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The Interaction of Location and Inventory in Designing Distribution Systems

The Interaction of Location and Inventory in Designing Distribution Systems. Stephen J. Erlebacher and Russell D. Meller. Presented By: Hakan Gultekin. Aim:. # of DCs. Their location. Which customers they serve. Reduces the cost of transporting product to retailers

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The Interaction of Location and Inventory in Designing Distribution Systems

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  1. The Interaction of Location and Inventory in Designing Distribution Systems Stephen J. Erlebacher and Russell D. Meller Presented By: Hakan Gultekin

  2. Aim: • # of DCs • Their location • Which customers • they serve

  3. Reduces the cost of transporting product to retailers Provide better service Reduces the cost of holding inventory via pooling effects Reduces the fixed costs associated with operating DCs Many DCs: Few DCs:

  4. The Location-Inventory Problem: Assumptions • Unit-square grid structure with C columns and R rows C= 5 1 2 3 4 5 1 2 3 4 R= 4

  5. The Location-Inventory Problem: Assumptions • Uniform customer demand across any grid 19 10 11 9 7 ...

  6. The Location-Inventory Problem: Assumptions • Rectilinear distances between plants and DCs and between • DCs and continuously represented customer locations . (a,b) . (x,y)

  7. The Location-Inventory Problem: Assumptions • Continuous review inventory system at DCs • Plant locations and capacities are known in advance • and fixed

  8. The Location-Inventory Problem: The Model Minimize Total Cost where, Total Cost = (Operating Cost) + (DC Inventory Cost) + (Transportation Cost)

  9. = Upper bound on the number of DCs The Location-Inventory Problem: The Model Operating Cost: F = Annual cost of operating a DC

  10. The Location-Inventory Problem: The Model Total Inventory Costs: z= Safety stock parameter A= Order cost s= Std. dev. of demand during lead time h= Holding cost For DC i, order cost + holding cost: , dj= Avg. demand for customer j

  11. The Location-Inventory Problem: The Model Total Inventory Costs: For 1 DC: For all DCs:

  12. The Location-Inventory Problem: The Model Transportation cost from plants to DCs: = Unit plant to DC transportation cost upi = Demand shipped from plant p to DC i qpi = Distance from plant p to DC i From plant p to DC i: From all plants to all DC s:

  13. The Location-Inventory Problem: The Model Transportation cost from DCs to customers: tij = Avg distance from DCi to customer grid j . . . (cj-1,y) (x,y) (cj,y)

  14. The Location-Inventory Problem: The Model Transportation cost from DCs to customers: tij = Avg distance from DCi to customer grid j . . . (cj,y) (cj,y) (x,y)

  15. The Location-Inventory Problem: The Model Transportation cost from DCs to customers: s = Unit DC to customer transportation cost From DC i customer j: For all DCs and all customers:

  16. The Location-Inventory Problem: The Model Constraints :: Each customer must be assigned to an open DC: One customer can be assigned to one DC:

  17. The Location-Inventory Problem: The Model Constraints : Each DC must be fully supplied: Capacity constraint for plants :

  18. The Location-Inventory Problem: The Model Min + + + s.t.

  19. The Location-Inventory Problem: Solution Method • Find the number of DCs, N • Find the location of these DCs and allocation of the • customers to these DCs

  20. Customer demand is entirely homogeneos Any amount of demand can be assigned to any DC Each DC serves an “optimally shaped region” (for rectilinear, diamond shaped region) Ignores different customer demands Discrete nature of the customer grid structure Impossible to have each DC serve an “optimally shaped region” The Location-Inventory Problem: Solution Method Finding N: Stylized model

  21. The Location-Inventory Problem: Solution Method Finding N: Stylized model Lemma 1: Given a number of DCs, N, any DCs that serve positive demand must serve the same size demand. Di= D/N

  22. The Location-Inventory Problem: Solution Method Finding N: Stylized model Let be the inventory parameter, be the transportation parameter and be the inbound logistics costs.

  23. The Location-Inventory Problem: Solution Method Finding N: Stylized model Lemma 2: Optimal N for stylized model can be found by (i) (ii) (iii) (iv)

  24. Optimal number of DCs Inventory Parameter The Location-Inventory Problem: Solution Method Actual Stylized

  25. Optimal number of DCs Transportation Parameter The Location-Inventory Problem: Solution Method Actual Stylized

  26. Optimal number of DCs Fixed Cost The Location-Inventory Problem: Solution Method Actual Stylized

  27. The Location-Inventory Problem: Solution Method Location Problems: • N-facility location problem NP-hard • N independent single facility location • Rectilinear mini-sum location problem

  28. The Location-Inventory Problem: Solution Method Allocation Heuristics v1:

  29. The Location-Inventory Problem: Solution Method Allocation Heuristics v2:

  30. Lower bound Relaxations: Separate inventory and transportation decisions Relax the actual customer locations Lemma 3:Lower bound on inventory costs are obtained by assigning the N-1 lowest demand customer grids to the first N-1 DCs and the remaining M-N+1 customer girds to DCN Sort customers from highest demand to lowest demand and assign them one at a time to a DC. Highest demand customers have more influence on the location of the DC which they are assigned

  31. Computational Results & Managerial Insight Two datasets considered: • Set I consists of 12 customers (on a 3X4 grid) • Set II consists of 16 customers (on a 4X4 grid) • 4 different ABC customer curves: • (80/20), (70/30), (60/40), (50/50) • v2 performed better than v1 in both sets

  32. Computational Results & Managerial Insight • Lower bound was between 4 and 36 % lower than the • optimal solution • Neither of the heuristics are guaranteed to terminate at a • local optimum. • Pairwise-exchange improvement procedure is added (v2+). • For dataset I, v2 found the optimum in 10/75 • while v2+ found in 62/75 • For 600 customers v2 solved in 2 minutes, • v1 solved in 30 hours and v2+ solved in 117 hours.

  33. Computational Results & Managerial Insight • As the skewness of ABC curve increases, N either stays the • same or decreases, since larger demand is concentrated in • fewer and fewer customers. • The customer layout also affects the optimal number of DCs. • Geary Ratio, is an autocorrelation factor that quantifies • spatial correlations • Tends to decrease when similarly-sized customer • demands are adjacent

  34. Computational Results & Managerial Insight . . . 0.23 1.08 0.41 . . . 0.26 0.23 411.9 . . . 0.39 0.31 0.63 . . . . . . . . . . . . 0.25 0.60 0.33 . . . 0.27 41.5 1.76 . . . 0.28 0.29 0.74 . . . . . . 0.23 1.08 0.410 . . . 0.26 41.5411.9 . . . 0.39 0.31 0.63 . . . . . . . . . . . . 0.25 0.60 0.33 . . . 0.27 0.23 1.76 . . . 0.28 0.29 0.74 . . . • Higher Geary Ratio • Smaller Geary Ratio • Higher number of DCs • Smaller number of DCs

  35. Future Research • Capacity limitations at the DCs • Different type of inventory policies • Multi-product environment

  36. The Interaction of Location and Inventory in Designing Distribution Systems Stephen J. Erlebacher and Russell D. Meller Presented By: Hakan Gultekin

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