Minimizing the Number of Turns in Arc Routing

1. Minimizing the Number of Turns in Arc Routing Benjamin Dussault Department of Mathematics, University of Maryland Bruce L. Golden Robert H. Smith School of Business, University of Maryland Edward Wasil Kogod School of Business, American University WARP 1 - Copenhagen May 2013

2. Introduction • Arc routing problems have many real-world applications • Street sweeping • Meter reading • Postal delivery • Snow plowing • Newspaper distribution • These problems are well-studied in the research literature

3. The Directed Rural Postman Problem (DRPP) • We focus on the DRPP in this talk • It has many applications (e.g., snow plowing) • Definitions • Let G = (V, A) be a directed graph • Let R denote the subset of required arcs in A • The goal is to find a minimum-cost cycle in G that traverses all arcs in R

4. An Example of the DRPP (a)Graph G with required arcs in bold (b) Balance-and-connect solution (c) Connect-and-balance solution

5. Avoiding Turns • In most applications, the objective is to minimize the total distance or travel time • In practice, drivers of trucks and snow plows prefer to travel straight ahead for as long as possible • U-turns are difficult • RH turns are quicker and safer than LH turns • LH turns push snow into the intersection • Even RH turns can be difficult with parked cars and smaller, faster vehicles on the road

6. Small Eulerian Graph Cycle 1: {0,1,2,0,2,1,0} Cycle 2: {0,1,0,2,1,2,0} Both cycles have the same total distance Cycle 1: one U-turn at node 0 Cycle 2: two U-turns at node 1 • We need to add turn penalties

7. Different Turns on a Small Graph • (b) The bold edge represents • a left turn {5, 3, 2} (a) A six-node graph • (c) The bold edge represents • traveling straight {5, 3, 1} • (d) The bold edge represents • a U- turn {5, 3, 5}

8. The Converted Graph • (a) Converted graph over • the original instance (b) Converted graph (c) Turning costs (bolded arcs) are the traversal (e.g., plowing or deadhead) costs for the first arc plus the turn penalty

9. Definitions in the Converted Graph • Arcin the converted graph • Think of arc as traveling from node i to node j and then turning in the direction of k (in the original graph) • Turn penalty • is the turn penalty associated with traversing • is a value between 0 and 1 that serves as a weighting factor between total distance and total turn penalty • When = 0, only turn penalties count

10. Integer Program (IP) • We formulate the DRPP with turn penalties on the converted graph as an IP • Objective function • Minimize • = the number of times arc is plowed • = the number of times arc is deadheaded • = the cost of deadheading arc

11. Solving the IP • After solving the IP (using CPLEX), we obtain an Eulerian (directed) graph with the number of traversals of each arc and each type of turn • We know the route length and sum of turn penalties • Our idea was to vary the parameter α in order to study the impact of turn penalties on the route length • How much would it cost to avoid or reduce turns? • Next, we see a cycle in the converted graph and its equivalent in the original graph

12. A Tale of Two Graphs (b) Cycle in original graph (a) Cycle in converted graph

13. A Computational Experiment • We generate instances using a 5 by 10 rectangular grid (50 nodes) • Each arc has a random plowing/service cost between 0 and 2, a random deadheading cost equal to (on average) half of the service cost, and a turn penalty of 1 for all (non-straight) turns • Next, we randomly delete each arc with prob = .2 • Finally, we randomly assign arcs to be required with prob = .8 • A small example is shown on the next slide

14. Generating a Random 2 by 3 Instance Random arcs are removed Random arcs are required A six-node graph

15. Details of Computational Experiment • We solve 100 randomly generated instances for= 0, 1, 10, 20, 30, 40, 50, 60, 70, 80, 90, 99, 100% • Using a single thread of a 1.8 GHz Intel i5 processor, the average run time per instance was 0.21 seconds using CPLEX 12.2 • From each solution, we can read off the total length and the number of turns of each type

16. Three-Phase Behavior for Aggregate results for 100 randomly generated instances. Dark lines indicate average percentage of straight turns. Light lines indicate average percentage deviation from optimality with respect to total length.

17. Observations with Respect to Turn Penalties • When turn penalties are ignored (= 100%), the average percentage of straight turns is 31.9% • For all other values of , i.e., whenever turn penalties are taken into account, the average percentage of straight turns increases to 60.9%

18. Observations with Respect to Route Length • When route length is ignored (= 0%), the average deviation of route length from optimality is 8.2% • For all other values of , i.e., whenever route length is taken into account, the route lengths are optimal • This three-phase behavior with respect to turn penalties and route length for = 0%, 0 < < 100%, and = 100% was observed for instances of many sizes, from 2 by 3 to 5 by 10

19. Tentative Conclusions • For the DRPP with turn penalties, it appears that you can optimize for turn penalties without any negative impact on route length • Alternatively, it seems that route length and turn penalties are not competing objectives

20. Focus on a Single 5 by 10 Instance • We selected one of the 100 instances • We solved it for = 100% using the IP and obtained an Eulerian (directed) graph • By randomizing the order in which arcs are investigated at each step in Fleury’s algorithm (FA), we can produce a new randomly generated circuit each time FA is applied • We sampled 1,000 random circuits, without repetition, from the set of all circuits that represent optimal solutions to the IP • We compute the percentage of straight turns and plot the resulting distribution

21. Distribution of Straight Turns • The average proportion of straight turns is 32.5% • The chance that the proportion of straight turns exceeds 40% is very small • You must actively minimize turn penalties to obtain a proportion of straight turns equal to 52.0%

22. Three-Phase Behavior for a Single Instance Results for a randomly generated instance. Dark lines indicate average percentage of straight turns. Light lines indicate average percentage deviation from the lower bound with respect to total length.

23. Possible Explanation of Three-Phase Behavior • In the instances we have examined, the number of optimal solutions w.r.t. route length is very large • Similarly, the number of optimal solutions w.r.t. number of straight turns is also very large • So, if you look for solutions that take both objectives into account, you can find some that are optimal w.r.t. both • If you focus on a single objective, you are likely to find one that is suboptimal w.r.t. the second objective

24. Reordering the Circuit to Promote Straight Turns (b) A different circuit that is route-length optimal with four straight turns (vertical through 1, vertical through 2, horizontal through 1, horizontal through 2) (a) A possible circuit that is optimal with respect to route length, but with no straight turns

25. Conclusions and Future Work • In exploring the trade off between route length and turn penalties for the DRPP, we performed a computational experiment and observed an unexpected three-phase behavior • Do the computational results rely on the fact that the instances are rectilinear and the node degrees are small, or are they more general? • Is there an analytical result to explain this behavior? • Is this behavior common in arc routing problems?