Vehicle Routing & Scheduling Cluster Algorithms Improvement Heuristics Time Windows

Cluster Algorithms Select certain customers as seed points for routes. –Farthest from depot. –Highest priority. –Equally spaced. Grow routes starting at seeds. Add customers: –Based on nearest neighbor or nearest insertion –Based on savings. –Based on minimum angle. Re-optimize each route (solve a TSP for customers in each route).

Cluster with Nearest Neighbor depot Suppose each vehicle capacity = 4 customers Select 3 seeds depot Add nearest neighbor

Cluster with Nearest Neighbor depot Suppose each vehicle capacity = 4 customers Add nearest neighbor depot Add nearest neighbor

VRP Improvement Heuristics Start with a feasible route. Exchange heuristics within a route. –Switch position of one customer in the route. –Switch 2 arcs in a route. –Switch 3 arcs in a route. Exchange heuristics between routes. –Move a customer from one route to another. –Switch two customers between routes.

K-opt Exchange Replace k arcs in a given route by k new arcs so the result is a route with lower cost. 2-opt: Replace 4-5 and 3-6 by 4-3 and 5-6. depot Original route depot Improved route

3-opt Exchange 3-opt: Replace 2-3, 5-4 and 4-6 by 2-4, 4-3 and 5-6. depot Original route depot Improved route

Improvement Heuristics depot Cluster with Nearest Neighbor depot “Optimized” routesStarting routes

Time Windows Problems with time windows involve routing and scheduling. depot 3,[2-4] 1, [9-12]2,[1-3] 6,[9-12] 4,[10-2] 5,[8-10] 3,[2-4] Customer number Start and end of time window (2:00 pm - 4:00 pm)

Clustering and Time Windows Cluster customers based on location and time window. Design routes for each cluster. depot 3,[2-4] 1, [9-12]2,[1-3] 6,[9-12] 4,[3-5] 5,[8-11]

Savings Method with Time Windows Start with n one stop routes from depot to each customer. Calculate all savings for joining two customers and eliminating a trip back to the depot. –S ij = C i0 + C 0j - C ij Order savings from largest to smallest. Form route by linking customers according to savings if time windows are satisfied.

Savings Method with Time Windows Order savings from largest to smallest. –S 35 –S 34 –S 45 –S 36 –S 56 –S 23 –S 46 –S 24 –S 25 –S 12 –S 26 –S 13 –etc.

Savings Method with Time Windows One hour travel time between any two customers. Half hour delivery time at each customer. depot 3,[2-4] 1, [9-12]2,[1-3] 6,[9-12] 4,[10-2] 5,[8-11] Largest savings = S 35 Next largest savings: S 34, S 45, S 36, S 56 Leave depot: 10:00 Arrive at 5: 11:00 Leave 5: 11:30 Arrive at 3: 2:00

Savings Method with Time Windows One hour travel time between any two customers. Half hour delivery time at each customer. depot 3,[2-4] 1, [9-12]2,[1-3] 6,[9-12] 4,[10-2] 5,[8-11] S 56 Next largest savings: S 23, S 46, S 24 Leave depot: 8:30 Arrive at 6: 9:30 Leave 6: 10:00 Arrive at 5: 11:00 Leave 5: 11:30 Arrive at 3: 2:00

Savings Method with Time Windows One hour travel time between any two customers. Half hour delivery time at each customer. depot 3,[2-4] 1, [9-12]2,[1-3] 6,[9-12] 4,[10-2] 5,[8-11] S 24 Need S 16 or S 14. Will customer 1 fit on black or red route? Leave depot: 12:30 Arrive at 4: 1:30 Leave 4: 2:00 Arrive at 2: 3:00 Leave depot: 8:30 Arrive at 6: 9:30 Leave 6: 10:00 Arrive at 5: 11:00 Leave 5: 11:30 Arrive at 3: 2:00

Savings Method with Time Windows One hour travel time between any two customers. Half hour delivery time at each customer. depot 3,[2-4] 1, [9-12]2,[1-3] 6,[9-12] 4,[10-2] 5,[8-11] S 14 Leave depot: 11:00 Arrive at 1: 12:00 Leave 1: 12:30 Arrive at 4: 1:30 Leave 4: 2:00 Arrive at 2: 3:00 Leave depot: 8:30 Arrive at 6: 9:30 Leave 6: 10:00 Arrive at 5: 11:00 Leave 5: 11:30 Arrive at 3: 2:00