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MIT1_204S10_lec23

MIT1_204S10_lec23 - 1.204 Lecture 23 Analytic...

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1.204 Lecture 23 Analytic approximations V hi l ti Vehicle routing Transit design Analytic approximations First spiral in developing problem solution Assist in requirements, prototyping, initial results, Assist in requirements, prototyping, initial results, review Many analytic approximations are visual, unlike almost all algorithms Recall role of visualization in finding roots of equations, and how poorly algorithms do without it Generally allow a broader treatment of the question, with more variables, more flexible obj ectives and constraints Provide guidance in framing heuristics Many real problems do not have optimal algorithms We have very few O(n) or O(n 2 ) algorithms for complex problems; most are O(2 n ) 1
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Vehicle routing Variables: Number of routes or employees Number of customers Time windows or appointments Capacity of vehicle or employee Whether customers are known at start of route And many others… Objectives Objectives Customer service (timeliness, appointments) Cost minimization Constraints Labor rules, … Dispatch routing options Work center Customer 2
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W rk center How to serve customers? o Area a “Linehaul” p Work center Area a Trip length L to visit n randomly distributed customers in area a and return: __ L = 2p + k na (Beardwood, Halton, Hammersley 1959) Shape of dispatch zones Which shape is better? 3
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Zone shape Let’s try to elongate them Tour length is the same for same number of points Tour length is the same for same number of points, same area, different shapes Comparison P1 Wk ctr P2 P3 Wk ctr P4 Elongated zones have shorter driving distance if there are a lot of customers in each zone 4
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P2 Wk ctr Elongated zones, many customers P1 P3 Wk ctr P4 With many customers, elongated zones are better Fat zones, few customers Wk ctr Wk ctr If there are only a few customers, shape matters more In this case,’fat’ zones are better 5
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t t Rules for building tours The “break point” for fat versus skinny zones is about 6 customers, based on simulation and geometric probability If
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