A location–allocation problem for a web services provider in a competitive market

Web Services have become a viable component technology in distributed e-commerce platforms. Due to the move to high-speed Internet communication and tremendous increases in computing power, network latency has begun to play a more important role in determining service response time. Hence, the locations of a Web Services provider’s facilities, customer allocation, and the number of servers at each facility have a significant impact on its performance and customer satisfaction. In this paper we introduce a location–allocation model for a Web Services provider in a duopoly competitive market. Demands for services of these servers are available at each node of a network, and a subset of nodes is to be chosen to locate one or more servers in each. The objective is to maximize the provider’s profit. The problem is formulated and analyzed. An exact solution approach is developed and the results of its efficiency are reported.     

An exact procedure and LP formulations for the leader—follower location problem

The leader—follower location problem consists of determining an optimal strategy for two competing firms which make decisions sequentially. The leader optimisation problem is to minimise the maximum market share of the follower. The objective of the follower problem is to maximise its market share. We describe linear programming formulations for both problems and analyse the use of these formulations to solve the problems. We also propose an exact procedure based on an elimination process in a candidate list.     

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.     

Perturbation resilience for the facility location problem

We analyze a simple local search heuristic for the facility location problem using the notion of perturbation resilience: an instance is $\gamma$-perturbation resilient if all costs can be perturbed by a factor of $\gamma$ without changing the optimal solution.We prove that local search for FLP succeeds in finding the optimal solution for $\gamma$-perturbation resilient instances for $\gamma \ge 3$, and we show that this is tight.     

Fujita exponents for a space–time weighted parabolic problem in bounded domains

This paper deals with a semilinear weighted parabolic problem in general bounded domains, subject to zero Dirichlet boundary conditions, where the weighted functions depend not only on space variable but also on time variable. Fujita exponents for blow-up and global existence of solutions are determined by using semigroup methods and the comparison principle, which are composed by the dimensions of the space domains and the eight exponents in nonlinear coupled sources and the space–time weighted functions.     

Tabu list management methods for a discrete–continuous scheduling problem

A problem of scheduling jobs on parallel, identical machines under an additional continuous resource to minimize the makespan is considered. Jobs are non-preemtable and independent and all are available at the start of the process. The total amount of the continuous resource available at a time is limited and the resource is a renewable one. Each job simultaneously requires for its processing a machine and an amount (unknown in advance) of the continuous resource. Processing rate of a job depends on the amount of the resource allotted to this job at a time. The problem is to find a sequence of jobs on machines and, simultaneously, a continuous resource allocation that minimizes the makespan. The tabu search (TS) metaheuristic is presented to attack the problem. Three different tabu list management methods: the tabu navigation method (TNM), the cancellation sequence method (CSM) and the reverse elimination method (REM) are discussed and examined. A computational experiment is described and the results obtained for the methods tested are compared to optimal solutions. A few conclusions and final remarks are presented.     

New lower bounds for the facility location problem with clients’ preferences

A bilevel facility location problem in which the clients choose suppliers based on their own preferences is studied. It is shown that the coopertative and anticooperative statements can be reduced to a particular case in which every client has a linear preference order on the set of facilities to be opened. For this case, various reductions of the bilevel problem to integer linear programs are considered. A new statement of the problem is proposed that is based on a family of valid inequalities that are related to the problem on a pair of matrices and the set packing problem. It is shown that this formulation is stronger than the other known formulations from the viewpoint of the linear relaxation and the integrality gap.     

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.     

Critical space for the parabolic-parabolic Keller–Segel model in Rd

We study the Keller–Segel system in ${\mathbb{R}}^d$ when the chemoattractant concentration is described by a parabolic equation. We prove that the critical space, with some similarity to the elliptic case, is that the initial bacteria density satisfies $n_0\in L^a({\mathbb{R}}^d)$, $a>d/2$, and that the chemoattractant concentration satisfies $\mathrm{?}{c}_0\in L^d({\mathbb{R}}^d)$. In these spaces, we prove that small initial data give rise to global solutions that vanish as the heat equation for large times and that exhibit a regularizing effect of hypercontractivity type. To cite this article: L. Corrias, B. Perthame, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

A discrete prey–predator model preserving the dynamics of a structurally unstable Lotka–Volterra model

Leslie's method to construct a discrete two dimensional dynamical system dynamically consistent with the Lotka–Volterra type of competing two species ordinary differential equations is applied in a newly extended manner for the Lotka–Volterra prey–predator system which is structurally unstable. We show that, independently of the time step size, the derived discrete prey–predator system is dynamically consistent with the continuous counterpart, keeping the nature of neutrally stable periodic orbit. Further, we show that the extended method to construct the discrete prey–predator system can provide a dynamically consistent model also for the logistic Lotka–Volterra one.     

The generalized Riemann problem for a scalar Chapman–Jouguet combustion model

The generalized Riemann problem for a scalar Chapman–Jouguet combustion model in a neighborhood of the origin (t > 0) on the (x, t) plane is studied. Under the entropy conditions, we obtain the solutions constructively. It is found that, for some cases, the perturbed Riemann solutions are essentially different from the corresponding Riemann solutions. The perturbation may transform a combustion wave CJDT into SDT in the neighborhood of the origin. Especially, it can be observed that burning happens although the corresponding Riemann solution doesn’t contain combustion waves, which exhibits the instability for unburnt states. This work is supported by NSFC 10671120     

Global Cauchy problem for a Leray-α model

Acta Mathematicae Applicatae Sinica, English Series - In this paper, we consider the Cauchy problem for the 3D Leray-α model, introduced by Cheskidov et al.[11]. We obtain the global solution...     

The branch and cut method for the facility location problem with client’s preferences

Numerical study is provided of the methods for solving the facility location problem when the clients choose some suppliers by their own preferences. Various formulations of this problem as an integer linear programming problem are considered. The authors implement a cutting plane method based on the earlier proposed family of valid inequalities which arises from connection with the problem for a pair of matrices. The results of numerical experiment are presented for testing this method. An optimal solution is obtained by the two versions of the branch and cut method with the suggested cutting plane method. The simulated annealing method is proposed for obtaining the upper bounds of the optimal solution used in exact methods. Numerical experiment approves the efficiency of the implemented approach in comparison with the previously available methods.     

A note on a discrete model for the harmonic oscillator Schrödinger equation in N-cartesian coordinates

We construct a discrete model for the time-independent harmonic oscillator Schrödinger partial differential equation and demonstrate that it can be separated into N ordinary difference equations for the case of N-cartesian space coordinates.     

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.