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|||||||WCR|||R|||| W5iJ"'0Wl"|||||DDReport on A dynamical programming solution for shortest path itineraries in robotics by M. Talbot
This paper concerns a problem in autonomous mobile robotics where one wants to find the shortest path from one location to another under the condition that certain other locations must be visited. As the author notes, this problem is similar to, but not exactly the same as, the well-known traveling salesman problem. A two-phase approach is proposed for this problem using integer programming techniques. Overall, the application is very interesting, and the paper could be acceptable, but major changes need to be made. As it is written, it should not be published. But, if the changes below are made, it should be considered again.
There is a problem with this approach. In the second-phase of the algorithm, a modified shortest path problem is solved (formulation C). The author claims that the solution to this problem will never have subtours, which is incorrect. If, in the example used, the arcs (11, 15) and (15, 11) each have time 4 instead of 6, an optimal solution is x1,8 = x8,7 = x11,13 = x13,11 = 1, giving a subtour. Thus, subtours need to be considered in this formulation.
The author also needs to be careful in how he solves his problems. First, in order to generate formulation C, the author goes through a number of shortest path calculations. One would not solve these problems using linear programming, because it is too slow. One would either use a combinatorial shortest path algorithm like Dijkstras method repeatedly, or would use an All-Pairs shortest path algorithm. In either case, the complexity is polynomial and performs much better than any LP solver; such combinatorial methods take advantage of the special structure of the problem while LP solvers generally dont. Second, formulation C cannot be solved in general by the simplex method. This is because there is no guarantee that the LP solution would be integral. Thus, some sort of IP algorithm such as branch-and-bound would be needed.
Some other minor points on the paper: (1) the author should forego the table of contents; (2) the author notes the given problem is NP-complete. He should give a reference to this. Because subtours are allowed, is it possible this problem can be solved in polynomial time? (3) Since most undergraduates may not have seen any linear programming, LP and IP should be introduced, and the simplex method mentioned as one algorithm to solve IP. (4) There is no need to mention sensitivity analysis and duality. (5) The MCNFP model needs summations in the first group of constraints. (6) Remove the section on the mechanics of the simplex algorithm. (7) Sections 10 and 11 can be combined into 1 section. This will make the paper flow better.
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