Simulation Tools and Setup

To simulate the locations of robots on a terrain the following setup was formulated:

• A two dimensional grid of size 12*12 was created, and normally distributed data sets consisting

of the x-y coordinates of the robot locations on the grid were generated.

• The values of mean and variance were varied from 0 ≤ µ ≤ 10 andσ= ±2 respectively for each

data set.

• The data sets with 35 nodes were generated, with 3 to 7 clusters per data set.

• The intra-partition similarity measures in terms of number of robots per partition were taken

into consideration. For example, a data set 5 7 would have 5 partitions, each partition having the number of robots ranging from_{b0.7 ∗ (35/5)c = 4 to b(0.3 ∗ (35/5)) + (35/5)c = 9.}

• Each experimental result was an average of 200 repetitions with random gateway location ini-

tializations.

• The Nash equilibrium (NE) solution to the n-person normal form game is identified using the

Simplical Subdivision algorithm, which has been identified to work consistently better than other existing NE methodologies available in literature. Based upon the simplex method, the al- gorithm starts with a given grid size, and converges to an approximate solution point by iterative labeling of the sub-simplexes.

• Gambit [108], an open source C library of game theory analyzer software toolkit for identifica-

tion of NE solution was used as a solution methodology.

6.5.2 Analysis

Experiments were conducted to study the performance of this method in simultaneously optimiz- ing the objective functions; the compaction measure (SSE) and the uniform power distribution measure (L). As shown in Figure 6.6, for a data set 6 7 that consists of 6 gateways, and 35 robots distributed among the gateways, our methodology identifies the partitions with the SSE of 58.86, and the L being 10.83, which contributes toward an improvement of 90.3% and 90.2% respectively from the initial values. However, for the KMeans algorithm, although the improvement in compaction is 1.3% higher

than game theoretic method, the power distribution objective is 45.1% worse. Overall, the game theory based multi-metric optimization method outperforms the KMeans algorithm in terms of simultaneous optimization of the multiple objectives.

Figure 6.6 Identification of optimum sizes of the clusters and the locations of the cluster centers using game theoretic algorithm, and KMeans algorithm. Total number of robots = 35, total number of gateways = 6, and name of data set = 6 7.txt.

The average performance of the new method was also compared with the KMeans algorithm. All 35 data sets were executed and average of the outputs for improvements in SSE and L were plotted on a graph as shown in Figure 6.7. As shown, the improvement in the compaction objective is higher for the KMeans algorithm. This is intuitive since KMeans performs the partitioning only on the basis of optimization of compaction objective. However, this adversely affects the uniform power distribution objective, and is evident from the graph.

In contrast, our algorithm simultaneously optimizes both the objectives. It is important to note that both the objectives are optimized with an average improvement of more than 50% in terms of results. An interesting observation is that as the number of partitions increase, the performance of this method improves and after certain limit it degrades. This is due to the increasing dimensionality of the problem. If the number of partitions are too few, the initial partitions identified by the initial iteration

Figure 6.7 Average performance of algorithms on artificial data sets. The KMeans and the game theoretic algorithm are compared for their performance on the cluster compaction (SSE) and uniform power distribution (L) metrics.

of KMeans optimally partitions the data, and a game is not formulated often. As the partitions increase, the KMeans initialization is unable to identify optimum clusters resulting in multiple iterations of game formulation and hence simultaneous optimization of objectives. However, as the partitions increase beyond a certain limit, the number of strategies per game increase and the game theoretic model in its current form prunes the strategy set to control the dimensionality of the problem. Due to this, occasionally the strategies that are not locally optimal but have a global effect may get pruned thereby affecting the performance.

The response time of a microeconomic model largely determines its practicability in an applica- tion domain. The parameters that largely govern the response time for game theoretic model in this context include the number of players, the number of partitions, and the total number of strategies of players. Table 6.2 shows the average values of these parameters for different number of gateways. For smaller number of partitions, the initial KMeans clustering is often optimal and a game is not required to be played, and hence average number of iterations of game is less than one. The results on the simulated data sets are promising because as the number of partitions increase, the number of strate- gies do not increase exponentially, which is a concern with most of the problems modeled in a game theoretic framework. The linear relationship between the size of strategy set and number of partitions is attributed to the novel definition of the strategy and the modeling of the game in this context of this

work. The strategy set for a player in this model does not depend on combinations of the number of resource locations that are availing the resources, but on the number of units a player may have to lose for keeping the resource in a consistent state.

Table 6.2 Parameters affecting the game theoretic model. The inter-relationship between the important attributes of a game such as the number of players, number of strategies, number of clusters, total number of game iterations, and the execution time of the algorithm is identified.

Partitions 3 4 5 6 7

Avg. Iterations of Game 0.23 0.80 1.52 1.96 2.02

Avg. Number of Players 0.22 0.60 1.39 1.60 2.13

Avg. Number of Strategies 0.32 2.08 4.71 5.27 6.77

Response Time (sec.) 0.0003 0.0627 0.1447 0.1615 0.1968

6.6 Discussion

A novel microeconomic approach for multi-objective robot team formation problem has been de- veloped in this research. It models the problem as a hybrid approach involving Kmeans and non- cooperative multi-player normal form game with Nash equilibrium based solution. The objective functions being considered in the model are compactness, and uniform power distribution. The sim- ulations have been conducted using normally distributed artificial data sets. The performance of this method as compared to the KMeans algorithm conforms to the claim that our model is better suited for robot aggregation than the existing partitioning methods. The average complexity of the system is non-exponential. This is the first successful attempt in the direction of robot team formation on the basis of multiple objectives. Currently, the model is simplistic, and optimizes only two objectives simultaneously. However, the practical implementation of the model may require more objectives, like improved radio communication, minimum inter-team communication, etc. to be considered. In such scenarios, the payoff modeling would need further investigation and refinement. Also, in practice, the capabilities of each robot are different, and such considerations must be reflected in the modeling. It is required to deploy robots in several real world test scenarios to efficiently and accurately evaluate the performance of the algorithm.

CHAPTER 7