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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]

- 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.