# Optimization of Mass Concrete Construction Using a Twofold Parallel Genetic Algorithm
## 3 Optimization Problem of the Construction Phase of a Massive Concrete Structure Using a GA
### Design Variables  
The design variables considered for the optimization of the construction phase of massive concrete structures can be described by the following discrete set : the type of concrete \((Ty_{c})\), the placing temperature \((P_{T})\), the height of the lifts \((H_{l})\), and the placing frequency \((P_{f})\). Thus, the vector of design variables is defined as:  
\[\mathbf{X}=(x_{1},x_{2},x_{3},x_{4})=(Ty_{c},P_{T},H_{l},P_{f}) \tag{4}\]  
The adequate choice of the type of material is very important because it directly influences the evolution of the hydration reaction and the development of the mechanical properties. As an example, the addition of pozzolans and lower cement consumption can greatly reduce the heat generated by the hydration reaction.  
The placing temperature of the concrete is directly related to the maximum temperature reached in the structure and also to thermo-activation, which is responsible for the kinetics of the hydration reaction. Otherwise, the lower the heating rate of the structure, the more the heat will dissipate through the boundaries. Therefore, the thermal gradients will be lower, and the risk of cracking will be lower. In this paper the maximum placing temperature \((P_{T_{max}})\) is limited to the average environmental temperature, and the minimum placing temperature \((P_{T_{min}})\) should be defined by the contractor according to the cooling facility available in the field.  
The height of the lifts influences the dissipation of the heat generated by the hydration reaction. Then it is related to the maximum temperature that the concrete will reach, as well as with the time of construction. For higher values of \(H_{l}\), the lower is the construction cost, due to the reduced construction time. However, the risk of cracking increases due to the reduction in heat dissipation in a more massive structure. For the sake of simplicity, this paper considers that the lifts have the same height, except for the highest lift whose height is the one required to reach the total height of the structure \((H_{l})\). Thus, the total number of lifts \((N_{l})\) is given by:  
\[N_{l}=\left\{\begin{array}{ll}\frac{H_{l}}{H_{l}},&\text{if $\frac{H_{l}}{H_{l}}$ is an exact division}\\ \\ int\left(\frac{H_{l}}{H_{l}}\right)+1,&\text{if $\frac{H_{l}}{H_{l}}$ is not exact} \end{array}\right. \tag{5}\]  
where \(int(x)\) represents the integer part of \(x\).  
The placing frequency is the time interval between the placing of a lift and the subsequent one. This variable is directly related to construction time and hence its cost. On the other hand, increasing theplacing frequency allows for the dissipation of hydration heat and the development of the mechanical properties of the lifts that had already been built.  
Therefore, the construction time \(t_{cons}\) can be calculated by the following expression:  
\[t_{cons}=(N_{l}-1)P_{f}+P_{f_{final}} \tag{6}\]  
where \(P_{f_{final}}\) is the time required for finishing the construction after placing the last lift.