## 4 The Parametric PERT/CPM Method for Finding the Optimum Schedule  
In this section, we present a fast scheduling algorithm that solves the problem using two conjugated algorithms, \(A\) and \(B\) (see Figure 3). The main algorithm, denoted \(A\), is the PERT/CPM project management method, which considers the case where the robot's route is fixed, and each processing time lies within a specified time interval known as a time window. We also consider the case where the number of processing operations is limited by a small constant, as is the case in practice in enzymatic assays. In this case, the total number of possible robot routes is a reasonable fixed number, and the proposed algorithm begins with generating all available robot routes. Then, the first algorithm can alternately process all available robot routes, one after another, and find, among them, the route that produces the minimum-time 1-cyclic schedule. For this purpose, we use and extend the parametric PERT/CPM algorithm on a directed weighted graph, which had been previously developed by the authors in [21]; the latter algorithm is modified to accommodate the specifics of HTS, and is described below. This procedure is repeated for each feasible robot route, and then, among all the obtained pathfinding solutions, the algorithm \(A\) selects the 1-cyclic route with the minimum cycle time.  
At the output of the first algorithm \(A\) we obtain the following results: (i) the optimal 1-cyclic robot path, and (ii) the optimal values of the processing times \(p_{k}\) and the variable \(\beta\) that satisfy the technological constraints of the process, as detailed below.  
Next, we introduce a second subalgorithm, denoted Algorithm \(B\), which takes as input the optimal processing times \(p_{k}\) of all screening operations provided by the aforementioned PERT/CPM subalgorithm \(A\) and finds a set of optimal 2-cyclic schedules. Note that 2-cyclic schedules are generally more efficient than 1-cyclic schedules, and have the following additional advantages: (1) their optimal average cycle time is usually strictly less than the optimal 1-cyclic time, and (2) the resulting set of different 2-cyclic optimal schedules (with each one having possibly different component lengths, but the same average cyclic time) allows us to obtain a robust problem solution. To do this, we modify and use the  
Figure 3: Schematic diagram of the PERT/CPM/PIM algorithm.  
above-mentioned prohibited intervals method (PIM) from our previous work [15]. Using real numerical data from [11; 14], we apply the combined PERT/CPM/PIM algorithm and efficiently solve the considered scheduling problem in polynomial time with respect to the number of operations.  
Let us start by describing how we generated all 1-cyclic robot routes. Recall that in the problem formulation in Section 2, it was stated that the robot SCARA moves microplates between four available workstations \(M_{0}\), \(M_{1}\), \(M_{2}\), and \(M_{3}\), which are the Cytomat2C, Multidrop-384, Envision, and Teleshake workstations, respectively. The sequence of these movements is uniquely determined by the sequence of moments of completion of the five following operations (\(O_{0}\), \(O_{1}\), \(O_{2}\), \(O_{3}\), \(O_{4}\)). Thus, the considered HTS system has \(4!=24\) different robot routes, which we denote by \(R=(O_{0}\), \(O_{[1]}\), \(O_{[2]}\), \(O_{[3]}\), \(O_{[4]})\). Obviously, these routes are determined by permutations of the numbers 1, 2, 3, and 4. The route \(R=(O_{0}\), \(O_{1}\), \(O_{2}\), \(O_{3}\), \(O_{4})\) is trivial. Thus, the general scheme of the solution procedure is as follows: we first need to consider the 23 remaining routes one by one, then solve the PERT/CPM problem for each robot route, which is presented in the next subsection, and finally select the solution with the minimum cycle time \(T\). In fact, as noted in [14], the specific input data of the enzymatic assay screening process can significantly reduce the number of possible robot routes to be tested.