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

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̃ [x] ≜ ℭ [( ) ∈ 0̅̅̅, ̅ ̅ ] ∈ ℛ + ( α , ( ) ∈ ̅0̅̅,̅ ̅ ) ∈ ̃ [x] and, moreover, it is defined the nonempty finite set (sol)[x] ≜ {( 0 , ( 0 ) ∈ 0̅̅̅, ̅ ̅ ) ∈ ̃ [x] | ℭ 0 [ ( 0 ) ∈ 0̅̅̅, ̅ ̅ ] = ̃ [x] } of all optimal solutions (for problem (15)). The complete routing problem is defined as ℭ α [ ( ) ∈ 0̅̅̅, ̅ ̅ ] → min , ( α , ( ) ∈ 1̅̅̅,̅ ̅ , x) ∈ D ;

(16)

(17)

(18)

for this problem the corresponding extremum is defined as ≜ min ℭ α [ ( ) ∈ 0̅̅̅, ̅ ̅ ] = min ̃ [x] ∈ R + ( α , ( ) ∈ 0̅̅̅, ̅ ̅ ,x) ∈ D ∈ 0 and, moreover, the nonempty finite set SOL ≜ { ( 0 , ( 0 ) ∈ 0̅̅̅, ̅ ̅ , 0 ) ∈ D | ℭ 0 [ ( 0 ) ∈ 0̅̅̅, ̅ ̅ ] = } of all optimal solutions for problem (18) is defined. Finally, we consider ̃ [x ] → min , x ∈ X 0 , as the problem of the starting point optimization; extremum of (21) coincides with and 0 ≜ { 0 ∈ X 0 | ̃ [ 0 ] = } (22) is the (nonempty finite) set of optimal starting points. If x 0 ∈ o0 pt and ( 0 , ( t 0 ) ∈ ̅0̅,̅ ̅ ) ∈ ሺ•‘Žሻሾš Ͳ ሿ then by (19), (22) ( α 0 , ( 0 ) ∈ 0̅̅̅, ̅ ̅ , x 0 ) ∈ SOL . (23) Our basic goal consists in determination and some solution (23) of the set (20). The problem (21) permits to use For solution of the problem (18), we use two-stage dynamic programming procedure. For separate consideration, we select ℳ 1 -problem (the service problem for megalopolises of ℳ 1 ) and ℳ 2 -problem (the analogous problem for ℳ 2 ). In the first problem, we introduce ̃ 1 ≜ { pr 1 ( z ) : z ∈ K 1 } and X 00 ≜ ⋃ (24) ∈ 1, ̅̅̅̅̅̅ \ ̃ 1 (19) (20) (21) series x-problems (15) for this goal. 2. Algorithm at the functional level