*3.5. Optimization*

Simulated annealing (SA) is the heuristic algorithm used to carry out the RDO. This algorithm has been used in a lot of research to solve optimization problems [32,33]. In the present paper, the method of Medina [34] is used to calibrate the initial temperature. This method suggests that the starting temperature is halved when the rate of acceptance is above 40% but doubled when it is below 20%. When a Markov chain ends, the temperature then drops in accordance with a cooling coe fficient *k* based on the equation *T* = *k*\**T*. In this study, the calibration showed that the length of the Markov chain of 1000 and a coe fficient of cooling of 0.8 are suitable. The algorithm stops after three Markov chains without finding improvements.

#### **4. Problem Design**

In this section, the robust design optimization problems proposed are discussed. Section 4.1 describes the structure considered and Section 4.2 defines the characteristics of the problem. Section 4.2 includes the initial uncertain parameters considered and the objective functions studied.

#### *4.1. Description of the Box-Girder Footbridge*

The structure is a concrete pedestrian bridge with three continuous spans of 40–50–40 m long. The box-girder cross-section has a uniform width of 3 m, and seven variables define the remaining geometric dimensions of the cross-section (Figure 3): depth (*h*), web inclination width (*d*), bottom slab width (*b*), bottom slab thickness (*ei*), top slab thickness (*es*), external cantilever section thickness (*ev*), and webs slab thickness (*ea*). The variables are restricted to a range as shown in Table 1. The haunch (*c*), is determined from the other variables (Equation (6)) as recommended by Schlaich and Sche ff [35]. Furthermore, the haunch must also allow space to enclose the ducts in the high and low points. This structure was used to compare the standard heuristic optimization and kriging-based heuristic optimization. In this work, the kriging-based heuristic optimization and RDO are applied to the same structure. More detailed information about this structure can be found in Penadés-Plà et al. [24].

$$t = \max\{\frac{b - 2 \cdot ea}{5}, ei\} \tag{6}$$

**Figure 3.** Box-girder cross-section.

**Table 1.** Main parameters of the analysis.


Spanish regulations [36,37] and the Eurocodes [38,39] are used to carry out the structural verification defined by the ultimate and service limit states: bending, vertical shear, longitudinal shear, punching shear, compression and tension stress, torsion, torsion combined with bending and shear, cracking, and vibration. Moreover, compliance with constructability and geometrical criteria must are also be checked.

#### *4.2. Description of the Robust Design Optimization Problem*

The robust design optimization proposed in this paper is defined by two objective functions: the first one is the mean cost, and the second one is the structural stability represented by the vertical displacement in the middle of the bridge [19]. The statistical parameters for both objective functions are obtained varying the initial uncertain parameter (modulus of elasticity, overload, and prestressing force) according to a uniform distribution with three di fferent levels of uncertainty (10%, 20%, and 30% for the modulus of elasticity and 10%, 20%, 30%, and 40% for the overload and prestressing force). These uncertain parameters were chosen after carrying out a sensitivity analysis of the vertical displacement and selecting the critical parameters.

In this way, the di fferences between the di fferent Pareto frontiers obtained for each problem can be studied. Therefore, the goal is to obtain the design with the best cost that has the best structural stability for each RDO problem. Equations (7) and (8) correspond to these objective functions assessed.

$$\mu\_{\rm COST} = \sum\_{i=1,\mu} \varepsilon\_i \times m\_i(\mathbf{x}\_1 \mathbf{x}\_2 \dots \mathbf{x}\_n) \tag{7}$$

$$\sigma\text{-}V\text{ENTICALDIISPLACEMENT}\left(\mathbf{x}\_1,\mathbf{x}\_2,\mathbf{x}\_3,\dots,\mathbf{x}\_n\right)\tag{8}$$

where *x*1, *x*2, *x*3, ... ... , *xn* are the design variables.

The conventional objective function assesses the cost for the construction units taking into account the placement and material used. The BEDEC ITEC database provides unit costs [40]. The compressive strength grade determines the cost of the concrete. The unit costs of the problem are shown in Table 2. The measurements ( *mi)* relating to the construction units are calculated as defined by the design variables. The variation of the vertical displacement in the middle of the bridge has been obtained according to the standard deviation of 20 di fferent cases varying the initial uncertain parameter. Each one of these vertical displacements has been calculated in accordance with Spanish regulations [36,37] along with the Eurocodes [38,39].

**Table 2.** Unit cost.


It is common that a multi-objective optimization problem is transformed into a mono-objective optimization where the objective function is an aggregation function [19] (Equation (9)).

$$\text{Agregation function} = w\_1 \cdot \mu\_{\text{COST}} + w\_2 \cdot \sigma\_{\text{VERITACALDSPLACEMENT}} \tag{9}$$

Here, the mean and the standard deviation are the normalized values of the objective functions, and *w*1 and *w*2 are weights with values in the range [0,1] such that *w*1 + *w*2 = 1.

In this work, 200 di fferent cases ( *N*) are considered in such a way that *w*1 runs from 0 to 1 with increasing 1/*N* and *w*2 corresponds to 1- *w*1. In this way, 200 di fferent optimizations are made and all the possible solutions of the Pareto frontier are covered.
