Route optimization for multiple searchers
Johannes Royset2010, Naval Research Logistics (NRL)
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Abstract
We consider a discrete time-and-space route-optimization problem across a finite time horizon in which multiple searchers seek to detect one or more probabilistically moving targets. The paper formulates a novel convex mixed-integer nonlinear program for this problem that generalizes earlier models to situations with multiple targets, searcher deconfliction, and target-and location-dependent search effectiveness. We present two solution approaches, one based on the cutting-plane method and the other on linearization. These approaches result in the first practical exact algorithms for solving this important problem, which arises broadly in military, rescue, law enforcement, and border patrol operations. The cutting-plane approach solves many realistically sized problem instances in a few minutes, while existing branch-and-bound algorithms fail. A specialized cut improves solution time by 50% in difficult problem instances. The approach based on linearization, which is applicable in important special cases, may further reduce solution time with one or two orders of magnitude. The solution time for the cutting-plane approach tends to remain constant as the number of searchers grows. In part, then, we overcome the difficulty that earlier solution methods have with many searchers.
Key takeaways
- In the case of a single target, the searchers seek to minimize the probability of not detecting the target across a time horizon.
- We derive SP1-LM from SP1 by introducing an "information state" P c,t which equals the probability that the target occupies cell c in time period t and that the target has not been detected prior to t. Given this information state and a search plan with j searchers occupying cell c in time period t, the probability of detection in cell c in time period t and no prior detection, is simply P c,t (1 − e −jαc,t ), where α c,t is the detection rate of each searcher in cell c in time period t. Suppose that a search plan is described by the binary variables V c,t,j , which is 1 if j searchers occupy cell c in time period t and is 0 otherwise.
- for single searcher problems (see [30]), we conclude that problem instances with 2-5 searchers tend to be the most difficult to solve.
- We note, however, that the optimality gaps obtained by Algorithm 2 are typically half of those of Algorithm 1 in the case of few searchers.
- In the case of a target moving according to a Markov chain, the solution time for solving a linearization of MINLP is longer but an instance with three searchers, 81 cells, and 10 time periods is still solved to optimality in less than three minutes.
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