Local Search Strategy to produce Schedules for a ... - Semantic Scholar

Local Search Strategy to produce Schedules for a GPS Surveying Network Hussain Aziz SALEH & Peter DARE Applied Geodesy Research Unit School of Surveying, University of East London, Longbridge Road, Dagenham, Essex RM8 2AS UK. E-mail:{h.saleh, p.j.dare} @uel.ac.uk. Tel: 020 8223 2545. Fax: 020 8223 2918.

Figure 1: GPS satellite constellation6.

Abstract This paper introduces a new formulation for GPS networks which is an important technique in Surveying. The objective is to find an effective receiver schedule at least cost using heuristic techniques within the field of Operational Research. To evaluate the solution, a comparison with known optimal solution is made. Large networks (with unknown optimal schedules) were solved using Simulated Annealing and Tabu Search heuristic techniques. Further work is to solve these large networks and compare with the solution from heuristic techniques such as Genetic Algorithms. Key words: Global Positioning System (GPS), Network, Simulated Annealing (SA), Tabu Search (TS), Travelling Salesman Problem (TSP) and Combinatorial Optimization Problem (COP)

The YOR11 Conference, University of Cambridge, Cambridge, UK 28-30 March, 2000

Tutorial and keynote papers

Introduction This paper describes the development of a user-friendly method based on an effective computer based heuristic technique that have been applied to analyse and improve the efficiency of schedules for a Global Positioning System (GPS) surveying network. GPS, which is a 24 satellite constellation (see Figure 1), provides the user with a highly accurate three dimensional position, velocity and time information on a 24-hour basis and in all weather conditions at any location on the globe. The impact of GPS on surveying work can be seen in determining locations and changes in locations with high accuracy within short observational periods and over long distances. This can be done without any need of a requirement of intervisiblity between stations which was essential with traditional surveying methods36. On the other hand, GPS equipment is very expensive compared with other equipment, and cost becomes crucial when the amount of work increases (i.e. for a large network). Designing GPS surveying networks to optimality (or close to it) can be formulated as Combinatorial Optimization Problem (COP)31, 32, 4. Exact methods can solve only small networks and are not practical as the size of the network increases. Hence, to provide a method to solve large network, Heuristic Techniques have been implemented. The paper is organized as follows. Firstly, the problem formulation and solution representation using a simplified small network will be analysed to highlight the stages of the scheduling problem and a proposed constructive procedure used leading to the initial schedule design. Secondly, a basic strategy of Local Search Descent (LSD) method is illustrated thoroughly and used as a search base in Simulated Annealing (SA) and Tabu Search (TS) heuristic techniques to obtain the cheapest schedule. Thirdly, the results of the known optimal schedule are compared with the obtained results from the developed heuristics applied to the same network. Also, some examples for large GPS networks with unknown optimal schedules are used to evaluate these developed techniques. Finally, the paper is concluded with a summary and directions for further research.

Problem formulation and solution representation A network in GPS can be defined as a set of stations (a, b, c, d .etc) which are coordinated by placing receivers (X, Y, Z. etc) on them to determine the sessions (ab, ac, dc. etc) between them as shown in Figure 2. The minimum number of receivers is two, and the problem of network design becomes crucial as the number of receivers increases. In general, if there are R receivers observing simultaneously then there are (R-1) independent sessions that can be used in the computation. The problem addressed is to determine the order in which each GPS receiver should be moved between the points to be surveyed in an efficient manner taking in consideration some important factors such as time, cost etc. From the above a proper design for a GPS survey network can be guaranteed by good organization of fieldwork components such as: 1. Location of stations: With GPS, stations are generally located where they are needed, unlike in the traditional methods, provided good tracking is ensured (i. e., the satellites are above a 15o elevation, ground stability, security etc). 2. Type and number of receivers: Depending on the manufacturer (Trimble, Leica, etc.). 3. Required sessions to be observed: A session (for example ab as shown in Figure 2) can be defined as a period of time which two or more receivers are simultaneously 88



recording satellite signals. If we have N points in the network and R receivers for use, then the maximum number of sessions is given by Umix=N!/(N-R)!, while the minimum number of sessions to make the network solvable is Umin=Integer (0.99+N)/(R-1)15. The actual number of session to satisfy the requirement is {(2-2.5) Umin}3. 4. Length of session observations: The length of time to observe a session is dependent on a number of factors including the geometry of satellites, day or night observation and distance between points etc34. 5. The initial starting schedule: Scheduling may be seen as the arrangement of objects into a pattern in time or space in such a way that some goals are achieved, or nearly achieved, and that constraints on the way the objects may be arranged are satisfied, or nearly satisfied. In GPS surveying a schedule will usually include objects that represent all the necessary information to carry out the observation of the whole network. This may include times at which session observations are to take place and work plans for individual personnel or receivers. The pattern may be an ordering of sessions defined in terms of observation time to be carried out during the periods as shown in Figure 2 or a structure of routes as shown in Figure 11. The formulation of a GPS network as a COP is generally specified by a universe of potential solutions Q. If V represents a schedule consisting of the required number of sessions U, then the number of possible schedules or the size of the solution space Q is given by V! (which is clearly a very large number for some of the networks). The challenge is to try to determine the optimal schedule giving the lowest cost. The following notations are used for representing the GPS network designing problem: N : the number of stations; R : the number of receivers; U : the number of observed sessions; PRi : the initial position of receiver i; Si : the set of stations visited by receiver i; Cab : the cost of moving receiver i from station a to station b; C(Si) : the total cost of the route of receiver i; V : the feasible starting schedule; C(V) : the total sum of each individual route C(Si) for all i ∈ R; Q : the universe of potential schedules Q={1,….,V}; I : size of the neighbourhood. The aim is to determine an optimal schedule or close to it that minimizes the total cost of observing the whole network and satisfies the heuristic and GPS requirements:

Minimise: Subject to:

C (V)

V ∈ I, r

US

i

I ⊆ Q;

≥ Ν;

i =1

C(V) =

∑ C(S i∈S i

89

i

)

∀i ∈ R ;



Data transmission by GPS satellites

Session observations by GPS receivers and creation of the GPS Network

No.

Session

Receiver X at: Station

Moving

Receiver Y at: Station

Cost

Receiver Z

Moving

Session Cost

Cost

1

ab

a

[- -]

b

[- -]

2

ac

a

[0]

c

[Cbc]

Cbc

3

dc

d

[Cad]

c

[0]

Cad

4

eg

e

[Cde]

g

[Ceg]

Cde + Ceg

The Total Cost of the Schedule

Figure 2: Sessions observation using GPS receivers. 90

--

Σ Cij



Determining the optimal schedule V with an objective function using the actual cost matrix is the minimization of the total cost (time spent) by all receivers in order to observe all required sessions. The original cost matrix represents the cost of moving a receiver from one point to another and computed based upon the criteria to be minimized. For example, as shown in Figure 2, the cost of measuring session ac is obtained by the cost associated by moving receiver Y from station b to station c where the receiver X is still at station a. Also, the cost of measuring session cd is obtained by the cost associated by moving receiver X from station a to station d where receiver Y still at station c. For more details about the evaluation of the cost matrix see Dare3. In this section, a detailed example with a manually generated schedule is given to demonstrate the concepts of GPS problem within the frame of COP. This example as shown in Figure 3 represents a small network coordinated by placing a number of receivers R={R1, R2} on a number of stations N={A, B, C, D} to determine sessions U={U1, U2, U3, U4, U5, U6} between them. In this example, R=2, N= 4, U=6, then the maximum number of sessions is given by Umax=4!/(4-2)! ⇒ Umax = 12 sessions, while the minimum number of sessions to make the network solvable is Umin = Integer (0.99+4)/(21) ⇒ Umin = 4 sessions. In practice, the actual number of sessions that would likely be used to satisfy the precision and reliability requirements would U =(2-2.5) Umin ⇒ U = 10 sessions. In this small network Q =12!, i.e., there are 47900160 schedules. U6 B

C U4

U2

U1

U5

A U3 D Figure 3: Simple four station network3. Session U1 U2 U3 U4 U5 U6

Receiver 1 A A A B C C

Receiver 2 B C D D D B

Table 1: Receiver schedule for two receivers3.

There are difficulties in the design of GPS surveying networks to establish good schedules. Currently, an experienced surveyor in GPS surveying creates the schedule 91



manually using intuition and experience on a day-to-day basis. In this paper, a constructive procedure is proposed to obtain an initial feasible schedule using an effective computer method. The aim of this procedure, which can be implemented as a greedy local search method, is to generate quickly an initial schedule. Greedy methods are an important class of single pass heuristics in which successive steps are taken so as to minimize the immediate cost. Further, a feasible schedule is not found until the end of the constructive procedure. To represent this proposed constructive procedure more realistically, the following stages are adopted to construct the initial schedule: Stage 1: Represents the initial situation for carrying out the survey and includes: D ={PR1,…,PRr}= r, the visited stations (the initial locations of receivers). D’={1,…,n}-{ PR1,…,PRr}= n-r, unvisited stations. Stage 2: Represents the procedure adopted in moving the receivers from PRr (current location) to the nearest stations (new locations) provided that the receivers work closely satisfying the requirements of surveying. D’={n-2r}. Stage 3: Repeat stage 2 to cover all stations and observe all the sessions. Stage 4: The resulting schedule represents a number of routes which have been selected to produce the initial solution containing all the required sessions to complete the survey. The above stages of the above proposed procedure can be illustrated in Figure 4. Receivers

All the possible movements

All the stations in the designed network.

All the possible remaining movements Figure 4: The hierarchical structure of receiver’s movements through the network.

Receiver movement is an integral part of a GPS’s total logistics in designing a surveying network. When studying a large GPS surveying networks, the corresponding mathematical model for the receiver movements can be very large and the uncertainty in the data which implies that a detailed model might be too ambitious etc. This can give rise to difficulties in computations and this in turn will effect the ability to obtain good results. Figures (5-10) show the large size of the obtained data through the stages of receiver’s movements during the observing of a small network depicted in Figure 11, where each receiver has 4 possible movements. To solve the above problem more effectively by reducing the size of computation, a typical multi-sessions observation system has been adopted in this research. This system, which enables consolidation of receiver’s movements into the chosen sessions, is a three steps hierarchical structure consisting of receivers, all the possible movements and stations as shown in Figure 4. Firstly, assembling of the receiver’s 92



movements in sessions giving the initial starting schedule as shown in Table 1. Secondly, using heuristics techniques to optimize the initial schedule into an optimal schedule. Thirdly, using the sessions of the obtained optimal schedule, produce the optimal routes for the receivers as shown in Figure 11. The stages of session observations applied to the initial schedule of Table 1 can be illustrated as shown in Figures (5-10) as follows: •

Measuring session U1 : The initial position of the two receiver R1(A), and R2(B) are at stations A and B respectively with all the possible movements as shown in Figure 5. R1 (A)

R2 (B)

NA NB NC ND (R1, A) (R1, B) (R1, C) (R1, D) (R2, A) (R2, B) (R2, C) (R2, D) Figure 5: The initial position of receivers. •

Measuring session U2: The two receiver’s movements R1(A), and R2(B-C) are assembled in session U2 and the current position of receivers with all the remaining possible movements. R1 (A)

R2 (B, C)

A B C D ( ) (R1, B) (R1, C) (R1, D) (R2, A) (R2, B) (R2, C) ( ) Figure 6: Measuring session U2. • Measuring session U3: The two receiver’s movements R1(A), and R2(B-C-D) are assembled in session U3 and the current position of receivers with all the remaining possible movements. R1 (A)

R2 (B,C,D)

A

B C D ( ) (R1, B) (R1, C) (R1, D) (R2, A) ( )( )( ) Figure 7: Measuring session U3. 93



• Measuring session U4: The two receiver’s movements R1(A-B), and R2(B-C-D) are assembled in session U4 and the current position of receivers with all the remaining possible movements. R1 (A,B) R2 (B,C,D)

A

B

( )( (R2, A) (

C D ) (R1, C ) (R1, D) ) ( )( )

Figure 8: Measuring session U4. • Measuring session U5: The two receiver’s movements R1(A-B-C), and R2(B-C-D) are assembled in session U5 and the current position of receivers with all the remaining possible movements. R1 (A,B,C)

R2 (B,C,D)

A B C D ( ) ( ) ( ) (R1, D) (R2, A) ( ) ( ) ( ) Figure 9: Measuring session U5.

• Measuring session U6: The two receiver’s movements R1(C-OB), and R2(D-OB) are assembled in session U6 and the current position of receivers with all the remaining possible movements. R1 (A,B,C) R2 (B,C,D,B)

A ( ) ( (R2, A) (

B

C ) ( ) (

D ) (R1, D) ) ( )

Figure 10: Measuring session U6. •

Finally, the above measurements of the required sessions can be sequenced in two routes: 94



S1 represents the stations visited by the receiver R1 and S2 represents the stations visited by R2 as shown below in Figure 11. In general, the term “sequence” is an order in which activities of problem are carried out. In GPS surveying, the sequence is the order in which sessions are observed using the receivers. The problem of sequencing sessions in these circumstances is known as a flow shop problem.

R1

A

A

A

B

C

C

B

C

D

D

D

B

Session 1

Session 2

Session 3

Session 4

Session 5

Session 6

R2

S1

S2

Figure 11: The initial sequences includes the two routes of receivers S1, S2 and observed sessions for the whole network.

Strategy of the Local Search Decent (LSD) method To understand the heuristic strategy search, the LSD method is introduced. This method forms a general class of the well-known heuristics based on the concept of searching the local neighbourhood of the current schedule35, 14. It attempts to improve an arbitrary initial schedule V to a GPS network by a series of local improving changes (swapping sessions)17. LSD (move-by-move method) perturbates the given initial schedule V to generate different neighbourhoods I(V). Neighbouring schedules are generated by “a move generation mechanism”. A move-generation is a transition from a schedule V to another one V’∈I(V) in one step (or iteration). These schedules are selected and accepted according to some pre-defined criteria25. The returned schedule V may not be optimal, but it is the best schedule in its local neighbourhood I(V). A local optimal schedule is a schedule with the local minimum possible cost value and it is called local optima43. The basic steps for the LSD are depicted in Figure 12, which returns a schedule V. • • • • •

Select a given schedule V∈ I(V) and compute its cost value C(V). Generate a schedule V’∈ I(V) and compute its cost value C(V’). If C(V’) < C(V) then, V’ replaces V as a current schedule. Otherwise, retain V and generate other moves until C(V’)