Edge covering problem under hybrid uncertain environments

Keywords: Uncertainty modelling Edge cover Random fuzzy programming Random fuzzy simulation Genetic algorithm a b s t r a c t Edge covering problem (ECP) is to find an edge cover with the minimum weight in a graph. By quantifying costs, time or opponent's payoffs as weights on edges, ECP is employed to model many real-life problems in engineering and management. Since various types of uncertainty always exist in real world, this paper considers ECP under hybrid uncertain environments where randomness and fuzziness coexist. We propose three decision models and present a hybrid intelligent algorithm to solve the proposed models where genetic algorithm and random fuzzy simulation are embedded. Numerical experiments are performed to show the robustness of the proposed algorithm. Edge covering problem (ECP) is to find an edge cover with the minimum weight in a weighted graph, where an edge cover is a set of edges such that every vertex in the graph is an endpoint of at least one edge in the set. Since first introduced in 1950s [1], ECP has attracted a lot of attention from many areas such as computer science [2], management engineering [3], combinational mathematics [4], to name a few. The reason why ECP is popular is due to its ability to model many real-life situations. In real-life applications, the weights on edges usually represent costs, time or opponent's payoffs. A simple example of the application of ECP is to monitor all the crossings in a transportation system. In order to keep a good transportation situation for each crossing in a city, police cars are assigned to some streets to monitor the crossings. Each police car is assigned to a single street to drive back and forth such that the two crossings as the endpoints of the street can be monitored. As assigning a police car to different streets leads to different costs, the problem is how to make the assignment such that the total costs are minimized. By representing each street as an edge, each crossing as a vertex and the cost of assigning a police car to each street as the weight on the corresponding edge, we can model the transportation system as a graph and this crossing monitoring problem is equivalent to an edge covering problem. Many researchers focused on how to solve ECP efficiently [5–10], but they only investigate cases where the weights of edges are determinate. In real …