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

Peter Dare and Hussain Saleh

School of Surveying

University of East London

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.
• 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.
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.