12/27/001 1-to-Many Distribution Vehicle Routing Part 2 John H. Vande Vate Spring, 2001

Spacefilling Curves zThere are no more points in the unit square than in the interval from 0 to 1!?

Proof zEach point (X,Y) on the map zExpress X = string of 0’s and 1’s yX = 16.5 = *2 4 +0*2 3 +0*2 2 +0*2 1 +0*2 0 +1* *2 -2 zExpress Y = string of 0’s and 1’s yY = 9.75 = *2 4 +1*2 3 +0*2 2 +0*2 1 +1*2 0 +1* *2 -2 zSpace Filling Number - interleave bits y  (X,Y) =

So,... zEach pair of points X = 16.5 = Y = 9.75 = maps to a unique point  (X,Y) =

An obsessive travelling salesman ! zWe can visualize the spacefilling curve as a route of a travelling salesman who wants to visit every point in the unit square !

How to Use this? zA mapping of  (X,Y) into the unit interval zThink of this as the inverse mapping of the unit interval onto the square (our super tour) zFor a given customer  (X,Y) is the fraction of the way along the super tour where it lies zVisit the customers in the order of  (X,Y) (short cut the super tour to visit our customers)

Sierpinski spacefilling curve

“Meals on Wheels” zSenior Citizen Services, Inc. is a private, nonprofit corporation in Atlanta, Georgia whose purpose is to provide social services for the elderly, especially the elderly poor, in Fulton County. zNeed to deliver meals to these people between 10 am and 2 pm everyday. zNormally use 4 drivers, each delivers meals to locations.

“Meals on Wheels” zAs for most charitable organizations, they have unstable, almost always insufficient, sometimes desparate funding. zCan’t afford computing resources zNeed bare-bones administrative costs - any extra funds are required for meals and other services. The manager is too busy with more important issues zList of clients is quite volatile

Why Spacefilling Curves ? zThe spacefilling curve heuristic (SFC) has many advantages, if you are willing to accept solutions that are about 25% longer than optimum (expected, for random point sets). These advantages include:  The SFC algorithm is fast: Only O(n log n) effort to construct a tour of n points and only O(log n) effort to update the solution by adding or removing points.  The SFC heuristic does not need explicit distances between points and so there is no need to compute these, as most other heuristics must.  The algorithm is parallelizable. (A comparable algorithm, Nearest Neighbor, is apparently not parallelizable.)  The length of links in the SFC tour of random points is expected to be small and so (1/k)-th of the stops account for about (1/k)-th of the travel time. This means that an SFC tour can easily be converted to tours for k vehicles simply by partitioning the SFC route into k contiguous pieces.

Some applications  To build a routing system for Meals-on-Wheels in Fulton County (Atlanta, GA), which delivers hundreds of meals daily to those too ill or old to shop for themselves. Built on two rolodex card files.  To route blood delivery to hospitals in the Atlanta metropolitan area by the American Red Cross.  To target a space-based laser (part of the Strategic Defense Iniative, or "Star Wars" program).  To control a pen-plotter for the drawing of maps. M. Iri and co-workers at the University of Tokyo showed how it could be used to reduce drawing time for large road maps by routing the pen efficiently.

The TSP zFor More on SpaceFilling Curves visit zThere are several books on the TSP z………

Our Approach zMinimize Transportation Cost (Distance) yTraveling Salesman Problem zRespect the capacity of the Vehicle yMultiple Traveling Salesmen zConsider Inventory Costs yEstimate the Transportation Cost yEstimate the Inventory Cost yTrade off these two costs.

Idea zIncreasing Service to the Stores yMore frequent deliveries xReduce inventory xIncrease transportation yHow often should we deliver? zHigh level approach yEstimate Transportation Cost as function of frequency of delivery yEstimate Inventory cost as function of frequency of delivery yTrade off the two

The Simple Story zTransportation costs are T now zWhat will they be if we deliver twice as frequently? z2T Duh      

Simple Story Continued zInventory Carrying Costs are C now zWhat will they be if we deliver twice as frequently? zC/2 Q Q/2

The EOQ zn = Number of times to deliver per year zTotal Cost = nT+C/n zHow often to dispatch? zn =  C/T