## 2 Materials and Methods
### The Construction Project Model
#### 2.3.2 The Optimization Model  
The construction project presented above has the following optimization model: Parameters:  
* The task of the contractor in the project is to construct a set of construction units: \[Z=\{Z_{1},Z_{2},Z_{3},...,Z_{i},...,Zn\}.\] (4)
* The set of working groups performing one type of work is as follows: \[B=\{B_{1},\,B_{2},\,B_{3},\,...,\,B_{j},\,...,\,B_{m}\}.\] (5)
* Each unit \(Z_{i}\in Z\) requires implementation of \(m\) activities which form the set: \[\text{O}_{i}=\{\text{O}_{i1},\,\text{O}_{i2},\,\text{O}_{i3},\,...,\,\text{O }_{ij},\,...,\,\text{O}_{im}\}.\] (6)
* \(D_{j}\)--set of possible execution modes (methods) for performing activity \(j\) by a working group \(B_{j}\),
- \(D_{jk}\)--\(k\) method of performing activity \(j\) by a working group \(B_{j}\), a \(k=1\,...\,p\). * It is assumed that the activity \(O_{ij}\in O_{i}\) can be done by the working group \(B_{j}\). The durations of activities from the set \(O_{i}\) are defined by a vector: \[t_{i}=[t_{i1},\,t_{i2},t_{i3},\,...,\,t_{ij},\,...,\,t_{im}],\,t_{ij}=[t_{ij1 },\,\ldots,\,t_{ijk},\,\ldots,\,t_{ip}].\] (8) The duration \(t_{ijk}\) is subject to the method selection from the set of \(D_{j}\) modes of activity \(O_{ij}\). * Similarly, as the above, it is assumed that the activity \(O_{ij}\in O_{i}\) can be implemented by the working group \(B_{j}\). The cost of realization of the activity \(O_{ij}\) by the working group \(\[t_{u\_ik}=n+r \tag{10}\]  
where \(n\) is the largest number such that:  
\[t_{ij}\geq\sum_{z=0}^{n}y(x,R_{x},z)_{i,j} \tag{11}\]  
and \(r\) is the time during which the remaining (less than 1 day) part of the work should be performed:  
\[r=\frac{t_{ij}-\sum_{0}^{n}y(x,R_{x},z)_{i,j}}{y(x,R_{x},z)_{i,j}}, \tag{12}\]  
- \(y(x,R_{x},z)_{i,j}\)--the efficiency (performance) of the \(j\)th working group in carrying out the \(i\)th job on the \(z\)th day in turn. Constraints:  
* The order of execution of the activities resulting from work technology is assumed such that:  
\[O_{i,j-1}<O_{i,j}<O_{i,j+1}. \tag{13}\]
* It is assumed that only one job can be performed at a time. * It is assumed that the activity cannot be stopped during its performance (preemption is not allowed). Decision variables:  
* the order \(\pi\) of execution of units (permutation):  
\[\pi=(\pi(1),\,\pi(2),\,\pi(3),\,...,\,\pi(j),\,...,\,\pi(n)), \tag{14}\]
* set of numbers of methods of activity execution (from \(k=1\) to \(k=p\)):  
\[R=(R_{1},\,R_{2},\,R_{3},\,...,\,R_{i},\,...,\,R_{n}), \tag{15}\]  
where \(R_{i}=(R_{i1},\,R_{i2},\,R_{i3},\,...,\,R_{ij},\,...,\,R_{im})\), and \(R_{ij}\) -the number of methods of performing (from \(k=1\) to \(k=p\)) of the work \(j\) in unit \(i\). The value of the number \(k\) of the way of realization \(R_{ij}\) enables allocation of the method \(D_{jk}\) from the set \(D_{j}\) to the work \(j\) in the unit \(i\). The form of decision variable \(R\) uniquely identifies the allocation of methods to realization of activities in the units. Therefore, using the decision variable \(R\) there are uniquely established not only the durations of individual works carried out in the units but also their cost. After the adoption of the decision variable \(R\) the durations of works \(t_{u\_i}\) and costs \(c_{i}\) of works from the set \(O_{i}\) are as follows:  
\[B=\{B_{1},\,B_{2},\,B_{3},\,...,\,B_{j},\,...,\,B_{m}\}.\]0  
\[B=\{B_{1},\,B_{2},\,B_{3},\,...,\,B_{j},\,...,\,B_{m}\}.\]1  
The deadlines for the activities can be determined from the formula:  
\[B=\{B_{1},\,B_{2},\,B_{3},\,...,\,B_{j},\,...,\,B_{m}\}.\]2  
- \(\pi(0)=0,\)  
- \(F_{0,j}=0,\)  
- \(F_{i,0}=0.\)  
The duration of the entire project \(F_{\pi,m}\) (duration of all works performed in the units) for \(\pi^{\star}\) and for the decision variable \(R^{\star}\) is  
\[B=\{B_{1},\,B_{2},\,B_{3},\,...,\,B_{j},\,...,\,B_{m}\}.\]3The deadlines for the performance of individual works and their cost can be found in time \(O(nm)\). The number of possible solutions to the presented model is \(n!\cdot p^{nm}\).