Transportation Logistics Professor Goodchild Spring 2011

Link Costs TSP and VRP assume cost of travel between 2 points problem input There are formulations where these costs can vary over time, or by vehicle type Cost = a*time + b*distance What else matters?

Link Travel Times Depend on volume of traffic Link performance functions Link cost functions T=f(flow on link)

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

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.