In order to avoid infeasibility when breaking and reorga nizing the groups one

In order to avoid infeasibility when breaking and

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In order to avoid infeasibility when breaking and reorga- nizing the groups, one can solve (3) after each modification. For efficiency, we have found that it was sufficient to apply both operators until either there was no change in the groups or the groups resulted in an infeasible schedule. input: A subset of groups G t ∈ G output: A fragmented set H t with same passengers 1 Set H t G t 2 foreach g G t do 3 if | H t | < |C| & MGWT ( g, g (1) ) > MPWT ( g (1) ) then 4 for i 2 , . . . , | g | do 5 if MPWT ( g ( i ) ) > MPWT ( g (1) ) then 6 Set g 0 ← { g (1) , . . . , g ( i - 1) } 7 Set g 00 ← { g ( i ) , . . . , g ( | g | ) } 8 Set H t ( H t \ { g } ) ∪ { g 0 , g 00 } 9 break 10 end 11 end 12 if | H t | = |C| then 13 break 14 end 15 end 16 end Algorithm 3: Breaks groups with co-passenger delays Experiments We present numerical experiments in the setting of Figure 1. The passengers originate from a set of 4 stations and desire to reach one of the buildings in B1-B10 within a specified time-window. U consists of MT services that reach T0 ev- ery 15 minutes and the travel time between stations is 5 min- utes. We assume that the CVs are all parked at T0 and return back to T0 after dropping off all passengers. The CVs take 1 minute to go between T0 -B1, T0 -B10, and all pairs of build-
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input: A subset of groups G b ∈ G output: A reorganized set H b with same passengers 1 Set H b ← ∅ , h g b 1 2 for i 2 , . . . , | G b | do 3 if | h | = CV max then 4 Set H b H b ∪ { h } , h g b i 5 end 6 if | h | < CV max then 7 for j 1 , . . . , min { CV max - | h | , | g b i |} do 8 Set p g b i ( j ) 9 if rel ( p ) ded ( h (1) ) & MGWT ( h ∪ { p } , h (1) ) = MGWT ( h, h (1) ) then 10 Set h h ∪ { p } 11 end 12 end 13 h 0 g b i \ h 14 if | h 0 | > 0 then 15 Set H b H b ∪ { h } , h h 0 16 end 17 end 18 end 19 H b H b ∪ { h } Algorithm 4: Reorganizes passenger groups ings that are adjacent on the shaded track, except for B5-B6 which takes 2 minutes. The CVs are restricted to move along the shaded region shown in Figure 1. We also assume that a CV spends 0.5 minutes at a building where they drop passen- gers. As a result, the time that a passenger reaches the des- tination depends on co-passengers in the CV that have prior destinations. The modeling of drop-off time is important in applications where the capacity of CV max is small, typi- cally 5 . Inspired by the application to corporate-campus settings, we use a small number of destinations. This may not be the case in all applications, where one might expect the number of destinations for the LMT service to be in the same order as the number of passengers, thus making the problem harder to solve. In those cases, however, we believe that the suggestion by Mah´eo, Kilby, and Hentenryck (2018) to treat last-mile stops as aggregations of several passenger destinations, such as bus stops, is a reasonable compromise. Hence, the chosen number of destinations is of minor impor- tance.
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