Abstract
In a connected simple graph, the weighted Roman domination problem is considered at which the cost of positioning at each vertex is imposed in addition to the costs of potential deployments from a vertex to some of its neighboring vertices. Proper decision in practice is prone to a high degree of indeterminacy, mostly raised by unpredictable events that do not obey the rules and prerequisites of the probability theory. In this study, we model this problem with such assumptions in the context of the uncertainty theory initiated by Liu (Uncertainty theory. Studies in fuzziness and soft computing, Springer, Berlin, 2007). Two different optimization models are presented, and a concrete example is provided for illustrative purposes. Weaknesses of the probability theory and fuzzy theory in dealing with this problem are also mentioned in detail.
Highlights
Roman domination problem has historical significance and dates back to the fourth century when the emperor of Rome, Constantine the Great, decreed that two types of legions should be positioned in Roman provinces
In the graph theory language, the problem has been originally introduced by Ian Stewart as the ‘‘Roman domination problem’’ Stewart (1999), each province was denoted by a vertex and linkages between provinces were depicted as edges
Our further computational experiences with different deployment costs revealed that when these costs are higher enough, regarding the belief rate, the optimal solution assigns all vertices to be stationed by only one legion
Summary
Roman domination problem has historical significance and dates back to the fourth century when the emperor of Rome, Constantine the Great, decreed that two types of legions should be positioned in Roman provinces. In addition to army placement, the same sort of mathematics is useful when a planner wants to know the best place in a town to construct a new public service facility such as hospital, fire station and emergency forces’ bases Such optimization problems can be modeled by Roman domination or its variants. Consider augmenting of temporary hospitals to the existing healthcare system as one solution for dealing with a surge of patients related to war, pandemic disease outbreaks or natural disasters These situations are almost always unprecedented and unrepeatability characteristics of the problem in question, and lack of historical data does not permit the use of probability theory in the model. Concluding remarks and outlook on the further works direction are given in the final section
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