Abstract: Mathematical human resource planning has been widely employed to represent inter-dependencies between stocks(current numbers) and flows(transfers) in manpower systems. This study proposes a multivariate non-homogeneous Markov system for human resources planning within the departments of Damietta University, taking into account the horizontal and vertical transitions within and between the different departments, fuzzy goal programing was used to determine the optimal allocation for new comers and transfers between and within different - 2 - departments, with merge the operational cost related to recruitment and retirement and transitions within and between the different departments in university. According to what has been obtained, the optimal solution showed clear results of the recruitment variable (r), which requires the administration to take into account the need for work necessary for this number of new employees and make estimates of the vectors of the costs of recruitment and transition, with regard to the transition probabilities, it is acceptable, as well as the estimates of the total cost vector, most of the cost estimates obtained are greater than the aspiration level and therefore the aspiration level by management should be increased for each functional grade at the level of each department..