Date of Award


Degree Type


Degree Name

Master of Science in Industrial Engineering (MSIE)


Industrial Engineering

First Advisor

David M. Shao


This thesis discusses a mathematical approach to solving the bearings only target motion analysis problem. The problem of solving a sonar contact's tract can be reduced to a two variable minimization problem of finding the values of initial and final ranges to contact that minimize the sum of squares of error between the actual sonar bearings and the computed bearings given the estimated initial and final ranges.

Because a submarine can use specific tactics when tracking a contact and because maximum detection ranges can be estimated, the general shape of the sum of squares of error function and its orientation are known. This research develops a search technique in which two conjugate gradient searches converge simultaneously from opposite sides of the optimum. Rules are developed to determine starting points which guarantee that the searches remain on opposite sides of the optimum in both variables. The stopping criterion for the search is the distance between the searches after each iteration. A measure of the progress of the search in terms of maximum distance from the optimum is guaranteed because either search is no further from the optimum at any iteration than the distance between the two searches. The research shows that a solution to the bearings only target motion analysis problem is obtained with the required accuracy more efficiently by searching simultaneously from opposite s ides of the optimum than by searching from one point only.

The advent of multi-processors and co-processors in small computers argues for exploring the concept of searching simultaneously from multiple starting points. Multiple search in itself is not a solution. Criteria are still needed for stopping the searches and choosing a solution from the results of the two searches. This study tests a number of criteria for stopping the search and also evaluation criteria for choosing a solution.