Finding the weighted stable set of a graph with uncertain weights

The inherent characteristics of real-world data is uncertainty. If data is generated in valid experiments or collected standard, probability theory or fuzzy theory is a powerful tool for analysis it in the uncertainty conditions. But data is not always reliable; especially when it is not possible to perform multiple tests or reliable data collection. In this context, referring to the beliefs of experts in the field in question is an alternative approach and uncertainty theory is a tool by which the beliefs of experts can be mathematically incorporated into the problem-solving structure. A stable set has a wide range of applications in many fields, while in most cases its problems are without reliable data. In this paper, we investigate the finding of stable weighted sets with uncertain weights. These weights have an uncertain distribution based on the degree of belief of the field expert. For this purpose, we offer two methods. In the first method, by introducing the concept of chance constraint, we come to an integer linear programming model with definite coefficients. The second method is based on the concept of uncertain expected value. Finally, a numerical example for these two methods is presented.