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

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

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

4 GPS Session u For a GPS session 2 or more receivers observe simultaneously. u For a network we have a number of sessions. u With 2 receivers, 6 sessions required for this network. u List of sessions is a schedule.

5 GPS Session u 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 u Given the list of sessions required, what is the optimum order of the sessions? u Need to define cost. u Cost can be defined, for example, by time of travel or shortest distance. u As optimum sought we aim to minimise the total cost incurred.

8 One receiver problem u Classic Travelling Salesman Problem (TSP) of Operational Research (OR). u Solved using Branch-and-Bound algorithm in Turbo Pascal to make use of pointers. u Limitations: Only one receiver; starts and ends at a point. u 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 Matrix: A B C D A B C D Least-cost Solution: A-D-B-C-A Cost: 14 units Cost to move between B and C

10 Two Receiver Problem u For 2 receivers, cost is maximum of individual movements if time is criteria. u For example, cost of changing from session AC to BD is: u A to B: 5 units C to D: 3 units u Total cost: 5 units. u If distance is criteria, sum costs (e.g., total 8 units).

11 Two Receiver Problem u Need to allow reversal of sessions e.g., AC to DB. Cost is: u A to D: 3 units C to B: 4 units Total cost: 4 units. u However, now need to prevent receiver swaps. u For example, AC to CA. u Prevented by setting cost to infinity.

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

13 Solution to Two Receiver Problem u Modifications needed to standard TSP algorithm. u Solution (costing 9 units) is: –Rec. 1 A A D B A –Rec. 2 B D C C B u However, first and last sessions are duplicates! u Concept of base station needed.

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

15 Example Survey - 1 u Cost matrix: 20* elements not shown here! u Observed schedule: u Rec. 1 Day 1: Day 2: u Rec. 2 Day 1: Day 2: u Total time: 180 minutes.

16 Example Survey - 2 u Optimal schedule: u Rec. 1 Day 1: Day 2: u Rec. 2 Day 1: Day 2: u Total time: 173 minutes. u But large cost matrix needed: 20*20. u To work with larger networks, approximate solutions (heuristics) needed.

17 Heuristics u Heuristics belong to the field of OR. u A Heuristic attempts to find near-optimal solutions in a reasonable amount of time. u The solution may be optimal but no guarantee. u Popular heuristics are: F Simulated annealing F Tabu search

18 Simulated Annealing (SA) u ‘Annealing’ - the cooling of material in a heat bath. F Solid material F Heated past melting point F Cooled back to a solid F Structure of new solid depends upon cooling rate

19 Application to Schedule Design - 1 No SA: u ‘Guess’ a schedule. u Change schedule to reduce cost. u Stop when no more improvements can be made. u 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: u ‘Guess’ a schedule. u Change schedule to reduce cost. u Allow some ‘uphill’ moves climb out of local optimum. u Stop when no more improvements can be made global optimum (hopefully!)

22 Recommendations and conclusions u Optimal solution obtainable for small networks. Heuristics for large networks. u Further development of non-optimal solutions: –simulated annealing; tabu search; genetic algorithms. u Incorporate with other aspects of network design.