Transportation Logistics Professor Goodchild Spring 2009

Traveling Salesman Problem Visit a set of cities and minimize total travel cost Applies to delivery routes Assume travel cost independent of order Individual traveler

Traveling Salesman Problem Can be formulated as an integer programming problem The time to find an optimal solution increases very quickly with N Requires location of each city (customer) to be visited

TSP approximation Is there a formula for L* (the optimum expected length) if N points are randomly scattered (with density δ) in a square region of area A? L*~k √(AN)=kN/√δ k depends on the metric (approximately 0.72 for L2 (Euclidean),.92 for L1 (grid)) Works well for large N Other formulae for different shapes, moderate N

Vehicle Routing Problem Assume given locations of N points, a depot, a matrix of costs to travel between locations, a demand for each point, a vehicle capacity Find an allocation of points to vehicles and a set of vehicle routes ending and beginning at the depot that minimizes either vehicle distance, number of vehicles, or a combination of the two Assumes number of vehicles known

VRP Can be formulated as an integer program in a variety of ways The time to find an optimal solution increases very quickly with N Faster solution methods have been developed that don’t find the optimum but find a good solution Local search methods (simulated annealing)

TSP approximation r: distance from depot to center of tour area D: total demand (units) v m : vehicle capacity L vrp ≤L tsp +2Dr/v m

Time windows A time window is an interval in time, provided for the delivery of some good A narrow time window is a short one, say 30 minutes in length A wide time window is a long one, say 8 hours in length How do time windows effect the vehicle routing problem?

Questions How does the length of a tour change with demand density? How does the number of drivers change with the length of a tour? How would you calculate the demand density with 30 minute time windows versus 2 hour time windows?

Tailored Strategies Tighter time windows for customers that are willing to pay more. Deliveries outside of peak travel periods. Allow transportation companies to expand their markets. Increase logistical complexity.

Link costs are a function of volume

Wardrop’s First Principal The journey times in all routes actually used are equal and less than those which would be experienced by a single vehicle on any unused route. Each user non-cooperatively seeks to minimize his cost of transportation. Referred to as "user equilibrium" (UE) flows. User-optimized equilibrium is reached when no user may lower his transportation cost through unilateral action.

Wardrop’s Second Principal At equilibrium the average journey time is minimum. This implies that each user behaves cooperatively in choosing his own route to ensure the most efficient use of the whole system. Referred to as "system optimal" (SO). Economists argue this can be achieved with marginal cost road pricing. marginal costroad pricing

O D1 D2 Link 1 Flow = X 1 Cost = Z 1 Link 2 Flow = X 2 Cost = Z 2 Link 3 Flow = X 3 Cost = X 3 Given OD flow from O to D2 is 4 Choose fraction of flow to travel link 1 and 3 or link 2 Identify UE solution, SO solution

O D1 D2 Link 1 Flow = X 1 Cost = Z 1 Link 2 Flow = X 2 Cost = Z 2 Link 3 Flow = X 3 Cost = X 3 X 1 =4(1+x), X 2 =4(1-x), X 3 =4x x is fraction of items for D2 sent through D1 Z 1 =1/X 1, Z 2 =X 2, Z 3 =1 Total Cost = X 1 Z 1 +X 2 Z 2 +X 3 Z 3 UE: travel cost on both links equivalent SO: minimize total travel cost

O D1 D2 Link 1 Flow = X 1 Cost = Z 1 Link 2 Flow = X 2 Cost = Z 2 Link 3 Flow = X 3 Cost = X 3 UE: x=.7 SO: x=.5

O D1 D2 Link 1 Flow = X 1 Cost = Z 1 Link 2 Flow = X 2 Cost = Z 2 Link 3 Flow = X 3 Cost = X 3 X 1 =4(1+x), X 3 =4x, X 2 =4(1-x) Z 1 =X 1 -1/2, Z 2 =3X 2 -1/2, Z 3 =1 Total Cost = X 1 Z 1 +X 2 Z 2 +X 3 Z 3 Total Cost = 2(1+x) 1/2 +6(1-x) 1/2 +4x

X 1 =4(1+x), X 3 =4x, X 2 =4(1-x) Total Cost = X 1 Z 1 +X 2 Z 2 +X 3 Z 3 Z 1 =X 1 -1/2, Z 2 =3X 2 -1/2, Z 3 =1 Total Cost = 2(1+x) 1/2 +6(1-x) 1/2 +4x Higher the flow, lower the per unit cost – economy of scale Which links have economies of scale?

x Total cost Total cost minimized at x=1 Total cost = 6.8 Want to send everything on the same route