Sungsoo Park

Algorithms for the variable sized bin packing problem

In this paper, we consider the variable sized bin packing problem where the objective is to minimize the total cost of used bins when the cost of unit size of each bin does not increase as the bin size increases. Two greedy algorithms are described, and analyzed in three special cases: (a) the sizes of items and bins are divisible, respectively, (b) only the sizes of bins are divisible, and (c) the sizes of bins are not divisible. Here, we say that a list of numbers a1,a2,…,am are divisible when aj exactly divides aj−1, for each 1<j⩽m. In the case of (a), the algorithms give optimal solutions, and in the case of (b), each algorithm gives a solution whose value is less than 119C(B*)+4119, where C(B*) is the optimal value. In the case of (c), each algorithm gives a solution whose value is less than 32C(B*)+1.

Optimization of Area Traffic Control for Equilibrium Network Flows

A bilevel programming approach is used to tackle an optimization problem for area traffic control and equilibrium flows. The signal timing plan is defined by common cycle time, and by starts and durations of greens. The system performance index is defined as the sum of a weighted linear combination of rate of delay and number of stops per unit time for all traffic streams, which is evaluated by the traffic model from TRANSYT. User equilibrium traffic assignment is formulated as a variational inequality problem. Approximate mathematical expressions for various components of the performance index and the average delay to a vehicle at the downstream junction in the TRANSYT model for both undersaturated and oversaturated links have been derived. For a locally optimal search, the gradient projection method is used in deciding whether feasible descent directions leading to a Karush-Kuhn-Tucker point, which is potentially a local optimum, can be identified. A global search heuristic is proposed in this paper by which successively better Karush-Kuhn-Tucker points can be found with reasonable computation effort. The mixed search procedure, including the locally optimal search and global search heuristic, is proposed. Encouraging results for Allsop and Charlesworths network have confirmed that the approximately optimizing mixed search procedure in solving the area traffic control optimization problem can achieve substantially better results than does the nonoptimizing calculations of mutually consistent signal timings and link flows.

Efficient solution procedure and reduced size formulations for p-hub location problems

In this paper we consider the uncapacitated p-hub location problems, where multiple and single allocation cases are considered. In the multiple allocation problem, a nonhub node may be allocated to more than one hub. We show that the multiple allocation problem can be solved efficiently by the shortest path algorithm when p is fixed. Also, this algorithm can be applied to the cases in which some other hubbing policies are used. In the single allocation problem, each nonhub node must be allocated to exactly one of the p hubs. We provide a reduced size formulation, and a mixed integer formulation for the model with fixed hub locations, where fixed costs for opening links are considered. Computational experience is provided using the data given in the literature and the instances having fixed costs for opening links.

Optimal routing and wavelength assignment in WDM ring networks

We consider the routing and wavelength assignment (RWA) problem on WDM ring networks without wavelength conversion. When the physical network and required connections are given, RWA is the problem to select a suitable path and wavelength among the many possible choices for each connection such that no two paths using the same wavelength pass through the same link. We give an integer programming formulation of the problem and propose an algorithm to solve it. Although the formulation has exponentially many variables, we solve the linear programming relaxation of it by using the column generation technique. We solve the column generation problem efficiently by decomposing the problem into several subproblems. After solving the linear programming relaxation, we apply the branch-and-price procedure to get an optimal solution. We test the proposed algorithm on some randomly generated data. Test results show that the algorithm gives optimal solutions to almost all instances under the given node limit of the branch-and-bound tree.

A linear program for the two-hub location problem

Abstract This paper considers the discrete two-hub location problem. We need to choose two hubs from a set of nodes. The remaining nodes are to be connected to one of the two hubs which act as switching points for internodal flows. A configuration which minimizes the total flow cost needs to be found. We show that the problem can be solved in polynomial time when the hub locations are fixed. Since there are at most 1 2 n(n−1) ways to choose the hub locations, the two-hub location problem can be solved in polynomial time. We transform the quadratic 0–1 integer program of the single allocation problem in the fixed two-hub system into a linear program and show that all extreme points of the polytope defined by the LP are integral. Also, the problem can be transformed into a minimum cut problem which can be solved efficiently by any polynomial time algorithm.

A polyhedral approach to edge coloring

We formulate the edge coloring problem on a simple graph as the integer program of covering edges by matchings. For the NP-hard case of 3-regular graphs we show that it is sufficient to solve the linear programming relaxation with the additional constraints that each odd circuit be covered by at least three matchings. We give an efficient separation algorithm for recognizing violated odd circuit constraints and a linear programming based constrained weighted matching algorithm for pricing. Computational experiments with the overall linear programming system are presented.

Robust Vehicle Routing Problem with Deadlines and Travel Time/Demand Uncertainty

In this article, we investigate the vehicle routing problem with deadlines, whose goal is to satisfy the requirements of a given number of customers with minimum travel distances while respecting both of the deadlines of the customers and vehicle capacity. It is assumed that the travel time between any two customers and the demands of the customer are uncertain. Two types of uncertainty sets with adjustable parameters are considered for the possible realizations of travel time and demand. The robustness of a solution against the uncertain data can be achieved by making the solution feasible for any travel time and demand defined in the uncertainty sets. We propose a Dantzig-Wolfe decomposition approach, which enables the uncertainty of the data to be encapsulated in the column generation subproblem. A dynamic programming algorithm is proposed to solve the subproblem with data uncertainty. The results of computational experiments involving two well-known test problems show that the robustness of the solution can be greatly improved.

The single allocation problem in the interacting three-hub network

We consider the single allocation problem in the interacting three-hub network with fixed hub locations. In the single allocation hub network, the hubs are fully interconnected and each nonhub node has to be connected to exactly one of the hubs. The flows between each pair of nodes are sent using the hubs as intermediate switching points. The problem is to find an optimal allocation of nonhub nodes to the hubs which minimizes the total flow cost. We show that the single allocation problem is NP-hard as soon as the number of hubs is three, although the problem in a two-hub system has polynomial time algorithms. This paper provides a mixed integer formulation of the problem and considers the polyhedral properties of it. The formulation can also be used for the single allocation problem with fixed costs for opening links, the three-terminal cut problem, and the three-processor distribution problem. Computational experiences are reported for data given in the literature and randomly generated problems.

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.