The paper deals with a version of the general non-permutation flow-shop scheduling problem in which the duration of each operation on certain machines is a linear function of the allotted part of a constrained resource (e.g. energy, fuel, catalyzer, oxygen, raw material, money) and the objective is to determine both a sequence of the operations on each machine and an allocation of resource to each operation in order to minimize the over-all completion time. The algorithm for solving this problem is based on the disjunctive graph theory and branch and bound technique. Due to the present elimination properties, the search tree is strongly restricted. It applies a special system of fixing precedence relations in each node of the search tree yielding strong lower and upper bounds. To quickly obtain a strong upper bound a special heuristic method is used (to calculate a good initial solution) and the descendants in the search tree are chosen in order of the non-decreasing lower bounds. Some computational results and concluding remarks are included as well.