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.     

The connected <Emphasis Type="Italic">p</Emphasis>-median problem on block graphs

In this paper, we study a variant of the p-median problem on block graphs G in which the p-median is asked to be connected, and this problem is called the connected p-median problem. We first show that the connected p-median problem is NP-hard on block graphs with multiple edge weights. Then, we propose an O(n)-time algorithm for solving the problem on unit-edge-weighted block graphs, where n is the number of vertices in G.     

Branch and Bound Algorithm for the <Emphasis Type="Italic">p</Emphasis>-Median Transportation Problem

The p-median transportation problem is to determine an optimal solution to a transportation problem having an additional constraint restricting the number of active supply points. The model is discussed as an example of a public sector location/allocation problem. A branch and bound procedure is proposed to solve the problem. Lagrangian relaxation is used to provide lower bounds. Computational results are given.     

<Emphasis Type="Italic">p</Emphasis>-Medians and Multi-Medians

The classical p-median problem is discussed, together with methods for its solution. The multi-median problem, a generalization of the p-median problem in which more than one type of facility is allowed, is introduced and methods of solution developed. Numerical results are presented.     

The pos/neg-weighted 2-medians in balanced trees with subtree-shaped customers

This paper deals with the pos/neg-weighted p-median problem on tree graphs where all customers are modeled as subtrees. We present a polynomial algorithm for the 2-median problem on an arbitrary tree. Then we improve the time complexity to O(n log n) for the problem on a balanced tree, where n is the number of the vertices in the tree.     

An Algorithm for Solving Capacitated Multicommodity <Emphasis Type="Italic">p</Emphasis>-median Transportation Problems

An algorithm for solving a special capacitated multicommodity p-median transportation problem (CMPMTP), which arises in container terminal management, is presented. There are some algorithms to solve similar kinds of problems. The formulation here is different from the existing modelling of the p-median or some related location problems. We extend the existing work by applying a Lagrangean relaxation to the CMPMTP. In order to obtain a satisfactory solution, a heuristic branch-and-bound algorithm is designed to search for a better solution, if one is possible. A comparison is also made with different algorithms.     

On the complexity of inverse convex ordered 1-median problem on the plane and on tree networks

An ordered median function is used in location theory to generalize a class of problems, including median and center problems. In this paper we consider the complexity of inverse ordered 1-median problems on the plane and on trees, where the multipliers are sorted nondecreasingly. Based on the convexity of the objective function, we prove that the problems with variable weights or variable coordinates on the line are NP-hard. Then we can directly get the NP-hardness result for the corresponding problem on the plane. We finally develop a cubic time algorithm that solves the inverse convex ordered 1-median problem on trees with relaxation on modification bounds.     

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.     

Approximate solution of the <Emphasis Type="Italic">p</Emphasis>-median minimization problem

A version of the facility location problem (the well-known p-median minimization problem) and its generalization—the problem of minimizing a supermodular set function—is studied. These problems are NP-hard, and they are approximately solved by a gradient algorithm that is a discrete analog of the steepest descent algorithm. A priori bounds on the worst-case behavior of the gradient algorithm for the problems under consideration are obtained. As a consequence, a bound on the performance guarantee of the gradient algorithm for the p-median minimization problem in terms of the production and transportation cost matrix is obtained.     

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 approximation algorithm for finding a <Emphasis Type="Italic">d</Emphasis>-regular spanning connected subgraph of maximum weight in a complete graph with random weights of edges

An approximation algorithm is suggested for the problem of finding a d-regular spanning connected subgraph of maximum weight in a complete undirected weighted n-vertex graph. Probabilistic analysis of the algorithm is carried out for the problem with random input data (some weights of edges) in the case of a uniform distribution of the weights of edges and in the case of a minorized type distribution. It is shown that the algorithm finds an asymptotically optimal solution with time complexity O(n ²) when d = o(n). For the minimization version of the problem, an additional restriction on the dispersion of weights of the graph edges is added to the condition of the asymptotical optimality of the modified algorithm.     

Approximation of an optimal control problem in coefficient for variational inequality with anisotropic <Emphasis Type="Italic">p</Emphasis>-Laplacian

We study an optimal control problem for a variational inequality with the so-called anisotropic p-Laplacian in the principle part of this inequality. The coefficients of the anisotropic p-Laplacian, the matrix A(x), we take as a control. The optimal control problem is to minimize the discrepancy between a given distribution \({y_d \in L^{2}(\Omega)}\) and the solutions \({y \in K \subset W^{1,p}_{0}(\Omega)}\) of the corresponding variational inequality. We show that the original problem is well-posed and derive existence of optimal pairs. Since the anisotropic p-Laplacian inherits the degeneracy with respect to unboundedness of the term \({|(A(x)\nabla y, \nabla y)_{\mathbb{R}^N}|^{\frac{p-2}{2}}}\), we introduce a two-parameter model for the relaxation of the original problem. Further we discuss the asymptotic behavior of relaxed solutions and show that some optimal pairs to the original problem can be attained by the solutions of two-parametric approximated optimal control problems.     

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.     

A new algorithm for minimizing makespan, <Emphasis Type="Italic">C</Emphasis><Subscript>max</Subscript>, in blocking flow-shop problem through slowing down the operations

In this paper, a new algorithm with complexity O(nm²) is presented, which finds the optimal makespan, C_max, for a blocking flow-shop problem by slowing down the operations of a no-wait flow-shop problem, F _m ∣no-wait∣C_max, for a given sequence where restriction on the slowing down is committed. However, the problem with performance measure makespan, C_max, in a non-cyclic environment, is a special case of cyclic problem with cycle time, C _t , as its performance measure. This new algorithm is much faster than the previously developed algorithms for cyclical scheduling problems.     

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.     

An accelerated augmented Lagrangian method for linearly constrained convex programming with the rate of convergence <Emphasis Type="Italic">O</Emphasis>(1/<Emphasis Type="Italic">k</Emphasis><Superscript>2</Superscript>)

In this paper, we propose and analyze an accelerated augmented Lagrangian method (denoted by AALM) for solving the linearly constrained convex programming. We show that the convergence rate of AALM is O(1/k ²) while the convergence rate of the classical augmented Lagrangian method (ALM) is O(1/k). Numerical experiments on the linearly constrained l ₁?l ₂ minimization problem are presented to demonstrate the effectiveness of AALM.     

Parametric Integer Programming Analysis: A Contraction Approach

An algorithm is presented for solving families of integer linear programming problems in which the problems are "related" by having identical objective coefficients and constraint matrix coefficients. The righthand-side constants have the form b + θd where b and d are conformable vectors and θ varies from zero to one.The approach consists primarily of solving the most relaxed problem (θ = 1) using cutting planes and then contracting the region of feasible integer solutions in such a manner that the current optimal integer solution is eliminated.The algorithm was applied to 1800 integer linear programming problems with reasonable success. Integer programming problems which have proved to be unsolvable using cutting planes have been solved by expanding the region of feasible integer solutions (θ = 1) and then contracting to the original region.     

Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control

Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for higher-order Lagrangian systems. Given a regular higher-order Lagrangian \(L:T^{(k)}Q\rightarrow {\mathbb {R}}\) with \(k\ge 1\), the resulting discrete equations define a generally implicit numerical integrator algorithm on \(T^{(k-1)}Q\times T^{(k-1)}Q\) that approximates the flow of the higher-order Euler–Lagrange equations for L. The algorithm equations are called higher-order discrete Euler–Lagrange equations and constitute a variational integrator for higher-order mechanical systems. The general idea for those variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the invariants of the original system, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. We construct an exact discrete Lagrangian \(L_d^e\) using the locally unique solution of the higher-order Euler–Lagrange equations for L with boundary conditions. By taking the discrete Lagrangian as an approximation of \(L_d^e\), we obtain variational integrators for higher-order mechanical systems. We apply our techniques to optimal control problems since, given a cost function, the optimal control problem is understood as a second-order variational problem.     

An exact algorithm for finding a vector subset with the longest sum

We consider the problem: Given a set of n vectors in the d-dimensional Euclidean space, find a subsetmaximizing the length of the sum vector.We propose an algorithm that finds an optimal solution to this problem in time O(n^d?1(d + logn)). In particular, if the input vectors lie in a plane then the problem is solvable in almost linear time.     

A Note on the Joint Replenishment Problem under Constant Demand

This note considers the joint replenishment inventory problem for N items under constant demand. The frequently-used cyclic strategy (T; k₁, …, k _N ) is investigated: a family replenishment is made every T time units and item i is included in each k _i th replenishment. Goyal proposed a solution to find the global optimum within the class of cyclic strategies. However, we will show that the algorithm of Goyal does not always lead to the optimal cyclic strategy. A simple correction is suggested.