1-to-Many Distribution Vehicle Routing

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1-to-Many DistributionVehicle Routing

• John H. Vande Vate
• Spring, 2005

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Shared Transportation Capacity

• Large shipments reduce transportation costs but
  increase inventory costs
• EOQ trades off these two costs
• Reduce shipment size without increasing
  transportation costs?
• Combine shipments on one vehicle

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TL vs LTL
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Shared Loads
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Inventory
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Issues

• Design Routes that
• Minimize the transportation cost
• Respect the capacity of the vehicle
• This may require several routes
• Consider inventory holding costs
• This may require more frequent visits

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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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The Traveling Salesman Problem

• Minimize Distance
• s.t.
• start at the depot,
• visit each customer exactly once,
• return to the depot
• A single vehicle no capacity
• Only issue is distance.

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IP Formulation

• Set Cities
• param dCities, Cities
• var xCities, Cities binary
• minimize Distance
• sumf in Cities, t in Citiesdf,txf,t
• s.t. DepartEachCityf in Cities
• sumt in Citiesxf,t 1
• s.t. ArriveEachCityt in Cities
• sumf in Citiesxf,t 1

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SubTours
3 cities 3 edges
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Eliminating Subtours

• S.t. SubTourElimination
• S subset Cities
• card(S) gt 0
  and
• card(S) lt
  card(Cities)
• sumf in S, t in S xf,t lt card(S) -
  1

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An Equivalent Statement
3 cities No edges out
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An Equivalent Formulation

• S.t. SubTourElimination
• S subset Cities
• card(S) gt 0
  and
• card(S) lt
  card(Cities)
• sumf in S, t not in S xf,t gt 1

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How Many Constraints?

• How many subsets of N items?
• 2N
• Omit 2 subsets
• All N items
• The empty set
• 2N - 2

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OK, Half of that...

• If we have an edge out of S, we must have an edge
  out of N\S.
• Why? The edge out of S is an edge into N\S. But
  there are exactly N\S edges into cities in N\S.
  Since one of them comes from S, not all the edges
  from cities in N\S can lead to cities in N\S. At
  least one must go to S.

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Still Lots!

• How many subsets of N items?
• 2N
• Omit 2 subsets
• All N items
• The empty set
• 2N - 2
• Half of that 2N-1 - 1

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How Many?
N 2N-1 - 1
Too Many!
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Optimization is Possible But...

• It is difficult
• Few 100 cities is the limit
• For more details see www.tsp.gatech.edu
• Is it appropriate?
• Other approaches.

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Heuristics

• The Strip Heuristic
• Partition the region into narrow strips
• Routing in each strip is easy 1-Dimensional
• Paste the routes together

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The Strip Heuristic
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Nearest Neighbor
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Clark-Wright

• Shortcut a tour by finding the greatest
  savings

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Clark-Wright

• Shortcut a tour by finding the greatest
  savings

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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Nearest Insertion
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Improvement Heuristics

• 2-Opt

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Local Minima

• No improvement found, but
• Tour still isnt good

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Probabilistic Methods

• Simulated Annealing
• With probability that reduces over time, accept
  an exchange that makes things worse (gets you out
  of local minima).

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Optimization-Base Heuristics

• Minimum Spanning Tree Heuristic
• Build a minimum spanning tree on the edges
  between customers
• Double the tree to get a Eulerian Tour (visits
  everyone perhaps several times and returns to the
  start)
• Short cut the Eulerian Tour to get a Hamilton
  Tour (Traveling Salesman Tour)

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The Spanning Tree

• Is Easy to construct
• Use the Greedy Algorithm
• Add edges in increasing order of length
• Discard any that create a cycle
• Is a Lower bound on the TSP
• Drop one edge from the TSP and you have a
  spanning tree
• It must be at least as long as the minimum
  spanning tree

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The Spanning Tree
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Double the Spanning Tree

• Duplicate each edge in the Spanning Tree
• The resulting graph is connected
• The degree at every node must be even
• Thats an Eulerian Graph (you can start at a
  city, walk on each edge exactly once and return
  to where you started)
• Its no more than twice the length of the
  shortest TSP

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The Spanning Tree
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Short Cut the Eulerian Tour
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Short Cut the Eulerian Tour
Short Cut the Eulerian Tour
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Spacefilling Curves

• There are no more points in the unit square than
  in the interval from 0 to 1!?

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Proof

• Each point (X,Y) on the map
• For illustration lets consider points in 0,
  322
• Express X string of 0s and 1s, 5 before the
  decimal
• X 16.5 10000.10
• 12402302202102012-1
  02-2
• Express Y string of 0s and 1s5 before the
  decimal
• Y 9.75 01001.11
• 02412302202112012-1
  12-2
• Space Filling Number - interleave bits and move
  the decimal
• ?(X,Y) 10010.000011101

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So,...

• Each pair of points
• X 16.5 10000.10
• Y 9.75 01001.11
• maps to a unique point
• ?(X,Y) 10010.000011101

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How to Use this?

• A mapping of ?(X,Y) into the unit interval, i.e.
• 18.056640625000 10010.000011101
• X 16.5 10000.10
• Y 9.75 01001.11
• Think of this as the inverse mapping of the unit
  interval onto the square (our super tour)
• For a given customer ?(X,Y) is the fraction of
  the way along the super tour where it lies
• Visit the customers in the order of ?(X,Y) (short
  cut the super tour to visit our customers)

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The TSP

• For More on SpaceFilling Curves visit
• http//www.isye.gatech.edu/faculty/John_Bartholdi/
  mow/mow.html
• There are several books on the TSP
• .

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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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Different Approaches

• Route First - Cluster Second
• Build a TSP tour
• Partition it to meet capacity
• Cluster First - Route Second
• Decide who gets served by each route
• Then build the routes

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Route First
Vehicle Cap 15
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Cluster First

• Sweep Heuristic

Vehicle Cap 15
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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.