Exact approaches for solving robust prize-collecting Steiner tree problems

In the Prize-Collecting Steiner Tree Problem (PCStT) we are given a set of customers with potential revenues and a set of possible links connecting these customers with fixed installation costs. The goal is to decide which customers to connect into a tree structure so that the sum of the link costs plus the revenues of the customers that are left out is minimized. The problem, as well as some of its variants, is used to model a wide range of applications in telecommunications, gas distribution networks, protein–protein interaction networks, or image segmentation.     

The time constrained maximal covering salesman problem

We introduce the time constrained maximal covering salesman problem (TCMCSP) which is the generalization of the covering salesman and orienting problems. In this problem, we are given a set of vertices including a central depot, customer and facility vertices where each facility can supply the demand of some customers within its pre-determined coverage distance. Starting from the depot, the goal is to maximize the total number of covered customers by constructing a length constrained Hamiltonian cycle over a subset of facilities. We propose several mathematical programming models for the studied problem followed by a heuristic algorithm. The developed algorithm takes advantage of different procedures including swap, deletion, extraction-insertion and perturbation. Finally, an integer linear programming based improvement technique is designed to try to improve the quality of the solutions. Extensive computational experiments on a set of randomly generated instances indicate the effectiveness of the algorithm.     

一类多商品设施选址问题的基于线性松弛解的启发式方法

多商品设施选址问题是众多设施选址问题中一类重要而困难的问题.在这一问题中,顾客的需求可能包含不止一种商品.对于大规模问题,成熟的商业求解器往往不能在满意的时间内找到高质量的可行解.研究了无容量限制的单货源多商品设施选址问题的一般形式,并给出了应用于此类问题的两个启发式方法.这两个方法基于原选址问题的线性规划松弛问题的最优解,分别通过求解紧问题和邻域搜索的方式给出了原问题的一个可行上界.理论分析指出所提方法可以实施于任意可行问题的实例.数值结果表明所提方法可以显著地提高求解器求解此类设施选址问题的求解效率.     

The gravity p-median model

In this paper we propose a new model for the p-median problem. In the standard p-median problem it is assumed that each demand point is served by the closest facility. In many situations (for example, when demand points are communities of customers and each customer makes his own selection of the facility) demand is divided among the facilities. Each customer selects a facility which is not necessarily the closest one. In the gravity p-median problem it is assumed that customers divide their patronage among the facilities with the probability that a customer patronizes a facility being proportional to the attractiveness of that facility and to a decreasing utility function of the distance to the facility.     

Public and private optimization at a service facility with approximate information on congestion

An M/GI/1 queueing model is considered, where the arrival rate to the facility is a continuous variable which depends, in the steady state, upon the average congestion at the facility. The population of customers arriving to the facility is partitioned into several classes dependent on the ratio of the value of time to the reward due to service but are served according to first-in-first-out rule. It is shown that under the privately optimal behavior of the individuals the facility will be dominated by the class with the highest net reward per value of time. The publicly optimal policy which maximizes the net reward due to service, after costs of waiting are deducted, is shown either to admit only a single class of customers to the facility, thus discriminating against the other classes or to be indifferent to the mix of classes. The class chosen for admission may not be the class which would have privately dominated the facility. When the expected delay experienced at the facility is fixed, a policy of tolls and rebates for the customers is obtained that will assure equal access to the facility for all customers irrespective of their classes. It is shown that the publicly optimal policy, under the condition of fixed aggregate arrival rate to the facility, is shown to be deversified.The optimal arrival rates desired by a single class are derived for two cases. When the proportions of arrivals from the classes are fixed, the aggregate arrival rate desired by a class is shown to be not greater than the equilibrium rate for the individuals from that class. Alternatively, when the aggregate arrival rate is fixed, conditions are obtained under which a class will prefer usage of the facility by several classes to its own domination.     

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.     

Lower bounds for the two-stage uncapacitated facility location problem

In the two-stage uncapacitated facility location problem, a set of customers is served from a set of depots which receives the product from a set of plants. If a plant or depot serves a product, a fixed cost must be paid, and there are different transportation costs between plants and depots, and depots and customers. The objective is to locate plants and depots, given both sets of potential locations, such that each customer is served and the total cost is as minimal as possible. In this paper, we present a mixed integer formulation based on twice-indexed transportation variables, and perform an analysis of several Lagrangian relaxations which are obtained from it, trying to determine good lower bounds on its optimal value. Computational results are also presented which support the theoretical potential of one of the relaxations.     

Center location problems on tree graphs with subtree-shaped customers

We consider the p-center problem on tree graphs where the customers are modeled as continua subtrees. We address unweighted and weighted models as well as distances with and without addends. We prove that a relatively simple modification of Handler’s classical linear time algorithms for unweighted 1- and 2-center problems with respect to point customers, linearly solves the unweighted 1- and 2-center problems with addends of the above subtree customer model. We also develop polynomial time algorithms for the p-center problems based on solving covering problems and searching over special domains.     

Location-allocation on a line with demand-dependent costs

This paper extends the location-allocation formulation by making the cost charged to users by a facility a function of the total number of users patronizing the facility. Users select their facility based on facility charges and transportation costs. We explore equilibria where each customer selects the least expensive facility (cost and transportation) and where the facility is at a point that minimizes travel costs for its customers. The problem in its general form is quite complex. An interesting special case is studied: facilities and customers are located on a finite line segment and demand is distributed on the line by a given density function.     

带惩罚的容错设施布局问题的近似算法

在带惩罚的容错设施布局问题中, 给定顾客集合、地址集合、以及每个顾客和各个地址之间的连接费用, 这里假设连接费用是可度量的. 每位顾客有各自的服务需求, 每个地址可以开设任意多个设施, 顾客可以被安排连接到某些地址的一些开设的设施上以满足其需求, 也可以被拒绝, 但这时要支付拒绝该顾客所带来的惩罚费用. 目标是确定哪些顾客的服务需求被拒绝并开设一些设施, 将未被拒绝的顾客连接到不同的开设设施上, 使得开设费用、连接费用和惩罚费用总和最小. 给出了带惩罚的容错设施布局问题的线性整数规划及其对偶规划, 进一步, 给出了基于其线性规划和对偶规划舍入的4-近似算法.     

Competitive facility location problem with attractiveness adjustment of the follower: A bilevel programming model and its solution

We are concerned with a problem in which a firm or franchise enters a market by locating new facilities where there are existing facilities belonging to a competitor. The firm aims at finding the location and attractiveness of each facility to be opened so as to maximize its profit. The competitor, on the other hand, can react by adjusting the attractiveness of its existing facilities with the objective of maximizing its own profit. The demand is assumed to be aggregated at certain points in the plane and the facilities of the firm can be located at predetermined candidate sites. We employ Huff’s gravity-based rule in modeling the behavior of the customers where the fraction of customers at a demand point that visit a certain facility is proportional to the facility attractiveness and inversely proportional to the distance between the facility site and demand point. We formulate a bilevel mixed-integer nonlinear programming model where the firm entering the market is the leader and the competitor is the follower. In order to find the optimal solution of this model, we convert it into an equivalent one-level mixed-integer nonlinear program so that it can be solved by global optimization methods. Apart from reporting computational results obtained on a set of randomly generated instances, we also compute the benefit the leader firm derives from anticipating the competitor’s reaction of adjusting the attractiveness levels of its facilities. The results on the test instances indicate that the benefit is 58.33% on the average.     

Heuristic algorithms for the multi-depot ring-star problem

In this paper, we consider the problem of designing urban optical networks. In particular, given a set of telephone exchanges, we must design a collection of ring-stars, where each ring-star is a cycle composed of a telephone exchange, some customers, some transition points used to save routing costs and customers not on the cycle connected to the cycle by a single edge. The ring topology is chosen in many fiber optic communication networks since it allows to prevent the loss of connection due to a single edge or even a single node failure. The objective is to minimize the total cost of the optical network which is mainly due to the excavation costs. We call this problem Multi-Depot Ring-Star Problem (MDRSP) and we formulate it as an optimization problem in Graph Theory. We present lower bounds and heuristic algorithms for the MDRSP. Computational results on randomly generated instances and real-life datasets are also presented.     

Single vehicle routing problems with a predefined customer sequence, compartmentalized load and stochastic demands

We consider the problem of finding the optimal routing of a single vehicle that delivers K different products to N customers according to a particular customer order. The demands of the customers for each product are assumed to be random variables with known distributions. Each product type is stored in its dedicated compartment in the vehicle. Using a suitable dynamic programming algorithm we find the policy that satisfies the demands of the customers with the minimum total expected cost. We also prove that this policy has a specific threshold-type structure. Furthermore, we investigate a corresponding infinite-time horizon problem in which the service of the customers does not stop when the last customer has been serviced but it continues indefinitely with the same customer order. It is assumed that the demands of the customers at different tours have the same distributions. It is shown that the discounted-cost optimal policy and the average-cost optimal policy have the same threshold-type structure as the optimal policy in the original problem. The theoretical results are illustrated by numerical examples.     

Optimal pricing for service facilities with self-optimizing customers

Many service industries (e.g., walk-in clinics, vehicle inspection facilities, and data-processing centers) have customers who choose among congested facilities, and select the facility with the lowest combination of travel cost plus congestion cost at the facility. In general, customers over-utilize attractive facilities, causing higher costs than if customers were assigned to facilities to minimize total costs. Optimal facility prices induce customers to select facilities that minimize total cost. We find optimal facility prices and show they equal charging customers for the impact (net costs and benefits) they cause for others. We explore a rich flexibility that allows a range of optimal prices, useful when negotiating the implementation of facility fees. Facility prices can be positive or negative (price discounts), and can be adjusted to be all positive, or to provide net subsidy or net revenue. We contribute to unifying and generalizing several disparate streams of research.     

Hop constrained Steiner trees with multiple root nodes

We consider a network design problem that generalizes the hop and diameter constrained Steiner tree problem as follows: Given an edge-weighted undirected graph with two disjoint subsets representing roots and terminals, find a minimum-weight subtree that spans all the roots and terminals so that the number of hops between each relevant node and an arbitrary root does not exceed a given hop limit H. The set of relevant nodes may be equal to the set of terminals, or to the union of terminals and root nodes. This article proposes integer linear programming models utilizing one layered graph for each root node. Different possibilities to relate solutions on each of the layered graphs as well as additional strengthening inequalities are then discussed. Furthermore, theoretical comparisons between these models and to previously proposed flow- and path-based formulations are given. To solve the problem to optimality, we implement branch-and-cut algorithms for the layered graph formulations. Our computational study shows their clear advantages over previously existing approaches.     

Greedy approaches for a class of nonlinear Generalized Assignment Problems

The traditional Generalized Assignment Problem (GAP) seeks an assignment of customers to facilities that minimizes the sum of the assignment costs while respecting the capacity of each facility. We consider a nonlinear GAP where, in addition to the assignment costs, there is a nonlinear cost function associated with each facility whose argument is a linear function of the customers assigned to the facility. We propose a class of greedy algorithms for this problem that extends a family of greedy algorithms for the GAP. The effectiveness of these algorithms is based on our analysis of the continuous relaxation of our problem. We show that there exists an optimal solution to the continuous relaxation with a small number of fractional variables and provide a set of dual multipliers associated with this solution. This set of dual multipliers is then used in the greedy algorithm. We provide conditions under which our greedy algorithm is asymptotically optimal and feasible under a stochastic model of the parameters.     

Algorithms for the generalized quadratic assignment problem combining Lagrangean decomposition and the Reformulation-Linearization Technique

In this paper, we propose two exact algorithms for the GQAP (generalized quadratic assignment problem). In this problem, given M facilities and N locations, the facility space requirements, the location available space, the facility installation costs, the flows between facilities, and the distance costs between locations, one must assign each facility to exactly one location so that each location has sufficient space for all facilities assigned to it and the sum of the products of the facility flows by the corresponding distance costs plus the sum of the installation costs is minimized. This problem generalizes the well-known quadratic assignment problem (QAP). Both exact algorithms combine a previously proposed branch-and-bound scheme with a new Lagrangean relaxation procedure over a known RLT (Reformulation-Linearization Technique) formulation. We also apply transformational lower bounding techniques to improve the performance of the new procedure. We report detailed experimental results where 19 out of 21 instances with up to 35 facilities are solved in up to a few days of running time. Six of these instances were open.     

Solving the dynamic capacitated location-routing problem with fuzzy demands by hybrid heuristic algorithm

In this paper, the dynamic capacitated location-routing problem with fuzzy demands (DCLRP-FD) is considered. In the DCLRP-FD, facility location problem and vehicle routing problem are solved on a time horizon. Decisions concerning facility locations are permitted to be made only in the first time period of the planning horizon but, the routing decisions may be changed in each time period. Furthermore, the vehicles and depots have a predefined capacity to serve the customers with altering demands during the time horizon. It is assumed that the demands of customers are fuzzy variables. To model the DCLRP-FD, a fuzzy chance-constrained programming is designed based upon the fuzzy credibility theory. To solve this problem, a hybrid heuristic algorithm (HHA) with four phases including the stochastic simulation and a local search method are proposed. To achieve the best value of two parameters of the model, the dispatcher preference index (DPI) and the assignment preference index (API), and to analyze their influences on the final solution, numerical experiments are carried out. Moreover, the efficiency of the HHA is demonstrated via comparing with the lower bound of solutions and by using a standard benchmark set of test problems. The numerical examples show that the proposed algorithm is robust and could be used in real world problems.     

‘Multidimensional’ extensions and a nested dual approach for the m-median problem

A general framework for modeling median type locational decisions, where (i) travel costs and demands may be stochastic, (ii) multiple services or commodities need to be considered, and/or (iii) multiple median type objectives might exist, is presented—using the concept of “multidimensional networks”. The classical m-median problem, the stochastic m-median problem, the multicommodity m-median problem and and multiobjective m-median problem are defined within this framework.By an appropriate transformation of variables, the multidimensional m-median problem simplifies to the classical m-median problem but with a K-fold increase in the number of nodes, where K is the number of dimensions of the network. A nested dual approach to solve the resulting classical m-median problem, that uses Erlenkotter's facility location scheme as a subroutine, is presented. Computational results indicate that the procedure may perhaps be the best available one to solve the m-median problem exactly.     

IP modeling of the survivable hop constrained connected facility location problem

We consider a generalized version of the rooted connected facility location problem which occurs in planning of telecommunication networks with both survivability and hop-length constraints. Given a set of client nodes, a set of potential facility nodes including one predetermined root facility, a set of optional Steiner nodes, and the set of the potential connections among these nodes, that task is to decide which facilities to open, how to assign the clients to the open facilities, and how to interconnect the open facilities in such a way, that the resulting network contains at least λ edge-disjoint paths, each containing at most H edges, between the root and each open facility and that the total cost for opening facilities and installing connections is minimal. We study two IP models for this problem and present a branch-and-cut algorithm based on Benders decomposition for finding its solution. Finally, we report computational results.