Solving the problem of service requests based on the algorithm for minimizing the maximum time offset

The article proposes an approximate algorithm for forming the optimal sequences of service requests for repair of ISS elements with a minimum absolute error of the target function. The algorithm includes two steps. In the first step, new policy deadlines are set for the completion of application servicing. In the second step, a well-known algorithm for solving the problem with new Directive deadlines is used to determine the optimal sequence from which the desired solution to the original problem will differ by no more than an absolute error.


Introduction
Solving the problem of optimizing the service of requests for repair of ISS elements, it is of scientific and practical interest to consider the problem of schedule theorythe problem of minimizing the maximum time offset.
The problem consists in finding the optimal maintenance schedule requirements on one device with the smallest value of the maximum offset requirement equal to the difference between the date of completion of service requirements and policy completion of its service.
Work on solving this problem began in the 50s of the last century and is still underway [1,2]. In [3], it is proved that the problem of minimizing the maximum time offset is-difficult.
To solve this class of problems, several approaches have been developed based on the use of reduced iteration methods (the method of branches and borders) [4,5,6], the method of dynamic programming [7,8,9], and the creation of heuristic, metaheuristic, and hybrid algorithms [10,11].
In the group of heuristic algorithms, approximations are singled out separately, for which error estimates of the resulting solution are found [12]. Let c( π) j -the moment when the request service was completed jN  in the sequence π ; L( π)=c (π)-d j j j , jN  -temporary offset of the request j in the sequence. Maximum time offset of the request service j in the sequence π :

Problem statement
Find the optimal sequence of service requests for repairs *  with the lowest value of the maximum time offset , Π(N) -a set of maintenance sequences for a set of repair requests N . This task is NP -a difficult problem, for which an approximate algorithm is proposed, the idea of which is in changing the directive deadlines for servicing repair requests and obtaining such a sequence of service requests that provides the minimum value of the absolute error of the optimal value of the target function (1).
The approximate algorithm includes two steps: Step Determine the desired sequence of service requests for repairs.

Description of the algorithm for changing policy deadlines
Let * π , ' π Π(N)  -optimal sequence to service requests for the repair of the many N when decision- For tasks that minimize the maximum time offset with directive deadlines [13]: . Hence, the value of the target function (1) of the optimal sequence of service requests with directive deadlines ' d j , jN  differs from the value of the target function (1) of the optimal sequence of service requests with directive deadlines d j , jN  no more than on ρ . Therefore, if you can choose a directive deadline for completing the service of applications ' d j so that the problem is solved by an effective algorithm, the resulting sequence of request servicing will be an approximate solution of the original problem with an estimate of the absolute error of the optimal value of the target function (1) not exceeding ρ .
The problem of minimizing the absolute error ρ it can be represented as a mathematical programming problem (5), (6) provided that the repair requests of the set N numbered by the nondecreasing moments of the start of service requests Algorithm for solving the problem (5), (6).

Description of the second step of the approximate algorithm for solving the request service problem
Sequence of maintenance requests for repairs π Π(N,t)  not earlier than the moment of time t we will consider it acceptable if F(π) γ  , where γ -real number. At the second step of the approximate request service algorithm the amount allowed for repairs is set relative to the specified value γ sequence of requests h π Π(N,t)  , or it is established that such a sequence does not exist.
Step 1. To determine Step 2. To construct is valid with respect to π =π k h .

Conclusion
The paper considers an approximate algorithm for forming an optimal sequence of maintenance requests for repairs, based on changes in the directive terms of the original task and minimizing the absolute error of the optimal value of the target function.
The complexity of the presented algorithm is 2 max O(n P+nρ P) operations. The solution of a large number of examples (~1000) showed that in 20% of the examples, optimal sequences of maintenance requests for repairs were obtained, in other cases, the average value of the absolute error differed from the theoretically known value by no more than 15%.