Des, which includes only the ports in the islands; N0 : Set of nodes, like ports as well as the depot i0 ; h : Set of nodes at each and every island h; h : Set of all probable combinations of ports for island h; Khs : Set of nodes belonging to island h that happen to be selected as collection web-sites under combination s h ; h : Variety of nodes at every single island h; MCij : MTC from node i to node j; Qh : Freight volume to be collected from each and every island h; GChs : Total GTC at island h, if the port mixture s is selected. This price parameter represents the total GTC incurred by the island inhabitants if port mixture s is selected; Zi: Binary variable indicating if node i is chosen to become visited; Yij : Binary variable indicating if node j is visited straight away soon after node i; Xhs : Binary variable indicating if island h is visited using mixture s; Fij : Fictitious flows from node i to node j. Accordingly, the Approximated Model is formulated as (1)13). Expressions (1) and (two) correspond to the MTC and GTC objective functions, respectively. Constraints (three) make sure the precise collection of a single port situation s for every single island h. Constraints (4) and (five) are logical relationships among the decision variables Z and X, ensuring that if a mixture of port s is chosen for each island h (Xhs = 1), only the selection variables for the associated ports are activated (Zi =1, for every single i Khs). Constraints (six) and (7) make sure that, for each and every chosen port (i.e., Zi = 1), the barge have to enter and exit precisely after, respectively. Constraints (8)11) would be the sub-tour elimination constraints ([74]). Lastly, Constraints (12) and (13) are of domain. Mini,j N0 ,j =iMCij Yij(1)Minh H shGChs Xhs(two)s Ehsubject to : h H Xhs =(three) (four) (five) (6) (7) (eight)Xhs Zi Xhs 1 – Zih H, s Eh , i h /i Khs h H, s Eh , i h /i Khs /j N:i = jYij = Zi Yji = Zii N i Nj N:j =ij NFi0 j = Zjj NjN : j iY ji = Z ii N(7) (8)Fj Ni0 j= Zjj NMathematics 2021, 9,jN 0 / i jFij =jN 0 / i jF ji Z ijii N(9) 7 of 33 (10)(9) (11)Fj N=Fij N YijFijFij =j N0 /i = jj N0 /i = ji i i Fji Z, j iN 0 ,N ji , j N 0 , i jZ , Yij , X hs 0,1j NFji0 =i , j N , h H , s Eh(12) (ten) (13)Fij | N | Yij i, j N0 , i = j (11) 2.three. An Precise Formulation for the BO-InTSP F 0 i, j N0 , i = j (12) The proposed novel precise ij mathematical formulation for the BO-InTSP considers a Z, Y relying on the i, j N, with the s Eh (13) disaggregated scheme, ij , Xhs 0, 1 information h H,realdemand places (e.g., households, restaurants, or Bestatin web hospitals) and their respective demand values. In this model, 2.3. Aninhabitants must travel to their nearest selected port for transporting their freight. island Precise Formulation for the BO-InTSP Therefore, when capacity constraints mathematical formulation for the BO-InTSP considers The proposed novel precise at the ports are certainly not relevant, the model will allocate each and every demand place to its nearest chosen know-how in the real 4. It could be shown that a disaggregated scheme, relying on theport, as shown in Figuredemand locations (e.g., any effective option is equivalent to a greedy assignment demand values. Within this model, households, restaurants, or hospitals) and their respectivescheme that all inhabitants might adhere to. Note that need to travel to their nearest chosen port for transporting their freight. island inhabitantsthis figure presents an instance of a part of a total Dehydroemetine Purity & Documentation feasible solution (ground transportation component). Hence, when capacity constraints in the ports will not be relevant, the model will allocate each de.