Continuous filling and emptying of storage systems in constraint-based scheduling

In constraint-based scheduling, storage systems are modelled by a class of resource called reservoir. Activities can either produce or consume a capacity in a reservoir. Most of the models presented in the literature assume that the production and the consumption is instantaneous but they are not good for some class of problems. This paper introduces the continuous reservoir model in which the activity fills or empties the reservoir at a constant rate from its start time to its end time. Then it presents how timetabling is adapted to this new model. 1. Modelling storage systems with reservoirs and continuous reservoirs Inventory management is of key importance in project management. Basically, there are two important constraints that must be satisfied when one encounter a storage system (e.g. tank of oil, depository of raw materials or storehouse of products that must be delivered to customers) in a process scheduling problem. The first one is to be sure that an activity whose process consumes some amount of the stored product can actually use this amount of product at the time it is processed. The second one states that when an activity produces some amount of the stored product, there must be enough room in the storage system to receive it. In this paper, we will use the generic term “reservoir” to refer to the storage system. We will call “level of the reservoir” the curve that represents at any time point the amount of capacity that is stored in the reservoir when all the activities that produce or consume a quantity stored in the reservoir are scheduled. In the entire schedule time interval, the level of the reservoir must neither be negative nor exceed the “capacity of the reservoir” that is the maximum capacity that can be stored in the reservoir. So, the reservoir is very similar to resources as they are considered in ‘resource scheduling’, the level of the reservoir being similar to the load of the resource. However, we remark that an unlimited number of producing and consuming activities can be processed at the same time even if the capacity of the reservoir is very low. In the literature [4, 5], the assumption that the action of filling or emptying the reservoir is instantaneous is generally made. Under this assumption, the level of the reservoir is a stepwise function, that is the level of the reservoir is modified at some time point, corresponding to one or several emptying and/or filling, and remains constant till the next discrete change. But these models with instantaneous emptying and filling may be too crude for some realworld process scheduling problems. For example, the filling or the emptying of a tank by a liquid is usually subject to a constant rate depending of the size and the number of the taps and siphons. A second typical example is the load and unload of batteries. In this paper, we introduce the “continuous reservoir”. In this model, the activity fills or empties the reservoir at a constant rate, that is the process is linear between the start time and the end time of the activity. For example, if an activity, that starts at time st and ends at time et, can fill the reservoir by a capacity C, the quantity put in the continuous reservoir by the activity at time t (also stated as contribution of the activity at time t) is: