Vehicle routing problem

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Vehicle routing problem
Nakorn Indra-Payoong Maritime College, Burapha
University
Last updated 15.08.04
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What is vehicle routing problem (VRP)

• VRP is a central problem in the areas of
  transportation, distribution and logistics
• Transport cost typically constitutes more than
  half of the total logistics costs
• Decreasing transport costs can be achieved
  through better utilisation of resources such as
  vehicles
• VRP is to design route for vehicles so as to meet
  the given constraints and optimised objectives

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Problem characteristics

• Depots (number, location)
• Vehicles (capacity, cost, time to leave, driver
  rest period, type and number of vehicles)
• Customers (types of demand, available time
  windows, pickup and delivery, accessibility,
  split demand, priority)
• Route (time, distance, delay, generalised cost on
  network)
• TSP VRP with one vehicle with no capacity, no
  depot, and customers with no demand

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Business goals

• Min. the total travel time (mins, hrs)
• Min. the total travel distance (kms)
• Min. the number of vehicles (units)
• Min. the generalised cost (convert time, distance
  and cost into a monetary unit)

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A variation of VRP

• Capacitated vehicle every vehicle must have
  uniform capacity of a single commodity
• Multiple depots a company that has many depots,
  in particular when customers and depots
  locations are mixed (or cannot be grouped)
• Periodic time in general a routing period is one
  day, but in this case vehicles take several days
  to service customers

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A variation of VRP

• Split delivery allow the same customer can be
  serviced by different vehicles
• Dynamic (stochastic VRP) several components of
  the problems are stochastic, e.g. customers,
  demand and time
• VRP with time windows vehicles must visit
  customers within the interval time given by
  customers

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A variation of VRP

• VRP with pick-up and deliveries delivery demand
  and pick-up demand can be possible
• This restriction makes the problem more difficult
  and can lead to bad utilisations of vehicle
  capacities, increase total distance or increase
  for more vehicles
• Solution is to assume that all delivery start
  from depot and all pick-up demands are brought
  back to the depot, so there are no interchanges
  of goods between the customers

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Pick up and delivery
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A variation of VRP

• Another way is to assume that all customers must
  be visited exactly once
• VRP with backhaul customers can demand or return
  some goods with assumption that all delivery must
  be made on each route before any pickups can be
  made
• This is a fact that the vehicles are rear-loaded
  and rearrangement of the loads at the delivery
  point is not considered economical

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A variation of VRP

• Mix fleet using a number of vehicle types, each
  vehicle is characterised by its capacity, fixed
  cost and variable travel cost
• Dynamic VRP more important due to availability
  of GPS and wireless communication
• Information available to design a set of routes
  is given dynamically to the decision maker order
  information (pickups), vehicle status (delays),
  exception handling (vehicle breakdown), etc.

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Mathematical formulation

• The problem is expressed mathematically, used for
  academic interest, but it looks scary
• Bridging the gap between problem owner and
  problem modeller
• Taking some advantages of existing math. methods,
  e.g. LR, column generation, etc but it is less
  appealed when the problem expresses the
  IP-formulation
• It is not always necessary in practice

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VRPTW formulation (Cordone, 1999)

• Let G (N, A) be a directed route (graph) N
  0, 1, , n) is the node set and A i, j) i,
  j ? N, i ? j is the arc set
• Node 0 is the depot. 1, 2, , n) is the
  set of nodes (customers) to be visited
• Each arc A (i, j) is associated with travel cost
  cij 0 and a travel time tij 0. Each node i
  has a service request (demand) qi, service time
  si, and a time window ei, li

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VRPTW formulation

• All vehicles have the same capacity Q and the
  same running cost .
• Assume and that h
  is high enough to guarantee that minimising the
  number of vehicles is the main objective
• We may shrink time windows by setting
  and
  in order to remove portions which cannot be
  feasibly used
• pi denotes the beginning of the service at node i
• yi is the load of the vehicle leaving node i

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Decision variables

• Decision variable
• xij 1, if arc (i, j) is used
• 0, otherwise
• Objective function .

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Constraints
(1)
(2)
(3)
(4)
(5)
(6)
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The implication of constraints

• Constraints (1) and (2) ensure that each customer
  be assigned exactly to a single route
• Constraints (3) and (4) represent time windows
  constraints
• Constraints (5) and (6) represent vehicle
  capacity constraints

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Solutions for VRP .

• Exact methods
• - Total enumeration
• - IP branch-and-bound, tree-based search
• 2) Heuristic methods
• - Set covering based algorithm
• - Construction algorithm
• - Improvement algorithm, local search, or
• neighbourhood search (not given in this
  course)

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A set partitioning model

• Each column corresponds to a feasible route,
  whilst the row corresponds to customer
• For each column the variable x is defined by
• xr 1, if route r is used in the solution
• 0, otherwise
• and cr denotes the cost (or distance) of the
  route r,

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A set partitioning model

• Columns (C) represent all feasible routes

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A set partitioning model

• Objective function

• Constraints

• where R is the set of all feasible routes
• ir 1, if customer i is serviced by route r
• 0, otherwise

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Example

• Depot 1, customers 4
• Each customer has a demand 1 unit
• The capacity of a vehicle 3 units
• Planning time horizon 0 100 or time windows at
  depot
• Time windows for 4 customers 2 32, 4 14,
  4 18 and 3 32 respectively
• Service time at each customer 10 units

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Graph of the VRP
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Feasible routes

• Minimum total distance 14 13 27
• x6, x8 1 x1, x2, x3, x5, x7 0

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Discussions

• Very flexible for various route generation
  designs
• Only solvable for small to medium-sized problems
• Problem ! How to generate all feasible routes
  ..how many of them

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Time windows

• Earliest and latest time te and tl that the
  customer allow the service to start/finish
• Hard time windows
• - No missing time window is permitted (e.g.
  school bus routing)
• - Same objective fn with additional constraint
  (e.g. )
• Soft time windows
• - Missing time window is allowed but additional
  penalties must be added (e.g. taxi routing)
• - Same constraint but different objective fn

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Time windows for VRP

• Blue and white bars are the time window, the
  white area represents when we can make a delivery
  at that customer. Red bars show when is the
  delivery made for this particular solution.

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Example truck routing

• A single truck departs from a single depot and
  makes deliveries to customers
• The objective is to design a set of routes that
  requires the shortest travel time and do not
  overload the truck

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Fractions of truck loads to be delivered
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Depot and delivery locations
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Feasible routes

• D-3-1-2-D 0.39 0.25 0.33 0.97 lt1.00
• D-4-16-13-D 0.40 0.38 0.16 0.94
• D-15-14-17-20-D 0.22 0.19 0.26 0.31
  0.98

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Construction algorithm (closest insertion)

• Select a customer farthest from the depot (route
  D-9-D, load 0.5)
• Add closest point (i.e. customer 8) (new load is
  0.50 0.43 0.93) this is the first route
• Repeat with the next farthest customer
• This will produce a feasible route - not
  necessarily the best (optimal)

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The procedure

• Route is D-9-8-D with a load factor of 0.93 - no
  other customer can be added
• Customer 10 is the farthest from the depot,
  current route is D-10-D with a load of 0.22
• Customer 7 is closest to this route, new route is
  D-10-7-D with a load of 0.22 0.28 0.50
• Customer 11 is closest to this route, new route
  is D- 10-11-D with a load of 0.22 0.28 0.21
  0.71
• Customer 5 is closest to this route, new route is
  D- 10-7-5-11-D with a load of 0.98

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Final solution
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The saving algorithm (Clarke Wright ,1964)

• A classic method, performs well and very flexible
• Initial sol. has each customer visited by a
  separate vehicle
• Calculate savings (sij) for all pairs of
  customers
• where c(i, j) is cost, time, or distance
  associated with routing from i to j (0 denotes
  the depot)
• Sort savings in decreasing order
• If the savings is positive (a positive sign), it
  is worth to combining the routes

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The savings
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The saving algorithm

• Choose the pair of customers with the largest
  savings and determine if it is feasible to link
  them together. The route is (i, j) feasible, such
  as
• - demand not exceed vehicle capacity
• - distance not exceed the max. length of the
  route
• - respect max. number of customer per route

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The saving algorithm

• If the addition of a further customer (demand)
  exceed the vehicle capacity, a depot (0) will be
  added to the route to represent a return to the
  depot
• Add route (i, j) and delete route (0, i) and (j,
  0)
• Continue the pair of customers with the next
  largest savings
• Stop until all positive saving have been
  considered

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A saving enhancement

• A route shape parameter ? (or a saving
  multiplier)
• The larger ?, the more emphasis is placed on the
  distance between customers being connected (e.g.
  ? is varied between 0 and 2 in steps of 0.1)

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A school bus routing

• Figure Map with school bus stops

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Problem descriptions

• 184 - 214 students are estimated to require the
  school buss service
• Students are clustered into groups for the bus
  stops, there are 24 nodes (one school and 23 bus
  stops)
• Two assignments are made
• - bus stop must be 0.25 miles or less from the
  residences
• - a maximum of ten students per bus stop
• Three bus sizes are used a 54-seat bus, a
  20-seat bus, and a 16-seat bus and they have
  operating costs of 60, 50, and 54 per hour
  respectively

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A saving algorithm

• Distance matrix create the distance matrix (sij)
  for matrix size 24 ? 24
• Two distance matrices are considered for
  arterials and highway
• - average speed for arterials 15 mph
• - average speed for highway 50 mph

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A saving algorithm

• Time matrix calculate the time matrix (tij),
  each element represent the travel time (tij)
  between any two nodes i and j
• where d is the distance between nodes, v is the
  average running speed of the bus

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A saving algorithm

• Time saving matrix (tsij) is show in Table 1
• Route generation

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Time saving matrix (tsij)
s (24, 23)
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Calculation

• The largest time savings is s(24, 23) 12.0,
  this means that bus stops 24 and 23 need to be
  connected
• The max. savings related to node 24 is s(24, 22)
  11.2, and the max. savings related to node 23
  is s(23, 22) 11.3 (now checking bus capacity
  and time windows constraints are not violated)
• So.. node 22 is added to node 23, the route
  becomes 24-23-22
• The growth nodes are now nodes 24 and 22 (node 23
  is dead)

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Calculation

• The max. savings related to node 24 is s(24, 21)
  7.6, and the max. savings related to node 22 is
  s(22, 21) 7.8 (now checking bus capacity and
  time windows constraints are not violated)
• So...node 21 is added to node 22 the new route
  become 24-23-22-21
• The growth nodes are now nodes 24 and 21 (nodes
  23 and 22 are dead)

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Calculation

• The max. savings related to node 24 is s(24, 20)
  6.1, the max. savings related to node 21 is
  s(21, 20) 6.5 (now checking bus capacity and
  time windows constraints are still not violated)
• So.. node 20 is added to node 21, the new route
  is 24-23-22-21- 20
• The growth nodes are now nodes 24 and 20 (nodes
  23, 22, 21 are dead)
• The max. savings related to node 24 is s(24, 19)
  5.4, the max. savings related to node 20 is
  s(20, 19) 6.3 (now checking bus capacity and
  time windows constraints are violated ?)
• Node 0 (school) is added, the first route is
  0-24-23-22-21-20-0

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Calculation

• Row and column corresponding to nodes 20 and 24
  are deleted for considering the next route

s(18, 17) is the next largest
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Optimisation results
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VRP in the grocery delivery industry

• Mix fleet vehicle routing (Burchett Campion
  (2002))
• A number of vehicle types, each vehicle is
  characterised by its capacity, fixed cost and
  variable travel cost
• A limited number of vehicles
• Stochastic customer demand and multiple trips

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Problem characteristics

• Two regions are studied
• - region A (3 vehicles and all of different
  types
• are utilised) (1 of 3 vehicles is an instant
  service van)
• - region B (13 vehicles of four different
  types)
• (6 of 13 are instant service vans)

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Problem characteristics

• Service vans follow a set route each week carry
  a base stock and allow orders to be filled from
  this stock upon arrival at each customer
• Compare routing decisions with the last twelve
  weeks of delivery operation
• Input data customer locations in GPS
  coordinates, types of vehicles, customer demand
  patterns

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Routing decisions

• Option 1 Shifting customers between days,
  delivering only once a week, and optimising the
  delivery routes with
• 1.1. No change to the current service van
  operation
• 1.2. Optimisation of the instant service vans
  delivery schedules

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Routing decisions

• Option 2 Mixing instant service van deliveries
  with other deliveries and changing customers
  between delivery days with
• 2.1. Providing one delivery per week
• 2.2. Providing two deliveries per week to those
  customers who are currently receiving multiple
  deliveries (either grocery or instant service)

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Routing decisions

• Option 3 Delivering only once a week, on one of
  the days that a customer is currently receiving
  deliveries, and optimising the delivery routes
  with
• 3.1. No change to the current service van
  operation
• 3.2. Optimisation of the instant service vans
  delivery schedules

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Routing decisions .

• Option 4 Mixing instant service van deliveries
  with other deliveries - all deliveries being
  carried out on one of the days that customers are
  currently receiving deliveries by
• 4.1. Providing one delivery per week
• 4.2. Providing two deliveries per week to those
  customers who are currently receiving multiple
  deliveries (either grocery or instant service)

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Demand data

• Over 16,000 products are distributed, varieties
  of packaging sizes with different loading
  systems, e.g. a size of tobacco carton 0.0315
  cubic metre
• So.. customer demands are approximated
• In practice is preferred to over estimate the
  delivery size, this leaves some additional space
  to cope with the variability in customer demand
• To over-estimate, e.g. use (mean standard
  deviation of product sizes) to cover the
  variability of demand

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Distribution of product sizes .
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Routing constraints

• The requirements of each customer (e.g. time
  windows) must be satisfied
• The capacity of each vehicle must not be violated
• The weekly and daily number of trips for each
  vehicle must not be exceeded
• If multiple deliveries to customers are possible,
  these can not occur on the same day
• The number of customers allocated to some classes
  of vehicle cannot exceed a predetermined number
• The capacity of each type of type of bulk
  packaging must not be violated
• Each vehicle route must start and terminate at
  the depot

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Solution technique

• Solution method involves 2 phases
• - a construction phase use a saving algorithm
  to generate good initial solution
• a search phase use the improvement algorithm
  (i.e. tabu search) to improve a solution
  iteratively
• Use neighbourhood scheme based on several move
  operators

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Improvement algorithm

• x, y, and z are decision variables taking a
  decision value, such as x 0 or x 1

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Move operator 2-change
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Move operator 1-relocate
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Move operator swap
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Summary of savings in each alternative
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Distribution strategies
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Direct shipping advantages

• Suitable for perishable items, high value of
  goods, high bulk items
• Less inventory in supply chain
• Less handling and opportunity for product damage
• Direct store delivery (DSD) is amongst the most
  profitable in store
• Higher service satisfaction
• Improved accuracy - invoice match receiving
  records, correct products enter to the store

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Direct shipping disadvantages

• More deliveries, paper works, activities
• No pooling benefit
• No safety stock in the event of a supplier
  (customer) problem (high variation events
  holiday, promotion)
• Transportation cost can be higher

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Cross docking

• Cross docking is a flow through concept, it does
  not want the product to stop elsewhere
• Taking goods from a plant, delivery it to the
  front door of distributor, and almost immediately
  take the item out the back door and load it to a
  truck headed for the customer .
• Cross docking changes the concept from supply
  chain to demand chain

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Cross docking advantages

• Increase inventory turns by speeding the flow of
  products from the supplier to the store
• Cross docking coupled with consolidating
  warehouse avoid less than truck load (LTL)
  deliveries
• It eliminates the costs of handling and storing
  inventory
• Today mass merchandisers, grocery companies, LTL
  trucking companies, and air cargo carriers are
  the leading cross dock users

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Cross-docking challenges

• Cross docking relies on strong IT such as EDI,
  bar code, real time information, RFID, etc .
• It works best if trading partners engage in
  collaborative planning, forecasting, and
  replenishment
• Cross docking may require new facility layout,
  product visibility as it moves through the system
• To support JIT, product availability, accuracy
  and quality are critical

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Transhipment

• Sharing inventory between facilities at the same
  level in supply chain
• Better customer service, fewer stockouts can be
  achieved
• Requirements
• - Inventory visibility
• - Cooperation
• - Shipping and delivery processes

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Pool distribution .

• Consolidate shipments at the origin into load
  destined for defined regions
• - Transporting the load to a central point in
  the region
• - Making local deliveries from the central point
• Reduce transportation costs, better service
• Requirements
• - Multiple delivery points within a region
• - Significant and consistent volume entering the
  region