Prakash Mirchandani

Modeling and Solving the Two-Facility Capacitated Network Loading Problem

This paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry. In this problem, given point-to-point communication demand in a network must be met by installing loading capacitated facilities on the arcs: Loading a facility incurs an arc specific and facility dependent cost. This paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost. We consider two approaches for solving the underlying mixed integer program: a Lagrangian relaxation strategy, and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem. We show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the Lagrangian relaxation bound. Our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation: from an average of 25% to an average of 8%. These results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry.

The convex hull of two core capacitated network design problems

The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed point-to-point demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost.This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP.

The Minimum Satisfiability Problem

This paper shows that a minimization version of satisfiability is strongly NP-hard, even if each clause contains no more than two literals and/or each clause contains at most one unnegated variable. The worst-case and average-case performances of greedy and probabilistic greedy heuristics for the problem are examined, and tight upper bounds on the performance ratio in each case are developed.

Shortest paths, single origin‐destination network design, and associated polyhedra

We study a specialized version of network design problems that arise in telecommunications, transportation, and other industries. The problem, a generalization of the shortest path problem, is defined on an undirected network consisting of a set of arcs on which we can install (load), at a cost, a choice of up to three types of capacitated facilities. Our objective is to determine the configuration of facilities to load on each arc that will satisfy the demand of a single commodity at the lowest possible cost. Our results (i) demonstrate that the single-facility loading problem and certain “common break-even point” versions of the two-facility and three-facility loading problems are polynomially solvable as a shortest path problem; (ii) show that versions of the two-facility loading problem are strongly NP-hard, but that a shortest path solution provides an asymptotically “good” heuristic; and (iii) characterize the optimal solution (i.e., specify a linear programming formulation with integer solutions) of the common break-even point versions of the two-facility and three-facility loading problems. In this development, we introduce two new families of facets, give geometric interpretations of our results, and demonstrate the usefulness of partitioning the space of the problem parameters to establish polyhedral integrality properties. Generalizations of our results apply to (i) multicommodity applications and (ii) situations with more than three facilities.

Designing Hierarchical Survivable Networks

As the computer, communication, and entertainment industries begin to integrate phone, cable, and video services and to invest in new technologies such as fiber-optic cables, interruptions in service can cause considerable customer dissatisfaction and even be catastrophic. In this environment, network providers want to offer high levels of service-in both serviceability (e.g., high bandwidth) and survivability (failure protection)-and to segment their markets, providing better technology and more robust configurations to certain key customers. We study core models with three types of customers (primary, primary but critical, and secondary) and two types of services/technologies (primary and secondary). The network must connect all primary customers using primary (high bandwidth) services and, additionally, contain a back-up path connecting the critical primary customers. Secondary customers require only single connectivity to other customers and can use either primary or secondary facilities. We propose a general multi-tier survivable network design model to configure cost effective networks for this type of market segmentation. When costs are triangular, we show how to optimally solve single-tier subproblems, with two critical customers, as a matroid intersection problem. We also propose and analyze the worst-case performance of tailored heuristics for several special cases of the two-tier model. Depending upon the particular problem setting, the heuristics have worst-case performance ratios ranging between 1.25 and 2.6. We also provide examples to show that the performance ratios for these heuristics are the best possible.

The Multi-Tier Tree Problem

This paper studies the multi-tier tree (MTT) problem, a generalization of the Steiner-tree problem, in which we are given a graph with its nodes partitioned into several tiers (grades), and grade dependent edge-costs. The MTT problem seeks the cost minimizing choice of grades for the edges such that every pair of nodes, say at tiers t′ and t″ ≥ t′, is connected by a path containing grade-t′ or better (grade) edges. The MTT problem arises in the telecommunication setting (where we must choose between competing communication technologies) end in the transportation setting (where the cost incurred determines the type of access provided between two points). In this paper, we develop two solution approaches for the problem. We first develop a recursive heuristic for the MTT problem, and obtain data-independent bounds for MTT problems with 3-tiers. For one case, the bound on the performance ratio of the recursive heuristic is 1.52241. Next, we develop a dual-based solution procedure for this problem, and conduc...

Projections of the capacitated network loading problem

Abstract Consider an undirected network and a set of commodities with specified demands between various pairs of nodes of the network. Given two types of capacitated facilities that can be installed (loaded) for arc dependent costs, we have to determine the integer number of facilities to load on each arc in order to send the required flow of all commodities at minimum total cost. We present a natural mixed-integer programming formulation of the problem and then consider its single commodity and multicommodity versions. We develop “equivalent” formulations in a lower-dimensional space by projecting out the flow variables and study the polyhedral properties of the corresponding projection cones. Our results strengthen an existing result for multicommodity flow problems. We also characterize several classes of facet defining inequalities for this lower-dimensional polyhedron, and conclude by identifying some open problems and future research directions.

Technical Note—New Results Concerning Probability Distributions with Increasing Generalized Failure Rates

The generalized failure rate of a continuous random variable has demonstrable importance in operations management. If the valuation distribution of a product has an increasing generalized failure rate (that is, the distribution is IGFR), then the associated revenue function is unimodal, and when the generalized failure rate is strictly increasing, the global maximum is uniquely specified. The assumption that the distribution is IGFR is thus useful and frequently held in recent pricing, revenue, and supply chain management literature. This note contributes to the IGFR literature in several ways. First, it investigates the prevalence of the IGFR property for the left and right truncations of valuation distributions. Second, we extend the IGFR notion to discrete distributions and contrast it with the continuous distribution case. The note also addresses two errors in the previous IGFR literature. Finally, for future reference, we analyze all common (continuous and discrete) distributions for the prevalence of the IGFR property, and derive and tabulate their generalized failure rates. Subject classifications: probability distributions; hazard rate functions; revenue management; supply chain management. Area of review: Operations and Supply Chains. History: Received November 2011; revisions received May 2012, November 2012, January 2013; accepted May 2013.

Bundling Strategies When Products Are Vertically Differentiated and Capacities Are Limited

We consider a seller who owns two capacity-constrained resources and markets two products (components) corresponding to these resources as well as a bundle comprising the two components. In an environment where all customers agree that one of the two components is of higher quality than the other and that the bundle is of the highest quality, we derive the sellers optimal bundling strategy. We demonstrate that the optimal solution depends on the absolute and relative availabilities of the two resources as well as upon the extent of subadditivity of the quality of the products. The possible strategies that can arise as equilibrium behavior include a pure components strategy, a partial-or full-spectrum mixed bundling strategy, and a pure bundling strategy, where the latter strategy is optimal when capacities are unconstrained. These conclusions are contrary to findings in the prior literature on bundling that demonstrated the unambiguous dominance of the full-spectrum mixed bundling strategy. Thus, our work expands the frontier of bundling to an environment with vertically differentiated components and limited resources. We also explore how the bundling strategies change as we introduce an element of horizontal differentiation wherein different types of customers value the available components differently.

Connectivity Upgrade Models for Survivable Network Design

Disruptions in infrastructure networks to transport material, energy, and information can have serious economic, and even catastrophic, consequences. Since these networks require enormous investments, network service providers emphasize both survivability and cost effectiveness in their topological design decisions. This paper addresses the survivable network design problem, a core model incorporating the cost and redundancy trade-offs facing network planners. Using a novel connectivity upgrade strategy, we develop several families of inequalities to strengthen a multicommodity flow-based formulation for the problem, and show that some of these inequalities are facet defining. By increasing the linear programming lower bound, the valid inequalities not only lead to better performance guarantees for heuristic solutions, but also accelerate exact and approximate solution methods. We also consider a heuristic strategy that sequentially rounds the fractional values, starting with the linear programming solution to our strong model. Extensive computational tests confirm that the valid inequalities, added via a cutting plane algorithm, and the heuristic procedure are very effective, and their performance is robust to changes in the network dimensions and connectivity structure. Our solution approach generates tight lower and upper bounds with average gaps that are less than 1.2% for various problem sizes and connectivity requirements.