Allocation of resources of modular sizes with an application to Internet Protocol (IP) address allocation

We consider a resource allocation problem, where resources of different capacities must satisfy multiple demands. The demand sizes and the resource capacities are limited to sizes that are power‐of‐two integers (i.e., 1, 2, 4, 8, …). The cost of the resources exhibit economies‐of‐scale savings, i.e., the cost per capacity unit is smaller for resources with larger capacity. The problem is to select the minimum‐cost set of resources that satisfies the demands, while each of the demands must be assigned to a single resource and the number of selected resources does not exceed a specified upper bound. We present algorithms that take advantage of the special structure of the problem and provide optimal solutions in a negligible computing effort. This problem is important for the allocation of blocks of Internet Protocol (IP) addresses, referred to as subnets. In typical IP networks, subnets are allocated at a large number of nodes. An effective allocation attempts to balance the volume of excess addresses that are not used versus fragmentation of addresses at nodes to too many subnets with a discontinuous range of addresses. Due to the efficiency of the algorithms, they can readily be used as valuable modules in IP address management systems. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

A heuristic for capacity expansion planning with multiple facility types

A capacity expansion model with multiple facility types is examined, where different facility types represent different quality levels. Applications for the model can be found in communications networks and production facilities. The model assumes a finite number of discrete time periods. The facilities are expanded over time. Capacity of a high-quality facility can be converted to satisfy demand for a lower-quality facility. The costs considered include capacity expansion costs and excess capacity holding costs. All cost functions are nondecreasing and concave. An algorithm that finds optimal expansion policies requires extensive computations and is practical only for small scale problems. Here, we develop a heuristic that employs so-called distributed expansion policies. It also attempts to decompose the problem into several smaller problems solved independently. The heuristic is computationally efficient. Further, it has consistently found near-optimal solutions.     

Equitable bandwidth allocation in content distribution networks

Applications for content distribution over networks, such as Video‐on‐Demand (VOD), are expected to grow significantly over time. Effective bandwidth allocation schemes that can be repeatedly executed must be deployed since new programs are often installed at various servers while other are deleted. We present a model for bandwidth allocation in a content distribution network that consists of multiple trees, where the root of each tree has a server that broadcasts multiple programs throughout the tree. Each network link has limited capacity and may be used by one or more of these trees. The model is formulated as an equitable resource allocation problem with a lexicographic maximin objective function that attempts to provide equitable service performance for all requested programs at the various nodes. The constraints include link capacity constraints and tree‐like ordering constraints imposed on each of the programs. We present an algorithm that provides an equitable solution in polynomial time for certain performance functions. At each iteration, the algorithm solves single‐link maximin optimization problems while relaxing the ordering constraints. The algorithm selects a bottleneck link, fixes various variables at their lexicographic optimal solution while enforcing the ordering constraints, and proceeds with the next iteration. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

An integer programming model for the weekly tour scheduling problem

We study a workforce planning and scheduling problem in which weekly tours of agents must be designed. Our motivation for this study comes from a call center application where agents serve customers in response to incoming phone calls. Similar to many other applications in the services industry, the demand for service in call centers varies significantly within a day and among days of the week. In our model, a weekly tour of an agent consists of five daily shifts and two days off, where daily shifts within a tour may be different from each other. The starting times of any two consecutive shifts, however, may not differ by more than a specified bound. Furthermore, a tour must also satisfy constraints regarding the days off, for example, it may be required that one of the days off is on a weekend day. The objective is to determine a collection of weekly tours that satisfy the demand for agents' services, while minimizing the total labor cost of the workforce. We describe an integer programming model where a weekly tour is obtained by combining seven daily shift scheduling models and days‐off constraints in a network flow framework. The model is flexible and can accommodate different daily models with varying levels of detail. It readily handles different days‐off rules and constraints regarding start time differentials in consecutive days. Computational results are also presented. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 607–624, 2001.     

A network flow approach for capacity expansion problems with two facility types

A deterministic capacity expansion model for two facility types with a finite number of discrete time periods is described. The model generalizes previous work by allowing for capacity disposals, in addition to capacity expansions and conversions from one facility type to the other. Furthermore, shortages of capacity are allowed and upper bounds on both shortages and idle capacities can be imposed. The demand increments for additional capacity of any type in any time period can be negative. All cost functions are assumed to be piecewise, concave and nondecreasing away from zero. The model is formulated as a shortest path problem for an acyclic network, and an efficient search procedure is developed to determine the costs associated with the links of this network.     

A multifacility capacity expansion model with joint expansion set-up costs

This article describes a multifacility capacity expansion model in which the different facility types represent different quality levels. These facility types are used to satisfy a variety of deterministic demands over a finite number of discrete time periods. Applications for the model can be found in cable sizing problems associated with the planning of communication networks. It is assumed that the cost function associated with expanding the capacity of any facility type is concave, and that a joint set-up cost is incurred in any period in which one or more facilities are expanded. The model is formulated as a network flow problem from which properties associated with optimal solutions are derived. Using these properties, we develop a dynamic programming algorithm that finds optimal solutions for problems with a few facilities, and a heuristic algorithm that finds near-optimal solutions for larger problems. Numerical examples for both algorithms are discussed.     

A capacity-expansion model for two facility types

This paper describes a deterministic capacity-expansion model for two facility types with a finite number of discrete time periods. Capacity expansions are initialed either by new construction or by the conversion of idle capacity from one facility type to the other. Once converted, the capacity becomes an integral part of the new facility type. The costs incurred include construction, conversion, and holding costs. All cost functions are assumed to be nondecreasing and concave. Using a network flow approach, the paper develops an efficient dynamic-programming algorithm to minimize the total costs when the demands for additional capacity are nonnegative in each period. Thereafter, the algorithm is extended for arbitrary demands. The model is applied to a cable-sizing problem that occurs in communication networks, and numerical examples are discussed.