## 2 Description of the Problem  
As explained in the previous section, the HTS system is a robotic cell in which substances to be screened are supplied to the cell on microplates, and then the microplates are transported by the robot between workstations according to the screening process flow. To demonstrate the validity and practical usefulness of the scheduling method developed below, this paper focuses on the theoretical and numerical analyses of the HTS process for enzymatic assays. Enzymatic assays are one of the typical applications of HTS systems. A typical HTS system for an enzymatic assay includes the following basic workstations [11; 12; 13; 14]:  
1. Cytomat2C, a microplate hotel designed to prepare microplates for screening. 2. Multidrop-384, a dispenser for dispensing liquid substances into the wells of microplates. 3. The Envision device, a reader for identifying and analyzing changes occurring in the biochemicals being analyzed. 4. Teleshake, a device that maintains the necessary environment to allow biochemical substances in the wells of a microplate to incubate. 5. A storage center, which holds the microplates for screening. 6. The transportation robot SCARA, used for delivering microplates among the above devices to performing the specified screening procedures. The microplates enter the robotic cell periodically, and the robot SCARA transports all of the microplates between the workstations. There is an auxiliary device, separate from the robot, that automatically delivers the microplates between the storage center and the Cytomat2C, such that the system can continuously operate without interruption. In the screening process under consideration, each microplate passes through workstations in the same predefined technological sequence, as follows: Cytomat2C \(\rightarrow\) Multidrop-384 \(\rightarrow\) Envision \(\rightarrow\) Teleshake \(\rightarrow\) Envision \(\rightarrow\) Cytomat2C. In what follows, this technological sequence of workstations is denoted by \(\mathbf{S}=(M_{0},\,M_{1},\,M_{2},\,M_{3},\,M_{2},\,M_{0})\), where \(M_{i}\), \(i=0,\,1,\,2\), \(3,\,2,\,0\), are the designations of the respective workstations. In Figure 1 below we have plotted a PERT graph that depicts a typical HTS screening process and its actual timing characteristics, as presented in [11; 14]. The operations shown in this figure are repeated for each individual microplate. This graph and Figure 2 clearly show that the Cytomat2C and Envision workstations perform two operations during each cycle in the technological sequence. Nodes represent repetitive operations performed by workstations during the screening of each microplate. Capital letters within nodes identify workstations as follows: \(S\) for  
Figure 1: HTS process flow diagram for an enzymatic assay. Figure 2: The Gantt chart for a 1-cyclic schedule. \(R=(O_{0},\,O_{4},\,O_{1},\,O_{2},\,O_{3})\), \(T=200.5\). Storage Centre, \(C\) for Cytomat2C, \(M\) for Multidrop-384, \(E\) for Envision, and \(T\) for Teleshake. The numbers inside the nodes indicate the required time (in seconds) for the corresponding activity on the workstation. The arcs between nodes and the corresponding weights of the arcs denote the robot's delivery operations and their durations (in seconds), respectively. The arcs between the Storage Centre and the Cytomat2C, and between the Cytomat2C and the Storage Centre, which are both of zero length, represent the delivery operations performed by the auxiliary transportation device. Next, for the convenience of the readers, we present a Gantt chart in Figure 2 to visualize how this data are used to determine the robot's path and movement of microplates between workstations in space and time. This chart will be constructed and used later, in Section 4, using a general PERT/CPM-based method to find the optimal cyclic schedule that minimizes cycle time.