# Evaluating Algorithm Efficiency for Optimizing Experimental Designs with Correlated Data
## Materials and Methods
### Evaluation of Algorithms  
The above four algorithms were evaluated under varying experimental conditions to assess their effectiveness to improve field designs. Conditions considered include narrow-sense heritabilities, \(m=1\) initial designs, \(h^{2}\), of 0.1, 0.3, and 0.6, where \(h^{2}=\sigma_{g}^{2}/(\sigma_{g}^{2}+\sigma_{e}^{2})\); unrelated individuals (independent), half-sib and full-sib families; and a spatial correlation of \(\rho=0.6\). Every combination of conditions was repeated \(\lambda=10\) times for \(p=5000\) iterations. All implementation and evaluation of algorithms was done using the statistical package R [22]. The following scenarios were considered: \(\Omega_{A}^{(30)}\), \(\Omega_{D}^{(30)}\) and \(\Omega_{A}^{(196)}\). The first two represent RCB designs with 30 genotypes generated using \(A\)- and \(D\)-optimality criteria, respectively. In these layouts, the designs had six blocks each of dimensions five rows by six columns. Here, pedigree from half-sib families consisted of five male parents each with six individuals, and full-sib families consisted on a half-diallel with five parents for a total of 10 families each with three individuals. Scenario \(\Omega_{A}^{(196)}\) represents an RCB design with 196 genotypes generated using \(A\)-optimality criterion with four blocks of dimensions 14 rows by 14 columns per block. Pedigree files for half-sib families had 32 known parents each with six offspring, whereas full-sib families had 30 parents with several half-diallels for a total of 68 families each with approximately three offspring. Note that, GP98 was implemented only for \(\Omega_{A}^{(196)}\) scenario to swap 50% of the total genotypes at every single iteration, and GP14 represents swapping about 50% of the genotypes for \(\Omega_{A}^{(30)}\) and \(\Omega_{D}^{(30)}\) scenarios. A detailed practical example was implemented with all algorithms in order to investigate the level of design efficiencies and rates of convergence that can be obtained for a specified condition with all algorithms having to improve the same initial RCB experimental design. This was done using \(A\)-optimality criterion for an experiment with 30 genotypes, 6 blocks of sizes 5 rows by 6 columns, and comprised of half-sib families with five male parents each with six individuals, for \(h^{2}=0.1\), \(\rho=0.6\), and an arbitrary nugget effect of 0.1. Initially, \(m=1000\) designs were randomly generated and the best one selected for optimization. All the proposed algorithms were made to improve this initial design by going through \(p=20\),000 iterations. Traces from both successful and unsuccessful swaps were observed together with the time taken for each algorithm. The practical example was run from a 64-bit windows operating system Intel(R) Core(TM) i7-4720 HQ [EMAIL_ADDRESS] GHz, RAM 8.0 GB. To evaluate the improvement of a design, relative overall design efficiency (ODE), that quantifies how efficient the improved design is relative to an initially non-improved design for \(A\)- and \(D\)-optimality was calculated as a proportion or percentage difference between the initial best criterion and the final optimal-value:  
\[\gamma_{ij}^{A}=\frac{\bar{A}_{ij}-A_{(opt)ij}}{\bar{A}_{ij}}; \gamma_{ij}^{D}=\frac{\bar{D}_{ij}-D_{(opt)ij}}{\bar{D}_{ij}}; \tag{2}\] \[\text{for}\quad i=1,2,\cdots,\xi\quad\text{conditions};\quad j=1,2, \cdots,\lambda\quad\text{replicates}\]  
where \(\bar{A}_{ij}\) and \(\bar{D}_{ij}\) are averages of \(m\) initial traces and log-determinants, respectively, for \(i\)-th condition and \(j\)-th replicate, \(A_{(opt)ij}\) and \(D_{(opt)ij}\) are the smallest trace and log-determinant, respectively, obtained from an improved design. Finally, ODE calculations over the \(\lambda=10\) replicates per condition were summarized. A schematic diagram that represents a summary of the procedure to improve a given randomized complete block design is displayed in Figure 1 and Table 1 shows the simulation conditions implemented for the motivating example.