Package Transportation
Scheduling
Albert Lee
Robert Z. Lee
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.
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
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 shortestpath 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 2nd best routes of any l-path of any node, and if the best
route is i-j-I, then the 2nd best is used.
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
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 Si (service time at
node i), where Si ~ N(Vi, θi2)
Model (cont’d)
 Time horizon [0, ∞] split into T segments s.t. I0 (=0) < I1
<…<IT-1<IT = ∞.
 Travel time D of the truck between nodes is time
dependent (traffic varies depending on the time)
 if the route starts in [It−1, It), t = 1, …, T, the travel time
D ~ N(δ + ρt, σt2), where δ is the constant least possible
(free flow) travel time and ρt is the random delay time
of starting within the tth time interval.
Elimination of Routes
 Must check if subroutes satisfy all time windows of each node
by checking known distribution arrival times Yi against some
variable ϒ representing the maximum allowable tardiness
probability, such that we reject the route if P (Yi ≥ ui) > 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.
Pseudocode for Solution
Algorithm
Equation References:
 Equation 6:
 Equation 7:
Equation References Cont’d:
 Equation 8:
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)