\end{enumerate}
-B. Meindl and M. Templ~\cite{ref212} studied the efficiency of above optimization solvers. They used a set of instances of difficult optimization problems called the attacker problems in order to achieve a performance comparison of GLPK, lp$\_$solve, CLP, GUROBI, and CPLEX optimization solvers. They considered a total of 200 problem instances for this study, 100 of these problem instances are based on problems with two dimensions, and 100 problem instances are three-dimensional.
+B. Meindl and M. Templ~\cite{ref212} studied the efficiency of above optimization solvers. They used a set of instances of a difficult optimization problem called the Attacker problems (formulated as a LP) \cite{ref240} in order to achieve a performance comparison of GLPK, lp$\_$solve, CLP, GUROBI, and CPLEX optimization solvers. %They considered a total of 200 problem instances for this study, 100 of these problem instances are based on problems with two dimensions, and 100 problem instances are three-dimensional.
-In tables~\ref{my-label1}, \ref{my-label2}, and \ref{my-label3} we report the result of their comparisons the running times of the five linear program solvers to find solutions to the 200 two-dimensional, 200 three-dimensional, and all 400 problem instances. In order to solve the attacker’s problem for a given problem instance, it is needed to both minimize and maximize any given problem. Therefore, a total of 400 problem instances had been solved when only 200 problem instances have been generated. The running time of the fastest solver has been scaled to one and the running times of the other linear solvers were scaled to reflect this scaling.
+In tables~\ref{my-label1} and \ref{my-label2}, we report the result of their comparisons the running times of the five linear program solvers to find solutions for instances related to the two-dimensional problem, and for instances related to the three-dimensional problem (with more variables and constraints).
+%to the 200 two-dimensional, 200 three-dimensional, and all 400 problem instances.
+%In order to solve the attacker’s problem for a given problem instance, it is needed to both minimize and maximize any given problem. Therefore, a total of 400 problem instances had been solved when only 200 problem instances have been generated.
+The running time of the fastest solver has been scaled to one and the running times of the other linear solvers were scaled to reflect this scaling.
\begin{table}[h]
\end{table}
-\begin{table}[h]
-\caption{Total (in seconds) and scaled running times for all problems (results of B. Meindl and M. Templ~\cite{ref212})}
-\label{my-label3}
-\resizebox{\textwidth}{!}{%
-\begin{tabular}{|c|c|c|c|c|c|}
-\hline
-\textbf{Optimization Solvers} & \textbf{GLPK} & \textbf{lp\_solve} & \textbf{CLP} & \textbf{Gurobi} & \textbf{CPLEX} \\ \hline
-\textbf{Total Running Time} & 48585 & 982737 & 667066 & 38708 & 257 \\ \hline
-\textbf{Scaled Running Time} & 189 & 3822 & 2594 & 151 & 1 \\ \hline
-\end{tabular}
-}
-\end{table}
+%\begin{table}[h]
+%\caption{Total (in seconds) and scaled running times for all problems (results of B. Meindl and M. Templ~\cite{ref212})}
+%\label{my-label3}
+%\resizebox{\textwidth}{!}{%
+%\begin{tabular}{|c|c|c|c|c|c|}
+%\hline
+%\textbf{Optimization Solvers} & \textbf{GLPK} & \textbf{lp\_solve} & \textbf{CLP} & \textbf{Gurobi} & \textbf{CPLEX} \\ \hline
+%\textbf{Total Running Time} & 48585 & 982737 & 667066 & 38708 & 257 \\ \hline
+%\textbf{Scaled Running Time} & 189 & 3822 & 2594 & 151 & 1 \\ \hline
+%\end{tabular}
+%}
+%\end{table}
-The results in tables~\ref{my-label1}, \ref{my-label2}, and \ref{my-label3} indicate that open source solvers perform worse than standard commercial solvers when applied to instances of the attacker’s problem. The GLPK outperforms the other free and open source solvers, but is still slower than CPLEX and GUROBI. We have decided to use the GLPK as an optimization solver in this dissertation to solve the proposed integer programs during the decision phase of the nodes. We motivate the use of the GLPK optimization solver for many reasons, including:
+The results in tables~\ref{my-label1} and \ref{my-label2} indicate that open source solvers perform worse than standard commercial solvers when applied to instances of the attacker’s problem. The GLPK outperforms the other free and open source solvers, but is still slower than CPLEX and GUROBI. We have decided to use the GLPK as an optimization solver in this dissertation to solve the proposed integer programs during the decision phase of the nodes. We motivate the use of the GLPK optimization solver for many reasons, including:
\begin{enumerate} [(i)]
