## 4 Experimental Results and Analysis
### Experiment Case I  
The first test case was designed to evaluate the effectiveness of equally additive effects on the quantitative trait taking synergistic or antagonist influence. In this problem seven QTLs of equal effect \(\forall\beta_{j}\in B_{a},\beta_{j}=\pm 0.76\) are considered, with all QTLs positioned exactly at marker loci. We located two QTLs on chromosome 1 at markers 4 and 8 linked in synergistic coupling (their effect has same sign). Two other QTLs were located on chromosome 2 at markers 4 and 8; these QTLs were coupled antagonistically so their effect had opposite signs. We located three further QTLs on chromosomes 3, 4 and 5 at markers 6, 4 and 1 respectively. The four remaining chromosomes were left with no QTLs. The environmental variation had a normal distribution with \(\sigma^{2}=1\) and the heritability of the trait was set to 80%. 100 random problem instances were created for target sample sizes and for each, the Multi-Objective Evolutionary Based method was applied 10 times. The relationship between solution quality and algorithm iteration is depicted in Figures 1 and 2. The graphs show the best solution at each step in an arbitrary randomized simulation. Initially, the population has a low quality fit and a high complexity since it is random. As the algorithm progresses the optimization criteria improves and the best solution eventually reaches a local minima and so will be included in the Pareto optimal set. In Figures 1 and 2 the solution identified has a complexity of three parameters (out of the seven), the \(R^{2}\) fit is low at \(0.38\). This example demonstrates the diversity of solutions found; some of these solutions under-fit the problem at the expense of accuracy. Figure 3 shows all points on Pareto front between solution complexity 2 and 10. We summarize the results of the different sample sizes by averaging the goodness of fit for each solution complexity. Note the classic trade-off is illustrated between complexity and accuracy. As the complexity increases, the goodness of fit tends to increase. Significantly there is a change in the form of the trade-off curve at the point when the complexity of the solution exactly matches the known complexity of the given problem. Note how once the solution begins to over-fit the correct solution, the improvement in goodness of fit  
Figure 1: **Algorithm Sample Run:** The \(-R^{2}\) value as a function of algorithm iteration of the best individual for an arbitrary simulation. diminishes with the increase in complexity. By determining the maximum rate of change of goodness of fit with respect to the complexity, the optimal complexity could be identified that minimizes over and under fitting. Figure 4 shows the markers identified for each solution complexity, with each figure the sum over the randomized program runs. Since a red marker means maximal identification, the solution with complexity 7 consistently finds the correct location of all 7 markers, while avoiding extraneous markers. Interestingly, the synergistically linked markers are consistently identified, irrespective of solution complexity. Solutions with complexity 3 through 5 find the five correct synergistic markers. QTLs that were coupled antagonistically (chromosome 2, position 4 and 8) proved much more difficult to  
Figure 3: **Case I Pareto Front:** showing the relation between \(-R^{2}\) and complexity for Case 1. The values are average of 10 runs of 100 simulation replicates. Figure 2: **Algorithm Sample Run:** The complexity as a function of algorithm iteration of the best individual for an arbitrary simulation. identify. A low proportion of runs with complexity 6 find these markers, increasing in proportion as solution complexity grows. However, when complexity exceeds the given 7 parameters, many nearby extraneous markers begin to be found. **Figure 4. Case I Identified Markers:** Identified markers found in different complexity Pareto optimal solutions for Case I. The color is the proportion of random executions that the specific marker was found.