Minimizing the passengers’ traveling time in the stop location problem

In this paper we consider the location of stops along the edges of an already existing public transportation network. The positive effect of new stops is given by the better access of the passengers to the public transport network, while the passengers’ traveling time increases due to the additional stopping activities of the trains, which is a negative effect for the passengers. The problem has been treated in the literature where the most common model is to cover all demand points with a minimal number of new stops. In this paper, we follow this line and seek for a set of new stops covering all demand points but instead of minimizing the number of new stops we minimize the additional passengers’ traveling time due to the new stops. For computing this additional traveling time we do not only take the stopping times of the vehicles but also acceleration and deceleration of the vehicles into account. We show that the problem is NP-hard, but we are able to derive a finite candidate set and two tractable IP formulations. For linear networks we show that the problem is polynomially solvable. We also discuss the differences to the common models from literature showing that minimizing the number of new stops does not necessarily lead to a solution with minimal additional traveling times for the passengers. We finally provide a case study showing that our new model decreases the traveling times for the passengers while still achieving the minimal number of new stops.     

A constant factor approximation algorithm for the fault-tolerant facility location problem

We consider a generalization of the classical facility location problem, where we require the solution to be fault-tolerant. In this generalization, every demand point j must be served by r_j facilities instead of just one. The facilities other than the closest one are “backup” facilities for that demand, and any such facility will be used only if all closer facilities (or the links to them) fail. Hence, for any demand point, we can assign nonincreasing weights to the routing costs to farther facilities. The cost of assignment for demand j is the weighted linear combination of the assignment costs to its r_j closest open facilities. We wish to minimize the sum of the cost of opening the facilities and the assignment cost of each demand j. We obtain a factor 4 approximation to this problem through the application of various rounding techniques to the linear relaxation of an integer program formulation. We further improve the approximation ratio to 3.16 using randomization and to 2.41 using greedy local-search type techniques.     

A model for setting services on auxiliary bus lines under congestion

In this paper, a mathematical programming model and a heuristically derived solution is described to assist with the efficient planning of services for a set of auxiliary bus lines (a bus-bridging system) during disruptions of metro and rapid transit lines. The model can be considered static and takes into account the average flows of passengers over a given period of time (i.e., the peak morning traffic hour). Auxiliary bus services must accommodate very high demand levels, and the model presented is able to take into account the operation of a bus-bridging system under congested conditions. A general analysis of the congestion in public transportation lines is presented, and the results are applied to the design of a bus-bridging system. A nonlinear integer mathematical programming model and a suitable approximation of this model are then formulated. This approximated model can be solved by a heuristic procedure that has been shown to be computationally viable. The output of the model is as follows: (a) the number of bus units to assign to each of the candidate lines of the bus-bridging system; (b) the routes to be followed by users passengers of each of the origin–destination pairs; (c) the operational conditions of the components of the bus-bridging system, including the passenger load of each of the line segments, the degree of saturation of the bus stops relative to their bus input flows, the bus service times at bus stops and the passenger waiting times at bus stops. The model is able to take into account bounds with regard to the maximum number of passengers waiting at bus stops and the space available at bus stops for the queueing of bus units. This paper demonstrates the applicability of the model with two realistic test cases: a railway corridor in Madrid and a metro line in Barcelona.     

On the complexity of locating linear facilities in the plane

We consider the computational complexity of linear facility location problems in the plane, i.e., given n demand points, one wishes to find r lines so as to minimize a certain objective-function reflecting the need of the points to be close to the lines. It is shown that it is NP-hard to find r lines so as to minimize any isotone function of the distances between given points and their respective nearest lines. The proofs establish NP-hardness in the strong sense. The results also apply to the situation where the demand is represented by r lines and the facilities by n single points.     

Deterministic flow-demand location problems

In flow-covering (interception) models the focus is on the demand for service that originates from customers travelling in the network (not for the purpose of obtaining the service). In contrast, in traditional location models a central assumption is that the demand for service comes from customers residing at nodes of the network. In this paper we combine these two types of models. The paper presents four new problems. Two of the four deal with the problem of locating m facilities so as to maximize the total number of potential customers covered by the facilities (where coverage does not necessarily imply the actual consumption of service). In the two other problems the attention is directed to the consumption of service and thus the criteria is to maximize (minimize) the number of actual users (distance travelled). It is shown in the paper that all four problems have similar structure to other known location problems.     

校车安排问题

探讨如何安排校车运行使得教师和工作人员尽量满意的问题.首先建立动态规划模型和选址规划模型,求出合理站点位置及其总距离.然后用归一法定义满意度与距离的函数关系,考虑各区域人数,建立选址规划模型.得到合理站点位置和总满意度.之后建立双目标非线性规划模型,利用量纲分析法给出权重,以此求出合理乘车位置和满意度.最后对问题进行推...     

An algorithm for generalized constrained multi-source Weber problem with demand substations

In this paper, we consider a multi-source Weber problem of m new facilities with respect to n demand regions in order to minimize the sum of the transportation costs between these facilities and the demand regions. We find a point on the border of each demand region from which the facilities serve the demand regions at these points. We present an algorithm including a location phase and an allocation phase in each iteration for solving this problem. An algorithm is also proposed for carrying out the location phase. Moreover, global convergence of the new algorithm is proved under mild assumptions, and some numerical results are presented.     

Price Expectations and Cobwebs Under Uncertainty

Classical approaches to location problems are based on the minimization of the average distance (the median concept) or the minimization of the maximum distance (the center concept) to the service facilities. The median solution concept is primarily concerned with the spatial efficiency while the center concept is focused on the spatial equity. The k-centrum model unifies both the concepts by minimization of the sum of the k largest distances. In this paper we investigate a solution concept of the conditional median which is a generalization of the k-centrum concept taking into account the portion of demand related to the largest distances. Namely, for a specified portion (quantile) of demand we take into account the entire group of the corresponding largest distances and we minimize their average. It is shown that such an objective, similar to the standard minimax, may be modeled with a number of simple linear inequalities. Equitable properties of the solution concept are examined.     

Optimal network configuration and capacity expansion of railroads

Railroads ship individual cars according to blocking plans that route the cars in groups (blocks) that share common intermediate stops. An individual shipment is regrouped (reclassified) two to three times along the way from its origin to destination. Yards are crucial facilities of the rail network where cars are reclassified according to such blocking plans. Therefore, yard locations and the blocking plan impose the detour and classification of cars over the physical network. Yards are capacitated with respect to number of blocks made and number of cars classified; rail lines between major stations are capacitated with respect to number of cars that pass through. These restrictions are accounted for in designing the blocking plans. Changing the yard locations and expanding associated capacities may result in dramatic changes in blocking plans saving tens of millions of dollars in railroad transportation costs. We develop a mathematical programming formulation and propose solution methods for the yard location problem and the capacity expansion problems. We demonstrate that the railroads can save significantly by reconfiguring their networks.     

The <Emphasis Type="Italic">p</Emphasis>-Centre Problem—Heuristic and Optimal Algorithms

The p-centre problem, or minimax location-allocation problem in location theory terminology, is the following: given n demand points on the plane and a weight associated with each demand point, find p new facilities on the plane that minimize the maximum weighted Euclidean distance between each demand point and its closest new facility. We present two heuristics and an optimal algorithm that solves the problem for a given p in time polynomial in n. Computational results are presented.     

Local search algorithm for universal facility location problem with linear penalties

The universal facility location problem generalizes several classical facility location problems, such as the uncapacitated facility location problem and the capacitated location problem (both hard and soft capacities). In the universal facility location problem, we are given a set of demand points and a set of facilities. We wish to assign the demands to facilities such that the total service as well as facility cost is minimized. The service cost is proportional to the distance that each unit of the demand has to travel to its assigned facility. The open cost of facility i depends on the amount z of demand assigned to i and is given by a cost function \(f_i(z)\). In this work, we extend the universal facility location problem to include linear penalties, where we pay certain penalty cost whenever we refuse serving some demand points. As our main contribution, we present a (\(7.88+\epsilon \))-approximation local search algorithm for this problem.     

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 polynomial time algorithm for solving a quality control station configuration problem

We study unreliable serial production lines with known failure probabilities for each operation. Such a production line consists of a series of stations; existing machines and optional quality control stations (QCS). Our aim is to simultaneously decide where and if to install the QCSs along the line and to determine the production rate, so as to maximize the steady state expected net profit per time unit from the system.We use dynamic programming to solve the cost minimization auxiliary problem where the aim is to minimize the time unit production cost for a given production rate. Using the above developed O(N²) dynamic programming algorithm as a subroutine, where N stands for the number of machines in the line, we present an O(N⁴) algorithm to solve the Profit Maximization QCS Configuration Problem.     

The location of median paths on grid graphs

In this paper we consider the location of a path shaped facility on a grid graph. In the literature this problem was extensively studied on particular classes of graphs as trees or series-parallel graphs. We consider here the problem of finding a path which minimizes the sum of the (shortest) distances from it to the other vertices of the grid, where the path is also subject to an additional constraint that takes the form either of the length of the path or of the cardinality. We study the complexity of these problems and we find two polynomial time algorithms for two special cases, with time complexity of O(n) and O(nℓ) respectively, where n is the number of vertices of the grid and ℓ is the cardinality of the path to be located. The literature about locating dimensional facilities distinguishes between the location of extensive facilities in continuous spaces and network facility location. We will show that the problems presented here have a close connection with continuous dimensional facility problems, so that the procedures provided can also be useful for solving some open problems of dimensional facilities location in the continuous case.     

Demand Point Aggregation for Planar Covering Location Models

The covering location problem seeks the minimum number of facilities such that each demand point is within some given radius of its nearest facility. Such a model finds application mostly in locating emergency types of facilities. Since the problem is NP-hard in the plane, a common practice is to aggregate the demand points in order to reduce the computational burden. Aggregation makes the size of the problem more manageable but also introduces error. Identifying and controlling the magnitude of the error is the subject of this study. We suggest several aggregation methods with a priori error bounds, and conduct experiments to compare their performance. We find that the manner by which infeasibility is measured greatly affects the best choice of an aggregation method.     

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.     

On solving the discrete location problems when the facilities are prone to failure

The classical discrete location problem is extended here, where the candidate facilities are subject to failure. The unreliable location problem is defined by introducing the probability that a facility may become inactive. The formulation and the solution procedure have been motivated by an application to model and solve a large size problem for locating base stations in a cellular communication network. We formulate the unreliable discrete location problems as 0–1 integer programming models, and implement an enhanced dual-based solution method to determine locations of these facilities to minimize the sum of fixed cost and expected operating (transportation) cost. Computational tests of some well-known problems have shown that the heuristic is efficient and effective for solving these unreliable location problems.     

Algorithms for a Facility Location Problem with Stochastic Customer Demand and Immobile Servers

This paper studies a facility location problem with stochastic customer demand and immobile servers. Motivated by applications to locating bank automated teller machines (ATMs) or Internet mirror sites, these models are developed for situations in which immobile service facilities are congested by stochastic demand originating from nearby customer locations. Customers are assumed to visit the closest open facility. The objective of this problem is to minimize customers' total traveling cost and waiting cost. In addition, there is a restriction on the number of facilities that may be opened and an upper bound on the allowable expected waiting time at a facility. Three heuristic algorithms are developed, including a greedy-dropping procedure, a tabu search approach and an -optimal branch-and-bound method. These methods are compared computationally on a bank location data set from Amherst, New York.     

The backup 2-median problem on block graphs

The backup 2-median problem is a location problem to locate two facilities at vertices with the minimum expected cost where each facility may fail with a given probability. Once a facility fails, the other one takes full responsibility for the services. Here we assume that the facilities do not fail simultaneously. In this paper, we consider the backup 2-median problem on block graphs where any two edges in one block have the same length and the lengths of edges on different blocks may be different. By constructing a tree-shaped skeleton of a block graph, we devise an O（n log n q- m）-time algorithm to solve this problem where n and m are the number of vertices and edges, respectively, in the given block graph.     

Solving the Passive Optical Network with Fiber Duct Sharing Planning Problem Using Discrete Techniques

Similar to the constrained facility location problem, the passive optical network (PON) planning problem necessitates the search for a subset of deployed facilities (splitters) and their allocated demand points (optical network units) to minimize the overall deployment cost. In this paper we use a mixed integer linear programming formulation stemming from network flow optimization to construct a heuristic based on limiting the total number of interconnecting paths when implementing fiber duct sharing. Then, a disintegration heuristic involving the construction of valid clusters from the output of a k means algorithm, reduce the time complexity while ensuring close to optimal results. The proposed heuristics are then evaluated using a real-world dataset, showing favourable performance.