# A New Approach to the Allocation of Multidimensional Resources in Production Processes
## 3 Problem Description  
The production systems consisting of various manufacturing cells/workstations (w) with any form of production organization (job-shop, flow-shop, open-shop, multi-project, etc.) [17; 18] are considered. The single manufacturing cell/workstation [19] can process more than one task/job in parallel. The number of parallel tasks is specified by the SAw parameter. The production system processes a set of tasks/jobs that can be organized as orders or projects (p). The orders/projects implementation schedule is given, which includes the time of their implementation and the sequence of execution. The schedule does not include assignments of jobs/tasks of orders/projects to manufacturing cells/workstations. Additional resources (m) are needed to complete the orders/projects. In the discussed problem, these are the so-called multidimensional resources. These resources are characterized by the fact that they can have many functionalities (f). Furthermore, these functionalities can be changed or supplemented, which enables the configuration and adjustment of a given resource. Examples of such resources are PLCs with dedicated software, tool magazines with specialized tools, as well as employees with their competencies, etc. On the other hand, similarly to classic resources, they are limited and have an assigned cost of their use, which may depend on the functionality used. In the systems under consideration, we also take into account the unavailability of certain resources and/or their functionalities. The unavailability of resources is modeled by introducing the parameter (u) defining the so-called state of unavailability. For example, the state u1 = {m1, m3} specifies that the resources m1 and m3 are unavailable, u2 = {m5} that the resource m5 is unavailable, etc. In the described manufacturing system, three levels can be logically distinguished: the level of additional multidimensional resources, the level of manufacturing cells/workstations and the level of orders/projects. This division is schematically shown in Figure 1.  
Figure 1: Logical levels in the considered production systems. The use of configurable multidimensional resources in the production system gives the possibility to increase the flexibility of the system under consideration, including through its self-organization and self-optimization. This, in turn, can improve the system's resistance to external factors, reduce production costs, shorten production cycles, etc. The most important questions that arise in the context of controlling, configuring and allocating multidimensional resources are:  
* Are there sufficient multidimensional resources to perform a given set of orders/projects according to a given schedule? (_Q_1)
* What and how many multidimensional resources and with what functionalities are missing to perform a set of orders/projects according to a given schedule? (_Q_2)
* Is it possible to complete a set of orders/projects according to a given schedule when the specific multidimensional resource is unavailable? (_Q_3)
* Which projects should be implemented and how (with what resources) to maximize the profit? (_Q_4)
* How should a set of multidimensional resources be configured to perform a set of tasks according to a given schedule in the absence of any resource? (_Q_5)  
Finding answers to the above questions is crucial in the context of the implementation of a given set of orders/projects in accordance with the imposed conditions. As can be seen, individual questions are of different nature. On the one hand, these are general questions (_Q_1, \(Q\)3) and specific questions (_Q_2, \(Q\)4, \(Q\)5). On the other hand, they are reactive (_Q_1, \(Q\)2, \(Q\)4) or proactive (_Q_5). Question \(Q\)3 can be asked in both forms. Obtaining proactivity is achieved thanks to the appropriate selection and configuration of a set of multidimensional resources, which allows for keeping the production schedule regardless of the lack of availability of the resource or its functionality.