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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 α ∈ [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 presentdoi:10.7494/opmath.2016.36.4.513 fatcat:zixo2ofnufhevmpmftcnf4xe6y