## 3 Results
Global optimum A and global optimum B are considered identical for  
Figure 9: Full MIPLIB results: number of distinct optima vs. run number for all MIPLIB instances tested. The plots are arranged in descending order of maximum number of unique optima found. the purposes of this test. Results with many choices for the number of runs on the \(x\)-axis and two individual datasets can be found in Figures 10 and 11. Results with fewer choices for the number of runs for nine datasets can be found in Figure 12.  
A practical motivation for looking at multiple optima of MadHitter instances is to identify which genes occur in any optimal solution or in all optima. An observed example of the impact of MORSE is conveyed by analyzing the brain cancer dataset GSE103322. With default MadHitter parameters and using the solution listing delineated above, the gene _CELSR1_ (short for cadmium EGF LAG seven-pass G-type receptor 1) appears (by chance) in multiple optima generated by MORSE, including the second optimum. However, when  
Figure 11: Number of distinct optima found (\(y\)-axis) vs. run number (\(x\)-axis) for GSE147082 (ovarian cancer) 1269-gene dataset, lower bound = 0.8, upper bound = 0.1. With MORSE, we find all 360 distinct optima in 959 runs. With uniform weights, we find 215 optima in 1000 runs. Figure 10: Number of distinct optima found (\(y\)-axis) vs. run number (\(x\)-axis) for GSE103322 (brain cancer) 1269-gene dataset, lower bound = 0.8, upper bound = 0.1. With MORSE, we find all 5 distinct optima in 9 runs. With uniform weights, we find these optima in 397 runs. using [PERSON]'s solution pool, _CELSR1_ appears for the first time in optimum number 397. The gene _CELSR1_ has previously been shown to have higher messenger RNA levels in brain cancer as compared to surrounding non-cancerous brain tissue by [46; 47], so this is a favorable gene to select as a possible target in an optimal treatment. To quantify bias in the solution space, we used Shannon entropy. A higher Shannon entropy value corresponds to a higher level of randomness and a lower level of bias in the distribution. In the present context, a high Shannon entropy value is desirable, as the algorithm should not favor any particular solution in the solution space. To find the Shannon entropy for each feasible permutation, we assigned an empirical normalized probability \(p_{i}\) to each gene solution in the solution set and applied the standard Shannon entropy formula (see Section 2.7). Full MadHitter Shannon entropy results can be found in Figure 13.  
For the MIPLIB instances tested, we also employed the Shannon entropy on the optima found with both MORSE and the uniform weights method. We calculated the Shannon entropy on the multiplicity of each optimum found (Figure 14) by assigning an empirical normalized probability \(p_{i}\) to each distinct optimum found. Additionally, for each instance tested, we calculated Shannon entropy on each variable by assigning \(p_{i}\) to each distinct variable value, then averaged the Shannon entropy score for each variable to find the  
Figure 12: Tabular results. Number of distinct optima found (\(y\)-axis) vs. run number (\(x\)-axis) for all 1269-gene cancer datasets tested, lower bound = 0.8, upper bound = 0.1. For GSE127465 (lower row, center), the results for MORSE (red) and uniform weights (blue) are identical  
average Shannon entropy (Figure 15). Both tests only considered the variables in the objective function. We found that, for both approaches, the Shannon entropies for optima found with MORSE generally exceeded those of the optima found with uniform weights, except for one instance (harp2) where the latter marginally exceeded the former. When computing Shannon entropies over variables and averaging, we assign more influence to variables whose values differ more often between different optimal solutions.