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Vehicle Routing and
Scheduling
Martin Savelsbergh
The Logistics Institute
Georgia Institute of Technology
Vehicle Routing and
Scheduling
Part I:
Basic Models and Algorithms
Introduction
Freight routing
routing of shipments
Service routing
dispatching of repair technicians
Passenger routing
tran...
Freight Routing (LTL)
pickup and delivery to and from end-of-line
terminal
shuttle from end-of-line terminal to regional
d...
LTL Linehaul Network
EOLs
DCs
Customers
Origin-Destination Route
Origin
EOL
DC
Destination
Linehaul
scheduling
Pickup and
delivery
Shuttle
scheduling
Routing and Scheduling
Routing and scheduling does not follow a
single “one-size-fits-all” formula. Routing
and scheduling...
Models
Traveling Salesman Problem (TSP)
Vehicle Routing Problem (VRP)
Vehicle Routing Problem with Time
Windows (VRPTW)
Pi...
Traveling Salesman Problem
In the TSP the objective is to find the shortest
tour through a set of cities, visiting each ci...
Traveling Salesman Problem
Vehicle Routing Problem
In the VRP a number of vehicles located at a
central depot has to serve a set of
geographically di...
Vehicle Routing Problem
Vehicle Routing Problem
with Time Windows
In the VRPTW a number of vehicles is located at
a central depot and has to serve...
Vehicle Routing Problem
with Time Windows
Pickup and Delivery Problem
with Time Windows
In the PDPTW a number of vehicles has to serve a
number of transportation re...
Pickup and Delivery Problem
with Time Windows
Delivery
Pickup
Pickup and Delivery Problem
with Time Windows
Delivery
Pickup
Routing and Scheduling
Objectives
minimize vehicles
minimize miles
minimize labor
satisfy service requirements
maximize or...
Routing and Scheduling
Practical considerations
Single vs. multiple depots
Vehicle capacity
homogenous vs. hetrogenous
vol...
Routing and Scheduling
Practical considerations (cont.)
Delivery windows
hard vs. soft
single vs. multiple
periodic schedu...
Routing and Scheduling
Practical considerations (cont.)
Fixed and variable delivery times
Fixed vs. variable regions/route
Recent Variants
Dynamic routing and scheduling problems
More and more important due to availability
of GPS and wireless co...
Recent Variants
Stochastic routing and scheduling
problems
Size of demand
Travel times
Algorithms
Construction algorithms
Improvement algorithms
Set covering based algorithms
Construction Heuristics
Savings heuristic
Insertion heuristics
Savings
Initial
i j
Intermediate Final
Savings s(i,j) = c(i,0) + c(0,j) – c(i,j)
Savings Heuristic (Parallel)
Step 1. Compute savings s(i,j) for all
pairs of customers. Sort savings.
Create out-and-back ...
Savings Heuristics (Sequential)
Step 1. Compute savings s(i,j) for all
pairs of customers. Sort savings. Create
out-and-ba...
Savings Heuristic -
Enhancement
Route shape parameter
s(i,j) = c(i,0) + c(0,j) - λ c(i,j)
The larger λ, the more emphasis ...
Insertion
Initial
i
j
k
Intermediate Final
Insertion Heuristics
Start with a set of
unrouted stops
Select an unrouted stop
Are there any
unrouted stops?
Insert selec...
Nearest addition
Selection:
If partial tour T does not include all cities,
find cities k and j, j on the tour and k not, f...
Nearest Insertion
Selection:
If partial tour T does not include all cities,
find cities k and j, j on the tour and k not, ...
Farthest Insertion
Selection:
If partial tour T does not include all cities,
find cities k and j, j on the tour and k not,...
Cheapest Insertion
Selection:
If partial tour T does not include all cities,
find for each k not on T the edge {i,j} of T
...
Worst case results
Nearest addition: 2
Nearest insertion: 2
Cheapest insertion: 2
Farthest insertion:
> 2.43 (Euclidean)
>...
Implementation
Priority Queue
insert(value, key)
getTop(value, key)
setTop(value, key)
k-d Tree
deletePt(point)
nearest(po...
Implementation
Tree->deletePt(StartPt)
NNOut[StartPt] := Tree->nearest(StartPt)
PQ->insert(Dist(StartPt, NNOut(StartPt)), ...
Improvement Algorithms
Start with a
feasible solution x
x
y
z
Is there an
improving neighbor
y in N(x)?
Define neighborhoo...
2-change
O(n2) possibilities
3-change
O(n3) possibilities
1-Relocate
O(n2) possibilities
2-Relocate
O(n2) possibilities
Swap
O(n2) possibilities
GENI
Vehicle Routing and
Scheduling
Vehicle Routing and
Scheduling
1-relocate
B-cyclic k-transfer
Set covering based
algorithms
Assignment decisions are often the most
important
Assignment decisions are often the most
di...
Vehicle routing and scheduling
Possible route
Vehicle routing and scheduling
Possible route
Vehicle routing and scheduling
Possible route
Vehicle routing and scheduling
Possible route
Vehicle routing and scheduling
Possible route
Vehicle routing and scheduling
Possible route
Vehicle routing and scheduling
Possible route
Set partitioning formulation
c1 c2 c3 c4 …
Cust 1 1 1 0 1 … = 1
Cust 2 1 0 1 0 … = 1
Cust 3 0 1 1 0 … = 1
…
Cust n 0 1 0 0...
Set covering based
algorithms
Advantage
very flexible
heuristics for route generation
complicating constraints in route ge...
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Supply chain logistics : vehicle routing and scheduling

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Vehicle routing and scheduling Models:
Travelling salesman problem
vehicle routing problem with time window
Pick up and delivery problem with time window

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Supply chain logistics : vehicle routing and scheduling

  1. 1. Vehicle Routing and Scheduling Martin Savelsbergh The Logistics Institute Georgia Institute of Technology
  2. 2. Vehicle Routing and Scheduling Part I: Basic Models and Algorithms
  3. 3. Introduction Freight routing routing of shipments Service routing dispatching of repair technicians Passenger routing transportation of elderly
  4. 4. Freight Routing (LTL) pickup and delivery to and from end-of-line terminal shuttle from end-of-line terminal to regional distribution center transportation between distribution centers rail sleeper teams single driver
  5. 5. LTL Linehaul Network EOLs DCs Customers
  6. 6. Origin-Destination Route Origin EOL DC Destination Linehaul scheduling Pickup and delivery Shuttle scheduling
  7. 7. Routing and Scheduling Routing and scheduling does not follow a single “one-size-fits-all” formula. Routing and scheduling software must usually be customized to reflect the operating environment and the customer needs and characteristics
  8. 8. Models Traveling Salesman Problem (TSP) Vehicle Routing Problem (VRP) Vehicle Routing Problem with Time Windows (VRPTW) Pickup and Delivery Problem with Time Windows (PDPTW)
  9. 9. Traveling Salesman Problem In the TSP the objective is to find the shortest tour through a set of cities, visiting each city exactly once and returning to the starting city. Type of decisions: routing
  10. 10. Traveling Salesman Problem
  11. 11. Vehicle Routing Problem In the VRP a number of vehicles located at a central depot has to serve a set of geographically dispersed customers. Each vehicle has a given capacity and each customer has a given demand. The objective is to minimize the total distance traveled. Type of decisions: assigning routing
  12. 12. Vehicle Routing Problem
  13. 13. Vehicle Routing Problem with Time Windows In the VRPTW a number of vehicles is located at a central depot and has to serve a set of geographically dispersed customers. Each vehicle has a given capacity. Each customer has a given demand and has to be served within a given time window. Type of decisions: assigning routing scheduling
  14. 14. Vehicle Routing Problem with Time Windows
  15. 15. Pickup and Delivery Problem with Time Windows In the PDPTW a number of vehicles has to serve a number of transportation requests. Each vehicle has a given capacity. Each transportation request specifies the size of the load to be transported, the location where it is to be picked up plus a pickup time window, and the location where it is to be delivered plus a delivery time window. Type of decisions: assigning routing scheduling
  16. 16. Pickup and Delivery Problem with Time Windows Delivery Pickup
  17. 17. Pickup and Delivery Problem with Time Windows Delivery Pickup
  18. 18. Routing and Scheduling Objectives minimize vehicles minimize miles minimize labor satisfy service requirements maximize orders maximize volume delivered per mile
  19. 19. Routing and Scheduling Practical considerations Single vs. multiple depots Vehicle capacity homogenous vs. hetrogenous volume vs. weight Driver availability Fixed vs. variable start times DoT regulations (10/1, 15/1, 70/8)
  20. 20. Routing and Scheduling Practical considerations (cont.) Delivery windows hard vs. soft single vs. multiple periodic schedules Service requirements Maximum ride time Maximum wait time
  21. 21. Routing and Scheduling Practical considerations (cont.) Fixed and variable delivery times Fixed vs. variable regions/route
  22. 22. Recent Variants Dynamic routing and scheduling problems More and more important due to availability of GPS and wireless communication Information available to design a set of routes and schedules is revealed dynamically to the decision maker Order information (e.g., pickups) Vehicle status information (e.g., delays) Exception handling (e.g., vehicle breakdown)
  23. 23. Recent Variants Stochastic routing and scheduling problems Size of demand Travel times
  24. 24. Algorithms Construction algorithms Improvement algorithms Set covering based algorithms
  25. 25. Construction Heuristics Savings heuristic Insertion heuristics
  26. 26. Savings Initial i j Intermediate Final Savings s(i,j) = c(i,0) + c(0,j) – c(i,j)
  27. 27. Savings Heuristic (Parallel) Step 1. Compute savings s(i,j) for all pairs of customers. Sort savings. Create out-and-back routes for all customers. Step 2. Starting from the top of the savings list, determine whether there exist two routes, one containing (i,0) and the other containing (0,j). If so, merge the routes if the combined demand is less than the vehicle capacity.
  28. 28. Savings Heuristics (Sequential) Step 1. Compute savings s(i,j) for all pairs of customers. Sort savings. Create out-and-back routes for all customers. Step 2. Consider route (0,i,…,j,0) and determine the best savings s(k,i) and s(j,l) with routes containing (k,0) and (0,l). Implement the best of the two. If no more savings exist for this route move to the next.
  29. 29. Savings Heuristic - Enhancement Route shape parameter s(i,j) = c(i,0) + c(0,j) - λ c(i,j) The larger λ, the more emphasis is placed on the distance between customers being connected
  30. 30. Insertion Initial i j k Intermediate Final
  31. 31. Insertion Heuristics Start with a set of unrouted stops Select an unrouted stop Are there any unrouted stops? Insert selected stop in current set of routes Yes Done No
  32. 32. Nearest addition Selection: If partial tour T does not include all cities, find cities k and j, j on the tour and k not, for which c(j,k) is minimized. Insertion: Let {i,j} be either one of the two edges involving j in T, and replace it by {i,k} and {k,j} to obtain a new tour including k.
  33. 33. Nearest Insertion Selection: If partial tour T does not include all cities, find cities k and j, j on the tour and k not, for which c(j,k) is minimized. Insertion: Let {i,j} be the edge of T which minimizes c(i,k) + c(k,j) - c(i,j), and replace it by {i,k} and {k,j} to obtain a new tour including k.
  34. 34. Farthest Insertion Selection: If partial tour T does not include all cities, find cities k and j, j on the tour and k not, for which c(j,k) is maximized. Insertion: Let {i,j} be the edge of T which minimizes c(i,k) + c(k,j) - c(i,j), and replace it by {i,k} and {k,j} to obtain a new tour including k.
  35. 35. Cheapest Insertion Selection: If partial tour T does not include all cities, find for each k not on T the edge {i,j} of T which minimizes c(T,k) = c(i,k) + c(k,j) - c(i,j). Select city k for which c(T,k) is minimized. Insertion: Let {i,j} be the edge of T for which c(T,k) is minimized, and replace it by {i,k} and {k,j} to obtain a new tour including k.
  36. 36. Worst case results Nearest addition: 2 Nearest insertion: 2 Cheapest insertion: 2 Farthest insertion: > 2.43 (Euclidean) > 6.5 (Triangle inequality)
  37. 37. Implementation Priority Queue insert(value, key) getTop(value, key) setTop(value, key) k-d Tree deletePt(point) nearest(point)
  38. 38. Implementation Tree->deletePt(StartPt) NNOut[StartPt] := Tree->nearest(StartPt) PQ->insert(Dist(StartPt, NNOut(StartPt)), StartPt) loop n-1 time loop PQ->getTop(ThisDist, x) y := NNOut[x] If y not in tour, then break NNOut[x] = Tree->nearest(x) PQ->setTop(Dist(x, NNOut[x]), x) Add point y to tour; x is nearest neighbor in tour Tree->deletePt(y) NNOut[y] = Tree->nearest(y) PQ->insert(Dist(y, NNOut[y]), y) Delayed update Find nearest point Update
  39. 39. Improvement Algorithms Start with a feasible solution x x y z Is there an improving neighbor y in N(x)? Define neighborhood N(x). Replace x by y Yes N(x) x is locally optimal No
  40. 40. 2-change O(n2) possibilities
  41. 41. 3-change O(n3) possibilities
  42. 42. 1-Relocate O(n2) possibilities
  43. 43. 2-Relocate O(n2) possibilities
  44. 44. Swap O(n2) possibilities
  45. 45. GENI
  46. 46. Vehicle Routing and Scheduling
  47. 47. Vehicle Routing and Scheduling 1-relocate
  48. 48. B-cyclic k-transfer
  49. 49. Set covering based algorithms Assignment decisions are often the most important Assignment decisions are often the most difficult
  50. 50. Vehicle routing and scheduling Possible route
  51. 51. Vehicle routing and scheduling Possible route
  52. 52. Vehicle routing and scheduling Possible route
  53. 53. Vehicle routing and scheduling Possible route
  54. 54. Vehicle routing and scheduling Possible route
  55. 55. Vehicle routing and scheduling Possible route
  56. 56. Vehicle routing and scheduling Possible route
  57. 57. Set partitioning formulation c1 c2 c3 c4 … Cust 1 1 1 0 1 … = 1 Cust 2 1 0 1 0 … = 1 Cust 3 0 1 1 0 … = 1 … Cust n 0 1 0 0 … = 1 y/n y/n y/n y/n …
  58. 58. Set covering based algorithms Advantage very flexible heuristics for route generation complicating constraints in route generation Disadvantage small to medium size instances

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