Optimal design provides the most efficient design to study dose-response functions. It is often observed to adopt the four-parameter logistic model to describe the dose-response relationships in many dose finding trials. Under the four-parameter logistic model, optimal design to estimate the EDp accurately is presented. The EDp is the dose achieving 100p% of the maximum treatment effect. C-optimal design works the best to estimate the EDp, but the value of p must be predetermined in order to obtain the c-optimal design. Here we investigate the efficiency of c-optimal design to estimate the EDp for different values of p and present robust c-optimal design that works well for the changes in the value of p. Five different values of p are considered in this study: ED10, ED30, ED50, ED70, and ED90. The performance of the robust c-optimal design is obtained and compared to the c-optimal designs and traditional uniform designs.