1 1 1-to-Many Distribution Vehicle Routing Part 2 John H. Vande Vate Spring, 2005

2 2 Our Approach Minimize Transportation Cost (Distance) –Traveling Salesman Problem Respect the capacity of the Vehicle –Multiple Traveling Salesmen Consider Inventory Costs –Estimate the Transportation Cost –Estimate the Inventory Cost –Trade off these two costs.

3 3 Idea High level approach –Estimate Transportation Cost as function of frequency of delivery –Estimate Inventory cost as function of frequency of delivery –Trade off the two

4 4 The Simple Story Transportation costs are T now What will they be if we deliver twice as frequently? 2T Duh      

5 5 Simple Story Continued Inventory Carrying Costs are C now What will they be if we deliver twice as frequently? C/2 Q Q/2

6 6 Look Familiar? n = Number of times to dispatch per year Total Cost = nT+C/n How often to dispatch? n =  C/T

7 7 System Design We don’t know the transportation cost How to estimate it? Assume we have estimates of –c m = $/mile (may include $/hr figures) –c s = $/stop (may include $/hr figures) –c i = $/item ….

8 8 The Easy Stuff Stops –Number of customers –Number of deliveries Items –Customer demand Miles? What might we be important to know?

9 9 Customer Distribution Is this rural North Dakota or Downtown Manhattan? Might estimate it from –Census information –Marketing information –GIS Customer Density –  customers per sq. mile

10 How Far between Customers?  = 9 customers per sq. mile 1 mile                    1/3 mile

11 Conclusion Customer density about  customers per sq. mile leads to average distance between customers of about  1/  miles What does this mean for transportation costs?

12 Extreme Cases N is the number of customers C is the number of customers per vehicle If there are “few” routes, e.g, No. of routes much less than customers/route N/C << C or N << C 2 If there are “many” routes, e.g, No. of routes much more than customers/route N/C >> C or N >> C 2

13 Few Routes Avoid “line hauls” x x x x x x x x x x x x x x x x

14 Total Distance Customer density about the same in each zone. Each zone visits C customers Each zone travels about kC  1/  Total Travel about kN  1/  k is a constant that depends on the metric x x x x x x x x x x x x x x x x

15 Many Routes If there are “many” routes, e.g, No. of routes much more than customers/route N/C >> C or N >> C 2 Can’t fit them all around the DC Approach more like the strip heuristic

16 x x x x x x x x x x x x x x x x Partition the Customers

17 The Partition Each partition Is  k’/  wide Is C/  k’  long Area is C/  C customers on average Effect on Travel? x x  k’/  C/  k’ 

18 The Line Haul Length of the route  2r + Ck  1/  With N/C routes… Transportation Costs  2E(r)N/C + NkE (  1/  ) x x x r

19 Example: Package Delivery Greater Atlanta (Hypothetical)  280,000 households  300 sq miles (17 miles x 17 miles)  18 Deliveries per year  14,000 deliveries per week  2,000 per day  1,000 per shift

20 Parcel Delivery Hypothetical Delivery Density (on shift basis)  300 sq miles  1,000 per shift  3.3 customers per sq. mile  0.55 miles between customers

21 Using Few Routes Customers per route?  Determined by driver schedule  7 hour shift  15 miles/hr avg. speed  0.55 miles between customers (2.2 minutes)  2 minutes per stop (4-5 minutes per customer)  customers/hr  customers per route  routes per shift

22 Consistent? C  80+ Customers per route N  1000 Customers Not Extreme –Neither N >> C 2 –Nor N << C 2

23 Using Many Routes What’s r roughly? Total Area 300 sq miles 17 miles by 17 miles 2r < 17 miles Line haul speed  miles/hr r costs  30 min. out of each 7 hr shift. (7%) x x x r

24 Realities Tiered Service What’s the impact of peak and off-peak times? Peak and off-peak seasons? Congestion!

25 Ford Service Parts Suppose Ford operated the delivery fleet What to do? –Deliver to all the Dealerships at once? –Stagger deliveries? What’s the trade-off? Proposals?