## Paper submitted to 2015 Mathematical Contest in Modeling for consideration

Optimal Search Strategy

——A generic model of plane searching

Summary

We mainly construct four models, analyzing the physical process of a plane from departing to eventually being found in open water, to assist "searchers" in planning a useful search theoretically.

Model 1 deals with the location where plane loses contact. We get airline equations with the time when plane loses contact, the coordinates of departure A and destination B by analytic geometry and coordinate conversion formula. We can work out the longitude and latitude of the location where plane loses contact.

Model 2 studies the location where plane plunges into water. The falling process is simplified to a projectile motion. We get complete equations to describe the projectile motion with Coriolis force (stems from earth rotation) and air resistance taken into consideration. The plunging point of any falling plane can be calculated by solving the equations. The example of Air France Flight 447 we cite in the verification part proves the effectiveness of the model.

Model 3 concentrates on the disturbance of wind and ocean currents towards wreckage plunging into water. As to the floating wreckages, we utilize Finite Volume Community Ocean Model and Navier‐Stokes equations. In order to simplify calculation and get detailed search area, we hypothesize the floating process is a Markov Chain and wind velocity obeys logarithmic normal distribution. We employ Lagrange‐Galerkin method and Monte Carlo method to numerically obtain the coordinate of final position and possible search radius. As to the sinking wreckages, by reasonably assuming ocean current is Couette flow, a model to locate the final position is demonstrated.

Model 4 is heuristically built to select a suitable search pattern. Once the floating time of wreckages and local wind velocity are given, we can evaluate a most appropriate search pattern by comparing several indexes, respectively representing time duration of searching, probability of being found and the comprehensive “search ability” of different search patterns.

We further research on the sensitivity of two parameters: wind velocity and floating time. When wind velocity fluctuates ±5%, most indexes fluctuates less than 3%, but some indexes varies more than 20%, which indicates some search patterns are in lack of stability.

1.      Introduction. 2

1.1       Background  knowledge. 2

1.2 Existing search  strategy. 2

1.2.1    Search equipment 2

1.2.2    Search patterns. 3

2. Problem Analysis. 5

2.1 Goals. 5

2.2 The Layout of Problem  Solving. 5

3. General Assumptions. 6

4. Model Design and  Solving. 7

4.1 Model 1--- Airline. 7

4.2 Model 2--- Projectile  motion. 8

4.3 Model 3--- The  disturbance of wind and ocean current 11

4.3.1 Floating Process. 11

4.3.2 Sinking Process. 14

4.4 Model 4--- Search  pattern. 16

4.4.1 Model of radar or  sonar system.. 17

4.4.2 Sector search  pattern. 17

4.4.3 Expanding square  search pattern. 19

4.4.4 Estimate effectiveness  and efficiency. 20

4.4.5 Comparison between  two search patterns. 21

5. Verifications of our  model 23

5.1 Verification of Model  1. 23

5.2 Verification of Model  2. 23

6. Sensitivity Analysis. 24

7. Weaknesses and  Strengths of the Model 27

8. References. 29

9. Appendix  (Non-technical Paper) 30

Whole paper and code are available below: paper.pdf matlab_code.rar

Problem:

PROBLEM B: Searching for a lost plane

Recall the lost Malaysian flight MH370. Build a generic mathematical model that could assist "searchers" in planning a useful search for a lost plane feared to have crashed in open water such as the Atlantic, Pacific, Indian, Southern, or Arctic Ocean while flying from Point A to Point B. Assume that there are no signals from the downed plane. Your model should recognize that there are many different types of planes for which we might be searching and that there are many different types of search planes, often using different electronics or sensors. Additionally, prepare a 1-2 page non-technical paper for the airlines to use in their press conferences concerning their plan for future searches.