A mathematical programming problem formulated on the basis of incomplete, fuzzy, or poor-quality information is a fuzzy optimization problem. This article considers a fuzzy assignment problem of a special type. In this optimization problem, it is assumed that the elements of the weight matrix in the assignment problem are not defined explicitly. The weight matrix itself belongs to a certain valid set of matrices. An extreme matrix is sought for which the maximum difference between the maximum and minimum values of the objective function in the corresponding assignment problem is achieved. The type of the extreme weight matrix and the value of the maximum difference are found and given in this paper. The conditions imposed on the elements of the weight matrix have a practical interpretation. The results of the study are applicable, for example, to the mathematical model for constructing an optimal schedule for processing raw materials. They allow us to evaluate the application benefits of an optimal processing schedule in industrial production.