operational research for engineering applications. optimization methods for logistics design of gps...

64th Meeting of the EURO Working Group on Multicriteria Decision Aiding", Larissa, Greece

September 28-30, 2006

Operational Research for Engineering Applications.

Optimization Methods for Logistics Design of GPS

Network Design.

Dr. Vassilios. N. Pagounis*, George Ch. Stavridis

**

*Associate Professor, Department of Surveying, TEI of Athens, Greece,

[email protected]

** Surveying Engineer TEI of Athens, Student of MSc Project Management City

University of Washington State & TEI of Piraeus, Greece, [email protected]

Abstract

This study is a tune up for the description of a technique for the cost reduction of carrying out

an epoch survey. The increased use of GPS (Global Positioning System) in surveying

engineering applications as an accurate, fast and cost saver method has led to techniques applications within the field of operational research. An epoch survey design consists of a list

of sessions to be observed. Cost saver sessions schedules are identified given the required sessions and the cost of moving receivers between points.

Two cases are being investigated. At the first case a small network is being studied, using

optimal solution algorithm and nearest neighbor algorithm. The second case considers a larger

network, using nearest neighbor algorithm. Furthermore, there are being proposed algorithms

capable to produce session schedules. The above methods are developed in the software

Voyager v1.0 programmed in C++.

Finally, conclusions are drawn and recommendations for further study are suggested.

Keywords: operations research, heuristics, GPS networks, nearest neighbor, optimization

problems

1. Introduction

The Problem of the study is a specialized Travelling Salesman Problem (TSP). The

TSP deals with the problem that a salesman is trying to resolve during the planning of

his visits to a definite number of cities (only one visit per city) following the optimal

route or path, beginning from an office (base) and ending to it.


September 28-30, 2006 6

For this study, two GPS receivers are used, and the points that have to be observed. In

order to accomplish a session, both receivers must observe simultaneously (one point

each receiver per session). If each pair of the points that have to be observed gets

connected with an imaginary line, then there are represented the sessions or the

cities of the problem (assuming each line as node. See ch. 2). The starting ending office (office / base or OB) represents additional options of movement of the

GPS receivers (from benchmark point 1 to OB, from benchmark point 2 to OB, etc.)

and not sessions. The receivers must both start and end to the OB. The pair of the

receivers represents the salesman.

Figure 1. A small network consisting of four benchmark points and OB.

2. Definition of Terms

Movement: the transmission of the resources (human and equipment) from one point

(or one pair of points, or office, or city) to another. For transmitting from a Session to another, the Movement 1234 means: transmission of the resources (people and GPS receiver I) from benchmark point 1 to 3, and transmission of the resources

(people and GPS receiver II) from benchmark point 2 to 4.

Cost of Movement (or just Cost): The value that represents the effort, the size, the financial cost etc. or the compilation of some or all of the above for the realization

of a movement. Example: C12C34 = C12+ C34

Schedule: The Schedule represents the solution to the problem. It consists of

movements with definite order. In other words it gives the order of the sessions.

Node or City: Wherever the two receivers stay for a period of time (session or OB)

3. Limitations of the Methodology

The software (and also this study) is limited to calculate schedules according to the

movements that derive from the sessions and the OB.

For example: let it be 3 benchmark points named 1, 2, 3, then, three sessions are

automatically chosen: 12, 13, 23 (it is assumed that if session 12 is carried out,

session 21 is excluded. The movements are calculated according to the paths that are

resulted by the above three cities. This means that the movements of the receivers

Dr. Vassilios N. Pagounis, George Ch. Stavridis / Operational Research for Engineering Applications.

Optimization Methods for Logistics Design of GPS Network Design. 7

resulted by 21, 31, 32 are excluded. (For example 1213 1231, Due to exclusion of 31, the possibility to use 1231 is missed.)

4. The Mathematic Formulation of the Problem / Logistics

Design Problem

Type1: When the cost is independed from the direction of the movement, e.g.:

13, 31 (C1,3 = C3,1) then, a network with symmetrical cost is formed, and the number of the movements is given by:

2

)1(

NNS

(1)

(N represents the number of the points (benchmark points) of a geodetic or

topographic network is represented).

Important: For this Network Type S equals to the number of the cities of the problem. For any Network Type, and no matter how the number of the movements is

calculated, the equation (1) always gives the number of the nodes / cities.

Type2: In order to construct the network more realistic, an office / base (OB) is

added. The reflecting cost because of the OB represents the movement of the GPS

receivers from the OB to the pair of points. Thus, the movements are given by:

2

)1(

NNS

(2)

Figure 2. A small network with symmetrical cost consisting of four benchmark points and OB.

Type3: A network with asymmetric cost is considered to be more realistic. For example: In case the movement 1 3 is faster, easier, or cheaper than 3 1, then there the following inequality is also valid: C1,3 < C3,1 (cost values). For this reason

the number of the movements is given by:

)1( NNS (3)


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Figure 3. A small network with asymmetric cost consisting of four benchmark points and OB.

General the number of the feasible session schedules (possible solutions) is given by

the simple metathesis of the elements (sessions):

!2

)1(

NNP (4)

5. Solving the Problem

5.1 Methodology Adoption

Many problems are categorized as NP Hard Problems, this means that if these problems are in a large scale, the optimal solution cannot be derived in feasible time

periods. This happens because of the very large number of the feasible solutions. For

the solution of such problems the methods that are used are: a) Optimal Solutions

Methods b) Near Optimal Solutions (heuristics) Methods.

Optimal Solution Method: The search for the solution is being applicated to the

totality of the feasible solutions. The solution is the one with the minimum cost.

These methods are using the Optimal Solution Algorithm.

Nearest Neighbor Algorithm: As an alternative method, this study adopts the

Nearest Neighbor Algorithm. This Algorithm starts from the OB and for every step it

examines only the neighbor cities and picks for the next movement, the one with the minimum cost. This repeats until all sessions are observed (or cities are visited).

5.2 Multi TSP / Software Limitations

For most geodetic topographic networks, many times it is not possible to carry out the whole survey in a single working period (day). The problem can be more realistic

and general, if at the end of each working period, the receivers would return to the

OB or to another temporary base OB2 (if the survey is carried out away from the OB)

until their return to the network for the next working period. These returns must be

also computed for the final solution, which includes sessions for all working periods.



With this addition the problem is being transformed into a Multi TSP (MTSP) were more than one salesman must also share the work (cities) in an optimal manner. For this study the salesman remains still one (the pair of GPS receivers) but the work

must break into periods in an optimal manner.

For the software Voyager v1.0 MTSP is not included.

One more software limitation for the optimal solution algorithm is that as the

benchmark points get more (check with mathematic expression (4)-ch.4), there is

enormous growth of the feasible solutions, and due to the lack of memory cleaning in the process, the maximum number of benchmark points is four. For the Nearest

Neighbor Algorithm, there are no limitations according to the number of the cities.

6 The two Topographic Epoch Surveys

6.1 Survey 1 Comparing Optimal Solution to Nearest Neighbor Algorithm

A small sized network (on Zakynthos island, Greece) with symmetrical cost consisting of 4 benchmark points and . Below are described the steps followed in order to produce session schedule.

Figure 5. Left - the network on map. Right - the network ready for calculations

Step 1: The software Voyager v1.0 gives the opportunity to assign each movement

costs by using a calculator. This option can be bypassed if desired (especially for big

networks), and assign the values directly to the next Table (see Table 2)

Table 1. Cost Assignment

Assigning Cost to the Movement OB_114026

Factor Value Weight Subtotal

Type of road, path, etc. 2 2 4

Length 3 1 3

Traffic 2 3 6


September 28-30, 2006 10

off - road 3 4 12

Custom factor 1 0 0 0

Custom factor 2 0 0 0

TOTAL 25

Step 2: Cost of Movements. Below all the possible movements are structured in

tabular format. (About the possible movements used below see: ch. 3)

Table 2. Cost of Movements

Node/Node OB 114026 114042 114020 114005

OB 0 25 16 18 23

114026 25 0 28 23 23

114042 16 28 0 16 19

114020 18 23 16 0 19

114005 23 23 19 19 0

Step 3: The coding of the Names of Benchmark Points is essential in order to create

more easy-to-read data.

Table 3. Coding the Names of Benchmark Points

Ben. Code Ben. Name

1 114026

2 114042

3 114020

4 114005

Step 4: Construction of the Cost of Movements or Cost Matrix (from one Session to

another. (About the possible movements used below see: ch. 3).

Table 4. Cost of Movements or Cost Matrix

Node/Node OB 1_2 1_3 1_4 2_3 2_4 3_4

OB 0 41 43 48 34 39 41

1_2 41 0 16 19 36 39 42

1_3 43 16 0 19 20 39 42



1_4 48 19 19 0 39 20 23

2_3 34 36 20 39 0 19 35

2_4 39 39 39 20 19 0 16

3_4 41 42 42 23 35 16 0

The element (3,1) of the Table 4 equals with 43. This value is the Cost of Movement

of 1_3OBase. This means: movement of the resources and GPS receiver I from the benchmark point with the code 1 (for full name see table 3) to the OBase, and

movement of the resources and GPS receiver II from the benchmark point with the

code 3 (for full name see table 3) to the OBase. The cost derives from the partial costs

of individual movements, summarizing the corresponding values from Table 2

(25+18 =43)

Step 5: Solving the problem / Session Schedule Production (Optimal Solution

Algorithm see ch. 5).

The software Voyager v1.0 represents the solution as shown below.

Figure 6. Session Schedule. Optimal Solution Algorithm

The first column contains the start of each movement. The first element is OB.

The second column contains the destination of each movement. The last element is

OB.

The third column contains the cost of each movement.

The fourth column contains the cumulative cost. The final Element represents the

total cost of the Schedule (166).

Step 6: Solving the problem / Session Schedule Production (Nearest Neighbor


The software Voyager v1.0 represents the solution as shown below:

Figure 7. Session Schedule. Nearest Neighbor Algorithm


September 28-30, 2006 12

Step 7 Conclusion: As expected the Optimal Solution Algorithm resulted better solution. The Nearest Neighbor Algorithm resulted a slightly worst solution varying

2,35%. This variation can be considered very small and acceptable, because of the

cost estimation error, which can be bigger. Empirical tests showed that the two

algorithms result solutions that vary approx. >15% for this Network Size.

6.2 Survey 2 Solving the Problem with Nearest Neighbor Algorithm

A normal sized network asymmetric cost consisting of 8 benchmark points and . Below are described the steps followed in order to produce session schedule.

Step 1: Cost of Movements. Below all the possible movements are structured in

tabular format. (About the possible movements used below see: ch. 3)

Table 5. Cost of Movements

Node/Node OB 150690 245245 54643 36456 7856932 5324523 67543 56754

OB 0 4 4 5 3 2 3 3 5

150690 5 0 4 3 2 3 3 3 3

245245 4 2 0 3 1 5 3 3 4

54643 3 2 5 0 4 1 5 1 2

36456 1 3 3 3 0 1 5 3 4

7856932 2 1 1 3 5 0 4 5 4

5324523 1 5 3 2 2 5 0 3 3

67543 2 1 1 4 2 1 3 0 4

56754 5 2 5 5 2 3 1 3 0

Step 2: The coding of the Names of Benchmark Points is essential in order to create

more easy-to-read data.

Table 6. Coding the Names of Benchmark Points

Ben. Code Ben. Name

1 150690

2 245245

3 54643

4 36456

5 7856932



6 5324523

7 67543

8 56754

Step 3: Construction of the Cost of Movements or Cost Matrix (from one Session to

another. (About the possible movements used below see: ch. 3).

Figure 8. Cost of Movements or Cost Matrix

Figure 9. Session Schedule. Nearest Neighbor Algorithm


September 28-30, 2006 14

Step 4: Solving the problem / Session Schedule Production (Nearest Neighbor


The software Voyager v1.0 represents the solution as shown in figure 9.

7. Conclusion / Suggestions for Further Study

The Optimal Solution Algorithm is very difficult (as the number of benchmark points

increase, the solution can be impossible) to give session schedules for large sized

geodetic networks in short time periods. Especially for this Study additional points,

equal a bigger number of additional cities for the problem (see ch. 4 exp. (1)). Techniques such branch and bound or other cleaning memory routines, can improve the results. Despite the Nearest Neighbor Algorithm computes very fast

session schedules (no matter the size of the network, it solves in a few seconds), as

the network grows, the quality varies more and more from the Optimal Solution. For

this reason even Nearest Neighbor Algorithm has a constraint relative to the size of

the network but for quality matters. Simulated Annealing [Dare (2000)] and Tabu

Search Algorithms [Dare (2001)] can provide very good results in acceptable time

periods. The limitations that are mentioned during this study could also represent very

interesting suggestions for further improvement and study.

8. References

Dare P., Saleh H. (2000). GPS Network Design: Logistics Solution Using, Optimal

and Near Optimal Methods. Journal of Geodesy, Springer Verlag, 74: 467 478

Dare P., Saleh H. (2001). Effective Heuristics for the GPS Survey Network of Malta:

Simulated Annealing and Tabu Search Techniques. Journal of Heuristics, Kluwer

Academic Publishes, 7: 533 549

Rossikopoulos D. (1999). Topographic Networks and Calculations, 2nd

Ed.,

Thessaloniki: Ziti

Stavridis G., Spinos D. (2004). Logistics for Surveys of GPS Geodetic Networks.

Thesis, Department of Surveying Engineering Technological Educational Institution of Athens

All the tables adopt the format of the software Voyager v1.0 (it is part of a thesis [Stavridis

(2004)]) and the figures 6-9 are screenshots of the same software. The references to the

software in this paper aim to support the study of this paper. For this reason the main concept

& the limitations of this study are the same for the software too. The software, the code and

the algorithms used are available for academic use. For any interest or additional information

please contact the Authors.

operational research for engineering applications. optimization methods for logistics design of gps...

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