A stochastic interval algebra for smart factory processes

This paper presents a stochastic interval algebra specifically developed to evaluate the time and cost properties of smart factories. This algebra models production tasks as intervals and treats allocation and scheduling as algebraic operations on these intervals, with the goal of analysing the impact of resource allocation decisions on total production time or economic cost. The theoretical foundations of this notation are introduced, and then several simple examples of their use are presented. The proposed algebra can be also applied to describe multi-stage production and service processes, recorded with an activity-on-arrow type of graphs, In addition, it was analysed a real-life application of the described technique to planning and scheduling the activities in restaurants preparing takeaway meals. The data was collected in 30 restaurants throughout Poland, using a bespoken software/hardware Kitchen Delivery System, in which over 65,000 orders were registered. Time criteria for the correctness of individual stages of meal preparation were proposed and, after filtering out incorrect orders, the appropriate probability distributions were fitted to the remaining measured activity durations. The resulting probabilities can then be used in practice to improve the accuracy of predicting the completeness of food preparation, which in turn should improve food delivery planning with greater accuracy and enable more accurate order delivery times to be provided to end customers