Stratified random sampling is an efficient sampling technique for estimating the characteristics of a population. Identifying stratum borders and allocating sample size to strata are two of the most critical components for increasing estimate accuracy. Most surveys are conducted under severe budget constraints, and the survey must be completed within a set time frame. Hence, the majority of surveys include cost and time considerations as highly important goals. Thus, they are necessitating to be under consideration. In many cases, auxiliary variables are utilized in the absence of the main study variable or missing values, indicating that this hypothesis was examined in the study. The paper proposed a mathematical goal programming model for determining optimum stratum boundaries and allocating sample size to different strata utilizing one auxiliary variable as stratification factor when cost and time are taken into consideration. It is not necessary to have all the data to get the optimum stratum boundary; rather, it is sufficient to know information about the parameters of the distribution based on the researcher's experience or other previous studies. Therefore, here the mathematical goal programming is proposed to make it easier for any researcher or statistician to make a prediction the optimum stratum boundary using the information represented by the parameters of the appropriate distribution that fit the nature of the data. In addition, the paper employed Covid-19 data to evaluate the performance of the proposed model using the auxiliary variable (population at old age), where it distributed exponential distribution and the results of the suggested model are satisfying.