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___________________________________________________________________________ Operations Research  Jan Fábry

Probabilistic Inventory Models. ___________________________________________________________________________ Operations Research  Jan Fábry. Inventory Models. Probabilistic Inventory Models. When to order? . How much to order? . How much to store in safety stock ? .

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___________________________________________________________________________ Operations Research  Jan Fábry

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  1. ProbabilisticInventory Models ___________________________________________________________________________ Operations Research  Jan Fábry

  2. Inventory Models Probabilistic Inventory Models • When to order? • How much to order? • How much to store in safety stock? ___________________________________________________________________________ Operations Research  Jan Fábry

  3. Probabilistic Inventory Models Model with Continuous Demand ___________________________________________________________________________ Operations Research  Jan Fábry

  4. Inventory Models Probabilistic Model withContinuous Demand Assumptions • Single item • Probabilistic distribution of demand(stationary demand) • Deterministic lead time (constant) • Continuous (but not uniform) depletion of the inventory ___________________________________________________________________________ Operations Research  Jan Fábry

  5. Inventory Models Probabilistic Model withContinuous Demand Assumptions • Purchasing cost is independent of the OQ • Unit holding cost is independent of the OQ • No additional cost in case of shortage • Replenishment - exactly on the point when the shipment arrives ___________________________________________________________________________ Operations Research  Jan Fábry

  6. InventoryLevel Placing Order q Shortage r d d Time 0 Inventory Models Probabilistic Model withContinuous Demand Cycle I Cycle II ___________________________________________________________________________ Operations Research  Jan Fábry

  7. μQ + σQ μQ – σQ μQ Demand Inventory Models Probabilistic Model withContinuous Demand Probability distribution of demand Mean of demand μQ Standard deviation σQ ___________________________________________________________________________ Operations Research  Jan Fábry

  8. Inventory Models Probabilistic Model withContinuous Demand Example – Brewery • Estimation of annual demand = 120 000 cases • Standard deviation of annual demand = 12 000 cases • Annual holding cost per case = 20 CZK • Ordering cost – transportation = 11 000 CZK per order – other = 1 000 CZK per order • Lead time = ½ of month • Objective: minimize total annual cost ___________________________________________________________________________ Operations Research  Jan Fábry

  9. Inventory Models Probabilistic Model withContinuous Demand Example – Brewery • Mean of annual demand μQ = 120 000 cases • Standard deviation of annual demand σQ = 12 000 cases • Annual holding cost c1 = 20 CZK per case • Ordering cost c2 = 12 000 CZK per order • Lead time d = 1/2 of month = 1/24 of year ___________________________________________________________________________ Operations Research  Jan Fábry

  10. Inventory Models Probabilistic Model withContinuous Demand Example – Brewery • Optimum order quantity ___________________________________________________________________________ Operations Research  Jan Fábry

  11. Inventory Models Probabilistic Model withContinuous Demand Example – Brewery • Mean of demand within the LEAD TIME = = Optimum reorder point • Standard deviation of demand within the LEAD TIME ___________________________________________________________________________ Operations Research  Jan Fábry

  12. Lead-Time Demand 4 500 5 500 4 000 5 000 6 000 Inventory Models Probabilistic Model withContinuous Demand Example – Brewery Standard deviation σd = 500 Mean μd = 5 000 ___________________________________________________________________________ Operations Research  Jan Fábry

  13. building of SAFETY STOCK Inventory Models Probabilistic Model withContinuous Demand Example – Brewery Deterministic model – planned shortages Probabilistic model – random occurance of shortages ___________________________________________________________________________ Operations Research  Jan Fábry

  14. Inventory Models Probabilistic Model withContinuous Demand Example – Brewery Service level - definition 1. Service Level is the PROBABILITY with which DEMAND will be MET within the inventory cycle. 2. Service Level is the PROBABILITY with which SHORTAGE WILL NOT OCCUR within the inventory cycle. 3. Service Level is the PERCENTAGE of TIME that all DEMAND is MET. ___________________________________________________________________________ Operations Research  Jan Fábry

  15. Safety stock level Optimum reorder point Inventory Models Probabilistic Model withContinuous Demand Example – Brewery • Implemented reorder point (for the given service level p) ___________________________________________________________________________ Operations Research  Jan Fábry

  16. InventoryLevel r* d d Time 0 Inventory Models Probabilistic Model withContinuous Demand ___________________________________________________________________________ Operations Research  Jan Fábry

  17. InventoryLevel rp r* w d d Time 0 Inventory Models Probabilistic Model withContinuous Demand ___________________________________________________________________________ Operations Research  Jan Fábry

  18. Holding cost ofsafety stock Inventory Models Probabilistic Model withContinuous Demand Example – Brewery • Mean of total cost • Objective: find such SAFETY STOCK level w that satisfies the given SERVICE LEVEL pandminimizes MEAN of TOTAL COST TC ___________________________________________________________________________ Operations Research  Jan Fábry

  19. SERVICELEVEL Real LEAD-TIME DEMAND Implemented REORDER POINT Inventory Models Probabilistic Model withContinuous Demand Example – Brewery Determination of optimum SAFETY STOCK level ___________________________________________________________________________ Operations Research  Jan Fábry

  20. ~ N (r*, σd) ~ N (0, 1) Inventory Models Probabilistic Model withContinuous Demand Example – Brewery Determination of optimum SAFETY STOCK level Real LEAD-TIME DEMAND Qd Transformation ___________________________________________________________________________ Operations Research  Jan Fábry

  21. Inventory Models Probabilistic Model withContinuous Demand Example – Brewery Determination of optimum SAFETY STOCK level ___________________________________________________________________________ Operations Research  Jan Fábry

  22. Inventory Models Probabilistic Model withContinuous Demand Example – Brewery Determination of optimum SAFETY STOCK level ___________________________________________________________________________ Operations Research  Jan Fábry

  23. Inventory Models Probabilistic Model withContinuous Demand Example – Brewery • Optimum SAFETY STOCK level p = 0.95 p = 0.99 ___________________________________________________________________________ Operations Research  Jan Fábry

  24. Inventory Models Probabilistic Model withContinuous Demand Example – Brewery • Optimum mean of total annual cost p = 0.95 p = 0.99 ___________________________________________________________________________ Operations Research  Jan Fábry

  25. Probabilistic Inventory Models Single-Period Decision Model ___________________________________________________________________________ Operations Research  Jan Fábry

  26. penalty !!! penalty !!! Inventory Models Single-Period Decision Model Assumptions • Only one order in time period • Probabilistic distribution of demand (continuous or discrete) • End of time period - surplus - stockout ___________________________________________________________________________ Operations Research  Jan Fábry

  27. Inventory Models Single-Period Decision Model Seasonal or perishable items • Newspapers – „Newsboy problem“ • Bread • Flowers • Fruits • Seasonal clothing • Christmas trees • Halloween pumpkins ___________________________________________________________________________ Operations Research  Jan Fábry

  28. Inventory Models Single-Period Decision Model Example – Happyland • Bakery department – optimize everyday order of rolls • Purchase price = 1 CZK per roll • Selling price = 2 CZK per roll • Remaining rolls are changed into crumbs 20 rolls in 1 sack of crumbs Selling price of crumbs = 12 CZK per sack ___________________________________________________________________________ Operations Research  Jan Fábry

  29. Inventory Models Single-Period Decision Model Example – Happyland • Daily demand – normal probabilistic distribution  = 10 000 rolls  = 500 rolls • Objective: determine optimum order quantity ___________________________________________________________________________ Operations Research  Jan Fábry

  30. Q < q Q > q Q = q Inventory Models Single-Period Decision Model Example – Happyland • Real daily demand for rolls – Q • Daily quantity of ordered rolls - q Evening ___________________________________________________________________________ Operations Research  Jan Fábry

  31. ( q – Q )rolls remain crumbs Inventory Models Single-Period Decision Model Example – Happyland Q < q • Marginal loss per 1 roll ML = purchase price – salvage value ___________________________________________________________________________ Operations Research  Jan Fábry

  32. shortage of ( Q – q )rolls Inventory Models Single-Period Decision Model Example – Happyland Q > q • Marginal profit loss per 1 roll MPL = selling price – purchase price Q = q • No loss ___________________________________________________________________________ Operations Research  Jan Fábry

  33. probability p probability (1 – p) Inventory Models Single-Period Decision Model Example – Happyland No stockout Expected ML = p(ML) Stockout Expected MPL = (1-p)MPL ___________________________________________________________________________ Operations Research  Jan Fábry

  34. Inventory Models Single-Period Decision Model Example – Happyland • Optimum expected cost • Probability with which no stockout occurs (optimum service level) ___________________________________________________________________________ Operations Research  Jan Fábry

  35. SERVICELEVEL REAL DEMAND ORDER QUANTITY Inventory Models Single-Period Decision Model Example – Happyland Determination of optimum order quantity ___________________________________________________________________________ Operations Research  Jan Fábry

  36. ~ N (, ) ~ N (0, 1) Inventory Models Single-Period Decision Model Example – Happyland Determination of optimum order quantity Real demand Q Transformation ___________________________________________________________________________ Operations Research  Jan Fábry

  37. Inventory Models Single-Period Decision Model Example – Happyland Determination of optimum order quantity ___________________________________________________________________________ Operations Research  Jan Fábry

  38. Inventory Models Single-Period Decision Model Example – Happyland • Optimum order quantity ___________________________________________________________________________ Operations Research  Jan Fábry

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