1. 1 1 1-to-Many Distribution with Transshipments John H. Vande Vate Spring, 2001

2. 2 2 Why Does Ford Transship?

3. 3 3 WebVan 9 stations serving Metro Atlanta Alpharetta Atlanta (Airport) Atlanta (Downtown) Marietta Norcross Roswell Suwanee (also DC) Tucker Woodstock

4. 4 4 Why Does WebVan Transship?

5. 5 5 Ford New Vehicle Distribution Consolidation of shipments –Reduces staging area at plants –Speeds delivery –Cuts pipeline inventory

6. 6 6 Intel Into Europe

7. 7 7 Transship for Taxes

8. 8 8 Transshipment for Freight Consolidation Vehicle Capabilities

9. 9 9 Ford & WebVan Customers Terminals The Depot Single Route to Terminals Single sourcing at Customers

10. 10 Transportation Costs Variable Total Cost Volume Shipped Total = fixed + variable*volume Concave!

11. 11 Areas of Influence

12. 12 Non-overlapping

13. 13 Questions How Many Terminals? Where? Whom to serve? Minimize –Inventory and –Transportation Costs

14. 14 “Analytic Approach” How large should influence areas be? “Distant” Terminal

15. 15 Where Should Terminals Go? In the Center? “Distant” Terminal Influence Area Why “Distant”?

16. 16 The Economics Distribution Cost/Item at a terminal location as a function of the size of the influence area I

17. 17 Transportation Costs Inbound

18. 18 Two Cases Load Driven: Depart when vehicle is full –Ford’s new vehicle distribution Schedule Driven: Depart to meet service reqs. –WebVan

19. 19 Load-Driven Capacity of vehicle: Cap Somewhat different for Schedule-Driven –Loads depend on frequency of schedule –Customers per route depends on time

20. 20 Estimating The Cost Customers = Terminals Density  = 1/I Total transport cost/headway 2E(r)N/C + NkE (  1/  ) Number N  R/I R is the total area we serve Route headway t Volume to each customer  tI Customers per route C  Cap/( tI) 2E(r)R t/Cap + Rk/  I

21. 21 Cost Per Item Total Transport Cost 2E(r)R t  Cap+ Rk/  I Total Items R t Cost Per Item  2E(r)/Cap+ k/( t  I) economies from less peddling

22. 22 Inbound Cost/Item

23. 23 Outbound Transportation Customers are original customers in an Influence area I. Customer density:  Number of customers:  I  Shipment size: v Vehicle capacity: cap Customers per route C = cap/v 2E(r)N/C + Nk  2E(r’)I  v/cap + Ik  per terminal

24. 24 Outbound Transportation 2E(r’)I  v/cap + Ik  per terminal E(r’) depends on I –probably proportional to  k’  2k’I 3/2  v/cap + Ik  R/I terminals 2k’R  I  v/cap + Rk  over all routes 2k’  I  v/( tcap) + k  /( t  per item Increasing outbound linehaul

25. 25 Outbound Cost per Item

26. 26 Terminal Cost “Fixed” Cost per terminal: f (per headway t) –Independent of volume, e.g. “rent” Unit costs per terminal: u –Depend on volume, e.g., unit handling cost Total Cost across all terminals –fR/I + u tR Terminal Costs per item f/( tI) + u Economies of scale for fixed investment

27. 27 Terminal Cost/Item

28. 28 Total Cost Total Avg. $/item Inbound –2E(r)/Cap+ k/( t  I) Outbound –2k’  I  v/( tcap) + k  /( t  Terminal –f/( tI) + u

29. 29 Total Cost per Item I*I*

30. 30 Dependence on Location Replace E(r) with r –How large should influence areas be by location Different customer densities, etc. Find I(r) or I(x,y) at several points Partition region into Influence areas with these properties

31. 31 Detailed Models Analytical Model suggests number of terminals and initial locations Location-Allocation model –positions terminals –assigns customers Routing Procedures Evaluate Results Crude Sensitivity Analysis

32. 32 Location Allocation in AMPL Project consists of –Model and parameter declarations in.mod file –Commands in.run file

33. 33 Parameters set Customers; param X{Customers}; param Y{Customers}; param NFacs; set Facs := 1..NFacs; param FacX{Facs}; param FacY{Facs}; param Distance{Customers, Facs}; param sqrtN; param MaxOuterIters; param MaxInnerIters; param MinX default 1e6; param MaxX default -1e6; param MinY default 1e6; param MaxY default -1e6;

34. 34 Model var Assign{Customers, Facs} binary; minimize TotalDistance: sum{cust in Customers, fac in Facs} Distance[cust, fac]*Assign[cust,fac]; s.t. OneFacPerCustomer{cust in Customers}: sum{fac in Facs} Assign[cust, fac] = 1;

35. 35 Commands /* -------------------------------------- Intial Locations --------------------------------------*/ for {cust in Customers} {let MinX := min(MinX, X[cust]); let MaxX := max(MaxX, X[cust]); let MinY := min(MinY, Y[cust]); let MaxY := max(MaxY, Y[cust]); }; let sqrtN := max(1, round(sqrt(NFacs),0)); let {fac in Facs} FacX[fac] := MinX + ((fac mod sqrtN)+1)*(MaxX-MinX)/(sqrtN+1); let {fac in Facs} FacY[fac] := MinY + ((fac mod sqrtN)+1)*(MaxY-MinY)/(sqrtN+1);

36. 36 let {cust in Customers, fac in Facs} Distance[cust, fac] := sqrt((X[cust] - FacX[fac])*(X[cust] - FacX[fac]) + (Y[cust] - FacY[fac])*(Y[cust] - FacY[fac])); for {outeriter in 1..MaxOuterIters} { solve; /* Assign Customers */ for {inneriter in 1..MaxInnerIters} /* Locate Facilities */ {for {fac in Facs: sum{cust in Customers} Assign[cust,fac] > 0} {let FacX[fac] := (sum{cust in Customers: Assign[cust, fac] >.5} X[cust]/Distance[cust, fac] )/ (sum{cust in Customers: Assign[cust, fac] >.5} 1/Distance[cust, fac]); let FacY[fac] := (sum{cust in Customers: Assign[cust, fac] >.5} Y[cust]/Distance[cust, fac])/ (sum{cust in Customers: Assign[cust, fac] >.5} 1/Distance[cust, fac]); }; /* end for fac in Facs loop */

37. 37 /* --------------------------------------- Re-Calculate distances ---------------------------------------*/ let {cust in Customers, fac in Facs} Distance[cust, fac] := sqrt((X[cust] - FacX[fac])*(X[cust] - FacX[fac]) + (Y[cust] - FacY[fac])*(Y[cust] - FacY[fac])); /* --------------------------------------- Check stopping criteria for location loop: ---------------------------------------*/ }; /* end of the for inneriter loop */ /* --------------------------------------- Check stopping criteria for outer loop: ---------------------------------------*/ }; /* end of the for outeriter loop */

38. 38