Vehicle Routing

1
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
• 1930s Menger, Travelling salesman problem
• 1800s Hamiltonian circuit

2
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

• First generation (1950-60s)
• greedy, local improvement heuristics
• Second generation (1970-80s)
• mathematical programming models
• Third generation ?
• Artificial intelligence?
• Human-machine interactive?
• on-line optimisation-based heuristics?
• Need robust approach

4
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
B
90 minutes
84
84
A
I
66
138
120
132
C
90
F
348
60
126
H
156
126
132
48
48
J
Note All link times are in minutes
D
150
DESTINATION Fort Worth
G
5
Dijkstras 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
ORIGIN Amarillo
Oklahoma City
E
B
90 minutes
84
84
A
I
66
138
120
132
C
90
F
348
60
126
H
156
126
132
48
48
J
Note All link times are in minutes
D
150
DESTINATION Fort Worth
G
7
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
Vehicle Routing
9
Vehicle Routing
10
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
dA,B
dO,B
Depot
Depot
dB,O
B
B
dB,O
11
Vehicle Routing
12

• 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
B
X
B
X
A
B
A
13
Other tour construction heuristics

• Nearest Addition
• 1. Start with a tour of a single city, s1
  i1.
• 2. Find the nearest city to the tour. Solve
• Let the minimum be ckj where k ?S, j ?S. Let
  (i, j) be an edge incident to j on TSP tour.
• Replace edge (i, j) in tour by (i, k) and (k,
  j).
• (k is added to tour next to j).
• 3. Repeat from Step 2 until tour completed.

14
Other tour construction heuristics

• Nearest Insertion
• Select city k as nearest addition method. Insert
  it between cities i and h which minimizes
• cik ckh ?cih (increase in tour length).
• (After selecting k (which is closest to j),
  insert k anywhere in the tour to minimise
  increase in tour length.)

15
Other tour construction heuristics

• Nearest Merge
• 1. Start with each city as a subtour.
• 2. Find 2 subtours that minimize cij i ? T1, j
  ?T2
• 3. Merge
• (i)
• (ii)
• (iii)

16
Other tour construction heuristics

• Furthest Insertion
• 1. Select k to maximize distance to current
  subtour.
• 2. Insert k to minimize cik ckj ?cij
• (to minimize additional length of tour).

17
Other tour construction heuristics

• Euclidean Problems
• 1. Greatest angle insertion
• 2. Convex hull insertion
• - select k not on subtour to minimize cik ckj
  ?cij
• - break tie by

18
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
19
Practical Vehicle Routing Scheduling Problems

• fleet of vehicles
• vehicles capacitated
• time restrictions

20
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

21
Two-phased HeuristicGillette Millers 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
Route 2 9,000 units
22
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

23

• Second Generation
• Mathematical Programming Models and Heuristics

24
Travelling Salesman Problem
25
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

26

• 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.

27
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.

28

• 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.

29
Third Generation

• Optimisation-based heuristics
• AI techniques
• human-machine interactive systems
• Preview-Solve-Review