In this work we analyse the fed-batch operation of a biochemical reactor. In the first part of the work we consider a repeated fed-batch operation. The reaction is assumed to follow Haldane kinetics, i.e. it is characterised by substrate inhibition. The feed rate of the substrate is chosen as the control variable. The entire duration of the operation is divided into different subintervals. We optimise the system performance by approximating the feed flow rate using (i) discrete pulses, (ii) a constant flow rate over different sub-intervals. In the first strategy the equations lend themselves to an analytical solution. For the second case we use a shooting method coupled with a sequential quadratic programming technique to obtain the constant flow rates in the different sub-intervals. This has been analysed for two scenarios: (i) equal duration of sub-intervals and (ii) unequal duration of sub-intervals. The objective here is to maximise the biomass production over one cycle. The effect of converting a non-linear constraint to a linear constraint by reformulating the problem on the numerical convergence has also been investigated. In the second part of this work we have extended our method to investigate a non-repeated fed-batch reactor operation. Here the objective is to maximise the product formed.