Process performance maximization through input par using taylor series expectation approximation

In several industrial settings or manufacturing units the process output is expressed as a function of several inputs and such relationships can be expressed in the form of the equation based on the prior data or known truth. It is desirable in such a setting that the output is within a specified threshold for acceptance of the quality of the product. It is not always possible to operate the industrial plant under several settings and then evaluate the one that maximizes the performance. Besides such experiments/tests can be expensive as well as time consuming. Given that the output-input relationship in the form of an equation and the probability distribution of the inputs is known from the prior data the process performance at the current setting can be evaluated by randomly sampling inputs as per the input distribution and then checking the proportion of samples for which the output value is outside the specified threshold. If the process performance is not satisfactory there are generally two ways that we can optimize the performance. We can a) optimize the input parameters mean/mode, etc. keeping the variance of the input distribution constant (or/and) b) optimize the variance of the inputs keeping the other parameters of the distribution constant. Generally, in such settings controlling the variance of the input is often difficult and harder to implement as opposed to varying the mean or other parameters that don’t influence the variance. For example, in a setting if temperature is one of the inputs following normal distribution with mean 25 and variance 1.5 it is easier to vary the mean by increasing the temperature rather than to vary the variance. This paper attempts to illustrate an efficient technique to find the best input parameters (keeping the input variances constant) that maximize or minimize the output mean so that the proportion of output beyond the threshold is minimized. One of the frequently used techniques is to do sampling at various combination of parameter values for the inputs in the user specified given range and then choosing the input parameter setting that provides the minimum proportion of output values beyond the user specified threshold. The problem with this sampling technique is if the number of input variables are high then the combination of parameter values at which sampling needs to be performed and output performance evaluated increases exponentially. This results in high processing time to do the optimization.