Create Presentation
Download Presentation

Download Presentation

A Milk Collection Problem with Incompatibility Constraints

Download Presentation
## A Milk Collection Problem with Incompatibility Constraints

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**A Milk Collection Problem with Incompatibility Constraints**Selin Atalay Süheyl Güleçyüz Serdal Hakan Akyüz**MILK COLLECTION Problem**• Collecting raw milk from farmers • Well known problem in rural areas • Some farms can be small and inaccessible by large vehicles • Different milk types might exist • One milk type can be carried by one compartment**Asso.la.c - Info**• An Italiandairycompany • Collectsrawmilkfrom 158 farmers • Services in 4 towns • Cosenza, Catanzaro, ViboValentia, Crotone • Has a warehouse in thetown of Castrovillari**ASSO.la.c - info**Distribution of farmersgetservicedforeachtown (%) • Cosenza: 80 • Catanzaro: 8.7 • ViboValentia: 6.6 • Crotone: 4.7**Asso.LA.c – milk collection**• 3 different milk types to be collected • Standard-quality, high-quality, sanitary-prescription • Milk selection based on milk quality • Prerefrigeration process of the selected milk • Transfer of milk into a tank by means of pumps capable of loading 1.1 tons in five minutes • Writing of a report to record information about the milk type and quantity picked up**Asso.la.c**Figure 1: A Milk Collection Tank Truck Consists of Different Compartments**ASSO.la.c – fleet**• A fleetcomprised of completetrucksandpuretrucks • Alltrucks start theirtoursfromCastrovillari • 8 hoursworkshift**Milk collection: VRP**• Milk collection problem is a specialized instance of vehicle routing problem (VRP) • VRP • A set of homogeneous vehicles serves customers • Objective is to minimize one or more objective functions • Minimize fleet size, total routing cost etc... • Each route starts and ends at a depot • Vehicle’s capacity may not be exceeded • NP hard, usually solved by (meta)heuristics**ttrp**• VRP assumes that customers are always reached by vehicles • Size of a truck,location of the customer is not relevant • In practice, some trucks may not be able to reach some locations • Complete vehicle & Pure truck • Known as Truck and Trailer Routing Problem ( TTRP )**Ttrp**• Assumptions of TTRP • S : Set of customers • Str : accessible with or without a trailer • Swt : only accessible without a trailer • Complete Vehicle Route (CVR) • Sum of all demands collected on the CVR may not exceed the vehicle total capacity (C + C’) • Transported commodities cannot be transferred between the truck and the trailer**TTrp**Figure 2: A TTRP solution is shown**TTRP**• TTRP’s objective is to find routes that minimize the total travel cost or time,with a possible limitation on the tour length • May also include finding the optimal number of subtours and the location of parking places • Farms are often small and inaccesible,which makes milk collection an important application of the TTRP**hvrp & HmChf**• Trucks and trailers have heteregenous capacities • It can be interpreted as heteregenous VRP (HVRP) • Heteregenous milk collection with heteregenous fleet (HMCHF)**CONSTRAINTS IN HMCHF**• Each node in the network can be a loading point or a parking area • The truck cannot contain a load greater than its capacity • Multiple trucks can pick up milk from a particular farmer • The time required for a tour cannot exceed the work shift • Only one milk type can be assigned to a compartment**FRAP & rdp**• FRAP: objective is to minimize the number of trucks by assigning farmers to trucks • FRAP is not followed directly by a route construction heuristic. Therefore, RDP is used. • RDP: It is used for defining routes according to results of the FRAP solution.**PROPOSED APPROACH**• First solves the Farmer Route Assignment Problem • Assign farmers to vehicles • Minimize number of trucks, satisfy capacity and demand constraints • Given a FRAP solution, using Route Definition Problem routes are determined • Assume a graph G=(S U {s0},A) with vertex set S U {s0} and arc set A • S0 represent the depot where m trucks and m’ trailers are parked • Trucks and trailers are considered homogeneous with capacity C and C’**PROPOSED APPROACH**• Each vertex si ЄS corresponds to a customer i with demand di ≥ 0 • A cost cij is associated with each arc (si,sj ), which represents the nonnegative travel time or distance from vertex si to sj**PROPOSED APPROACH**• In the FRAP, • I=I1 u ı2 , I1 is set of complete vehicles and I2 is the set of pure trucks • For farmer s and milk type j , the farmer demand QJS must be satisfied • If a positive amount of milk type j is loaded in the compartment k of tank truck i, this quantity must not be larger than the capacity cik of compartment k of tank truck i • At least one truck must serve each farmer • If a certaşn quantity of a milk type is loaded from farmer s in a tank truck i, then tank truck i serves farmer s • At most one milk type per compartment is allowed • Complete vehicles must visit at least one farmer sЄStr**PROPOSED APPROACH**• In the RDP, • R* : number of routes defined by the FRAP • Si: subsets of farmers assigned to each vehicle Iє • Si’: set of customers served by truck and trailers • Si’’: set of customers served by truck only • RDP formulation works on R* routes by seperately solving a relaxation of TSP • If tour i is associated with a pure truck, then Si=Si’’ • Contraints ensure that a tour starts and ends at the warehouse • If a truck arrives at a farmer site, it must leave the site**PROPOSED APPROACH**• If tour i is associated with a complete vehicle,the related RDP constraints split into two subsets of constraints • One for Si’ and one for Si’’ • RDP has two drawbacks • The minimization of the total tour time might return a duration for some tours that exceeds the time T allowed for a working shift • Subtours might be present**solution**• Tools used in the solution: • Algorithm: C Language • Mathematical Models: AMPL Language and solver CPLEX 7.0 • Aggregated farmers closer than 2 km. in a single node to decrease the problem size and improve tractability • Assumption: Preparation time tsis known for each farmer s.**complete vehicles**Table 1: Complete Vehicle Characteristics in Quintals**pure trucks**Table 2: Pure Truck Characteristics in Quintals**Case study computational results**• Proposed algorithm: • Tank trucks start the tour from Castrovillari and come back to Castrovillari within the work shift • Three complete vehicles CV1, CV2 farmers in Crotone and Vibo Valentina • CV4 serves Crotone, Vibo Valentina and Catanzaro • All eight pure trucks collect milk from farmers in Cosenza • Avg. filling ratio between the load and its capacity: 95% (85% in prior solution)**results**Table 3: Case Study Results for the Prior Solution Table 4: Case Study Results for Optimized Solution**Effectiveness of Constraints**• 20 iterations are used to compare effectiveness of two constraints. • In 3 cases: both constraint is used, none is used and only constraint 1 is used • It satisfy an improvement about no of tours, total tour length and fulfilment of the maximum shift duration for each tour.**Thank You for Your Attention**Any Questions?