Inverse 1-median Problem Research Articles

Unified approach to the inverse median location problem under the sum and the max objectives applying ordered median function

ABSTRACT In this paper, we propose a unified approach to solving the inverse 1-median problem on trees under both the sum and max objectives by using the ordered median function. For the problem under the k-centrum cost function, we investigate the structure of an optimal solution and transform the problem into a univariate optimization problem where the parameter equals the k largest cost. We take advantage of the convexity of the objective function to efficiently find the optimal solution in O ( nlog ⁡ n ) time, where n is the number of vertices in the tree. For the problem under the centdian cost function, we parameterize it based on the maximum cost item and prove that the induced cost function is convex. We leverage this insight to develop a combinatorial algorithm that solves the corresponding problem in O ( nlog ⁡ n ) time.

A generalized interval type-2 fuzzy random variable based algorithm under mean chance value at risk criterion for inverse 1-median location problems on tree networks with uncertain costs

In this paper, we propose a generalized interval type-2 fuzzy random variable based algorithm under mean chance value at risk criterion. First, we introduce the interval type-2 fuzzy random variable and then we propose a scalar expected value of the interval type-2 fuzzy random variables. Also the new concepts of mean chance value at risk and mean chance conditional value at risk are discussed. An application to the inverse 1-median location problem on tree network with uncertain costs proves that this algorithm solves this problem in O(n2logn) time.

Computational aspects of the inverse single facility location problem on trees under lk-norm

We consider in this paper the inverse versions of the two popular problems in location theory, say the 1-median and the 1-center problems on trees. The cost for modifying vertex weights is measured under lk-norm for a positive integer k and hence the cost function is nonlinear. For the inverse 1-median problem, we develop a linear time algorithm based on the optimal solution of the induced unconstrained problem. For the inverse 1-center problem, we prove that the problem can be decomposed into linearly many subproblems and the objective function in each subproblem is piecewise-convex. Furthermore, we also discuss an O(nlog⁡n) time algorithm for the inverse 1-center problem under l2-norm based on the convexity of the cost function of each subproblem.

Classical and Inverse Median Location Problems under Uncertain Environment

In this paper, we first consider the classical p-median location problem on a network in which the vertex weights and the distances between vertices are uncertain variables. The uncertainty distribution of the optimal objective value of the p-median problem is given and the concepts of the α-p-median, the most p-median and the expected p-median are introduced. Then, it is shown that the uncertain p-median problem is NP-hard on general networks. However, if the underlying network is a tree, an efficient algorithm for the uncertain 1-median problem with linear time complexity is proposed. Finally, we investigate the inverse 1-median problem on a tree with uncertain vertex weights and present a programming model for the problem. Then, it is shown that the proposed model can be reformulated into a deterministic programming model.

Inverse stable point problem on trees under an extension of Chebyshev norm and Bottleneck Hamming distance

ABSTRACT In the inverse optimization problem, we modify parameters of the original problem at minimum total cost so as to make a prespecified solution optimal with respect to new parameters. We extend in this paper a class of inverse single facility problems on trees, including inverse balance point, inverse 1-median and inverse 1-center problem, and call it the inverse stable point problem. For the general situation where variables are both edge lengths and vertex weights under an extension of Chebyshev norm and bottleneck Hamming distance, we first derive an algorithm that reduces the corresponding problem to the one under either Chebyshev norm or bottleneck Hamming distance and then develop an approximation approach for the problem. Special cases concerning the problem under this extension with strongly polynomial time algorithms are also discussed.

Inverse group 1-median problem on trees

<p style='text-indent:20px;'>In location theory, group median generalizes the concepts of both median and center. We address in this paper the problem of modifying vertex weights of a tree at minimum total cost so that a prespecified vertex becomes a group 1-median with respect to the new weights. We call this problem the inverse group 1-median on trees. To solve the problem, we first reformulate the optimality criterion for a vertex being a group 1-median of the tree. Based on this result, we prove that the problem is <inline-formula><tex-math id="M1">\begin{document}$ NP $\end{document}</tex-math></inline-formula>-hard. Particularly, the corresponding problem with exactly two groups is however solvable in <inline-formula><tex-math id="M2">\begin{document}$ O(n^2\log n) $\end{document}</tex-math></inline-formula> time, where <inline-formula><tex-math id="M3">\begin{document}$ n $\end{document}</tex-math></inline-formula> is the number of vertices in the tree.

The inverse 1-median location problem on uncertain tree networks with tail value at risk criterion

In an inverse 1-median location problem on a tree network, the goal is to modify the vertex weights of the underlying tree network at the minimum total cost such that a predetermined vertex becomes the 1-median. This paper investigates the case that the vertex weights and modification costs are considered as uncertain variables. We first obtain a necessary and sufficient condition for the α-1-median on uncertain trees. Based on this condition, we transform the problem into a linear programming model with deterministic constraints. Finally, we consider the proposed model with tail value at risk objective under the weighted l1 norm and present a solution algorithm for the problem with time complexity of O(nlog n).

Inverse 1-median problem on trees under mixed rectilinear and Chebyshev norms

We consider in this paper the inverse 1-median problem on trees with variable vertex weights, which are partitioned into groups. While the modifications in each group are measured by Chebyshev norm, rectilinear norm is applied for computing the cost of groups; and vice versa. As a result, it yields the sum of max and the max of sum objective functions. For the sum of max objective, we develop an O(Mlog⁡M) time algorithm based on a parameterization approach, where M is the number of vertices in the tree. On the other hand, the problem under the max of sum objective can be solved in linear time by an algorithm that prunes a half of indices in each iteration.

Resource allocation in rooted trees for VLSI applications

ABSTRACTSeveral optimization problems of modifying the weight of vertices in rooted trees, some of which are special cases of the inverse 1-median problem, are solved. Such problems arise in Very Large Scale Integration (VLSI) design of hardware security circuits, circuit synchronization, and analog-to-digital converters. These problems require assigning costly hardware (demands) to the leaves of rooted trees. One common property of these problems is that a resource allocated to an internal node can be shared by the entire sub-tree emanated at the node. Two types of problems are studied here. (1) A tree node employs an addition operation and the demands at the leaves are obtained by summing the resources allocated to nodes along the root-to-leaf paths. A linear-time bottom-up algorithm is shown to minimize the total resources allocated to tree nodes. (2) A tree’s node employs a multiplication operation and the demands at the leaves are obtained by multiplying the resources allocated to nodes along the root-to-leaf paths. A bottom-up dynamic programming algorithm is shown to minimize the total resources allocated to the tree’s nodes. While the above problems are usually solved by design engineers heuristically, this paper offers optimal solutions that can be easily programmed in CAD tools.

Linear Time Optimal Approaches for Max-Profit Inverse 1-Median Location Problems

This paper is concerned with a new variant of the inverse 1-median location problem in which the aim is to modify the customer weights such that a predetermined facility location becomes a 1-median location and the total profit obtained via the weight improvements is maximized. We develop novel combinatorial approaches with linear time complexities for solving the problem on tree networks and in the plane under the rectilinear and Chebyshev norms.

Intuitionistic fuzzy inverse 1-median location problem on tree networks with value at risk objective

The inverse p-median location problem on networks is to modify the parameters of the original problem at minimum total cost with respect to given modification bounds such that a prespecified set of p vertices becomes a p-median with respect to new parameters. Therefore, the inverse p-median location problem is a decision-making problem in which decision-makers’ knowledge of the modification costs may be vague and imprecise. In this paper, we investigate the inverse 1-median location problem on tree networks with intuitionistic fuzzy weight modification costs. We first propose the new concepts of the credibilistic value at risk and conditional value at risk metrics in an intuitionistic fuzzy environment. Then we prove that these metrics satisfy in the harmonious risk metric properties. Finally, we solve the inverse 1-median location problem with intuitionistic fuzzy weight modification costs on tree networks and obtain its value at risk function in $$O(n^2 \log n)$$ time.

Some polynomially solvable cases of the inverse ordered 1-median problem on trees

We consider the problem of modifying the edge lengths of a tree at minimum cost such that a prespecified vertex become an ordered 1-median of the perturbed tree. We call this problem the inverse ordered 1-median problem on trees. Gassner showed in 2012 that the inverse ordered 1-median problem on trees is, in general, NP-hard. We, however, address some situations, where the corresponding inverse 1-median problem is polynomially solvable. For the problem on paths with n vertices, we develop an O(n3) algorithm based on a greedy technique. Furthermore, we prove the NP-hardness of the inverse ordered 1- median problem on star graphs and propose a quadratic algorithm that solves the inverse ordered 1-median problem on unweighted stars.

A model for the inverse 1-median problem on trees under uncertain costs

We consider the problem of justifying vertex weights of a tree under uncertain costs so that a prespecified vertex become optimal and the total cost should be optimal in the uncertainty scenario. We propose a model which delivers the information about the optimal cost which respect to each confidence level \(\alpha \in [0,1]\). To obtain this goal, we first define an uncertain variable with respect to the minimum cost in each confidence level. If all costs are independently linear distributed, we present the inverse distribution function of this uncertain variable in \(O(n^{2}\log n)\) time, where \(n\) is the number of vertices in the tree.

Inverse 1-Median Problem on Block Graphs with Variable Vertex Weights

This paper addresses the problem of modifying the vertex weights of a block graph at minimum total cost so that a prespecified vertex becomes a 1-median of the perturbed graph. We call this problem the inverse 1-median problem on block graphs with variable vertex weights. For block graphs with equal edge lengths in each block, we can formulate the problem as a univariate optimization problem. By the convexity of the objective function, the local optimizer is also the global one. Therefore, we use the convexity to develop an $$O(M\log M)$$O(MlogM) algorithm that solves the problem on block graphs with M vertices.

Inverse $$k$$ k -centrum problem on trees with variable vertex weights

This paper considers a generalization of the inverse 1-median problem, the inverse $$k$$ -centrum problem, on trees with variable vertex weights. In contrast to the linear time solvability of the inverse 1-median problem on trees, we prove that the inverse $$k$$ -centrum problem on trees is $$\textit{NP}$$ -hard. Particularly, the inverse 1-center problem, a special case of the mentioned problem with $$k=1$$ , on a tree with $$n$$ vertices can be solved in $$O(n^{2})$$ time.

An 𝒪(n log n) algorithm for the Inverse 1-median problem on trees with variable vertex weights and edge reductions

We investigate the inverse 1-median problem on trees where it is possible to change the weights of vertices or to reduce the length of an edge to zero. An algorithm is designed for solving this problem in case of symmetric bounds on the vertex weights.

The Inverse 1-Median Problem on Tree Networks with Variable Real Edge Lengths

Location problems exist in the real world and they mainly deal with finding optimal locations for facilities in a network, such as net servers, hospitals, and shopping centers. The inverse location problem is also often met in practice and has been intensively investigated in the literature. As a typical inverse location problem, the inverse 1-median problem on tree networks with variable real edge lengths is discussed in this paper, which is to modify the edge lengths at minimum total cost such that a given vertex becomes a 1-median of the tree network with respect to the new edge lengths. First, this problem is shown to be solvable in linear time with variable nonnegative edge lengths. For the case when negative edge lengths are allowable, the NP-hardness is proved under Hamming distance, and strongly polynomial time algorithms are presented underl1andl∞norms, respectively.

2-balanced flows and the inverse 1-median problem in the Chebyshev space

In this paper, we consider the 1-median problem in Rd with the Chebyshev-norm. We give an optimality criterion for this problem which enables us to solve the following inverse location problem by a combinatorial algorithm in polynomial time: Given n points P1,…,Pn∈Rd with non-negative weights wi and a point P0 the task is to find new non-negative weights w̃i such that P0 is a 1-median with respect to the new weights and ‖w−w̃‖1 is minimized. In fact, this problem reduces to a 2-balanced flow problem for which an optimal solution can be obtained by solving a fractional b-matching problem.

Inverse 1-median problem on trees under weighted Hamming distance

The inverse 1-median problem consists in modifying the weights of the customers at minimum cost such that a prespecified supplier becomes the 1-median of modified location problem. A linear time algorithm is first proposed for the inverse problem under weighted l ? norm. Then two polynomial time algorithms with time complexities O(n log n) and O(n) are given for the problem under weighted bottleneck-Hamming distance, where n is the number of vertices. Finally, the problem under weighted sum-Hamming distance is shown to be equivalent to a 0-1 knapsack problem, and hence is $${\mathcal{NP}}$$ -hard.

The inverse 1-median problem on a tree and on a path

Abstract This article considers the inverse 1-median problem on a tree and on a path. The aim is to change the vertex weights at minimum total cost with respect to given modification bounds such that a prespecified vertex becomes 1-median. The inverse 1-median problem on trees with nonnegative weights can be formulated as a continuous knapsack problem and therefore the problem is solvable in O ( n ) -time. For a path with pos/neg weights the 1-median lies on one of the vertices with positive weights or lies on one of the end points. This property leads us to solve the inverse 1-median problem on a path with negative weights (the weight of endpoints are arbitrary) in linear time.