Algorithm of composing the schedule of construction and installation works

An algorithm for scheduling works is developed, in which the priority of the work corresponds to the total weight of the subordinate works, the vertices of the graph, and it is proved that for graphs of the tree type the algorithm is optimal. An algorithm is synthesized to reduce the search for solutions when drawing up schedules of construction and installation works, allocating a subset with the optimal solution of the problem of the minimum power, which is determined by the structure of its initial data and numerical values. An algorithm for scheduling construction and installation work is developed, taking into account the schedule for the movement of brigades, which is characterized by the possibility to efficiently calculate the values of minimizing the time of work performance by the parameters of organizational and technological reliability through the use of the branch and boundary method. The program of the computational algorithm was compiled in the MatLAB-2008 program. For the initial data of the matrix, random numbers were taken, uniformly distributed in the range from 1 to 100. It takes 0.5; 2.5; 7.5; 27 minutes to solve the problem. Thus, the proposed method for estimating the lower boundary of the solution is sufficiently accurate and allows efficient solution of the minimax task of scheduling construction and installation works.


Introduction
The purpose of this study was to consider the task of constructing an optimal maintenance schedule for devices in a given, specific sequence. The processes of servicing requirements by each of the devices are considered to be indivisible, i.e., having started the process of servicing the i-th requirement by the k-th device, it is necessary to bring it to the end without interrupting. Let the service times for each requirement be set for each device and the final time i T be set for completion of the process for servicing the i-th requirement by the last device. It is necessary to build a schedule that ensures the end of the process of servicing all requirements in the shortest possible time. We will focus only on the consideration of cases when the times of transportation of requirements and readjustments of devices can be neglected. The proposed problem formulation is quite general and is one of the core problems in the production organization. The models of the theory of schedules also deal with the tasks of organizing the computing process and many others.
The problems of scheduling theory are given considerable attention to in monographs and periodicals.In the literature, the network coverage of a set of admissible schedules and the construction of approximate methods for solving a problem using generators of acceptable schedules and certain preference rules received the widest coverage. These approaches allow us in some cases to find good approximations to the solution of problems using static modeling methods. However, the restrictions on the deadlines for completing the process of servicing the i-th requirement make it difficult for algorithms to formulate acceptable schedules and, and using known methods increases the time of solving the problem. Some other approaches are based on constructing linear integer models of the problem and applying integer linear programming methods. The proposed models were reduced to problems of linear programming with Boolean variables of very large dimension, which did not allow the practical problems of scheduling theory to be solved when using these methods. The attempts to build algorithms for obtaining exact solutions to the problem on the basis of branch and boundary methods were unsuccessful, either.

Problem definition
Resource allocation in the problems of scheduling is a fairly well-developed branch of science, with an overall focus on meeting the requirements of organizational and technological reliability in design, construction and installation by the criterion of minimizing the time of work execution and waste of resources in each sector. Unfortunately, in the performance of works there may be deviations from the standards which would require alteration. Also, the problems of stochastic uncertainty arise because of the very nature of the construction industry. Therefore, reducing construction time due to new approaches to drawing up calendar schedules and spending resources is the most pressing task.

Algorithm for solving the problem
To compile a schedule of construction and installation work, it is necessary to consider the minimax algorithm. Let there be one brigade and n number of works that are performed by it. The duration of the j-th work is determined by the expression:  is the time for the brigade to move from performing the i-th work to the j-th work, j  is the time of performing the j-th work.
It is required to determine the sequence of work execution, in which the maximum time for execution of each individual work is minimal.
The problem considered here reduces to the well-known problem of the travelling salesman in its minimax formulation.
Mathematically, the problem can be represented in the following form. It is required to determine: max min (1) under the following restrictions: ( 1) , , 0, , , where: Practical problems that reduce to (1) -(4) can have a very different physical meaning. For example, to (1) -(4) there reduces the problem of minimizing the system resource consumption when monitoring parameters [6]. For the solution of problem (1) -(4), the branch and boundary method can be used. In this case, it is necessary to determine the way to estimate the lower bound and the conditions for constructing a tree of possible variants.
We denote: U is a set of variables We assume that inequality (9)  If inequality (10) is satisfied for l = n, we get a new record solution Sn T , which is used later to verify inequality (9), (10). The computational process terminates if condition (9) is not satisfied when l=n-1,n-2,...,0. In this case, the last record solution 0 t and the corresponding set of variables n S is optimal.
Let's consider the presented way of solving the problem (1) -(4) on a concrete example, with its initial data being shown in    numbers uniformly distributed in the range from 1 to 100 were taken. The time of solving the problem for n=20, 30, 40, 50, is 0.5; 2.5; 7.5; 27 minutes respectively. Thus, the proposed method for estimating the lower boundary for the solution is sufficiently accurate and allows efficient solution of the minimax task of scheduling construction and installation works. Example 1. Let the number of works is 4. The initial data are given in Table 1 Then for the duration of the work we have: Suppose that the duration of consecutive works is 15, therefore, it is required to reduce the work package by 7 units. We select 7 times the smallest numbers Ki, and the following cases are possible: -Variant 1.  Then, Figure 3 shows the network graph obtained by the aggregated transformation, and the aggregation tree itself is shown in Figure 4:  Step 1.We generate the resulting table for aggregated work 5. We get: Step 2.The same is done for aggregated work 6. Weget: 6  4 3 2 S6 8 140 14 Step 3.The same thing goes for aggregated work 7. Weget: Step 4.We finish for aggregated work 8. We get: We define a minimax version of the work duration. This option is the only one: Т = 9, S = 18.
Thus, an algorithm for scheduling construction and installation work was obtained, taking into account the schedule for the movement of brigades, which is characterized by the ability to efficiently calculate the values of minimizing the work performance time in the parameters of organizational and technological reliability through the use of the branch and boundary method

Conclusions
On the basis of the studies carried out, models have been created that allow, subject to contractual terms, to select options for performing works that ensure the minimization of additional funds directed to shorten the period of work performance.