1. Vehicle Routing & Scheduling
Vehicle Routing & Scheduling - Janny Leung
Vehicle Routing & Scheduling
Find the best routes and schedule to deliver goods to a set of customers (with specified demands) from a central depot using a fleet of (identical) vehicles
1959 Dantzig and Ramser, ARCO gasoline delivery to gas stations
1964 Clarke and Wright, consumer goods delivery for Coop. Wholesale Society in Midlands, England
1930’s Menger, Travelling salesman problem
1800’s Hamiltonian circuit

2. Practical Complications
Vehicle Routing & Scheduling - Janny Leung
Practical Complications
Travel costs not symmetric, not known
Heterogeneous fleet, fleet size variable
Multiple capacity restriction
weight, volume, time
different for different products
Customer-vehicle compatibility
Delivery time-windows
Pickup & delivery on same route
Capacity? Precedence?
Multiple depots
Optional deliveries (e.g. replenishment), periodic deliveries
Complex or multiple objective(s)

3. Vehicle Routing: Theory and Practice
Vehicle Routing & Scheduling - Janny Leung
Vehicle Routing: Theory and Practice
First generation ( ’s)
greedy, local improvement heuristics
Second generation ( ’s)
mathematical programming models
Third generation ?
Artificial intelligence?
Human-machine interactive?
“on-line” optimisation-based heuristics?
Need “robust” approach

4. Vehicle Routing & Scheduling - Janny Leung
Shortest Path Problem
Given a network with (non-negative) costs on the arcs, find a “shortest-path” from a given origin node to a destination node.
ORIGIN
Amarillo
Oklahoma
City E
90 minutes B 84 84 A I
138 66
120
132 C 90
348 F 60
126 H
156
126
132 48 48
Note: All link times
are in minutes J D
150
DESTINATION
Fort Worth G

5. Dijkstra’s algorithm (1959)
Vehicle Routing & Scheduling - Janny Leung
Dijkstra’s algorithm (1959)
1. Initially, set e0 = 0 and ej= ¥ for all other nodes j. Let R= f.
2. Choose node k among nodes in N\R that minimises ej.Let R ¬ R U {k}.
3. If destination node in R, stop.
4. Update: for each arc (k,j) adjacent to k,
ej ¬ min{ ej , ej + dkj }
5. Repeat from Step 2.
Fast O(n2)
Easy to understand

6. Vehicle Routing & Scheduling - Janny Leung
ORIGIN
Amarillo
Oklahoma
City E
90 minutes B 84 84 A I
138 66
120
132 C 90
348 F 60
126 H
156
126
132 48 48
Note: All link times
are in minutes J D
150
DESTINATION
Fort Worth G

7. Travelling Salesman Problem
Vehicle Routing & Scheduling - Janny Leung
Travelling Salesman Problem
Starting from the depot, find a shortest “tour” that visits all other nodes exactly once and returns to the depot.
Very difficult (NP-complete)
No “quick” method to find a “guaranteed” optimal solution

8. Lin-Kernighan (1965, 1973) Local improvement heuristics
Vehicle Routing & Scheduling - Janny Leung
Lin-Kernighan (1965, 1973) Local improvement heuristics
2-opt
3-opt
k-opt k j m i
(a) n l k j m i
(b) n l
(a) Current tour
(b) Tour after exchange

9. Vehicle Routing Clarke-Wright (1964)
Vehicle Routing & Scheduling - Janny Leung
Vehicle Routing Clarke-Wright (1964)
Initially, each (customer) is served by a separate route from the depot.
Consider merging routes to nodes i and j:
savings= sij = di0 + d0j - dij
Merge routes with maximum (positive) savings.
dA,O
dO,A A A
dO,A O O
dO,B
dA,B
Depot
Depot
dB,O B B
dB,O

10. Vehicle Routing & Scheduling - Janny Leung
Re-calculate savings for current set of routes:
insert at beginning: savings= dX0 + d0A - dXA
insert at end: savings= d0X + dB0 - dBX
insert in middle: savings= d0X + dX0 + dAB - dAX - dXB
Repeat merging until no positive savings. X A X B B A X A B

11. Practical Vehicle Routing & Scheduling Problems
Vehicle Routing & Scheduling - Janny Leung
Practical Vehicle Routing & Scheduling Problems
fleet of vehicles
vehicles capacitated
time restrictions

12. Modified Clarke-Wright Savings methods
Capacitated homogeneous fleet
calculate C-W savings, but only merge routes if vehicle capacity not exceeded
Capacitated non-homogeneous fleet
consider vehicle one at a time, merge routes if capacity not exceeded
? Order of vehicles to consider?
Delivery time windows
only merge routes if delivery time (and/or vehicle capacity) restrictions met

13. Two-phased Heuristic Gillette & Miller’s Sweep Method (1974)
Vehicle Routing & Scheduling - Janny Leung
Two-phased Heuristic Gillette & Miller’s Sweep Method (1974)
First assigns nodes to vehicles (cluster), then find best route for each cluster.
(a) Pickup stop data
(b) “Sweep” method solution
Geographical
region
Route #1
10,000 units
Pickup
points
Route #3
8,000 units
1,000
1,000
4,000
4,000
2,000
2,000
3,000
3,000
2,000
2,000
3,000
3,000
3,000
3,000
2,000
1,000
Depot
2,000
1,000
Depot
2,000
2,000
2,000
2,000
Route #2
9,000 units
2,000
2,000

14. Heuristic Principles for Good Routing and Scheduling (Ballou)
Vehicle Routing & Scheduling - Janny Leung
Heuristic Principles for Good Routing and Scheduling (Ballou)
Customers on a route should be in close proximity (“clustered”)
Routes for different days should give tight clusters and avoid overlap
Build routes beginning from farthest customer from depot
Use largest vehicle first
Pickups should be mixed into delivery routes
Consider alternate means for a customer isolated from others on same route
Tight time-windows should be avoided

15. Vehicle Routing & Scheduling - Janny Leung
Second Generation
Mathematical Programming Models and Heuristics

16. Travelling Salesman Problem

17. Vehicle Routing & Scheduling - Janny Leung
Generalised Assignment Model for Vehicle Routing (Fisher & Jaikumar [1981])
The vehicle routing problem can be represented exactly by the following nonlinear generalized assignment problem. Defining

18. Vehicle Routing & Scheduling - Janny Leung
Of course, we lack a closed form expression for f(yk). The generalized assignment heuristic replaces f(yk) with a linear approximation Si dik yik and solves the resulting linear generalized assignment problem to obtain an assignment of customers to vehicles.

19. Set Partitioning Model for Vehicle Routing
Vehicle Routing & Scheduling - Janny Leung
Set Partitioning Model for Vehicle Routing
The set partitioning heuristic begins by enumerating a number of candidate vehicle routes. A candidate route is defined by a set S Í {1,…,n} of customers to be delivered by a single vehicle and a delivery sequence for these customers. We index the candidate routes by j and define the following parameters.

20. Vehicle Routing & Scheduling - Janny Leung
There are many effective optimization algorithms for set partitioning.
The set partitioning approach will find an optimal solution if the candidate route list contains all feasible routes. In most situations, this would result in a set partitioning problem too large to be solvable, so one instead heuristically generate routes that are likely to be near-optimal for consideration.

21. Vehicle Routing & Scheduling - Janny Leung
Third Generation
Optimisation-based heuristics
AI techniques
human-machine interactive systems
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