The median and its extensions

We review various representations of the median and related aggregation functions. An advantage of the median is that it discards extreme values of the inputs, and hence exhibits a better central tendency than the arithmetic mean. However, the value of the median depends on only one or two central inputs. Our aim is to design median-like aggregation functions whose value depends on several central inputs. Such functions will preserve the stability of the median against extreme values, but will take more inputs into account. A method based on graduation curves is presented.

Aggregation for Atanassov's intuitionistic and interval valued fuzzy sets: the median operator

Atanassov's intuitionistic fuzzy sets (AIFS) and interval valued fuzzy sets (IVFS) are two generalizations of a fuzzy set, which are equivalent mathematically although different semantically. We analyze the median aggregation operator for AIFS and IVFS. Different mathematical theories have lead to different definitions of the median operator. We look at the median from various perspectives: as an instance of the intuitionistic ordered weighted averaging operator, as a Fermat point in a plane, as a minimizer of input disagreement, and as an operation on distributive lattices. We underline several connections between these approaches and summarize essential properties of the median in different representations.

Upgrading arc median shortest path problem for an urban transportation network

In this paper, we propose an algorithm for an upgrading arc median shortest path problem for a transportation network. The problem is to identify a set of nondominated paths that minimizes both upgrading cost and overall travel time of the entire network. These two objectives are realistic for transportation network problems, but of a conflicting and noncompensatory nature. In addition, unlike upgrading cost which is the sum of the arc costs on the path, overall travel time of the entire network cannot be expressed as a sum of arc travel times on the path. The proposed solution approach to the problem is based on heuristic labeling and exhaustive search techniques, in criteria space and solution space, respectively. The first approach labels each node in terms of upgrading cost, and deletes cyclic and infeasible paths in criteria space. The latter calculates the overall travel time of the entire network for each feasible path, deletes dominated paths on the basis of the objective vector and identifies a set of Pareto optimal paths in the solution space. The computational study, using two small-scale transportation networks, has demonstrated that the algorithm proposed herein is able to efficiently identify a set of nondominated median shortest paths, based on two conflicting and noncompensatory objectives.

Solving the median shortest path problem in the planning and design of urban transportation networks using a vector labeling algorithm

This paper proposes an alternative algorithm to solve the median shortest path problem (MSPP) in the planning and design of urban transportation networks. The proposed vector labeling algorithm is based on the labeling of each node in terms of a multiple and conflicting vector of objectives which deletes cyclic, infeasible and extreme-dominated paths in the criteria space imposing cyclic break (CB), path cost constraint (PCC) and access cost parameter (ACP) respectively. The output of the algorithm is a set of Pareto optimal paths (POP) with an objective vector from predetermined origin to destination nodes. Thus, this paper formulates an algorithm to identify a non-inferior solution set of POP based on a non-dominated set of objective vectors that leaves the ultimate decision to decision-makers. A numerical experiment is conducted using an artificial transportation network in order to validate and compare results. Sensitivity analysis has shown that the proposed algorithm is more efficient and advantageous over existing solutions in terms of computing execution time and memory space used.

An algorithm to median shortest path problem (MSPP) in the design of urban transportation networks

This paper proposes an efficient solution algorithm for realistic multi-objective median shortest path problems in the design of urban transportation networks. The proposed problem formulation and solution algorithm to median shortest path problem is based on three realistic objectives via route cost or investment cost, overall travel time of the entire network and total toll revenue. The proposed solution approach to the problem is based on the heuristic labeling and exhaustive search technique in criteria space and solution space of the algorithm respectively. The first labels each node in terms of route cost and deletes cyclic and infeasible paths in criteria space imposing cyclic break and route cost constraint respectively. The latter deletes dominated paths in terms of objectives vector in solution space in order to identify a set of Pareto optimal paths. The approach, thus, proposes a non-inferior solution set of Pareto optimal paths based on non-dominated objective vector and leaves the ultimate decision to decision-makers for purpose specific final decision during applications. A numerical experiment is conducted to test the proposed algorithm using artificial transportation network. Sensitivity analyses have shown that the proposed algorithm is advantageous and efficient over existing algorithms to find a set of Pareto optimal paths to median shortest paths problems.