A Milk Collection Problem with Incompatibility Constraints

1. A Milk Collection Problem with Incompatibility Constraints Selin Atalay Süheyl Güleçyüz Serdal Hakan Akyüz

2. 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

3. Asso.la.c - Info • An Italiandairycompany • Collectsrawmilkfrom 158 farmers • Services in 4 towns • Cosenza, Catanzaro, ViboValentia, Crotone • Has a warehouse in thetown of Castrovillari

4. ASSO.la.c - info Distribution of farmersgetservicedforeachtown (%) • Cosenza: 80 • Catanzaro: 8.7 • ViboValentia: 6.6 • Crotone: 4.7

5. 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

6. Asso.la.c Figure 1: A Milk Collection Tank Truck Consists of Different Compartments

7. ASSO.la.c – fleet • A fleetcomprised of completetrucksandpuretrucks • Alltrucks start theirtoursfromCastrovillari • 8 hoursworkshift

8. ASSO.LA.C – Service area

9. 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

10. 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 )

11. 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

12. TTrp Figure 2: A TTRP solution is shown

13. 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

14. hvrp & HmChf • Trucks and trailers have heteregenous capacities • It can be interpreted as heteregenous VRP (HVRP) • Heteregenous milk collection with heteregenous fleet (HMCHF)

15. 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

16. 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.

17. 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’

18. 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

19. 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

20. 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

21. 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

22. Proposed approach

23. heurISTICS

24. 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.

25. complete vehicles Table 1: Complete Vehicle Characteristics in Quintals

26. pure trucks Table 2: Pure Truck Characteristics in Quintals

27. 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)

28. results Table 3: Case Study Results for the Prior Solution Table 4: Case Study Results for Optimized Solution

29. 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.

30. Similar application:

31. Farmer Route Assignment problem formulation

32. Farmer Route Assignment problem formulation

33. Farmer Route Assignment problem formulation

34. Farmer Route Assignment problem formulation

35. The route Definition problem formulation

36. Thank You for Your Attention Any Questions?