1. Package Transportation Scheduling Albert Lee Robert Z. Lee

2. Problem Summary UPS – Scheduling of deliveries, set number of trucks (machines), a set list of locations (nodes), and trying to optimize routes with consideration to stochastic nature of traffic To simplify, we approach the problem with one delivery truck and one depot, making deliveries to multiple delivery locations with stochastic travel time. Difficulty: What is ideal, the path with lowest mean travel time or with lowest variance? Difficulty: Stochastic nature of travel times means that optimal path is constantly changing. What is optimal one second may not be optimal the next.

3. Proposed Solution Stochastic dynamic traveling salesman problem with time windows (SDTSPTW). Assumption of travel time between nodes being approximated by normal distribution If delivery window is too small for a feasible solution with one truck, more trucks need to be added Heuristic algorithm using n-path relaxation of TSP and convolution-propagation approach

4. N-path Relaxation and Loop Elimination For a traveling salesman problem to minimize the total transportation cost for n customers (i.e., nodes), find a lower bound of the objective function by solving shortest- path problems of at most n links. Partial route (l-path) is a route built by adding links – it is an l-path at node j if the route of l links ends at node j To eliminate links that loop (i-j-i) we will track both the best and 2 nd best routes of any l-path of any node, and if the best route is i-j-I, then the 2 nd best is used.

5. Convolution-Propagation Approach Used as a mechanism to predict the arrival time of the vehicle at a node on the specified delivery route. CPA provides the normal distribution (mean, variance) of the arrival time at the node, which is helpful as we are looking at a stochastic model

6. Model N := {n nodes} and A := {m links} s.t. graph G = (N,A) is connected. Node 0 ∈ N denotes the depot from which the delivery truck originates Truck begins from Node 0 and must visit all other n-1 nodes, then return to Node 0. For Node i ∈ N, there exists some time window restriction in which the truck must arrive within in order to successfully make the delivery. Delivery processing time is denoted as S i (service time at node i), where S i ~ N(V i, θ i 2 )

7. Model (cont’d) Time horizon [0, ∞] split into T segments s.t. I 0 (=0) < I 1 <…<I T-1 <I T = ∞. Travel time D of the truck between nodes is time dependent (traffic varies depending on the time) if the route starts in [I t−1, I t ), t = 1, …, T, the travel time D ~ N(δ + ρ t, σ t 2 ), where δ is the constant least possible (free flow) travel time and ρ t is the random delay time of starting within the tth time interval.

8. Elimination of Routes Must check if subroutes satisfy all time windows of each node by checking known distribution arrival times Y i against some variable ϒ representing the maximum allowable tardiness probability, such that we reject the route if P (Y i ≥ u i ) > ϒ In the case of two routes arriving at one node, we consider if both the mean and variance of the arrival time of route A (B) are smaller than the corresponding values of route B (A), route A (B) is more efficient than route B (A); the inefficient route is discarded. If the mean of one route is smaller but the variance is larger than the other route, the two routes do not dominate each other; both are efficient and both are kept.

9. Pseudocode for Solution Algorithm

10. Equation References: Equation 6: Equation 7:

11. Equation References Cont’d: Equation 8:

12. Sample Network Illustration Starting at node 0, travelling to one of 5 nodes in step 1, from there travelling to one of remaining 4 nodes in step 2, so on and so forth until all nodes have been visited and return to node 0. TSP transformed to SP (shortest path)