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.     

Efficient heuristics for the rectilinear distance capacitated multi-facility Weber problem

In this paper, we consider the capacitated multi-facility Weber problem with rectilinear distance. This problem is concerned with locating m capacitated facilities in the Euclidean plane to satisfy the demand of n customers with the minimum total transportation cost. The demand and location of each customer are known a priori and the transportation cost between customers and facilities is proportional to the rectilinear distance separating them. We first give a new mixed integer linear programming formulation of the problem by making use of a well-known necessary condition for the optimal facility locations. We then propose new heuristic solution methods based on this formulation. Computational results on benchmark instances indicate that the new methods can provide very good solutions within a reasonable amount of computation time.     

Heuristic algorithms for general <Emphasis Type="Italic">k</Emphasis>-level facility location problems

In a general k-level uncapacitated facility location problem (k-GLUFLP), we are given a set of demand points, denoted by D, where clients are located. Facilities have to be located at a given set of potential sites, which is denoted by F in order to serve the clients. Each client needs to be served by a chain of k different facilities. The problem is to determine some sites of F to be set up and to find an assignment of each client to a chain of k facilities so that the sum of the setup costs and the shipping costs is minimized. In this paper, for a fixed k, an approximation algorithm within a factor of 3 of the optimum cost is presented for k-GLUFLP under the assumption that the shipping costs satisfy the properties of metric space. In addition, when no fixed cost is charged for setting up the facilities and k=2, we show that the problem is strong NP-complete and the constant approximation factor is further sharpen to be 3/2 by a simple algorithm. Furthermore, it is shown that this ratio analysis is tight.     

Approximation algorithms for the robust facility leasing problem

In this paper, we consider the robust facility leasing problem (RFLE), which is a variant of the well-known facility leasing problem. In this problem, we are given a facility location set, a client location set of cardinality n, time periods \(\{1, 2, \ldots , T\}\) and a nonnegative integer \(q < n\). At each time period t, a subset of clients \(D_{t}\) arrives. There are K lease types for all facilities. Leasing a facility i of a type k at any time period s incurs a leasing cost \(f_i^{k}\) such that facility i is opened at time period s with a lease length \(l_k\). Each client in \(D_t\) can only be assigned to a facility whose open interval contains t. Assigning a client j to a facility i incurs a serving cost \(c_{ij}\). We want to lease some facilities to serve at least \(n-q\) clients such that the total cost including leasing and serving cost is minimized. Using the standard primal–dual technique, we present a 6-approximation algorithm for the RFLE. We further offer a refined 3-approximation algorithm by modifying the phase of constructing an integer primal feasible solution with a careful recognition on the leasing facilities.     

An efficient algorithm for the single facility location problem with polyhedral norms and disk-shaped demand regions

The single facility location problem with demand regions seeks for a facility location minimizing the sum of the distances from n demand regions to the facility. The demand regions represent sales markets where the transportation costs are negligible. In this paper, we assume that all demand regions are disks of the same radius, and the distances are measured by a rectilinear norm, e.g. \(\ell _1\) or \(\ell _\infty \). We develop an exact combinatorial algorithm running in time \(O(n\log ^c n)\) for some c dependent only on the space dimension. The algorithm is generalizable to the other polyhedral norms.     

An improved approximation algorithm for the <Emphasis Type="Italic">k</Emphasis>-level facility location problem with soft capacities

We consider the k-level facility location problem with soft capacities (k-LFLPSC). In the k-LFLPSC, each facility i has a soft capacity u _i along with an initial opening cost f _i ≥ 0, i.e., the capacity of facility i is an integer multiple of u _i incurring a cost equals to the corresponding multiple of f _i . We firstly propose a new bifactor (ln(1/β)/(1 ?β),1+2/(1 ?β))-approximation algorithm for the k-level facility location problem (k-LFLP), where β ∈ (0, 1) is a fixed constant. Then, we give a reduction from the k-LFLPSC to the k-LFLP. The reduction together with the above bifactor approximation algorithm for the k-LFLP imply a 5.5053-approximation algorithm for the k-LFLPSC which improves the previous 6-approximation.     

The general facility location problem with connectivity on trees

In this note we study the general facility location problem with connectivity. We present an O(np ²)-time algorithm for the general facility location problem with connectivity on trees. Furthermore, we present an O(np)-time algorithm for the general facility location problem with connectivity on equivalent binary trees.     

Planar maximal covering location problem under block norm distance measure

This paper introduces a new model for the planar maximal covering location problem (PMCLP) under different block norms. The problem involves locating g facilities anywhere on the plane in order to cover the maximum number of n given demand points. The generalization, in this paper, is that the distance measures assigned to facilities are block norms of different types and different proximity measures. First, the PMCLP under different block norms is modelled as a maximum clique partition problem on an equivalent multi-interval graph. Then, the equivalent graph problem is modelled as an unconstrained binary quadratic problem (UQP). Both the maximum clique partition problem and the UQP are NP-hard problems; therefore, we solve the UQP format through a genetic algorithm heuristic. Computational examples are given.     

Probabilistic Formulation of the Emergency Service Location Problem

The problem of locating emergency service facilities is studied under the assumption that the locations of incidents (accidents, fires, or customers) are random variables. The probability distribution for rectilinear travel time between a new facility location and the random location of the incident P _i is developed for the case of P _i being uniformly distributed over a rectangular region. The location problem is considered in a discrete space. A deterministic formulation is obtained and recognized to be a set cover problem. Probabilistic variation of the central facility location problem is also presented.An example and some computational experience are provided to emphasize the impact of the probabilistic formulation on the location decision.     

A new heuristic approach for the P-median problem

The Euclidean p-median problem is concerned with the decision of the locations for public service centres. Existing methods for the planar Euclidean p-median problems are capable of efficiently solving problems of relatively small scale. This paper proposes two new heuristic algorithms aiming at problems of large scale. Firstly, to reflect the different degrees of proximity to optimality, a new kind of local optimum called level-m optimum is defined. For a level-m optimum of a p-median problem, where m<p, each of its subsets containing m of the p partitions is a global optimum of the corresponding m-median subproblem. Starting from a conventional local optimum, the first new algorithm efficiently improves it to a level-2 optimum by applying an existing exact algorithm for solving the 2-median problem. The second new algorithm further improves it to a level-3 optimum by applying a new exact algorithm for solving the 3-median problem. Comparison based on experimental results confirms that the proposed algorithms are superior to the existing heuristics, especially in terms of solution quality.     

Covering continuous demand in the plane

Continuous demand is generated in a convex polygon. A facility located in the area covers demand within a given radius. The objective is to find the locations for p facilities that cover the maximum demand in the area. A procedure that calculates the total area covered by a set of facilities is developed. A multi start heuristic approach for solving this problem is proposed by applying a gradient search from a randomly generated set of p locations for the facilities. Computational experiments for covering a square area illustrate the effectiveness of the proposed algorithm.     

Heuristic Solution Methods for Two Location Problems with Unreliable Facilities

In this paper, the p-median and p-centre problems are generalized by considering the possibility that one or more of the facilities may become inactive. The unreliable p-median problem is defined by introducing the probability that a facility becomes inactive. The (p, q)-centre problem is defined when p facilities need to be located but up to q of them may become unavailable at the same time. An heuristic procedure is presented for each problem. A rigorous procedure is discussed for the (p, q)-centre problem. Computational results are presented.     

An Optimal Algorithm for the <Emphasis Type="Italic">k</Emphasis>-Fixed-Endpoint Path Cover on Proper Interval Graphs

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

Operations policy for a supply chain system with fixed-interval delivery and linear demand

This research addresses a production-supply problem for a supply-chain system with fixed-interval delivery. A strategy that determines the optimal batch sizes, cycle times, numbers of orders of raw materials, and production start times is prescribed to minimize the total costs for a given finite planning horizon. The external demands are time-dependent following a life-cycle pattern and the shipment quantities follow the demand pattern. The shipment quantities to buyers follow various phases of the demand pattern in the planning horizon where demand is represented by piecewise linear model. The problem is formulated as an integer, non-linear programming problem. The model also incorporates the constraint of inventory capacity. The problem is represented using the network model where an optimal characteristic has been analysed. To obtain an optimal solution with N shipments in a planning horizon, an algorithm is proposed that runs with the complexity of Θ(N²) for problems with a single-phase demand and O(N³) for problems with multi-phase demand.     

A Polynomial Time Algorithm for the Resource Allocation Problem with a Convex Objective Function

Consider the resource allocation problem:minimize ∑ⁿ_i=1 f_i(x_i) subject to ∑ⁿ_i=1 x_i = N and x_i's being nonnegative integers, where each f_i is a convex function. The well-known algorithm based on the incremental method requires O(N log n + n) time to solve this problem. We propose here a new algorithm based on the Lagrange multiplier method, requiring O[n²(log N)²] time. The latter is faster if N is much larger than n. Such a situation occurs, for example, when the optimal sample size problem related to monitoring the urban air pollution is treated.     

An approximate solution algorithm for the one-dimensional online median problem

The online median problem consists in finding a sequence of incremental solutions of the k-median problem with k increasing. A particular case of the problem is considered: the clients and facilities are located on the real line. The best algorithm available for the one-dimensional case has competitive ratio 8. We give an improved 5.83-competitive algorithm.     

An alternate linear algorithm for the minimum flow problem

We present an alternate linear algorithm for finding the minimum flow in (s, t)-planar networks using a new concept of minimal removable sets developed here. The iterative nature of the algorithm facilitates the adjustment of solutions for systems in developmental stages. The minimum flow algorithm presented here requires O(|V|) time, where V denotes the set of vertices. The minimum flow problem arises in many transportation and communication systems.     

Common due window size and location determination in a single machine scheduling problem

We consider a single machine static and deterministic scheduling problem in which jobs have a common due window. Jobs completed within the window incur no penalties, other jobs incur either earliness or tardiness penalties. The objective is to find the optimal size and location of the window as well as an optimal sequence to minimise a cost function based on earliness, tardiness, window size, and window location. We propose an O(n log n) algorithm to solve the problem.