A robust model for a leader–follower competitive facility location problem in a discrete space

This paper aims at determining the optimal locations for the leader’s new facilities under the condition that the number of the follower’s new facilities is unknown for the leader. The leader and the follower have some facilities in advance. The first competitor, the leader, opens p new facilities in order to increase her own market share. On the other hand, she knows that her competitor, the follower, will react to her action and locate his new facilities as well. The number of the follower’s new facilities is unknown for the leader but it is assumed that the leader knows the probability of opening different numbers of the follower’s new facilities. The leader aims at maximizing her own market share after the follower’s new facilities entry. The follower’s objective is also to maximize his own market share. Since the number of the follower’s new facilities is unknown for leader, “Robust Optimization” is used for maximizing the leader’s market share and making the obtained results “robust” in various scenarios in terms of different numbers of the follower’s new facilities. The optimal locations for new facilities of both the leader and the follower are chosen among pre-determined potential locations. It is assumed that the demand is inelastic. The customers probabilistically meet their demands from all different facilities and the demand level which is met by each facility is computed by Huff rule. The computational experiments have been applied to evaluate the efficiency of the proposed model.     

A splitter location–allocation problem in designing fiber optic access networks

This paper deals with a physical access network design problem of fiber-to-the-home passive optical network (FTTH-PON). The design of FTTH-PON access network seeks the cost effective location of optical splitters that provide optical connectivity from central office to subscribers in a given service area. We formulate the problem as a multi-level capacitated facility location problem on a tree topology with nonlinear link cost. Dealing with the nonlinear link cost, we propose an objective function relaxation approach to obtain tight upper and lower bounds. We develop valid inequalities that enhance the lower bound and propose a local search heuristic procedure that improves the upper bound. Valid inequalities force integrality condition on the number of splitters placed at nodes. Local search heuristic improves the initial greedy solution by placing splitters on the sub-root nodes of a given tree network. Computational results demonstrate the effectiveness of the proposed solution procedures.     

Multi-source facility location–allocation and inventory problem

We consider a joint facility location–allocation and inventory problem that incorporates multiple sources of warehouses. The problem is motivated by a real situation faced by a multinational applied chemistry company. In this problem, multiple products are produced in several plants. Warehouse can be replenished by several plants together because of capabilities and capacities of plants. Each customer in this problem has stochastic demand and certain amount of safety stock must be maintained in warehouses so as to achieve certain customer service level. The problem is to determine number and locations of warehouses, allocation of customers demand and inventory levels of warehouses. The objective is to minimize the expected total cost with the satisfaction of desired demand weighted average customer lead time and desired cycle service level. The problem is formulated as a mixed integer nonlinear programming model. Utilizing approximation and transformation techniques, we develop an iterative heuristic method for the problem. An experiment study shows that the proposed procedure performs well in comparison with a lower bound.     

A location–allocation heuristic for the capacitated multi-facility Weber problem with probabilistic customer locations

The capacitated multi-facility Weber problem is concerned with locating m facilities in the Euclidean plane, and allocating their capacities to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a non-convex optimization problem and difficult to solve. In this work, we focus on a probabilistic extension and consider the situation where the customer locations are randomly distributed according to a bivariate distribution. We first present a mathematical programming formulation, which is even more difficult than its deterministic version. We then propose an alternate location–allocation local search heuristic generalizing the ideas used originally for the deterministic problem. In its original form, the applicability of the heuristic depends on the calculation of the expected distances between the facilities and customers, which can be done for only very few distance and probability density function combinations. We therefore propose approximation methods which make the method applicable for any distance function and bivariate location distribution.     

Hierarchical location–allocation models for congested systems

In this paper we address the issue of locating hierarchical facilities in the presence of congestion. Two hierarchical models are presented, where lower level servers attend requests first, and then, some of the served customers are referred to higher level servers. In the first model, the objective is to find the minimum number of servers and their locations that will cover a given region with a distance or time standard. The second model is cast as a maximal covering location (MCL) formulation. A heuristic procedure is then presented together with computational experience. Finally, some extensions of these models that address other types of spatial configurations are offered.     

Hub location–allocation in intermodal logistic networks

Within the context of intermodal logistics, the design of transportation networks becomes more complex than it is for single mode logistics. In an intermodal network, the respective modes are characterized by the transportation cost structure, modal connectivity, availability of transfer points and service time performance. These characteristics suggest the level of complexity involved in designing intermodal logistics networks. This research develops a mathematical model using the multiple-allocation p-hub median approach. The model encompasses the dynamics of individual modes of transportation through transportation costs, modal connectivity costs, and fixed location costs under service time requirements. A tabu search meta-heuristic is used to solve large size (100 node) problems. The solutions obtained using this meta-heuristic are compared with tight lower bounds developed using a Lagrangian relaxation approach. An experimental study evaluates the performance of the intermodal logistics networks and explores the effects and interactions of several factors on the design of intermodal hub networks subject to service time requirements.     

A path to the Arrow–Debreu competitive market equilibrium

We present polynomial-time interior-point algorithms for solving the Fisher and Arrow–Debreu competitive market equilibrium problems with linear utilities and n players. Both of them have the arithmetic operation complexity bound of )) for computing an -equilibrium solution. If the problem data are rational numbers and their bit-length is L, then the bound to generate an exact solution is O(n ⁴ L) which is in line with the best complexity bound for linear programming of the same dimension and size. This is a significant improvement over the previously best bound )) for approximating the two problems using other methods. The key ingredient to derive these results is to show that these problems admit convex optimization formulations, efficient barrier functions and fast rounding techniques. We also present a continuous path leading to the set of the Arrow–Debreu equilibrium, similar to the central path developed for linear programming interior-point methods. This path is derived from the weighted logarithmic utility and barrier functions and the Brouwer fixed-point theorem. The defining equations are bilinear and possess some primal-dual structure for the application of the Newton-based path-following method. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday. This author was supported in part by NSF Grants DMS-0306611 and DMS-0604513. The author would like to thank Curtis Eaves, Osman Güler, Kamal Jain and Mike Todd for insightful discussions on this subject, especially on their mathematical references and economic interpretations of the fixed-point model presented in this paper.     

Time-consistent mean–variance proportional reinsurance and investment problem in a defaultable market

In this paper, we consider an optimal time-consistent reinsurance-investment problem incorporating a defaultable security for a mean–variance insurer under a constant elasticity of variance (CEV) model. In our model, the insurer’s surplus process is described by a jump-diffusion risk model, the insurer can purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset, a defaultable bond and a risky asset whose price process is assumed to follow a CEV model. Using a game theoretic approach, we establish the extended Hamilton–Jacobi–Bellman system for the post-default case and the pre-default case, respectively. Furthermore, we obtain the closed-from expressions for the time-consistent reinsurance-investment strategy and the corresponding value function in both cases. Finally, we provide numerical examples to illustrate the impacts of model parameters on the optimal time-consistent strategy.     

A singularly perturbed convection–diffusion problem in a half-plane

A singularly perturbed convection–diffusion equation with constant coefficients is considered in a half plane, with Dirichlet boundary conditions. The boundary function has a specified degree of regularity except for a jump discontinuity, or jump discontinuity in a derivative of specified order, at a point. Precise pointwise bounds for the derivatives of the solution are obtained. The bounds show both the strength of the interior layer emanating from the point of discontinuity and the blowup of the derivatives resulting from the discontinuity, and make precise the dependence of the derivatives on the singular perturbation parameter.     

A heuristic Lagrangean algorithm for the capacitated plant location problem — A comment

This is a comment on the paper “A heuristic Lagrangean relaxation algorithm for the capacitated plant location problem” by J. Barcelo and J. Casanovas. The authors have a single source version of the problem where they provide a necessary and sufficient condition for the feasibility of the problem. We show that the condition as shown by the authors is only a necessary condition and not a sufficient condition using an example.     

Multiple solutions for a nonlocal problem in Orlicz–Sobolev spaces

Using the mountain pass theorem combined with the minimum principle, we obtain a multiplicity result for a nonlocal problem in Orlicz–Sobolev spaces. To our knowledge, this is the first contribution to the study of nonlocal problems in this class of spaces.     

A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem

In this paper, we develop an a posteriori error analysis of a mixed finite element method for a fluid–solid interaction problem posed in the plane. The media are governed by the acoustic and elastodynamic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the normal displacements of the solid and the fluid. The coupling of primal and dual-mixed finite element methods is applied to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the elastic body. The finite element subspaces consider continuous piecewise linear elements for the pressure and a Lagrange multiplier defined on the interface, and PEERS for the stress and rotation in the solid domain. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Clément interpolant and Raviart–Thomas operator are the main tools for proving the reliability of the estimator. Then, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, some numerical results confirming the reliability and efficiency of the estimator are reported.     

A Riemann–Hilbert problem for skew-orthogonal polynomials

We find a local (d+1)×(d+1)$(d+1)\times (d+1)$ Riemann–Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d  . Our Riemann–Hilbert problem is similar to a local d×d$d\times d$ Riemann–Hilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polynomials. This gives more motivation for finding methods to compute asymptotics of high order Riemann–Hilbert problems, and brings us closer to finding full asymptotic expansions of the skew-orthogonal polynomials.     

A Dido problem for domains in ℝ2 with a given inradius

We find which are the simply connected domains in ² satisfying the Dido condition for a straight shoreline, with a given area A and a fixed inradius , which minimize the length of the free boundary. There are three different cases according to the values of A and .Research partially supported by grant PS87-0115-C03-01 of the DGICYT.     

A constrained nonlinear 0–1 program for data allocation

This paper analyzes the problem of allocating copies of relations from a global database to the sites of a geographically distributed communication network. The objective of the allocation is to minimize the total cost due to transmissions generated by queries from the various sites, including queries that access multiple relations. This allocation problem is modeled as a constrained nonlinear 0–1 subproblems generated during subgradient optimization are solved as optimization. Some of the unconstrained quadratic 0–1 subproblems generated during subgradient optimization are solved as maximum flow problems, while the others require implicit enumeration, depending on the nature of the objective function coefficients of the subproblems. Our solution approach is tested extensively on data allocation problems with as many as 100 sites and 20 relations. On a set of randomly generated test problems our approach was close to two orders of magnitude faster than the general purpose integer programming code OSL.     

A note on optimal investment–consumption–insurance in a Lévy market

In Shen and Wei (2014) an optimal investment, consumption and life insurance purchase problem for a wage earner with Brownian information has been investigated. This paper discusses the same problem but extend their results to a geometric Itô–Lévy jump process. Our modelling framework is very general as it allows random parameters which are unbounded and involves some jumps. It also covers parameters which are both Markovian and non-Markovian functionals. Unlike in Shen and Wei (2014) who considered a diffusion framework, ours solves the problem using a novel approach, which combines the Hamilton–Jacobi–Bellman (HJB) and a backward stochastic differential equation (BSDE) in a Lévy market setup. We illustrate our results by two examples.     

A multiplicity result for a generalized Sturm–Liouville problem on time scales

By applying the fixed point theorem in cones, a new and general result on the existence of positive solutions to second-order generalized Sturm–Liouville boundary value problem on a time scale 𝕋 is obtained. The first order Δ-derivative is involved in the nonlinear term f explicitly.     

Solution of the Bitsadze–Samarskii problem for a parabolic system in a semibounded domain

We consider a nonlocal initial–boundary value Bitsadze–Samarskii problem for a spatially one-dimensional parabolic second-order system in a semibounded domain with nonsmooth lateral boundary. The boundary integral equation method is used to construct a classical solution of this problem under the condition that the vector function on the right-hand side in the nonlocal boundary condition only has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.