## 4 The Parametric PERT/CPM Method for Finding the Optimum Schedule
### The Prohibited Intervals Method B
Instead, we will now focus on finding an optimal 2-cyclic solution, which can, in general, be better than an optimal 1-cyclic schedule. **Proposition 1. ** A 2-cyclic schedule with time intervals \(T_{1}=151\) and \(T_{2}=250\) is feasible. Proof. : It is easy to see that conditions (10) are satisfied. Indeed, we have:  
\[T_{1}=151\in[136,210]\in U,\,T_{2}=250\in[250,250]\in U,\]  
\[Z_{j}\leq T+Z_{j}-p_{j},\,\text{i.e.,}\,p_{j}\leq T.\]0  
\[Z_{j}\leq T+Z_{j}-p_{j},\,\text{i.e.,}\,p_{j}\leq T.\]1  
\[Z_{j}\leq T+Z_{j}-p_{j},\,\text{i.e.,}\,p_{j}\leq T.\]2  
which proves the feasibility of the considered 2-cyclic schedule. \(\square\)  
The average cycle time of this 2-cyclic schedule, \(T_{avr}\), is \((T_{1}+T_{2})/2=(151+250)/2=200.5\). **Proposition 2. ** (i). All 2-cyclic schedules obtained by choosing time intervals \(T_{1}=191+x\) and \(T_{2}=210-x\), where \(0\leq x\leq 9.5\), are feasible. (ii). Any of these 2-cyclic schedules is optimal. (iii). The 2-cyclic schedule with cycle times \((T_{1},\,T_{2})=(151,\,250)\) is optimal. Proof. : (i). It is evident that the conditions in (10) for a schedule with \((T_{1},\,T_{2})=(191+x,\,210-x)\), where \(0\leq x\leq 9.5\) are satisfied. This is proven by the below:  
\[Z_{j}\leq T+Z_{j}-p_{j},\,\text{i.e.,}\,p_{j}\leq T.\]3  
\[Z_{j}\leq T+Z_{j}-p_{j},\,\text{i.e.,}\,p_{j}\leq T.\]4  
\[Z_{j}\leq T+Z_{j}-p_{j},\,\text{i.e.,}\,p_{j}\leq T.\]5  
\[Z_{j}\leq T+Z_{j}-p_{j},\,\text{i.e.,}\,p_{j}\leq T.\]6  
Hence, any two-cyclic schedule in the considered set is feasible. The average cycle time of these schedules is: \(T_{avr}=(T_{1}+T_{2})/2=[(191+x)+(210-x)]/2=200.5\). (ii). Suppose that there is a two-cyclic schedule with a cycle time \(T^{\prime}=T^{\prime}_{1}+T^{\prime}_{2}<401\). Then, from the definition of \(U\), it follows that:  
\(T^{\prime}=T^{\prime}_{1}+T^{\prime}_{2}\) either belongs to the allowed interval [97; 97], the allowed interval [136; 210], or the allowed interval [250; 250]. Assume that \(T^{\prime}_{1}\leq T^{\prime}_{2}\), then \(T^{\prime}_{1}\leq 250/2=125\), and \(T^{\prime}+T^{\prime}_{1}\leq 250+125=375\). From the definition of \(U\), it also follows that \(97\leq T^{\prime}_{1}\) and \(97\leq T^{\prime}_{2}\), therefore  
\[Z_{j}\leq T+Z_{j}-p_{j},\,\text{i.e.,}\,p_{j}\leq T.\]7  
Thus, \(T^{\prime}+T^{\prime}_{1}\
otin U\), i.e., a feasible two-cyclic schedule with cycle time \(T^{\prime}<401\) does not exist, which proves the Proposition, specifically cases (ii) and (iii). \(\square\)  
Computing two-cyclic robot route.For a given two-cyclic schedule with time intervals \((T_{1},\,T_{2})\) and cycle time \(T=T_{1}+T_{2}\), let us calculate completion times for operations \(O_{i}\) (\(i=0,\,1,\,2,\,3,\,4\)) in the time interval [0, \(T\)). Note that in the two-cyclic schedule each operation \(O_{i}\) appears twice,  
\[Z_{j}\leq T+Z_{j}-p_{j},\,\text{i.e.,}\,p_{j}\leq T.\]8  
Recall that, given two numbers \(a\) and \(b\), the modulo operation \(a\bmod b\) returns the remainder after dividing \(a\) by \(b\). For example, \(10\bmod 3=1\). The following Proposition 3 determines the optimal two-cyclic robot route \(R=(O_{[1]},\,O_{[2]},\,\ldots,O_{[10]})\). **Proposition 3**. : _Given real numbers \(T\) and \(T_{1}\), arrange the numbers \(t_{i}\) and \(t^{\prime}{}_{i}\) in increasing order._  
\[Z_{j}\leq T+Z_{j}-p_{j},\,\text{i.e.,}\,p_{j}\leq T.\]9  
_where the symbol \(t^{\bullet}{}_{i}\) denotes either \(t_{i}\) or \(t^{\prime}{}_{i}\)._  
The following sequence of operations  
\[Max\,p_{j}\leq T,\,j=0,\,1,\,2,\,4,\,5. \tag{5}\]0  
induced by the above ordering of the numbers \(t^{\bullet}{}_{i}\) is the only possible optimal 2-cyclic robot route with period \(T\). This fact immediately follows from two observations: (i) the robot handles not more than one microplate at a time, and (ii) the robot starts its \(k\)th move within \([0,T)\), exactly at time \(t=t^{\bullet}{}_{k}\), \(k=1,\ldots,\) 10 computed above. More details are given in [21].