Introducing the MCHF/OVRP/SDMP: Multicapacitated/Heterogeneous Fleet/Open Vehicle Routing Problems with Split Deliveries and Multiproducts

Table 1

Sets, indices, parameters, and variables for MIP.

Indices (for sets see below):
𝑖,𝑗	Index of customers, 𝑖,𝑗∈𝐼.
𝑘	Index of vehicles, 𝑘∈𝐾.
𝑚	Index of product types, 𝑚∈𝑀.
Parameters:
𝑅𝑖𝑗	0-1 matrix for connections of customers. 𝑅𝑖𝑗 is 1, if there is a connection between customer 𝑖 and customer 𝑗.
𝐷𝑗𝑚	The demand of product m for customer 𝑗.
𝐶𝑗𝑘	Transportation cost for vehicle 𝑘 while traveling to target city 𝑗.
𝐹𝑖𝑘	Stopping cost for vehicle 𝑘 on customer 𝑖.
𝑊𝑚	Weight coefficient for product 𝑚.
𝑉𝑚	Volume coefficient for product 𝑚.
𝐺𝑘	Weight capacity for vehicle 𝑘.
𝑃𝑘	Volume capacity for vehicle 𝑘.
𝐿	Large number.
Sets:
𝐼	The set of customers, 𝐼={1,2,…,𝐼max}.
𝐾	The set of vehicles, 𝐾={1,2,…,𝐾max}.
𝑀	The set of product types, 𝑀={1,2,…,𝑀max}.
Set1	{(𝑖,𝑗,𝑘)∣𝑖∈𝐼,𝑗∈𝐼,𝑘∈𝐾,𝑅𝑖𝑗=1}.
Decision variables:
𝑥𝑖𝑗𝑘𝑚	The quantity of product 𝑚 that is left on customer 𝑖, while vehicle 𝑘 is travelling to target customer 𝑗.
𝑦𝑖𝑗𝑘	1,ifvehicle𝑘stopsoncustomer𝑖whiletravelingtotargetcustomer𝑗;0,otherwise.