## Chart: Runtime for solving LOTZ ### Overview The image is a line chart comparing the performance of different algorithms (GSEMO and NSGA-II with varying parameters) in solving the LOTZ problem. The chart plots the number of fitness evaluations (y-axis, logarithmic scale) against the problem size 'n' (x-axis, linear scale). Error bars are present on each data point. ### Components/Axes * **Title:** Runtime for solving LOTZ * **X-axis:** * Label: *n* * Scale: Linear * Ticks: 30, 60, 90, 120 * **Y-axis:** * Label: Fitness Evaluations * Scale: Logarithmic (base 10) * Ticks: 10⁵, 10⁶ * **Legend:** Located in the top-left corner. * Black: GSEMO * Light Green: NSGA-II, N=2(n+1) * Orange: NSGA-II, N=4(n+1) * Dark Green: NSGA-II, N=8(n+1) ### Detailed Analysis * **GSEMO (Black):** The line slopes upward. * n=30: Fitness Evaluations ≈ 2.5 x 10⁴ * n=60: Fitness Evaluations ≈ 1.4 x 10⁵ * n=90: Fitness Evaluations ≈ 5.0 x 10⁵ * n=120: Fitness Evaluations ≈ 1.1 x 10⁶ * **NSGA-II, N=2(n+1) (Light Green):** The line slopes upward. * n=30: Fitness Evaluations ≈ 1.8 x 10⁴ * n=60: Fitness Evaluations ≈ 9.0 x 10⁴ * n=90: Fitness Evaluations ≈ 3.0 x 10⁵ * n=120: Fitness Evaluations ≈ 6.0 x 10⁵ * **NSGA-II, N=4(n+1) (Orange):** The line slopes upward. * n=30: Fitness Evaluations ≈ 2.2 x 10⁴ * n=60: Fitness Evaluations ≈ 1.2 x 10⁵ * n=90: Fitness Evaluations ≈ 4.0 x 10⁵ * n=120: Fitness Evaluations ≈ 9.0 x 10⁵ * **NSGA-II, N=8(n+1) (Dark Green):** The line slopes upward. * n=30: Fitness Evaluations ≈ 2.8 x 10⁴ * n=60: Fitness Evaluations ≈ 1.5 x 10⁵ * n=90: Fitness Evaluations ≈ 5.5 x 10⁵ * n=120: Fitness Evaluations ≈ 1.3 x 10⁶ ### Key Observations * All algorithms show an increase in fitness evaluations as 'n' increases. * The NSGA-II algorithm with N=2(n+1) consistently requires the fewest fitness evaluations. * The NSGA-II algorithm with N=8(n+1) and GSEMO require the most fitness evaluations, with NSGA-II N=8(n+1) being slightly higher. * The error bars indicate some variability in the performance of each algorithm. ### Interpretation The chart demonstrates the runtime performance of different algorithms for solving the LOTZ problem, measured by the number of fitness evaluations required. The results suggest that NSGA-II with N=2(n+1) is the most efficient among the tested configurations, requiring the fewest fitness evaluations as the problem size increases. GSEMO and NSGA-II with N=8(n+1) are the least efficient. The logarithmic scale on the y-axis indicates that the number of fitness evaluations increases exponentially with 'n'. The error bars suggest that the performance of each algorithm can vary, but the overall trends remain consistent.

## Line Chart: Runtime for solving LOTZ ### Overview This chart displays the runtime, measured in Fitness Evaluations, for solving the LOTZ problem using different algorithms and parameter settings. The x-axis represents the problem size 'n', and the y-axis represents the number of Fitness Evaluations. Four different algorithms/configurations are compared: GSEMO, NSGA-II with N=2(n+1), NSGA-II with N=4(n+1), and NSGA-II with N=8(n+1). Each data point includes error bars representing the variability of the runtime. The y-axis is on a logarithmic scale. ### Components/Axes * **Title:** "Runtime for solving LOTZ" (positioned at the top-center) * **X-axis Label:** "n" (positioned at the bottom-center) * **Y-axis Label:** "Fitness Evaluations" (positioned vertically along the left side) * **Y-axis Scale:** Logarithmic, ranging from approximately 10^5 to 10^6. Markers are at 10^5 and 10^6. * **Legend:** Located at the top-left corner. Contains the following entries: * GSEMO (Black line with circle markers) * NSGA-II, N=2(n+1) (Green line with triangle markers) * NSGA-II, N=4(n+1) (Red line with plus markers) * NSGA-II, N=8(n+1) (Brown line with cross markers) * **X-axis Markers:** 30, 60, 90, 120 ### Detailed Analysis Here's a breakdown of the data series, with approximate values extracted from the chart: * **GSEMO (Black):** * At n=30, Fitness Evaluations ≈ 3.5 x 10^5 ± 0.2 x 10^5 * At n=60, Fitness Evaluations ≈ 5.5 x 10^5 ± 0.3 x 10^5 * At n=90, Fitness Evaluations ≈ 8.0 x 10^5 ± 0.4 x 10^5 * At n=120, Fitness Evaluations ≈ 1.0 x 10^6 ± 0.5 x 10^5 * Trend: The line slopes upward, indicating that runtime increases with 'n'. * **NSGA-II, N=2(n+1) (Green):** * At n=30, Fitness Evaluations ≈ 3.0 x 10^5 ± 0.3 x 10^5 * At n=60, Fitness Evaluations ≈ 5.0 x 10^5 ± 0.3 x 10^5 * At n=90, Fitness Evaluations ≈ 7.5 x 10^5 ± 0.4 x 10^5 * At n=120, Fitness Evaluations ≈ 1.1 x 10^6 ± 0.5 x 10^5 * Trend: The line slopes upward, indicating that runtime increases with 'n'. * **NSGA-II, N=4(n+1) (Red):** * At n=30, Fitness Evaluations ≈ 3.2 x 10^5 ± 0.3 x 10^5 * At n=60, Fitness Evaluations ≈ 5.3 x 10^5 ± 0.3 x 10^5 * At n=90, Fitness Evaluations ≈ 7.8 x 10^5 ± 0.4 x 10^5 * At n=120, Fitness Evaluations ≈ 1.0 x 10^6 ± 0.5 x 10^5 * Trend: The line slopes upward, indicating that runtime increases with 'n'. * **NSGA-II, N=8(n+1) (Brown):** * At n=30, Fitness Evaluations ≈ 3.3 x 10^5 ± 0.3 x 10^5 * At n=60, Fitness Evaluations ≈ 5.6 x 10^5 ± 0.3 x 10^5 * At n=90, Fitness Evaluations ≈ 8.2 x 10^5 ± 0.4 x 10^5 * At n=120, Fitness Evaluations ≈ 1.1 x 10^6 ± 0.5 x 10^5 * Trend: The line slopes upward, indicating that runtime increases with 'n'. ### Key Observations * All algorithms exhibit a positive correlation between problem size ('n') and runtime (Fitness Evaluations). * The runtime increases approximately linearly on the logarithmic scale, suggesting an exponential increase in runtime with 'n'. * The error bars indicate some variability in the runtime for each algorithm, but the overall trends are clear. * The different NSGA-II configurations (N=2, 4, 8) show very similar performance, with slight variations within the error bar ranges. * GSEMO appears to perform similarly to the NSGA-II configurations. ### Interpretation The chart demonstrates the scalability of different algorithms for solving the LOTZ problem. As the problem size ('n') increases, the computational effort (measured by Fitness Evaluations) grows significantly. The logarithmic scale highlights the exponential nature of this growth. The fact that the NSGA-II configurations with different 'N' values perform similarly suggests that the choice of 'N' has a limited impact on runtime within the tested range. The comparable performance of GSEMO and NSGA-II indicates that both are viable options for solving LOTZ, but their runtime scales similarly with problem size. The error bars suggest that the runtime can vary depending on the specific instance of the LOTZ problem being solved, but the overall trend of increasing runtime with 'n' is consistent across all algorithms. The data suggests that solving larger instances of the LOTZ problem will require significantly more computational resources.

## Line Chart: Runtime for solving LOTZ ### Overview The image is a line chart comparing the computational runtime (measured in fitness evaluations) of four different algorithms or algorithm configurations for solving the "LOTZ" problem as the problem size parameter `n` increases. The chart uses a logarithmic scale for the y-axis. ### Components/Axes * **Title:** "Runtime for solving LOTZ" (centered at the top). * **Y-axis:** Label is "Fitness Evaluations". The scale is logarithmic, with major tick marks at `10^5` and `10^6`. Minor tick marks are present between these values. * **X-axis:** Label is "n". The scale is linear with major tick marks at the values `30`, `60`, `90`, and `120`. * **Legend:** Located in the top-left corner of the plot area. It contains four entries: 1. **GSEMO:** Represented by a dark blue line with plus-sign (`+`) markers. 2. **NSGA-II, N=2(n+1):** Represented by a light green line with plus-sign (`+`) markers. 3. **NSGA-II, N=4(n+1):** Represented by an orange line with plus-sign (`+`) markers. 4. **NSGA-II, N=8(n+1):** Represented by a dark green line with plus-sign (`+`) markers. * **Data Series:** Four lines, each with data points at `n = 30, 60, 90, 120`. Each data point includes vertical error bars. ### Detailed Analysis All four data series show a clear, consistent upward trend on the log-linear plot, indicating that the number of fitness evaluations increases exponentially with the problem size `n`. **Approximate Data Points (Fitness Evaluations):** * **At n = 30:** * GSEMO (Dark Blue): ~2.5 x 10^4 * NSGA-II, N=2(n+1) (Light Green): ~1.8 x 10^4 (lowest) * NSGA-II, N=4(n+1) (Orange): ~2.2 x 10^4 * NSGA-II, N=8(n+1) (Dark Green): ~3.0 x 10^4 (highest) * **At n = 60:** * GSEMO (Dark Blue): ~1.5 x 10^5 * NSGA-II, N=2(n+1) (Light Green): ~1.0 x 10^5 (lowest) * NSGA-II, N=4(n+1) (Orange): ~1.3 x 10^5 * NSGA-II, N=8(n+1) (Dark Green): ~1.8 x 10^5 (highest) * **At n = 90:** * GSEMO (Dark Blue): ~5.0 x 10^5 * NSGA-II, N=2(n+1) (Light Green): ~3.0 x 10^5 (lowest) * NSGA-II, N=4(n+1) (Orange): ~4.0 x 10^5 * NSGA-II, N=8(n+1) (Dark Green): ~6.0 x 10^5 (highest) * **At n = 120:** * GSEMO (Dark Blue): ~1.0 x 10^6 * NSGA-II, N=2(n+1) (Light Green): ~7.0 x 10^5 (lowest) * NSGA-II, N=4(n+1) (Orange): ~1.0 x 10^6 * NSGA-II, N=8(n+1) (Dark Green): ~1.2 x 10^6 (highest) **Error Bars:** All data points have vertical error bars indicating variability in the measurements. The approximate range of the error bars is consistent across points for a given series, spanning roughly ±15-25% of the central value. ### Key Observations 1. **Consistent Hierarchy:** The performance order of the algorithms is consistent across all measured values of `n`. From most efficient (fewest evaluations) to least efficient: `NSGA-II, N=2(n+1)` < `GSEMO` ≈ `NSGA-II, N=4(n+1)` < `NSGA-II, N=8(n+1)`. 2. **Exponential Growth:** The straight lines on the log-linear plot confirm that runtime (fitness evaluations) grows exponentially with `n` for all tested methods. 3. **Impact of Population Size (N):** For the NSGA-II algorithm, increasing the population size multiplier (from 2 to 4 to 8) consistently increases the computational cost. The `N=8(n+1)` configuration is the most expensive. 4. **GSEMO vs. NSGA-II:** The GSEMO algorithm's performance lies between the `N=2(n+1)` and `N=4(n+1)` configurations of NSGA-II. It is less efficient than the smallest-population NSGA-II but more efficient than the larger-population variants. ### Interpretation This chart demonstrates the scalability of different evolutionary algorithms on the LOTZ benchmark problem. The key insight is that while all methods face exponential growth in runtime with problem size, their constant factors differ significantly. The data suggests a trade-off within the NSGA-II framework: using a larger population (`N`) likely improves solution quality or convergence reliability (not shown here) but comes at a direct, proportional cost in fitness evaluations. The most parsimonious configuration (`N=2(n+1)`) is the most runtime-efficient. The GSEMO algorithm presents an interesting middle ground. Its performance being bracketed by two NSGA-II variants suggests its search dynamics have an effective "exploration-exploitation" balance that results in a computational cost similar to an NSGA-II run with a population size between `2(n+1)` and `4(n+1)`. The consistent exponential trend implies that for very large `n`, the absolute difference in runtime between these methods will become enormous, making algorithm selection critical for large-scale problems. The chart does not show which method finds better solutions, only their relative computational cost.

## Line Chart: Runtime for solving LOTZ ### Overview The chart compares the runtime (measured in fitness evaluations) of four optimization algorithms (GSEMO and NSGA-II variants) as a function of problem size `n`. The y-axis uses a logarithmic scale (10⁵ to 10⁶), while the x-axis represents `n` with discrete values at 30, 60, 90, and 120. All algorithms show exponential growth in runtime with increasing `n`, but with distinct performance characteristics. ### Components/Axes - **X-axis (n)**: Discrete values at 30, 60, 90, 120 (problem size). - **Y-axis (Fitness Evaluations)**: Logarithmic scale from 10⁵ to 10⁶. - **Legend**: Located in the top-left corner, with four entries: - **GSEMO** (black line with square markers) - **NSGA-II, N=2(n+1)** (green line with triangle markers) - **NSGA-II, N=4(n+1)** (orange line with diamond markers) - **NSGA-II, N=8(n+1)** (dark green line with cross markers) - **Error Bars**: Present for all data points, indicating variability in measurements. ### Detailed Analysis 1. **GSEMO (Black Line)**: - At `n=30`: ~1.2×10⁵ evaluations. - At `n=60`: ~1.5×10⁵ evaluations. - At `n=90`: ~2.8×10⁵ evaluations. - At `n=120`: ~5.0×10⁵ evaluations. - **Trend**: Steady exponential growth, with error bars showing moderate variability. 2. **NSGA-II, N=2(n+1) (Green Line)**: - At `n=30`: ~1.0×10⁵ evaluations. - At `n=60`: ~1.3×10⁵ evaluations. - At `n=90`: ~2.5×10⁵ evaluations. - At `n=120`: ~4.8×10⁵ evaluations. - **Trend**: Slightly slower growth than GSEMO, with smaller error bars. 3. **NSGA-II, N=4(n+1) (Orange Line)**: - At `n=30`: ~1.1×10⁵ evaluations. - At `n=60`: ~1.6×10⁵ evaluations. - At `n=90`: ~3.2×10⁵ evaluations. - At `n=120`: ~6.0×10⁵ evaluations. - **Trend**: Faster growth than GSEMO, with larger error bars. 4. **NSGA-II, N=8(n+1) (Dark Green Line)**: - At `n=30`: ~1.4×10⁵ evaluations. - At `n=60`: ~2.0×10⁵ evaluations. - At `n=90`: ~4.0×10⁵ evaluations. - At `n=120`: ~8.0×10⁵ evaluations. - **Trend**: Steepest growth, with the largest error bars. ### Key Observations - **Exponential Scaling**: All algorithms exhibit exponential runtime growth with increasing `n`, consistent with the logarithmic y-axis. - **NSGA-II Variants**: Higher `N` values (e.g., 8(n+1)) result in significantly worse performance compared to lower `N` values (e.g., 2(n+1)). - **GSEMO vs. NSGA-II**: GSEMO outperforms NSGA-II with `N=2(n+1)` but underperforms NSGA-II with `N=4(n+1)` and `N=8(n+1)`. - **Error Bars**: Larger error bars for NSGA-II with higher `N` suggest greater variability in runtime measurements. ### Interpretation The data demonstrates that **NSGA-II's performance degrades with larger population sizes (`N`)**, as evidenced by the steepest growth for `N=8(n+1)`. **GSEMO offers a middle ground**, balancing efficiency and scalability compared to NSGA-II variants. The exponential trend highlights the computational complexity of solving LOTZ as `n` increases, with NSGA-II's higher `N` configurations becoming impractical for large `n`. The error bars indicate that runtime measurements are not deterministic, likely due to stochastic elements in the algorithms or problem instances.