# Hybrid Harmony Search for Stochastic Scheduling of Chemotherapy Outpatient Appointments
## 4 Numerical Experiments and Analysis of Results  
To prove the effectiveness of the proposed HS in solving the COS problem under investigation, several data sets of numerical instances were generated. We considered three classes of problems at varying problem sizes, namely small (\(P=15\)), medium (\(P=40\)) and large (\(P=70\)). For each of them, a Design of Experiments (DOE) was performed by involving four factors (i.e., number of oncologists \(O\), number of chairs \(C\), drug batch size \(CAP\)) and the interval of the uniform distribution related to the batch delivery time \(U(TD)\) varied at two levels, low (L) and high (H), respectively. Hence, a total amount \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Vectors** & **Values** & & & \\ \hline \(x\) & 0.45 & 0.62 & \(-\)0.12 & 0.33 \\ \(x_{perm}\) & 3 & 4 & 1 & 2 \\ \(x_{r}\) & 0.61 & 0.27 & 0.82 & \(-\)0.23 \\ \(x_{rein}\) & 0.61 & 0.82 & \(-\)0.23 & 0.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Example of Harmony Reinitialization. \(2^{4}\)\(=16\) configurations for each class of problem were considered. Since ten instances were randomly generated for each configuration, \(3\cdot 16\cdot 10=480\) runs were executed. Table 4 reports the different factors/levels, whose values are fixed based on the benchmark problems addressed in the literature so far (please see [11; 16; 23; 25; 30; 35; 45] among others). Indeed, since both the drug transportation- and the batch size-related issues have never been investigated in the literature so far, their values were set based on a brief survey involving the medical staff. The number of nurses \(N\) was set by considering that the ratio \(N\)/\(C\) is usually fixed to \(1/4\) to respect the assumption that one nurse can monitor at most four patients simultaneously [2; 6]. Finally, to balance the pharmacy capability on the problem size, a different number of pharmacist technicians \(D\) = [1,2,3] was assigned to each class of problems, respectively. Due to the stochastic feature of the proposed COS problem, the number of scenarios \(\Omega\) to evaluate any candidate solution was set to 300. The stochastic parameters of the COS problem under investigation were set as follows. At each scenario \(\omega\), every patient \(p\) was randomly assigned to a referee oncologist \(Op_{p}\). The medical consultation time \(Dec^{\omega}_{p}\) of patient \(p\) at scenario \(\omega\) depends on an \(N\)(22.83, 3.19) normal distribution. The deferral coefficient \(\delta\,^{\omega}_{p}\) of patient \(p\) at scenario \(\omega\) arises from \(\lambda\,^{\omega}_{p}\,\in\,\,U\,\,[0,\,1]\) with \(\overline{\lambda}\) equal to 0.20 [25]. The drug preparation time \(Dt\,^{\omega}_{p}\) was extracted from an uniform distribution \(U\)(3, 7). The delivery time \(TD^{\omega}_{p}\) of batch \(p\) at scenario \(\omega\) was also derived from a uniform distribution as in Table 4. The set-up time \(Ds^{\omega}_{p}\) of patient \(p\) at scenario \(\omega\) was derived from \(U\)(5, 15). Finally, the treatment time \(D^{\omega}_{p}\) was drawn from a gamma distribution \(\Gamma\)(1.9, 52.37). The type of statistical distribution mentioned above refers to the results of an extensive time study carried out in a chemotherapy unit located in Southern Italy. Each metaheuristic algorithm was coded in Matlab(r)R2021b and executed on a 4 GB RAM-2 processors virtual machine embedded on a workstation equipped with an INTEL i9-9900 3.6 GHz 10 core CPU, 32 Gb DDR4 2,666 MHz RAM and Win 10 PRO OS. The exit criteria were set based on preliminary tests as follows: \(Max\_ev=3000\) and \(Max\_ct=300\) s for \(P=15\); \(Max\_ev=9000\) and \(Max\_ct=1000\) s for \(P=40\); \(Max\_ev=9000\) and \(Max\_ct=2500\) s for \(P=70\). Although the expected total flow time \(E(F)\) was the objective function of the COS problem under investigation, the Relative Percentage Deviation (\(RPD\)) function was handled to compare the results obtained by the tested optimization techniques.