorithm with column generation where an efficient branching rule is adopted. Computational results show that the algorithm can solve practically-sized problems to optimality within a reasonable time.

Optimal multicast route packing

This paper considers the hop-constrained multicast route packing problem with a bandwidth reservation to build QoS-guaranteed multicast routing trees with a minimum installation cost. Given a set of multicast sessions, each of which has a hop limit constraint and a bandwidth requirement, the problem is to determine the set of multicast routing trees in an arc-capacitated network with the objective of minimizing the cost. For the problem, we propose a branch-and-cut-and-price algorithm, which can be viewed as a branch-and-bound method incorporating both the strong cutting plane algorithm and the column generation method. We implemented and tested the proposed algorithm on randomly generated problem instances with sizes up to 30 nodes, 570 arcs, and 10 multicast sessions. The test results show that the algorithm can obtain the optimal solution to practically sized problem instances within a reasonable time limit in most cases.

A branch-and-price algorithm for the Steiner tree packing problem

This paper deals with the Steiner tree packing problem. For a given undirected graph G = (V, E) with positive integer capacities and non-negative weights on its edges, and a list of node sets (nets), the problem is to find a connection of nets which satisfies the edge capacity limits and minimizes the total weights. We focus on the switchbox routing problem in knock-knee model and formulate this problem as an integer programming using Steiner tree variables. We develop a branch-and-price algorithm. The algorithm is applied on some standard test instances and we compare the performances with the results using cutting-plane approach. Computational results show that our algorithm is competitive to the cutting plane algorithm presented by Grotschel et al. and can be used to solve practically sized problems.

Design of local networks using USHRs

We consider the problem of designing a local network in a two‐level telecommunication network. Given one or two hub nodes, central offices (COs) and conduits, the problem is to find a set of unidirectional self‐healing rings (USHRs) which covers all COs and satisfies all demands at minimum cost. The solution approach used is the decomposition and column generation. Master problem and subproblem are modeled as integer programming models. After the optimal solution to linear programming relaxation of the master problem is obtained, a branch‐and‐bound algorithm is used to get an integer solution. A set of valid inequalities for a subproblem is given and a branch‐and‐cut algorithm is used to find an optimal solution to the subproblem. Computational results using real data are reported.

Optimal Multicast Routing and Wavelength Assignment on WDM Ring Networks Without Wavelength Conversion

We consider the multicast routing and wavelength assignment (MC-RWA) problem on WDM bidirectional ring networks without wavelength conversion. We give an integer programming formulation of the problem and propose an algorithm to solve it optimally. The algorithm is based on column generation and branch-and-price. We test the proposed algorithm on randomly generated data and the test results show that the algorithm gives optimal solutions to all of the test problems.

Economic spare capacity planning for DCS mesh-restorable networks

Abstract This paper considers an integer programming (IP) based optimization algorithm to solve the Spare Channel Assignment Problem (SCAP) for the new synchronous transmission networks that use a Digital Cross-Connect System (DCS) for each node of the network. Given predetermined working channels on each link of the network, the problem is to determine the spare capacity that should be added on each link to ensure rerouting of the traffic in case of a link failure. We propose an IP model which determines not only the spare capacity on each link but also the number of each link facility needed to be installed on each link to meet the aggregated requirements of working and spare channels. The objective is to minimize the total installation cost. We propose a branch-and-cut algorithm to solve the SCAR To solve the linear programming (LP) relaxation of the problem, an efficient constraint generation routine was devised. Moreover, some strong valid inequalities were found and used to strengthen the formulation. Computational results show that the algorithm can solve real world problems to optimality within a reasonable time.