# The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses
## 6 Conclusions  
The present paper presents the Mixed-Integer Non-linear Programming (MINLP) approach to truss synthesis. The MINLP approach enables the topology, shape and discrete/standard cross-sectional dimensions to be optimized simultaneously. In the mixed continuous and discrete type of optimization problem, the continuous variables for continuous parameters (nodal coordinates, stresses, displacements) as well as the discrete binary variables for discrete decisions are defined. Binary 0-1 variables define the existence (1) or non-existence (0) of structural elements and are also subjected to the choice of the discrete/standard dimensions of cross-sections. The proposed MINLP truss synthesis is performed through three steps. The first involves the generation of a truss superstructure of different topology and discrete dimension alternatives; the second presents the development of a special MINLP model formulation for truss superstructures (MINLP-TS); and the last contains a solution for the defined MINLP problem. The problems are solved by the Modified OA/ER algorithm. \begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Synthesis**} & \multicolumn{2}{c}{**MINLP (NLP/MILP, NLP/LP) Solver**} \\ \cline{2-4}  & **DICOPT** & **BARON** & **MIPSYN** \\  & **(CONOPT/CPLEX)** & **(MINOS/CPLEX)** & **(CONOPT/CPLEX)** \\ \hline TC10a & 19,266.5406 & 19,266.5406 & 19,266.5406 \\ TC10b & 17,078.4080 & 16,968.0110 & 16,936.2314 \\ TC10c & 32,590.3776 & 32,204.1317 & 32,204.1317 \\ TC10d & 26,516.4615 & 26,573.4506 & 26,357.4575 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Results obtained by using different MINLP solvers. \begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Synthesis**} & \multicolumn{2}{c}{**MINLP (NLP/MILP, NLP/LP) Solver**} \\ \cline{2-4}  & **DICOPT** & **BARON** & **MIPSYN** \\  & **(CONOPT/CPLEX)** & **(MINOS/CPLEX)** & **(CONOPT/CPLEX)** \\ \hline TC10a & 1.01 & 14.00 & 9.657 \\ TC10b & 1.00 & 2513.00 & 13.951 \\ TC10c & 3.67 & 8909.00 & 1.485 \\ TC10d & 16.24 & 6205.00 & 4.324 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: The comparison of the CPU times (in seconds). Besides several non-linearities and non-convexities involved, the main difficulty of truss synthesis is that a very high number of discrete variables, particularly those associated with discrete/standard dimensions, may be included in the optimization. Multi-level MINLP strategies were thus developed and applied to make the solving of such comprehensive optimization problems possible. The main advantage of multi-level strategies is that the sub-set of binary variables, activated at each level, is considerably smaller than the full set. The additional reduction in the number of currently active binary variables for discrete dimensions was attained by the use of a special pre-screening procedure. Two new types of multi-level strategies were presented, namely the Modified Two-Phase (MTP) and the Sequential Two-Phase (STP) MINLP strategies. Both strategies were developed on the basis of the classical Two-Phase (TP) strategy. The motive of developing new strategies was the fact that the TP strategy does not allow the topology to be changed after the discrete sizing optimization and therefore does not consider any interaction between topology and discrete sizing optimization levels. Out of the two new strategies, the MTP strategy proved to be the more convenient option regarding the MINLP iterations required and the CPU times spent. Considering the final/optimal result, however, the basic TP strategy proved also to be a useful tool for obtaining good solutions by spending short CPU times. The proposed MINLP synthesis was applied to the simple and well-known numerical example of the ten-bar truss cantilever. The simultaneous topology and discrete sizing optimization was performed with/without the inclusion of shape optimization and with/without buckling constraints.