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1-to-Many DistributionVehicle Routing

Part 2

John H. Vande Vate

Spring, 2005

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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.

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IdeaHigh level approach

– Estimate Transportation Cost as function of frequency of delivery

– Estimate Inventory cost as function of frequency of delivery

– Trade off the two

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The Simple Story

• Transportation costs are T now

• What will they be if we deliver twice as frequently?

• 2T

• Duh

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Simple Story Continued

• Inventory Carrying Costs are C now

• What will they be if we deliver twice as frequently?

• C/2

QQ/2

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Look Familiar?

• n = Number of times to dispatch per year

• Total Cost = nT+C/n

• How often to dispatch?

• n = C/T

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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 ….

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The Easy Stuff

• Stops– Number of customers– Number of deliveries

• Items– Customer demand

• Miles?

• What might we be important to know?

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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

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How Far between Customers?

= 9 customers per sq. mile

1 mile

1 mile

1/3 mile

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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?

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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 << C2

• If there are “many” routes, e.g, • No. of routes much more than customers/route• N/C >> C or N >> C2

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Few Routes• Avoid “line hauls”

x

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Total Distance

• Customer density about the same in each zone.

• Each zone visits C customers

• Each zone travels about kC1/• Total Travel

about kN1/• k is a constant that

depends on the

metric

x

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x x

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Many Routes• If there are “many” routes, e.g,

• No. of routes much more than customers/route• N/C >> C or N >> C2

• Can’t fit them all around the DC

• Approach more like the strip heuristic

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Partition the Customers

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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’

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The Line Haul

• Length of the route 2r + Ck1/• With N/C routes…• Transportation Costs

2E(r)N/C + NkE(1/)x

x

x

r

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Example: Package Delivery

• Greater Atlanta (Hypothetical)280,000 households300 sq miles (17 miles x 17 miles)18 Deliveries per year14,000 deliveries per week2,000 per day1,000 per shift

2020

Parcel Delivery Hypothetical

• Delivery Density (on shift basis)300 sq miles1,000 per shift3.3 customers per sq. mile0.55 miles between customers

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Using Few Routes• Customers per route?

Determined by driver schedule7 hour shift15 miles/hr avg. speed0.55 miles between customers (2.2 minutes)2 minutes per stop (4-5 minutes per customer)12-15 customers/hr80-100 customers per route10-12 routes per shift

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Consistent?

• C 80+ Customers per route

• N 1000 Customers

• Not Extreme– Neither N >> C2 – Nor N << C2

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Using Many Routes

• What’s r roughly?

• Total Area 300 sq miles

• 17 miles by 17 miles

• 2r < 17 miles

• Line haul speed 30-35 miles/hr

• r costs 30 min. out of each 7 hr shift. (7%)

x

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r

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Realities

• Tiered Service

• What’s the impact of peak and off-peak times?

• Peak and off-peak seasons?

• Congestion!

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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?