# An Automated Design Method for Plane Trusses Based on User Preference Information
## 4 Cost Function Based on Modified Physical Programming (MPP)
### Coordination of Multiple Objectives with MPP  
The prediction model of user preference is integrated into the STSA method to guide the generation of a satisfactory structure. The STSA method combines structural grammar and performance evaluation with a stochastic optimization technique (simulated annealing,  
Figure 8: A class function of 1-S.  
SA) [1]. It can be used for topology generation and optimization of discrete structures. Structural grammar defines the basic units (e.g., triangles in 2D, tetrahedrons in 3D) and rules for the structural modification in sectional size, global shape, and topology, which can modify and transform the structures more flexibly. Similarly to human languages, the basic units can form a structure based on structural grammar. STSA improves the initial design continuously with structural grammar based on the feedback from performance evaluation until the final design meets the design goals of safety, efficiency, and economy. The workflow is shown in Figure 9[1]. Generally, many design factors, such as weight, displacement, stiffness, stress, frequency, etc., can be employed as the design objectives when STSA is applied to design structures. In this study, weight and displacement are selected as the first and second design objectives. Meanwhile, user preference is introduced as a third objective based on the prediction model. These objective functions are defined as follows:  
\[g_{1}=\min\sum_{i=1}^{n}\rho l_{i}s_{i}, \tag{17}\]  
\[g_{2}=\min\left[\max_{1\leq j\leq m}(disp_{j})\right], \tag{18}\]  
\[g_{3}=\min pref, \tag{19}\]  
where \(\rho\) = material density, \(n\) = number of members, \(l_{i}\) = length of the \(i\)th member, \(s_{i}\) = sectional area of the \(i\)th member, \(m\) = number of nodes, \(disp_{j}\) = displacement of the \(j\)th node, and \(pref\) = preference prediction value of the structure. Structural weight (Equation (17)) is a classical design objective in structural optimization. The smaller the weight, the better the structure. The objective of node displacement (Equation (18)) minimizes the maximum displacement at all the structural nodes. Similarly, the user preference objective (Equation (19)) minimizes the preference prediction value of the generated structures. A smaller value means a higher user satisfaction with the structural appearance. All three objectives conform to the feature of \"smaller is better\", which can be represented by 1-S.  
For the objectives of structural weight and node displacement, the class function of the PP method is redesigned and shown in Figure 10.  
Figure 9: Flow chartof STSA. The number of ranges for each objective is simplified to four (D, T, U, UNA), in which D, T, and U are pertinent to the evaluation options \"Preferred\", \"Acceptable\" and \"Disliked\" in the interactive system, respectively. The curve of the new class function expands to the UNA range, which is originally regarded as the \"infeasible region\" in the PP method. This is beneficial for the full exploration of the optimal structure, especially in the early optimization phase. The new class function contains three expressions. The first one is represented by a continuous quadratic function in the U and UNA ranges, where the objective value is expected to enter the U range from the UNA range quickly at a larger slope. Then the slope at the left boundary of the U range becomes smaller to reduce the impact of the current objective. It is conducive to the uniform optimization of all objectives. The second expression in the T range is set to be a linear function to keep a high slope throughout the range. This makes the objective values move quickly to the D range. The third expression in the D range is represented by an exponential function. The curves in the D and T ranges are continuous and have the same slope at the boundary. The new class function is defined as:  
\[\overline{g}_{i}(x)=\left\{