Location and allocation of service units on a congested network (original) (raw)
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We consider two capacity choice scenarios for the optimal location of facilities with fixed servers, stochastic demand, and congestion. Motivating applications include virtual call centers, consisting of geographically dispersed centers, walk-in health clinics, motor vehicle inspection stations, automobile emissions testing stations, and internal service systems. The choice of locations for such facilities influences both the travel cost and waiting times of users. In contrast to most previous research, we explicitly embed both customer travel/connection and delay costs in the objective function and solve the location–allocation problem and choose facility capacities simultaneously. The choice of capacity for a facility that is viewed as a queueing system with Poisson arrivals and exponential service times could mean choosing a service rate for the servers (Scenario 1) or choosing the number of servers (Scenario 2). We express the optimal service rate in closed form in Scenario 1 and the (asymptotically) optimal number of servers in closed form in Scenario 2. This allows us to eliminate both the number of servers and the service rates from the optimization problems, leading to tractable mixed-integer nonlinear programs. Our computational results show that both problems can be solved efficiently using a Lagrangian relaxation optimization procedure.
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We analyze the problem of optimal location of a set of facilities in the presence of stochastic demand and congestion. Customers travel to the closest facility to obtain service; the problem is to determine the number, locations, and capacity of the facilities. Under rather general assumptions (spatially distributed continuous demand, general arrival and service processes, non-linear location and capacity costs) we show that the problem can be decomposed and construct an efficient optimization algorithm. The analysis yields several insights, including the importance of "equitable facility configurations", the behavior of optimal and near-optimal capacities and robust class of solutions that can be constructed for this problem.
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We consider a class of location-allocation problems with immobile servers, stochastic demand and congestion that arises in several planning contexts: location of emergency medical clinics; preventive healthcare centers; refuse collection and disposal centers; stores and service centers; bank branches and automated banking machines; internet mirror sites; web service providers (servers); and distribution centers in supply chains. The problem seeks to simultaneously locate service facilities, equip them with appropriate capacities, and allocate user demand to these facilities such that the total cost, which consists of the fixed cost of opening facilities with sufficient capacities, the access cost of users' travel to facilities, and the queuing delay cost, is minimized. Under Poisson user demand arrivals and general service time distributions, the problem is set up as a network of independent M/G/1 queues, whose locations, capacities and service zones need to be determined. The resulting mathematical model is a non-linear integer program. Using simple transformation and piecewise linear approximation, the model is linearized and solved to -optimality using a constraint generation method. Computational results are presented for instances up to 400 users, 25 potential service facilities, and 5 capacity levels with different coefficient of variation of service times and average queueing delay costs per customer. The results indicate that the proposed solution method is efficient in solving a wide range of problem instances.
Optimal location of multi-server congestible facilities operating as M/E r /m/N queues
Most models for location of immobile congested facilities assume exponentially distributed service time at the facilities. Although the resulting formulations are tractable, they do not adequately represent service time distributions with small variances, as often occur in practice. In a recent paper, the authors utilized an order r Erlang distribution for the service time, applied to the simple case of single-server facilities. We generalize this approach to multiple-server facilities, which need a different mathematical treatment. The constraint on service availability is cast as a linear constraint on the proportion of time the servers are busy, and its righthand side parameter is provided for different situations. Extensive analysis is offered on the influence of the parameters of the service time and the capacity of the facilities on the performance of the system. Numerical results are given for a data set relating to the municipality of Rio de Janeiro.
A congested capacitated location problem with continuous network demand
RAIRO - Operations Research
This paper presents a multi-objective mixed-integer non-linear programming model for a congested multiple-server discrete facility location problem with uniformly distributed demands along the network edges. Regarding the capacity of each facility and the maximum waiting time threshold, the developed model aims to determine the number and locations of established facilities along with their corresponding number of assigned servers such that the traveling distance, the waiting time, the total cost, and the number of lost sales (uncovered customers) are minimized simultaneously. Also, this paper proposes modified versions of some of the existing heuristics and metaheuristic algorithms currently used to solve NP-hard location problems. Here, the memetic algorithm along with its modified version called the stochastic memetic algorithm, as well as the modified add and modified drop heuristics are used as the solution methods. Computational results and comparisons demonstrate that althoug...
Allocating servers to facilities, when demand is elastic to travel and waiting times
2005
Public inoculation centers are examples of facilities providing service to customers whose demand is elastic to travel and waiting time. That is, people will not travel too far, or stay in line for too long to obtain the service. The goal, when planning such services, is to maximize the demand they attract, by locating centers and staffing them so as to reduce customers' travel time and time spent in queue. In the case of inoculation centers, the goal is to maximize the people that travel to the centers and stay in line until inoculated. We propose a procedure for the allocation of multiple servers to centers, so that this goal is achieved. An integer programming model is formulated. Since demand is elastic, a supply-demand equilibrium equation must be explicitly included in the optimization model, which then becomes nonlinear. As there are no exact procedures to solve such problems, we propose a heuristic procedure, based on Heuristic Concentration, which finds a good solution to this problem. Numerical examples are presented.
The authors propose a model for locating a fixed number of multiple-server service centers or facilities that may become congested. Customers arriving at these centers must wait in line until served. The locations of the facilities and the allocations of customers to them are chosen by the planner, so to minimize both travel costs and system-wide congestion (or queuing) at centers. All the customers arriving at the facilities must be served, up to a certain maximum line length. The travel cost in which customers incur is a function of the length of the trip to the facility, while the congestion cost at a facility is a general function of the number of customers waiting on line or being served at the facility. The resulting model is a nonlinear p-median formulation. A solution method for this nonlinear model is proposed, and computational experience is presented.
Social Science Research Network, 2003
We propose a model and solution methods, for locating a fixed number of multiple-server, congestible common service centers or congestible public facilities. Locations are chosen so to minimize consumers' congestion (or queuing) and travel costs, considering that all the demand must be served. Customers choose the facilities to which they travel in order to receive service at minimum travel and congestion cost. As a proxy for this criterion, total travel and waiting costs are minimized. The travel cost is a general function of the origin and destination of the demand, while the congestion cost is a general function of the number of customers in queue at the facilities.
Facility location under service level constraints for heterogeneous customers
Annals of Operations Research, 2016
We study the problem of locating service facilities to serve heterogeneous customers. Customers requiring service are classified as either high priority or low priority, where high priority customers are always served on a priority basis. The problem is to optimally locate service facilities and allocate their service zones to satisfy the following coverage and service level constraints: (1) each demand zone is served by a service facility within a given coverage radius; (2) at least α h proportion of the high priority customers at any service facility should be served without waiting; (3) at least α l proportion of the low priority cases at any service facility should not have to wait for more than τ l minutes. For this, we model the network of service facilities as spatially distributed priority queues, whose locations and user allocations need to be determined. The resulting integer programming problem is challenging to solve, especially in absence of any known analytical expression for the service level function of low priority customers. We develop a cutting plane based solution algorithm, exploiting the concavity of the service level function of low priority customers to outer-approximate its non-linearity using supporting planes, determined numerically using matrix geometric method. Using an illustrative example of locating emerging medical service facilities in Austin, Texas, we present computational results and managerial insights.