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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

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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)

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Vehicle Routing: Theory and Practice• First generation (1950-60’s)

– greedy, local improvement heuristics

• Second generation (1970-80’s)– mathematical programming models

• Third generation ?– Artificial intelligence?– Human-machine interactive?– “on-line” optimisation-based heuristics?

Need “robust” approach

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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.

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OklahomaCity

Note: All link timesare in minutes

66

84

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48

150

126

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Dijkstra’s algorithm (1959)

1. Initially, set e0 = 0 and ej= for all other nodes j. Let R= .

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

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AB

C

FH

D G

I

J

E90 minutesORIGINAmarillo

DESTINATIONFort Worth

OklahomaCity

Note: All link timesare in minutes

66

84

138

348

120

156

84

132

132

60

48

150

126

48

126

90

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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

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Lin-Kernighan (1965, 1973)Local improvement heuristics• 2-opt • 3-opt

• k-opt

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Vehicle RoutingClarke-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.

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• 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 AB A

BX

A B

X

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Practical Vehicle Routing & Scheduling Problems

• fleet of vehicles

• vehicles capacitated

• time restrictions

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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

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Two-phased HeuristicGillette & Miller’s Sweep Method (1974)First assigns nodes to vehicles (cluster), then find

best route for each cluster.

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(a) Pickup stop data (b) “Sweep” method solution

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Pickuppoints

Route #110,000 units

Route #29,000 units

Route #38,000 units

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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

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Second Generation Second Generation

Mathematical Programming Models and Mathematical Programming Models and HeuristicsHeuristics

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Travelling Salesman Problem

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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

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• Of course, we lack a closed form expression for f(yk). The generalized assignment heuristic replaces f(yk) with a linear approximation i dik yik and solves the resulting linear generalized assignment problem to obtain an assignment of customers to vehicles.

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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 {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.

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• 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.

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Third Generation

• Optimisation-based heuristics

• AI techniques

• human-machine interactive systems

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