Most of the chemical processes with significant noise in the measured variables can be controlled using proportional-integral controllers. It is always important to determine the optimum control parameters of these proportional integral controllers depending on the different objectives. In this article, correlations which relate the optimum proportional integral controller parameters to process parameters for different types of process models are developed. Both servo and regulatory control correlations for proportional integral controllers are obtained for the process model types such as first order plus time delay (FOPTD) and second order plus time delay (SOPTD) with the objective of minimizing different performance criteria such as integral of absolute value of the error (IAE), integral of the time-weighted absolute value of the error (ITAE), integral of the squared value of the error (ISE) and integral of the time - weighted squared value of the error (ITSE). The corresponding performance of these proposed correlations are compared with that of the well-known tuning methods: Ziegler-Nichols continuous cycling method, Ziegler-Nichols reaction curve method, Cohen-Coon method and other proposed tuning methods in the literature in terms of values of overshoot, rise time, settling time and integral performance criteria and the advantages and disadvantages of the proposed correlations are discussed. It is found that using correlations obtained for first order plus time delay and second order plus time delay processes, several performance characteristics such as overshoot and settling time are reduced compared to those obtained using other tuning methods. Further, the regulatory control correlations proposed for first order plus time delay processes leads to minimum values of integral performance criteria than some of the other existing methods.