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The use of Heuristics in the Design of GPS Networks

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

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  1. 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

  2. Topics • Aim • GPS Sessions and Schedule • Problem description • Formulation as a Travelling Salesman Problem • Examples • Simulated Annealing • Recommendations and conclusions

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

  4. 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.

  5. GPS Session • Sessions Required • A B • C B • C D • A D • A C • B D

  6. Sessions and Schedule Schedule: ab-ac-dc

  7. 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.

  8. 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.

  9. 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

  10. 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).

  11. 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.

  12. Two receiver problem: example • Four sessions: AB-BC-CD-DA

  13. 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.

  14. Further Developments • To incorporate base, introduce dummy point. • To allow observations over more than one working day: • Extra dummy points. • Connect dummy points.

  15. 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.

  16. 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.

  17. 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

  18. 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

  19. 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.

  20. Local and global optimum Cost Start Global optimum Local optimum Iterations

  21. 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!)

  22. 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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