Modeling Rich Vehicle Routing Problems TIEJ601 Postgraduate Seminar Tuukka Puranen October 19 th 2009

Contents A tour on combinatorial optimization problems relevant to logistics design Vehicle Routing Problem and its variants The proposed model for VRP – Analysis Effects and possibilities of the new model I will not talk about implementation, system design, optimization results, algorithms

A Tour on Combinatorial Optimization Modeling Rich Vehicle Routing Problems

Computational Logistics Computational – Of or related to computation Logistics – Management of the flow of goods, information, energy, and people between the points of origin and the points of consumption Computational Logistics – Information system assisted planning based on formulating, solving and analyzing computational problems in logistics

Examples of Logistic Problems Shortest Path Problem Traveling Salesman Problem Vehicle Routing Problem Logistics Network Design Problem – Production and distribution, linear programming Network flow problem K-means problem, coverage problem Inventory management, storage design Job scheduling

Traveling Salesman Problem Given a list of locations (e.g., cities) and distances between them, find the shortest tour that visits each location Mathematically formulated in 1930; one of the most intensively studied problems in optimization Note also that usually in our context, TSP contains SPP as a subproblem when solved in a graph, e.g., road network

TSP

Vehicle Routing Problem Given a list of customers, distances between them and a set of vehicles, find tours that minimize the total length of the tours, such that one vehicle visits each location Formulated in 1959 Typically, one has to serve a scattered set of customers from a single central depot, such that each vehicle has a limited capacity

VRP

Vehicle Routing Problem Variants VRP with time windows (VRPTW) Fleet size and mix VRP (FSMVRP) Open VRP (OVRP) Multi-depot VRP (MDVRP) Periodic VRP (PVRP) VRP with backhauls (VRPB) Pickup and delivery problem (PDP) Dynamic VRP (DVRP) VRP with stochastic demands (VRPSD)

Pickup and Delivery Problem Each task consists of two parts – Pickup – Delivery VRP (and MDVRP) a special case of PDP Can be combined with other aspects – Time windows, capacity, fleet size and mix,... Real-life examples include oil transportation, school buses, courier services, …

PDP

A New Way of Describing Vehicle Routing Problems Modeling Rich Vehicle Routing Problems

Real-life Models In theory, these simple models work But if you would ever want to create a system for solving these problems, you would like to have a bit more expressiveness The ‘messy real-life’ Driver breaks, QoS limitations, compartments, special equipment, service restrictions, … COMDFSMPDPTW – Hence the name ’Rich VRP’

Real-life Objectives Minimize distance Maximize profit Minimize time Minimize CO 2 emissions Minimize effects on congestion Maximize customer satisfaction Minimize employee workload

Motivation A number of different cases have to be modeled and solved Time to build only a single solver A modeling language for describing the problem in a way that requires no changes on the solution space exploring system – Meaning that feasible region can be defined without modifying the solver itself – Algorithms and objectives can be tailored, but not necessarily require it

The Proposed Model Based partially on an idea of General Pickup and Delivery Problem (GPDP) – Each vehicle starts from and ends at an arbitrary point Combines concepts from constraint programming and automata theory In essence, a labeled network formulation Objecive is to be able to utilize combinatorial metaheuristic local and global search

Actors and Activities Actors and activities are described as nodes in a network Each actor corresponds to a vehicle Each activity corresponds to a task – Usually order pickup and delivery points – Can be used to other tasks, e.g., fleet selection A solution is formulated by selecting, ordering and assigning the activities to actors

Actors and Activities Illustrated

Labels Each node can have a set of labels that have an adjoining integral value There are two rules – Each label must have a nonnegative accumulated value – Each label must have zero value at the end A +1A -1B +1B

Example: Vehicle Capacities 1133 A +1 C +10 A -1 C -10 X +1 C -2 X -1 C Y +1 C -5 Y -1 C +5

Metrics If actors and activities are nodes in a network, we need a way to describe their relation, i.e., arcs – These relations include, for example, distance The model can have any number of metrics Metrics can also be assigned to nodes 1133 time = 5 dist = 3 time = 6 dist = 4 time = 2 dist = 1

time = 2 dist = 1 Situation A situation in given point is defined by – The set of labels and their accumulated values at that point – The values of each accumulated metric for each label 1133 A +1A -1B -1 SituationA = 1 time = 0 dist = 0 A = 1 time = 5 dist = 3 B = 1 time = 0 dist = 0 A = 1 time = 7 dist = 4 B = 0 time = 2 dist = 1 A = 0 time = 13 dist = 8 time = 5 dist = 3 time = 6 dist = 4 B +1

Constraints Constraints impose lower and upper bounds on metrics Assigned to given label-metric pair If that label is present in a situation, its given accumulated metric value must fall between the defined bounds Can be used to model time windows, breaks, QoS requirements

Example: Maximum Travel Time Assume that we need to – Restrict the length of the shift of the driver – Ensure that the customer sits in the vehicle no more than given number of minutes time = 2 dist = 1 13 A +1A -1B -1 A = 1 time = 0 dist = 0 A = 1 time = 5 dist = 3 B = 1 time = 0 dist = 0 A = 1 time = 7 dist = 4 B = 0 time = 2 dist = 1 A = 0 time = 13 dist = 8 time = 5 dist = 3 time = 6 dist = 4 B +1 (A, time) < 15 (B, time) < 5 13

Feasibility A route is feasible when – All labels in every situation are nonnegative – Labels have zero sum at the end – Metric values are within constraints in every situation A solution is feasible when – All routes are feasible Note that this does not require visit on every node

Example: Capacity Feasibility 1133 A +1 C +10 A -1 C -10 X +1 C -2 X -1 C Y +1 C -5 Y -1 C +5 A = 1 X = 1 Y = 1 C = -2 A +1 C +10 A -1 C -10 X +1 C -2 X -1 C +2 Y +1 C -5 Y -1 C +5 Z +1 C

Objective Function Objective function becomes just a single metric that has no constraints – Simple multiobjective optimization becomes natural feature of the system: change an objective to constrained metric and vice versa As usual, is used to evaluate the solution at given situation A penalty must be assigned for not visiting the nodes since feasibility does not require this

Dynamic Metrics Sometimes metrics change depending on the solution structure Label dependent – Trailers, special equipment,... – Also in objective function: complex cost structures – Assigning a metric transformation to labels – Keeping track of the active transformation Situation dependent – DVRP

Example: Trailer Affects Speed 1173 A +1 T +1 C +10 A -1 T -1 C -10 B +1 T -1 C +5 X -1 C X +1 C -3 B -1 T +1 C -5 (dist, A) = f( p1, p2 ) (time, A) = dist * 1,0 (time, B) = dist * 1,1 time = A dist = A time = B, A dist = A time = B, A dist = A time = B, A dist = A time = A dist = A time = dist =

Analysis on the Proposed Model Modeling Rich Vehicle Routing Problems

Benefits More expressive model – Expandable More implementation friendly formulation – Less work per modeled case – Visual Per aspect analysis – Easier to evaluate the cost on complexity – Generate only relevant aspects Multidisciplinary research

Variants VRP with time windows (VRPTW) Fleet size and mix VRP (FSMVRP) Open VRP (OVRP) Multi-depot VRP (MDVRP) Periodic VRP (PVRP) VRP with backhauls (VRPB) Pickup and delivery problem (PDP) Dynamic VRP (DVRP) VRP with stochastic demands (VRPSD)

Objectives Minimize distance Maximize profit Minimize time Minimize CO 2 emissions Minimize effects on congestion Maximize customer satisfaction Minimize employee workload

The ‘Messy Real-life’ Driver breaks QoS limitations – Maximum waiting time – Maximum ride time Fleet selection, special equipment Service restrictions, preferences Multiple capacities Compartment loading decisions Time dependent continuous demand

Future Research & Conclusions Modeling Rich Vehicle Routing Problems

Future Research Continuing implementation Modeling – Compartments – Stochastic metrics, labels – Interroute dependencies, e.g., assisting drivers Testing – Modeling complex cases – Benchmarking solution methods Multiobjective optimization

Conclusions A number of combinatorial optimization problems, starting from TSP, are important in designing logistic operations In practice, a more detailed model is often needed We proposed a new way for modeling VRPs, which should make it easier to incorporate difficult real-life aspects into optimization problems