PSI - Issue 40

A.G. Chentsov et al. / Procedia Structural Integrity 40 (2022) 105–111 Chentsov A.G., Chentsov P.A./ Structural Integrity Procedia 00 (2022) 000 – 000 of course, in [10], we obtain the nonempty subset of ℙ : P ≠ ∅ and P ⊂ ℙ . In [10], the basic singularity of our setting is realized: at first, megalopolises of ℳ 1 are serviced and later megalopolises of ℳ 2 are serviced too. We consider P as the set of all admissible routes. Of course, our routes of P are permutations of 1̅̅̅, ̅ ̅ . With every route of P , trajectories compatible with this route are connected. For corresponding definition some auxiliary notions are required. So, under j ∈ ̅1̅̅, ̅ ̅ ( ≜ {pr 1 (z) : z ∈ }) & ( M j ≜ {pr 2 (z) : z ∈ }) (11) are two nonempty subsets of M j . In this terms, we introduce ( ≜ ( ⋃ =1 ) ∪ 0 ) & ( ≜ (⋃ =1 ) ∪ 0 ) ; (12) in (11) and (12), all sets are finite. We consider × X as phase space in which our trajectories are realized. Now, we introduce the set ℤ of all mappings from ̅0̅̅, ̅ ̅ into × X . Under x ∈ X 0 and α ∈ P , we have the nonempty finite set α [x] ≜ { ( ) ∈ 0̅̅̅, ̅ ̅ ∈ ℤ | (z 0 = (x, x)) & (z t ∈ M α (t) ∀ t ∈ 1̅̅,̅̅ ̅ )} (13) of all trajectories corresponding to starting point x and coordinated with α ; in correspondence with [5, 7], we consider trajectories as motions of ordered pairs. Then, under x ∈ X 0 ̃ [x] ≜ {( α , ( ) ∈ 0̅̅̅, ̅ ̅ ) ∈ P × ℤ | ( ) ∈ 0̅̅̅, ̅ ̅ ∈ α [x]} is the set of all admissible solutions for the problem with fixed starting point x . Finally, we introduce nonempty finite set D ≜ {( α , ( ) ∈ 0̅̅̅, ̅ ̅ , x) ∈ P × ℤ × X 0 | ( α , ( ) ∈ 0̅̅̅, ̅ ̅ ) ∈ ̃ [x]} of all (admissible) routing processes of complete routing problem. We consider routing processes as triplets with elements in the form of routes, trajectories, and starting points. These processes should be admissible in the sense of precedence conditions defined by K 1 and K 2 and realize service of ℳ 1 earlier that service of ℳ 2 . Let is the family of all nonempty subsets of 1̅̅,̅̅ ̅ и ̅ is the union of all sets M N + j , j ∈ 1̅̅,̅ ̅̅̅−̅̅̅N̅ . For every nonempty set H, by ℛ + [ H ] we denote the set of all nonnegative real-valued functions defined on H. Then, we fix c ∈ ℛ + [ X × × ], c 1 ∈ ℛ + [ 1 × ], . . . , c n ∈ ℛ + [ × ],, f ∈ ℛ + [ ̅ ]. We use c for estimation exterior movements; c 1 , . . . , c N are used for estimation of works connected with the megalopolises visiting. The function f is used for estimation of terminal state. We suppose that under x ∈ X 0 , α ∈ P ( ) ∈ 0̅̅,̅ ̅ ∈ α [ x ] ℭ [ ( ) ∈ 0̅̅̅, ̅ ̅ ] ≜ ∑ [ ( 2 =1 ( −1 ), 1 ( ), 1 ( ̅̅,̅̅ ̅ )) + ( ) ( , 1 ( ̅̅,̅̅ ̅ ))] + f ( 2 ( )), (14) where α 1 (·) is the imaging operation. By (14) additive criterion is defined. Now, we introduce a series of extremal problems. So, for x ∈ X 0 , we consider the problem ℭ α [ ( ) ∈ 0̅̅̅, ̅ ̅ ] → min , ( α , ( ) ∈ 0̅̅̅, ̅ ̅ ) ∈ ̃ [x] ; (15) for this problem, extremum is defined as 4

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