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Vehicle Routing Part 2 John H. Vande Vate Fall, 2002 1 1

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. 2 2

Idea • If Ford owned the dealerships. . . – More frequent deliveries • Reduce inventory • Increase transportation – How often should Ford deliver? • High level approach – Estimate Transportation Cost as function of frequency of delivery – Estimate Inventory cost as function of frequency of delivery 3 – Trade off the two 3

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

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

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

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

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

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 9 9

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

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? 11 11

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, 1. No. of routes much more than customers/route 2. N/C >> C or N >> C 2 12 12

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

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

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 15 15

Partition the Customers x x x x 16

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

The Line Haul • Length of the route 2 r + Ck 1/ • With N/C routes… x • Transportation Costs 2 E(r)N/C + Nk. E( 1/ ) r x x 18 18

Example: Web. Van • Greater Atlanta (Hypothetical) 280, 000 households 300 sq miles (17 miles x 17 miles) 5% market share 14, 000 deliveries per week 2, 000 per day 1, 000 per shift 19 19

Web. Van Hypothetical • Delivery Density (on shift basis) 300 sq miles 1, 000 per shift 3. 3 customers per sq. mile 0. 55 miles between customers 20 20

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) 12 -15 customers/hr 84 -105 customers per route 10 -12 routes per shift 21 21

Consistent? • C 80+ Customers per route • N 1000 Customers • Middle ground – Neither N >> C 2 – Nor N << C 2 22 22

Using Many Routes • • • What’s r roughly? Total Area 300 sq miles x 17 miles by 17 miles r 2 r < 17 miles Line haul speed 30 -35 miles/hr • r costs 30 min. out of each 7 hr shift. (7%) x x 23 23

Double Market Share • Delivery Density (on shift basis) 300 sq miles 1, 000 per shift ==> 2, 000 per shift 3. 3 ==> 7 customers per sq. mile 0. 55 ==> 0. 38 miles between customers 2. 2 ==> 1. 5 minutes drive 4 -5 ==> 3 -4 minutes per customer 12 -15 ==> 15 -20 customers/hr 84 -105 ==>100 -140 customers per route 24 24

Realities • What’s the impact of peak and off-peak times? • What’s the impact of Scheduled deliveries? – How might we estimate the average distance per customer? 25 25

Ford vs Web. Van • Suppose Ford operated the delivery fleet • What to do? – Deliver to all the Dealerships at once? – Stagger deliveries? • What’s the trade-off? • Proposals? 26 26