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The use of Heuristics in the Design of GPS Networks - PowerPoint PPT Presentation

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: Peter@uel.ac.uk. Topics. Aim GPS Sessions and Schedule Problem description

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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

Dagenham, Essex, England

Email: Peter@uel.ac.uk

• Aim

• GPS Sessions and Schedule

• Problem description

• Formulation as a Travelling Salesman Problem

• Examples

• Simulated Annealing

• Recommendations and conclusions

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

• 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.

• Sessions Required

• A B

• C B

• C D

• A D

• A C

• B D

Schedule: ab-ac-dc

• 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.

• 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.

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

• 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).

• 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.

• Four sessions: AB-BC-CD-DA

• 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.

• To incorporate base, introduce dummy point.

• To allow observations over more than one working day:

• Extra dummy points.

• Connect dummy points.

• 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.

• 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 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

• ‘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

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.

Cost

Start

Global optimum

Local optimum

Iterations

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!)

• 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.