Locating flow-capturing units on a network with multi-counting and diminishing returns to scale

In the paper we consider the problem of locating flow-capturing units (facilities) on a transportation network, where the level of consuming service by customers depends on the number of facilities that they encounter on their pre-planned tour (the effect of multi-counting). Two location problems are considered: Problem 1 — minimizing the number of facilities required to ensure the maximal level of consumption, and Problem 2 — maximizing the total consumption given a restriction on the number of facilities. Both problems are NP-hard on general networks. Integer programming formulations of the problems are given. For Problem 2, a heuristic with worst-case analysis is presented. It is shown that Problem 2 is NP-hard even on a tree (and even without multi-counting). For Problem 1 on a tree a polynomial algorithm is presented. If it is required additionally that at most one facility can be located at each node and locations are restricted to nodes, then both problems are NP-hard on trees.     

An Iterated Local Search for the Budget Constrained Generalized Maximal Covering Location Problem

Capacitated covering models aim at covering the maximum amount of customers’ demand using a set of capacitated facilities. Based on the assumptions made in such models, there is a unique scenario to open a facility in which each facility has a pre-specified capacity and an operating budget. In this paper, we propose a generalization of the maximal covering location problem, in which facilities have different scenarios for being constructed. Essentially, based on the budget invested to construct a given facility, it can provide different service levels to the surrounded customers. Having a limited budget to open the facilities, the goal is locating a subset of facilities with the optimal opening scenario, in order to maximize the total covered demand and subject to the service level constraint. Integer linear programming formulations are proposed and tested using ILOG CPLEX. An iterated local search algorithm is also developed to solve the introduced problem.     

Optimizing capacity,pricing and location decisions on a congested network with balking

In this paper, we consider the problem of making simultaneous decisions on the location, service rate (capacity) and the price of providing service for facilities on a network. We assume that the demand for service from each node of the network follows a Poisson process. The demand is assumed to depend on both price and distance. All facilities are assumed to charge the same price and customers wishing to obtain service choose a facility according to a Multinomial Logit function. Upon arrival to a facility, customers may join the system after observing the number of people in the queue. Service time at each facility is assumed to be exponentially distributed. We first present several structural results. Then, we propose an algorithm to obtain the optimal service rate and an approximate optimal price at each facility. We also develop a heuristic algorithm to find the locations of the facilities based on the tabu search method. We demonstrate the efficiency of the algorithms numerically.     

Improving solution of discrete competitive facility location problems

We consider discrete competitive facility location problems in this paper. Such problems could be viewed as a search of nodes in a network, composed of candidate and customer demand nodes, which connections correspond to attractiveness between customers and facilities located at the candidate nodes. The number of customers is usually very large. For some models of customer behavior exact solution approaches could be used. However, for other models and/or when the size of problem is too high to solve exactly, heuristic algorithms may be used. The solution of discrete competitive facility location problems using genetic algorithms is considered in this paper. The new strategies for dynamic adjustment of some parameters of genetic algorithm, such as probabilities for the crossover and mutation operations are proposed and applied to improve the canonical genetic algorithm. The algorithm is also specially adopted to solve discrete competitive facility location problems by proposing a strategy for selection of the most promising values of the variables in the mutation procedure. The developed genetic algorithm is demonstrated by solving instances of competitive facility location problems for an entering firm.     

The uncapacitated facility location problem with demand-dependent setup and service costs and customer-choice allocation

We consider a generalization of the uncapacitated facility location problem, where the setup cost for a facility and the price charged for service may depend on the number of customers patronizing the facility. Customers are represented by the nodes of the transportation network, and facilities can be located only at nodes; a customer selects a facility to patronize so as to minimize his (her) expenses (price for service + the part of transportation costs paid by the customer). We assume that transportation costs are paid partially by the service company and partially by customers. The objective is to choose locations for facilities and balanced prices so as to either minimize the expenses of the service company (the sum of the total setup cost and the total part of transportation costs paid by the company), or to maximize the total profit. A polynomial-time dynamic programming algorithm for the problem on a tree network is developed.     

Planning and approximation models for delivery route based services with price-sensitive demands

Classical vehicle routing problems typically do not consider the impact of delivery price on the demand for delivery services. Existing models seek the minimum sum of tour lengths in order to serve the demands of a given set of customers. This paper proposes approximation models to estimate the impacts of price on delivery services when demand for delivery service is price dependent. Such models can serve as useful tools in the planning phase for delivery service providers and can assist in understanding the economics of delivery services. These models seek to maximize profit from delivery service, where price determines demand for deliveries as well as the total revenue generated by satisfying demand. We consider a variant of the model in which each customer’s delivery volume is price sensitive, as well as the case in which customer delivery volumes are fixed, but the total number of customers who select the delivery service provider is price sensitive. A third model variant allows the delivery service provider to select a subset of delivery requests at the offered price in order to maximize profit.     

General network design: A unified view of combined location and network design problems

This paper presents a unified framework for the general network design problem which encompasses several classical problems involving combined location and network design decisions. In some of these problems the service demand relates users and facilities, whereas in other cases the service demand relates pairs of users between them, and facilities are used to consolidate and re-route flows between users. Problems of this type arise in the design of transportation and telecommunication systems and include well-known problems such as location-network design problems, hub location problems, extensive facility location problems, tree-star location problems and cycle-star location problems, among others. Relevant modeling aspects, alternative formulations and possible algorithmic strategies are presented and analyzed.     

The plant location problem with demand-dependent setup costs and centralized allocation

Suppose that customers are situated at the nodes of a transportation network, and a service company plans to locate a number of facilities that will serve the customers. The objective is to minimize the sum of the total setup cost and the total transportation cost. The setup cost of a facility is demand-dependent, that is, it depends on the number of customers that are served by the facility. Centralized allocation of customers to facilities is assumed, that is, the service company makes a decision about allocation of customers to facilities. In the case of a general network, the model can be formulated as a mixed integer programming problem. For the case of a tree network, we develop a polynomial-time dynamic programming algorithm.     

Facility location for market capture when users rank facilities by shorter travel and waiting times

A firm wants to locate several multi-server facilities in a region where there is already a competitor operating. We propose a model for locating these facilities in such a way as to maximize market capture by the entering firm, when customers choose the facilities they patronize, by the travel time to the facility and the waiting time at the facility. Each customer can obtain the service or goods from several (rather than only one) facilities, according to a probabilistic distribution. We show that in these conditions, there is demand equilibrium, and we design an ad hoc heuristic to solve the problem, since finding the solution to the model involves finding the demand equilibrium given by a nonlinear equation. We show that by using our heuristic, the locations are better than those obtained by utilizing several other methods, including MAXCAP, p-median and location on the nodes with the largest demand.     

Applying the flow-capturing location-allocation model to an authentic network: Edmonton, Canada

Traditional location-allocation models aim to locate network facilities to optimally serve demand expressed as weights at nodes. For some types of facilities demand is not expressed at nodes, but as passing network traffic. The flow-capturing location-allocation model responds to this type of demand and seeks to maximize one-time exposure of such traffic to facilities. This new model has previously been investigated only with small and contrived problems. In this paper, we apply the flow-capturing location-allocation model to morning-peak traffic in Edmonton, Canada. We explore the effectiveness of exact, vertex substitution, and greedy solution procedures; the first two are computationally demanding, the greedy is very efficient and extremely robust. We hypothesize that the greedy algorithm's robustness is enhanced by the structured flow present in an authentic urban road network. The flow-capturing model was derived to overcome flow cannibalization, wasteful redundant flow-capturing; we demonstrate that this is an important consideration in an authentic network. We conclude that real-world testing is an important aspect of location model development.     

Stop location design in public transportation networks: covering and accessibility objectives

We consider the location of new stops along the edges of an existing public transportation network. Examples of StopLoc include the location of bus stops along some given bus routes or of railway stations along the tracks in a railway system. In order to evaluate the decision assume that potential customers in given demand facilities are known. Two objectives are proposed. In the first one, we minimize the number of stations such that any of the demand facilities can reach a closest station within a given distance of r. In the second objective, we fix the number of new stations and minimize the sum of the distances between demand facilities and stations. The resulting two problems CovStopLoc and AccessStopLoc are solved by a reduction to a classical set covering and a restricted location problem, respectively. We implement the general ideas in two different environments, the plane, where demand facilities are represented by coordinates, and in networks, where they are nodes of a graph.     

Location–Allocation of Multiple-Server Service Centers with Constrained Queues or Waiting Times

Recently, the authors have formulated new models for the location of congested facilities, so to maximize population covered by service with short queues or waiting time. In this paper, we present an extension of these models, which seeks to cover all population and includes server allocation to the facilities. This new model is intended for the design of service networks, including health and EMS services, banking or distributed ticket-selling services. As opposed to the previous Maximal Covering model, the model presented here is a Set Covering formulation, which locates the least number of facilities and allocates the minimum number of servers (clerks, tellers, machines) to them, so to minimize queuing effects. For a better understanding, a first model is presented, in which the number of servers allocated to each facility is fixed. We then formulate a Location Set Covering model with a variable (optimal) number of servers per service center (or facility). A new heuristic, with good performance on a 55-node network, is developed and tested.     

A note on the allocation of queuing facilities using a minisum criterion

This paper suggests a formulation and a solution procedure for resource allocation problems which consider a central planner, m static queuing facilities providing a homogeneous service at their locations, and a known set of demand points or customers. It is assumed that upon a request for service the customer is routed to a facility by a probabilistic assignment. The objective is to determine how to allocate a limited number of servers to the facilities, and to specify demand rates from customers to facilities in order to minimize a weighted sum of response times. This sum measures the total time lost in the system due to two sources: travel time from customer to facility locations and waiting time for service at the facilities. The setting does not allow for cooperation between the facilities.     

Directional approach to gradual cover: a maximin objective

The objective of original cover location models is to cover demand within a given distance by facilities. Locating a given number of facilities to cover as much demand as possible is referred to as max-cover, and finding the minimum number of facilities required to cover all the demand is referred to as set covering. When the objective is to maximize the minimum cover of demand points, the maximin objective is equivalent to set covering because each demand point is either covered or not. The gradual (or partial) cover replaces abrupt drop from full cover to no cover by defining gradual decline in cover. Both maximizing total cover and maximizing the minimum cover are useful objectives using the gradual cover measure. In this paper we use a recently proposed rule for calculating the joint cover of a demand point by several facilities termed “directional gradual cover”. The objective is to maximize the minimum cover of demand points. The solution approaches were extensively tested on a case study of covering Orange County, California.

     

A network location-allocation model trading off flow capturing andp-median objectives

The flow capturing and thep-median location—allocation models deal quite differently with demand for service in a network. Thep-median model assumes that demand is expressed at nodes and locates facilities to minimize the total distance between such demand nodes and the nearest facility. The flow-capturing model assumes that demand is expressed on links and locates facilities to maximize the one-time exposure of such traffic to facilities. Demand in a network is often of both types: it is expressed by passing flows and by consumers centred in residential areas, aggregated as nodes. We here present a hybrid model with the dual objective of serving both types of demand. We use this model to examine the tradeoff between serving the two types of demand in a small test network using synthetic demand data. A major result is the counter-intuitive finding that thep-median model is more susceptible to impairment by the flow capturing objective than is the flow capturing model to thep-median objective. The results encourage us to apply the model to a real-world network using actual traffic data.     

The minmax regret gradual covering location problem on a network with incomplete information of demand weights

The gradual covering location problem seeks to establish facilities on a network so as to maximize the total demand covered, allowing partial coverage. We focus on the gradual covering location problem when the demand weights associated with nodes of the network are random variables whose probability distributions are unknown. Using only information on the range of these random variables, this study is aimed at finding the “minmax regret” location that minimizes the worst-case coverage loss. We show that under some conditions, the problem is equivalent to known location problems (e.g. the minmax regret median problem). Polynomial time algorithms are developed for the problem on a general network with linear coverage decay functions.     

Optimizing pricing and location decisions for competitive service facilities charging uniform price

In this paper, we present the problem of optimizing the location and pricing for a set of new service facilities entering a competitive marketplace. We assume that the new facilities must charge the same (uniform) price and the objective is to optimize the overall profit for the new facilities. Demand for service is assumed to be concentrated at discrete demand points (customer markets); customers in each market patronize the facility providing the highest utility. Customer demand function is assumed to be elastic; the demand is affected by the price, facility attractiveness, and the travel cost for the highest-utility facility. We provide both structural and algorithmic results, as well as some managerial insights for this problem. We show that the optimal price can be selected from a certain finite set of values that can be computed in advance; this fact is used to develop an efficient mathematical programming formulation for our model.     

Minisum collection depots location problem with multiple facilities on a network

We investigate the problem of locating a set of service facilities that need to service customers on a network. To provide service, a server has to visit both the demand node and one of several collection depots. We employ the criterion of minimizing the weighted sum of round trip distances. We prove that there exists a dominating location set for the problem on a general network. The properties of the solution on a tree and on a cycle are discussed. The problem of locating service facilities and collection depots simultaneously is also studied. To solve the problem on a general network, we suggest a Lagrangian relaxation imbedded branch-and-bound algorithm. Computational results are reported.     

Siting and Sizing of Facilities under Probabilistic Demands

In this paper a discrete location model for non-essential service facilities planning is described, which seeks the number, location, and size of facilities, that maximizes the total expected demand attracted by the facilities. It is assumed that the demand for service is sensitive to the distance from facilities and to their size. It is also assumed that facilities must satisfy a threshold level of demand (facilities are not economically viable below that level). A Mixed-Integer Nonlinear Programming (MINLP) model is proposed for this problem. A branch-and-bound algorithm is designed for solving this MINLP and its convergence to a global minimum is established. A finite procedure is also introduced to find a feasible solution for the MINLP that reduces the overall search in the binary tree generated by the branch-and-bound algorithm. Some numerical results using a GAMS/MINOS implementation of the algorithm are reported to illustrate its efficacy and efficiency in practice.