A Study of Shape Penalties in Vehicle Routing

Transcription

1 A Study of Shape Penalties in Vehicle Routing Charles Gretton and Phil Kilby 10 June 2013 NICTA Funding and Supporting Members and Partners 1/37

2 Capacitated Vehicle Routing Problem with Time Windows Service a number of customers with a fleet of vehicles Minimise total distance travelled Capacity constraints Vehicles have bounded capacity Customers each have a demand Time windows Each customer determines when they can be serviced NP-Complete e.g., TSP is a subproblem Industrial instances can only be solved approximately using metaheuristics: e.g. savings and insertion based We use a large-neighbourhood search because it is naturally coupled with other general constraints processing procedures 2/37

3 Sketch Start with an Incumbent Solution i.e. valid and probably suboptimal 3/37

4 Sketch of LNS Remove Customers i.e. incomplete solution 4/37

5 Sketch Put Customers Back i.e. valid and usually also suboptimal 5/37

6 LNS Background Objective is to minimise a weighted sum of quantities such as: Classical: Distance, Time, Overtime, Opportunity Costs, etc. Non-Classical: Dispatch-Time Penalty, Slack Penalties, Surplus Penalties, Visual Unattractiveness, etc. While satisfying all hard constraints: Vehicle cannot be overloaded Customer must be serviced according to the given time windows LNS: Customers are removed according to a removal rule More-or-less greedy wrt customer contribution to a weighted fragment of the objective value of the incumbent (and ease of exchange) LNS: Customers are added according to an insertion rule More-or-less greedy wrt regret scores Sum k = 2..N of differences between objective values of best and k th most promising insertion 6/37

7 Visually Appealing Routes LNS is exceptionally good at quickly producing very short solutions Those solutions are typically not robust And they are typically not visually appealing Long narrow tours are bad (cannot swap pairs of customers on the fly) Drivers are not familiar with all customers in their vicinity Difficult for human planners to reason about on-the-fly modifications to overlapping routes (e.g. callbacks) Therefore, our clients are sometimes reluctant to implement optimised plans produced by using LNS 7/37

8 Example of Robust Visual Appeal Routes Route Hulls Unattractive Km Relatively Attractive Km 8/37

9 Penalty for Robustness and Visual Appeal Route Median Penalty i and j are visits, d(i, j) is the symmetric distance between i and j Given a route R, a visit i is a good candidate for the geographical centre of R if it has a relatively low score: V (i, R) = j R d(i, j) (1) The median of route R is a visit C R R with the lowest V score, in other words: C R = arg min V (i, R) (2) i R Route Median solution penalty is equal to the sum of distances of assigned visits to their route medians 9/37

10 Route Median Penalty in the Context of LNS... What will the routes look like? 10/37

11 Routing for 400 requests distributed uniformly 11/37

12 Routing for 400 requests distributed uniformly 12/37

13 Routing for 600 requests distributed uniformly 13/37

14 Routing for 600 requests distributed uniformly 14/37

15 Characterising Shape via Bending Energy Assume a route is a contour formed using a linear thin shelled medium. Where K (p) is the curvature at point p, the stored energy of a route is therefore: K (p) 2 dp (3) Nontrivial contours with minimum energy are circles i.e. are very visually appealing... 15/37

16 Curvature and Polygonal Curves We define the visual curvature at a point as follows: K N, S (p) = π N 1 i=0 #[H α i (S(p))] N S (4) H αi is a height function rotated by angle α i = π i N #[.] counts extreme points in the neighbourhood S(p) of size S. In the limiting case this gives us a curvature familiar to differential geometry (Liu etal., IJCV 2008): K (p) = lim θ(p) S 0 S (5) 16/37

17 Curvature and Polygonal Curves We define the visual curvature at a point as follows: K N, S (p) = π N 1 i=0 #[H α i (S(p))] N S (6) H αi is a height function rotated by angle α i = π i N #[.] counts extreme points in the neighbourhood S(p) of p of size S. In the limiting case this gives us a curvature familiar to differential geometry: K (p) = lim S 0 θ(p) S (7) = lim S 0 lim N K N, S (p) In the polygonal case, for small enough S visual curvature gives us turn angles i.e., where θ(p) is the turn angle at vertex p: θ(p) = S lim K N, S(p) (8) N 17/37

18 Bending Energy as a Penalty Minimise the sum of turn angles. Both bending and median penalties can be added to the objective. The median penalty guides LNS to produce non-overlapping routes. The bending penalty mitigates the tendency of LNS to select for long and messy routes given the median penalty. 18/37

19 A Short Route No Shape Penalties -Traveling Salesperson -LNS traces a short contour 19/37

20 Route Minimising Bending Energy -Traveling Salesperson -LNS guided by Bending Energy traces concentric contours 20/37

21 Bending Energy is a Measure of Shape -Scale Invariant -Invariant to Visit Count -Straight Lines -Circles -Spirals 21/37

22 Routes Demonstration of Both Penalties Median No Median Bending No Bending 22/37

23 Combining Both Penalties in Practice 23/37

24 Combining Both Penalties in Practice 24/37

25 Combining Both Penalties in Practice 25/37

26 Combining Both Penalties in Practice 26/37

27 Combining Both Penalties in Practice 27/37

28 Combining Both Penalties in Practice 28/37

29 Combining Both Penalties in Practice 29/37

30 Summary of Experimental Evaluation Benchmarks: Solomon and Extended Solomon instances with 10 2 to 10 3 customers 10 4 iterations of LNS on each instance Fixed the number of trucks equal to that of optimised solutions Forbid dropping of any customers (via very large penalty) Using the median penalty only, the area intersected by convex hulls induced by routes is reduced by 43% on average, relative to solutions found without shape penalties. Total solution distance increases by 12 13% on average By 9% when we also use the bending energy penalty (effect on hull intersection is marginal) LNS can compute visually attractive routes 30/37

31 Computational Considerations Median Penalty Compute a route median for each route encountered by the search For each route prefix encountered during an insertion: We sort the customers in the prefix lexicographically, and Map that to the local median score of each customer in the prefix. Novel median calculations use results cached as above. This trick enables us to guide an LNS according to the median penalty in practice. 31/37

32 Computational Considerations Median Penalty We investigated the effect of varying how often route medians are calculated During LNS insertion, a new median should be calculated for each route encountered A parameter ρ gives the probability that the penalty is computed according to a true median, rather than according to the previous median computed for the route We investigate the impact of different values of ρ 32/37

33 CPU and Memory Usage Varying ρ Memory Time (a) Average CPU-Time and Memory Usage of Indigo Average Memory Usage (KB) Average CPU Usage (Sec) Probability that a new median is calculated during an insertion (0%..100%) 70 33/37

34 Probability of a More Compact Solution Varying ρ 0.7 (b) Probability of Improving the Volume of Hull Intersections Relative to the Baseline 0.6 Probability of Lower Volume of Intersections Probability that a new median is calculated during an insertion (0%..100%) 34/37

35 Solution Length Varying ρ 1.3 (c) Average Increase in Distance Relative to the Baseline Factor of Default Score Probability that a new median is calculated during an insertion (0%..100%) 35/37

36 Graphical Conclusions Bending and Median Penalties Median Penalty Only Row 1 problem has 400 requests distributed uniformly, and Row 2 has 600. Under each heading, routes are depicted on the left, and the convex hulls induced by those routes on the right. 36/37

37 Textual Conclusions We introduce the idea of using bending energy to guide attractive-route construction We also use the notion of a route median a visit that is geographically central to the route to penalise a route according to how distant the visits are from the median. Together those penalties guide LNS to visually attractive solutions. We have some empirical evidence that bending energy helps identify optimised and attractive routes. We also investigate the effect of changing the frequency with which true (versus inherited) route medians are used in evaluating penalties during search. We have seen empirically that increased frequency leads to more attractive solutions which feature compact routes with comparatively little inter-route overlap. 37/37