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 Provide better service Reduces the cost of holding inventory via pooling effects Reduces the fixed costs associated with operating DCs Many DCs:Few DCs:

The Location-Inventory Problem: Assumptions Unit-square grid structure with C columns and R rows C= 5 R=

The Location-Inventory Problem: Assumptions Uniform customer demand across any grid

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

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

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

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

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

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

The Location-Inventory Problem: The Model Transportation cost from plants to DCs: From plant p to DC i: = Unit plant to DC transportation cost u pi = Demand shipped from plant p to DC i q pi = Distance from plant p to DC i From all plants to all DC s:

The Location-Inventory Problem: The Model Transportation cost from DCs to customers: t ij = Avg distance from DC i to customer grid j. (x,y)(c j -1,y).. (c j,y)

The Location-Inventory Problem: The Model Transportation cost from DCs to customers: t ij = Avg distance from DC i to customer grid j. (x,y) (c j,y)..

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

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

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

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

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

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

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. D i = D/N

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

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

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

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

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

The Location-Inventory Problem: Solution Method Location Problems: N-facility location problemNP-hard N independent single facility location Rectilinear mini-sum location problem

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

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

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 DC N 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

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

Neither of the heuristics are guaranteed to terminate at a local optimum. Computational Results & Managerial Insight 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. Lower bound was between 4 and 36 % lower than the optimal solution

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

Computational Results & Managerial Insight Higher Geary Ratio Smaller Geary Ratio Higher number of DCs Smaller number of DCs

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

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