## 4 Illustrative Example
### Elapsed Runtime and Iterations  
The efficiency of solving a problem is also an important criterion for algorithm comparisons; a good algorithm should solve problems within an acceptable time. This section compares the number of iterations of implementations of the CCM and LINGO to solve the WPM of multi-criteria ABC classification. The main difference between implementing the CCM algorithm and the LINGO solver was that we did not need to specify a starting point or moving step size. When the solution region is a polyhedron, determining the first basic solution (the starting point) would be vital, as the local optimal solution is usually located near the basic feasible solution (BFS); the quality of the solution is highly related to the location of the BFS. An unsuitable starting point could lead to a worse local optimal solution. The search region of an algorithm is related to the step size of the line search, and the search region decides whether a feasible solution can be found. The step size also determines the quality of the final solution. A large step size within a line search could jump over the optimal solution, whereas smaller search distances could trap in and require a significant amount of time to reach the local optimal solution. A good starting point and step size for a search can help researchers find acceptable solutions in less time. Table 4 illustrates the process of tuning the step size in order to reach a feasible solution. The CCM algorithm provides more flexibility than other solvers from commercial package software, which presents a higher probability of finding better solutions. In this study, using CCM to solve the problem required more time to achieve the local optimal solution than using LINGO because the step size of the line search determined whether the CCM could find feasible solutions. In this study, LINGO took mere seconds to find a feasible solution, whereas CCM took more time--sometimes nearly a minute--to find a solution.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{Item 4} & \multirow{2}{*}{Step size} & \multicolumn{2}{c}{\(\sum\limits_{j=1}^{l}w_{j}^{2}\)} & \multirow{2}{*}{Item 5} & \multirow{2}{*}{Step size} & \multicolumn{2}{c}{\(\sum\limits_{j=1}^{l}w_{j}^{2}\)} \\ \cline{2-2} \cline{5-7}  & 0.0003 & & & 1.000009 & 0.00045 & 1.000062 \\ \cline{2-2} \cline{5-7}  & 0.000295 & & 0.999993 & 0.000448 & 1.000059 \\ \cline{2-2} \cline{5-7}  & 0.000298 & & 1.000003 & 0.00044 & 1.000046 \\ \cline{2-2} \cline{5-7}  & 0.000297 & & 1 & 0.00041 & 0.999997 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Tuning the step size to reach a feasible solution.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{Item 4} & \multirow{2}{*}{Step size} & \multicolumn{2}{c}{\(\sum\limits_{j=1}^{l}w_{j}^{2}\)} & \multirow{2}{*}{Step size} & \multirow{2}{*}{\(\sum\limits_{j=1}^{l}w_{j}^{2}\)} \\ \cline{2-2} \cline{5-7}  & 0.0003 & & & & 0.00009 & 0.00045 & 1.0000062 \\ \cline{2-2} \cline{5-7}  & 0.000295 & & 0.999993 & 0.000448 & 1.000059 \\ \cline{2-2} \cline{5-7}  & 0.000298 & & 1.000003 & 0.00044 & 1.000046 \\ \cline{2-2} \cline{5-7}  & 0.000297 & & 1 & 0.00041 & 0.999997 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The 10 items reclassified using the WPM.  
When looking at the number of iterations each algorithm needs, CCM requires more iterations to solve the problem, even with a larger step size. As presented in Table 5, LINGO can solve most problems in only 60 iterations, whereas CCM might need approximately one thousand iterations. In conclusion, LINGO is more efficient than CCM in this study, which contradicts the results of the previous study, which stated that CCM can solve nonlinear problems more quickly than other software packages. The result of the current study might stem from the fact that the problem in this study is too simple to demonstrate the power of CCM.