An Integrated Location Inventory Model

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An Integrated Location Inventory Model

  1. 1. An Integrated Location/Inventory Model
  2. 2. Problem <ul><li>Develop a model that better captures strategic facility location and tactical inventory management decisions </li></ul>
  3. 3. Traditional Models <ul><li>Location </li></ul><ul><ul><li>Ignores inventory </li></ul></ul><ul><li>Inventory </li></ul><ul><ul><li>Assumes number and location of sites given </li></ul></ul><ul><li>Need for integrated model </li></ul>
  4. 4. Outline <ul><li>Model structure </li></ul><ul><li>Model formulation </li></ul>
  5. 5. Model Structure
  6. 6. Problem <ul><li>Given </li></ul><ul><ul><li>Plant, customer, and candidate DC locations </li></ul></ul><ul><ul><li>DC fixed cost </li></ul></ul><ul><ul><li>Shipment costs from DC to customers </li></ul></ul><ul><ul><li>Distribution of customer demands </li></ul></ul><ul><ul><li>Inventory costs at DC dependent on customer assignments </li></ul></ul><ul><ul><li>Shipment costs from plant to DC dependent on customer assignments </li></ul></ul>
  7. 7. Problem <ul><li>Find </li></ul><ul><ul><li>Number and location of DCs </li></ul></ul><ul><ul><li>Assignment of customers to DCs </li></ul></ul><ul><li>To minimize </li></ul><ul><ul><li>Facility location costs </li></ul></ul><ul><ul><li>Shipment costs </li></ul></ul><ul><ul><ul><li>From plant to DC </li></ul></ul></ul><ul><ul><ul><li>From DC to customers </li></ul></ul></ul><ul><ul><li>Inventory costs at DC </li></ul></ul><ul><li>Risk Pooling Effects in </li></ul><ul><ul><li>Safety stock </li></ul></ul><ul><ul><li>Working inventory </li></ul></ul><ul><ul><li>Shipments from plant to DC </li></ul></ul>
  8. 8. Costs and Inputs N(  i ,  i 2 ) Working Inventory Safety Stock Inventory Customer Set Customer served by another DC Transport Cost Fixed Volume dependent (constant + mileage related) Lead time Fixed Cost Transport Cost
  9. 9. Key Issues <ul><li>Working inventory, safety stock inventory, frequency and cost of shipments from plant to DC depend (non-linearly) on expected demand in customer set served by DC </li></ul><ul><li>But, customer set is endogenously determined and is not known a priori </li></ul><ul><li>Ignoring inventory issues at end customers </li></ul>
  10. 10. If we knew customer set … <ul><li>Let D be the total annual (expected) demand served by a particular DC and  2 L be the variance of demand during a lead time </li></ul>
  11. 11. Inventory at DC Time Inventory Safety Stock Reorder Point Lead Time Lead Time Reorder Size Reorder Size Reorder Size
  12. 12. Inventory Costs <ul><li>Safety Stock = </li></ul><ul><li>where </li></ul>
  13. 13. Order Frequency, n <ul><li>Fixed, Shipment to DC and Avg. Inventory Costs </li></ul><ul><li>where </li></ul>
  14. 14. With Linear Transport Costs… And…
  15. 15. Key Point <ul><li>Inventory costs depend on square root of UNKNOWN demand assigned to DC </li></ul><ul><li>Safety stock costs depend on UNKNOWN variance of assigned demand </li></ul>
  16. 16. Decision Variables
  17. 17. Additional Notation
  18. 18. Model Form Fixed + local delivery + volume based ship to DC But only to open DCs Each demand assigned Working inventory, (transport to DC), Safety stock Integrality

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