The use of heuristics in the design of gps networks
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The use of Heuristics in the Design of GPS Networks. Peter Dare and Hussain Saleh School of Surveying University of East London Longbridge Road Dagenham, Essex, England Email: [email protected] Topics. Aim GPS Sessions and Schedule Problem description

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The use of Heuristics in the Design of GPS Networks

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The use of Heuristics in the Design of GPS Networks

Peter Dare and Hussain Saleh

School of Surveying

University of East London

Longbridge Road

Dagenham, Essex, England

Email: [email protected]


Topics

  • Aim

  • GPS Sessions and Schedule

  • Problem description

  • Formulation as a Travelling Salesman Problem

  • Examples

  • Simulated Annealing

  • Recommendations and conclusions


Aim

  • To develop a method to determine the cheapest schedule given the sessions to be observed.


GPS Session

  • For a GPS session 2 or more receivers observe simultaneously.

  • For a network we have a number of sessions.

  • With 2 receivers, 6 sessions required for this network.

  • List of sessions is a schedule.


GPS Session

  • Sessions Required

    • A B

    • C B

    • C D

    • A D

    • A C

    • B D


Sessions and Schedule

Schedule: ab-ac-dc


Problem Description

  • Given the list of sessions required, what is the optimum order of the sessions?

  • Need to define cost.

  • Cost can be defined, for example, by time of travel or shortest distance.

  • As optimum sought we aim to minimise the total cost incurred.


One receiver problem

  • Classic Travelling Salesman Problem (TSP) of Operational Research (OR).

  • Solved using Branch-and-Bound algorithm in Turbo Pascal to make use of pointers.

  • Limitations: Only one receiver; starts and ends at a point.

  • Developments: 2 or more receivers; start and end at non-survey point; allow for more than one observing day.


Example with one receiver

Cost to move

between B and C

Cost Matrix:

A B C D

A 0 5 6 3

B 5 0 4 1

C 6 4 0 3

D 3 1 3 0

Least-cost Solution: A-D-B-C-A

Cost: 14 units


Two Receiver Problem

  • For 2 receivers, cost is maximum of individual movements if time is criteria.

  • For example, cost of changing from session AC to BD is:

  • A to B: 5 units C to D: 3 units

  • Total cost: 5 units.

  • If distance is criteria, sum costs (e.g., total 8 units).


Two Receiver Problem

  • Need to allow reversal of sessions e.g., AC to DB. Cost is:

  • A to D: 3 units C to B: 4 units Total cost: 4 units.

  • However, now need to prevent receiver swaps.

  • For example, AC to CA.

  • Prevented by setting cost to infinity.


Two receiver problem: example

  • Four sessions: AB-BC-CD-DA


Solution to Two Receiver Problem

  • Modifications needed to standard TSP algorithm.

  • Solution (costing 9 units) is:

    • Rec. 1 A A D B A

    • Rec. 2 B D C C B

  • However, first and last sessions are duplicates!

  • Concept of base station needed.


Further Developments

  • To incorporate base, introduce dummy point.

  • To allow observations over more than one working day:

    • Extra dummy points.

    • Connect dummy points.


Example Survey - 1

  • Cost matrix: 20*20 400 elements not shown here!

  • Observed schedule:

  • Rec. 1 Day 1: 2 2 1 Day 2: 2 3 5 6 6

  • Rec. 2 Day 1: 3 4 4 Day 2: 1 4 4 5 3

  • Total time: 180 minutes.


Example Survey - 2

  • Optimal schedule:

  • Rec. 1 Day 1: 1 1 2 2 Day 2: 3 4 6 6

  • Rec. 2 Day 1: 2 4 4 3 Day 2: 4 5 5 3

  • Total time: 173 minutes.

  • But large cost matrix needed: 20*20.

  • To work with larger networks, approximate solutions (heuristics) needed.


Heuristics

  • Heuristics belong to the field of OR.

  • A Heuristic attempts to find near-optimal solutions in a reasonable amount of time.

  • The solution may be optimal but no guarantee.

  • Popular heuristics are:

    • Simulated annealing

    • Tabu search


Simulated Annealing (SA)

  • ‘Annealing’ - the cooling of material in a heat bath.

    • Solid material

    • Heated past melting point

    • Cooled back to a solid

    • Structure of new solid depends upon cooling rate


Application to Schedule Design - 1

No SA:

  • ‘Guess’ a schedule.

  • Change schedule to reduce cost.

  • Stop when no more improvements can be made.

  • Problem - local optimum often found - need global optimum.


Local and global optimum

Cost

Start

Global optimum

Local optimum

Iterations


Application to Schedule Design - 2

With SA:

  • ‘Guess’ a schedule.

  • Change schedule to reduce cost.

  • Allow some ‘uphill’ moves climb out of local optimum.

  • Stop when no more improvements can be made global optimum (hopefully!)


Recommendations and conclusions

  • Optimal solution obtainable for small networks. Heuristics for large networks.

  • Further development of non-optimal solutions:

    • simulated annealing; tabu search; genetic algorithms.

  • Incorporate with other aspects of network design.


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