As discussed in the paper, when defining a schedule to process a query we want to control the scheduling behavior concerning the usage of computational resources, i.e., processing and network capacity. Our evaluation showed that attempting to minimize both processing and network utilization creates conflicting objectives, in the sense that to achieve a better balance in the query execution (and consequently a reduced makespan), an extra cost is incurred to transfer partitions to machines that are underloaded. To address this issue, we introduced a parameter in the model, termed f , to specify the degree of preference for a more balanced execution or for a lower usage of network resources.From a practical perspective, if the only concern for query execution is the communication cost, selecting a value for f is trivial (f =0). Analogously, if the only concern is the makespan, we can readily select a sufficiently large value for f so as to ignore the communication cost in the objective function. Often, however, the computational environment imposes constraints in both makespan and communication costs, and selecting a value for f that achieves a balance between these two objectives results in a reasonable saving of computational resources. Each numerical instance of the SM model has a particular given value for f but determining a suitable value of f is a non-trivial task, as it depends on the values of the processing and communication costs (w j and c ij ).We address in this supplementary material the question of determining an appropriate value for f by studying the effects of variations on this value on the optimal schedule for SM. We demonstrate how to determine intervals of f values for which an optimal schedule for SM remains unchanged. By establishing these intervals, we can identify the shape of the objective function (1.1) and provide answers to some practical questions that arise when scheduling queries using the SM model, such as: i) is it possible to reduce the makespan of a query even further?, ii) if yes, what is the minimum makespan and the additional communication cost incurred?, iii) analogously, what is the increase in makespan if we have to reduce the communication cost on a low bandwidth network?, and iv) what is the maximum value for f for which a change in makespan occurs?For linear programming, a fully-developed theory exists to determine valid intervals for the parameters of a model, for which an optimal schedule remains unchanged. It is known as postoptimality analysis (Bertsimas and Tsitsiklis 1997). This theory, however, is not valid for integer linear programming, and it turned out that a similar theory for the integer case is inherently more challenging and yet not fully developed (Blair 1997). Notwithstanding, by adapting the CONTACT Thiago Borges de Oliveira.