# 4 Vehicle Routing 1

8/2/2019 # 4 Vehicle Routing 1

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Material Handling & Transportation

Issues


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Shared Transportation Capacity

Large shipments reduce transportationcosts but increase inventory costs

EOQ trades off these two costs

Reduce shipment size withoutincreasing transportation costs?

Combine shipments on one vehicle


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TL vs LTL

Inventory

Transport

Transport

Inventory


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Shared Loads

Inventory

Transport

Transport?

Inventory

?


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Issues

Design Routes that

Minimize the transportation cost

Respect the capacity of the vehicle

This may require several routes

Consider inventory holding costs

This may require more frequent visits


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Vehicle Routing Problem


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Classical VRP

ncustomers must be served from a singledepot utilizing vehicles with capacity Qfor

delivering goods Each customer requires a quantity qi Qof

goods

Customer orders cannot be split


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Graphical Representation

Depot


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Additional Features

Depots Multiple locations

Vehicles Multiple vehicle

types and capacities

Release, maximumand down times

Customers Time windows (soft

or hard)

Accessibilityrestrictions

Priority Pickup and delivery

Routes Maximum time Link costs

Objective Functions Minimize total

traveled distance Minimize total

traveled time Minimize number of

vehicles

Maximize quality ofservice

Multiple objectivefunctions


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Vehicle Routing & Scheduling

Model

Problem Variety Pure Pickup or Delivery Problems

Mixed pickups and deliveries Pickup-Delivery Problems

Backhauls

Complications

Simplest Model: TSP


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Vehicle Routing

Find best vehicle route(s) to serve a set of ordersfrom customers.

Best route may be

minimum cost, minimum distance, or

minimum travel time.

Orders may be Delivery from depot to customer.

Pickup at customer and return to depot.

Pickup at one place and deliver to another place.


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Model

Nodes: physical locations

Depot.

Customers.

Arcs or Links

Transportation links.

Number on each arc

represents cost, distance,or travel time.

depot

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4

6

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5

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4

8

3


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Pure Pickup or Delivery

Delivery: Load vehicle at depot. Design route todeliver to many customers (destinations).

Pickup: Design route to pickup orders from

many customers and deliver to depot.

Examples:

UPS, FedEx, etc.

Manufacturers & carriers.

Carpools, school buses, etc.

depot


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TSP & VRP

TSP: Travelling Salesman Problem One vehicle can deliver all orders.

VRP: Vehicle Routing Problem

More than one vehicle is required to serve all orders.

VRP

depot

TSP

depot


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Mixed Pickup & Delivery

Interspersed

depot

Pickup

Delivery

depot

Not Interspersed

depot

Separate routes


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Interspersed Routes

depot

Pickup

Delivery

For clockwise trip:

Load at depot

Stop 1: Deliver A

Stop 2: Pickup B

Stop 3: Deliver C

Stop4: Deliver D

etc.

A

B

ED

C

F

G

H

I

J

K

L

ACDFIJK

CDFIJK

BCDFIJK

BDFIJK Delivering C requires moving B

BFIJK Delivering D requires moving B


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Pickup-Delivery Problems

Pickup at one or more origin and delivery to oneor more destinations.

Often long haul trips.

depot

A

B

C

Pickup

Delivery

A

B

C


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Intersperse Pickups and Deliveries?

Can pickups and deliveries be interspersed?

depot

A

B

C

Pickup

Delivery

A

B

C

Interspersed

depot

A

B

C

A

B

C

Not Interspersed


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Backhauls

If vehicle does not end at depot, should it returnempty (deadhead) or find a backhaul?

How far out of the way should it look for a backhaul?

Pickup

Delivery

depot

A

B

CA

B

C

DD


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Backhauls

Compare profit from deadheading and carryingbackhaul.

Pickup

Delivery

depot

A

B

CA

B

C

D D

Empty Return

Backhaul


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More Complications

Time windows for pickup and delivery. Hard vs. soft

Compatibility

Vehicles and customers. Vehicles and orders.

Order types.

Drivers and vehicles.

Driver rules (DOT)

Max drive duration = 10 hrs. before 8 hr. break.

Max work duration = 15 hrs. before 8 hr break.

Max trip duration = 144 hrs.


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Simple Models

Homogeneous vehicles.

One capacity (weight or volume).

Minimize distance.

No time windows or one time window percustomer.

No compatibility constraints.

No DOT rules.


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Simplest Model: TSP

Given a depot and a set of n customers, find atour (route) starting and ending at the depot,that visits each customer once and is of minimumlength.

One vehicle.

No capacities.

Minimize distance.

No time windows.

No compatibility constraints.

No DOT rules.


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Multiple Routes

Capacitated VRP: vehicles have capacities. Weight, Cubic feet, Floor space, Value.

Deadlines force short routes.

Pickup at end of day. Deliver in early morning.

Time windows

Pickup. Delivery.

Hard or Soft.


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Multiple Route Solution Strategies

Find feasible routes.

Cluster first, route second.

Cluster orders to satisfy capacities.

Create one route per cluster. (TSP for each cluster)

Route first, cluster second.

Create one route (TSP).

Break route into pieces satisfying capacities.

Build multiple routes simultaneously.

Improve routes iteratively.


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Heuristics that GrowFragments Strip Heuristic

Nearest neighbor

Double-endednearest neighbor

Heuristics that GrowTours Nearest addition

Farthest addition

Random addition

Heuristics Based onTrees Minimum spanning

tree

Christofides heuristic

Fast recursivepartitioning

Construction Heuristics


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The Strip Heuristic

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The Strip Heuristic Partition the region into narrow strips Routing in each strip is easy ~ 1-Dimensional Paste the routes together


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Sweep Algorithm

Draw a ray starting from the depot.

Sweep clockwise (or counter-clockwise) and addcustomers to the route as encountered.

Start a new route when vehicle is full.

Re-optimize each route (solve a TSP forcustomers in each route).


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1. Randomly select a starting node

2. Add to the last node the closest node until no morenodes are available

3. Connect the last node with the first node

O(n2) running time

Nearest Neighbor

Double-ended Nearest Neighbor

Conceptually the same as nearest neighbor heuristic

The fragment is allowed to grow from both ends


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Nearest Neighbor

Add nearest customer to end of the route.

depot

1

depot

2

depot

3


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Nearest Neighbor

Add nearest customer to end of the route.

depot

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depot

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depot

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Nearest Neighbor

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Nearest Neighbor Algorithm

depot

Suppose each vehicle capacity = 4 customers

depot

First route


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Third routedepot

Nearest Neighbor Algorithm


depot

Second route


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1. Select a node and thenode nearest to it andbuild a two-node tour

2. Insert in the tour theclosest node yuntil nomore nodes areavailable

O(n2) running time

x

y

z

Minimize d(x,y) + d(y,z)d(x,z)

Nearest Insertion


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Nearest Insertion

Insert customer closest to the route in the bestsequence.

depot

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depot

2

depot

3


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Nearest Insertion

Insert customer closest to the route in the bestsequence.

depot

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depot

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depot

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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Nearest Insertion

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1. Select a node and the node farthest toit and build a two-node tour

2. Insert in the tour the farthest node yuntil no more nodes are available

O(n2) running time

Farthest Insertion


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This heuristic adds points to the tour ina random order

Random Insertion


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Minimum Spanning Tree

Construct a minimum spanning tree

Traverse the tree to build a tour byeliminating edges from vertices with degree

three and adding edges to vertices withdegree one


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Build a minimum spanning tree on theedges between customers

Double the tree to get an Eulerian Tour(visits everyone perhaps several times andreturns to the start)

Short cut the Eulerian Tour to get aHamilton Tour (Traveling Salesman Tour)



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Is Easy to construct

Use the Greedy Algorithm

Add edges in increasing order of length

Discard any that create a cycle

Is a Lower bound on the TSP

Drop one edge from the TSP and you

have a spanning tree It must be at least as long as the minimum

spanning tree



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Double the Spanning Tree

Duplicate each edge in the Spanning Tree

The resulting graph is connected

The degree at every node must be even

Thats an Eulerian Graph (you can start at a

city, walk on each edge exactly once and

return to where you started) Its no more than twice the length of the

shortest TSP



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Short Cut the Eulerian Tour

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S


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Short Cut the Eulerian Tour

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Christofides Heuristic

Construct an MST[Min. Spanning Tree]

Construct a matching of all vertices with odd

degree Combine the matching edges with the MST

edges, so that now all vertices have an evendegree

Compute an Eulerian tour through the graph


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Clarke G. and J. W. Wright (1964) Scheduling of vehicles from a central depot to

a number of delivery points, Operations Research, vol. 12, pp. 568-581.

Start with an initial solution where each customer is serviced

individually from the depot

1

2

3

0

Savings Method


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Savings Calculation

i j

0

c0i cj0

cij

sij= ci0 + c0j- cij

Savings Method


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Procedure

1. Compute savings sij for all pairs2. Choose the pair with the largest savings

and join customers in a new route iffeasible.

3. If all positive savings have been examined,stop. Otherwise go to 2.

Savings Method


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Savings Method

Start with separate one stop routes from depot toeach customer.

Calculate all savings for joining two customers

and eliminating a trip back to the depot. Sij= Ci0 + C0j- Cij

Order savings from largest to smallest.

Form route by linking customers according tosavings.


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Savings Method

depot

1 2 3

4

Remove Add

depot

1 23

4

Savings = S12 = C10+C02 -C12


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Savings Method

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depot

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3

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5

depot

S121 2

3

4

6

depot

S13

1 2

3

4

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S14

depot

1 24

6

3

depot

S15

1 24

6

3

5

depot

S16

1 24

6

3

5

Small savings


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Savings Method

5 5

depot

S241 2

3

4

6

depot

S25

1 2

3

4

6

5

5

depot

S26

1 24

6

3

Large savings

depot

1 2

3

4

6

S23

Large savings

3

5

depot

S34

1 24

6depot

S35

1 24

6

3

5


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Savings Method

depot depot depot

depot

S451 2

3

4 5

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1 2

3

45

6

S46

S56

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3

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6

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1 24

5

6

3

Large savingsS36

In general, with n customers thereare n(n-1)/2 savings to calculate.


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Savings Method

Order savings from largest to smallest. S35

S34

S45

S36

S56

S23

S46

S24

S25

S12

S26

S13

etc.

h d


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Savings Method

depot

1 2

3

4 5

6

Form route by linking customers according tosavings.

S35:link 3&5

0-3-5-0

depot

1 2

3

45

6

S34:link 3&4 (keep 3-5)

0-4-3-5-0

S i h d


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Savings Method

Form route by linking customers according tosavings. S35 0-3-5-0

S34 0-4-3-5-0

S45

S36

S56

S23

S46

S24

S25

S12

S26

S13

etc.


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S i M th d


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Savings Method

depot

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3

4

6

5

S23: skip

depot

S46: skip

1 2

3

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6

5

depot

1 2

3

4

6

5

S24:link 2&4


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S i M th d


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Savings Method

Form route by linking customers according tosavings. S35 0-3-5-0

S34 0-4-3-5-0

S45 skip

S36 skip

S56 0-4-3-5-6-0

S23 skip

S46 skip

S24 0-2-4-3-5-6-0

S25 skip

S12 0-1-2-4-3-5-6-0

Diff t A h


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Different Approaches

Route First - Cluster Second

Build a TSP tour

Partition it to meet capacity

Cluster First - Route Second

Decide who gets served by eachroute

Then build the routes

R t Fi t


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Route First

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Vehicle Cap: 15

Cl t Fi t


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Cluster First

Sweep Heuristic

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Vehicle Cap: 15

Cluster Algorithms


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Cluster Algorithms

Select certain customers as seed points forroutes.

Farthest from depot.

Highest priority.

Equally spaced.

Grow routes starting at seeds. Add customers:

Based on nearest neighbor or nearest insertion

Based on savings.

Based on minimum angle.

Re-optimize each route (solve a TSP forcustomers in each route).

Cluster with Nearest Neighbor


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depot


Select 3 seeds

depot

Add nearestneighbor



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depot


Add nearestneighbor

depot

Add nearestneighbor

Cluster with Minimum Angle


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depot


Select 3 seeds

depot

Add customers withminimum angle



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depot



depot


Improvement Heuristics


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depot

Starting routes

depot

Improved routes

Intra-route

improvementInter-route

improvement

VRP Route Improvement Heuristics


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VRP Route Improvement Heuristics

Start with a feasible route.

Make changes to improve route. Exchange heuristics within a route

Switch position of one customer in the route. Switch 2 arcs in a route. Switch 3 arcs in a route.

Exchange heuristics between routes. Move a customer from one route to another. Switch two customers between routes.

Local search methods. Simulated Annealing. Tabu Search. Genetic Algorithms.

K opt Exchange


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K-opt Exchange

Replace k arcs in a given route by k new arcs sothe result is a route with lower cost.

2-opt: Replace 4-5 and 3-6 by 4-3 and 5-6.

depot

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Original route

depot

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Improved route

2 opt Exchange


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2-Opt

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2-opt Exchange

3 opt Exchange


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3-opt Exchange

3-opt: Replace 2-3, 5-4 and 4-6 by 2-4, 4-3 and5-6.

depot

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Original route

depot

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Improved route



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depot


depot

Optimized routesStarting routes

Documents

# 4 Vehicle Routing 1