1-to-Many Distribution Vehicle Routing

1-to-Many DistributionVehicle Routing John H. Vande Vate Spring, 2005 1

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 Shared Transportation Capacity 2

Transport Inventory Inventory Transport TL vs LTL         3

Transport Inventory Transport? Shared Loads      ?    Inventory 4

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 Issues 5

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. Our Approach 6

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. The Traveling Salesman Problem 7

Set Cities; param d{Cities, Cities}; var x{Cities, Cities} binary; minimize Distance: sum{f in Cities, t in Cities}d[f,t]x[f,t]; s.t. DepartEachCity{f in Cities}: sum{t in Cities}x[f,t] = 1; s.t. ArriveEachCity{t in Cities}: sum{f in Cities}x[f,t] = 1; IP Formulation 8

SubTours 3 cities 3 edges        9

S.t. SubTourElimination {S subset Cities: card(S) > 0 and card(S) < card(Cities)}: sum{f in S, t in S} x[f,t] <= card(S) - 1; Eliminating Subtours 10

An Equivalent Statement 3 cities No edges out        11

S.t. SubTourElimination {S subset Cities: card(S) > 0 and card(S) < card(Cities)}: sum{f in S, t not in S} x[f,t] >= 1; An Equivalent Formulation 12

How many subsets of N items? 2N Omit 2 subsets: All N items The empty set 2N - 2 How Many Constraints? 13

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. OK, Half of that... 14

How many subsets of N items? 2N Omit 2 subsets: All N items The empty set 2N - 2 Half of that: 2N-1 - 1 Still Lots! 15

How Many? N 2N-1 - 1 Too Many! 16

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…. 17

The Strip Heuristic Partition the region into narrow strips Routing in each strip is easy ~ 1-Dimensional Paste the routes together Heuristics 18

The Strip Heuristic x x x x x x x x x x x x x x x x 19

Nearest Neighbor x x x x x x x x x x x x x x x x 20

Shortcut a “tour” by finding the greatest “savings” Clark-Wright x x x x x x 21

Clark-Wright • Shortcut a “tour” by finding the greatest “savings” x x x x x x 22

Nearest Insertion x x x x x x x x x x x x x x x x 23

2-Opt Improvement Heuristics x x x x x x x x x x x x x x x x 38

No improvement found, but… Tour still isn’t good Local Minima 39

Simulated Annealing With probability that reduces over time, accept an exchange that makes things worse (gets you out of local minima). Probabilistic Methods 40

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) Optimization-Base Heuristics 41

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 The Spanning Tree 42

The Spanning Tree x x x x x x x x x x x x x x x x 43

Duplicate each edge in the Spanning Tree The resulting graph is connected The degree at every node must be even That’s an Eulerian Graph (you can start at a city, walk on each edge exactly once and return to where you started) It’s no more than twice the length of the shortest TSP Double the Spanning Tree 44

The Spanning Tree 14 15 13 x x 12 11 x 19 x 20 10 18 9 x x 17 x x 5 16 6 x 21 8 4 22 x x 7 23 3 27 x 26 28 x 25 2 24 29 x 1 x x 45 30

Short Cut the Eulerian Tour 14 15 13 x x 12 11 x 19 x 20 10 18 9 x x 17 x x 5 16 6 x 21 8 4 22 x x 7 23 3 27 x 26 28 x 25 2 24 29 x 1 x x 46 30

Short Cut the Eulerian Tour Short Cut the Eulerian Tour 14 15 13 x x 12 11 x 19 x 20 10 18 9 x x 17 x x 5 16 6 x 21 8 4 22 x x 7 23 3 27 x 26 28 x 25 2 24 29 x 1 x x 47 30

There are no more points in the unit square than in the interval from 0 to 1!? Spacefilling Curves 48

Each point (X,Y) on the map For illustration let’s consider points in [0, 32]2 Express X = string of 0’s and 1’s, 5 before the decimal X = 16.5 = 10000.10 1*24+0*23+0*22+0*21+0*20+1*2-1 +0*2-2 Express Y = string of 0’s and 1’s5 before the decimal Y = 9.75 = 01001.11 0*24+1*23+0*22+0*21+1*20+1*2-1 +1*2-2 Space Filling Number - interleave bits and move the decimal (X,Y) = 10010.000011101 “Proof” 49

Each pair of points X = 16.5 = 10000.10 Y = 9.75 = 01001.11 maps to a unique point (X,Y) = 10010.000011101 So,... 50

1-to-Many Distribution Vehicle Routing