# A New Approach to the Allocation of Multidimensional Resources in Production Processes
## 6 Illustrative Example and Computational Experiments  
In an illustrative example, a production system is given (Figure 4) which consists of 5 cells/workstations (w = w01... w05). The individual cell/workstation can process from two to four tasks simultaneously (SA\({}_{\text{w}}\)) and has a specific unit production cost (CR\({}_{\text{w}}\)). There are 14 additional resources in the system (m = m01... m14), which can have up to 12 functionalities (f = f01... f12) and six projects/orders (p = p01... p06) consisting of 20 tasks (o = o01... o20) that should be completed in the system. There is also a projects/orders execution schedule that defines the time and order of execution of tasks from the orders (Figure 5). The schedule does not define assignments of tasks or resources to cells/workstations. Individual projects are marked with different colors according to the principle as in Figure 5.  
Figure 3: Procedure for reducing the size of the modeled problem (pseudocode). All calculation examples were carried out using a computer with the following parameters: Intel(R) Core(TM) i5-4200 M processor, CPU @ 2.50 GHz, 800 GB RAM. Computational experiments with the use of the proposed model (Section 4) were carried out in two stages. In the first stage, the answers to questions \(Q\)1... \(Q\)5_ were searched for the illustrative example data instance (Appendix A). During the computational experiments, the AMPL model and the GUROBI solver were used. The first answer was to question \(Q\)1 (Are there sufficient additional resources to perform a given set of tasks according to a given schedule? ), which was YES. The responding cost (Cost_1) of completing the tasks was 238,000. The corresponding system configuration (which means the allocation of multidimensional resources with specific features to the appropriate workstations) at a selected point in time (\(\tau\) = 2) is presented in Figure 6, whereas the detailed implementation schedule with multidimensional resources and their features allocated to tasks is presented in Figure 7.  
Figure 4: The layout of production system for illustrative example. Figure 5: Execution schedule for illustrative example. The answer to question \(Q\)1 (What and how many additional resources and with what qualities are missing to perform a set of tasks according to a given schedule?) was 'not missing any resources or functionality', see \(Q\)1. Meanwhile, the answer to question \(Q\)3 (Is it possible to perform a set of tasks according to a given schedule when the specific additional resource is unavailable?) was considered for each resource. The answers are presented in Table 6.  
Figure 6: System configuration for a moment of time \(\tau=2\) (concerns question \(Q\)1). Figure 7: Project implementation schedule with the assignment of tasks and multidimensional resources to workstations for \(Q\)1 (individual projects have different colors as shown in Figure 5). The answer to question Q_4 (What is the minimum cost of resource allocation to perform a set of tasks according to a given schedule?) is as follows: the cost (Cost_1) of completing the tasks was 244,520. The corresponding system configuration at a selected point in time (\(\tau\) = 2) is presented in Figure 8, while the detailed implementation schedule with allocated multidimensional resources and features for tasks is presented in Figure 9.  
Whereas the answer to question Q_5 is the aggregation of all answers to question Q_3. Such a supplemented configuration with missing resource functionalities (Table 6) enables the implementation of a set of tasks according to the given schedule when any of the resources m1... m14 is unavailable.