Probabilistic Inventory Models Quantitative Methods of Management Jan

Probabilistic Inventory Models ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Inventory Models Ø When to order? Ø How much to store in safety stock? ______________________________________ Quantitative Methods of Management Jan Fábry

Probabilistic Inventory Models Model with Continuous Demand ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Assumptions Ø Single item Ø Probabilistic distribution of demand (stationary demand) Ø Deterministic lead time (constant) Ø Continuous (but not uniform) depletion of the inventory ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Assumptions Ø Purchasing cost is independent of the OQ Ø Unit holding cost is independent of the OQ Ø No additional cost in case of shortage Ø Replenishment - exactly on the point when the shipment arrives ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Inventory Level Cycle I Placing Order Cycle II q Shortage r 0 d d Time ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Probability distribution of demand Mean of demand μQ Standard deviation σQ μQ – σQ μQ μQ + σQ Demand ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Ø Estimation of annual demand = 120 000 cases Ø Standard deviation of annual demand = 12 000 cases Ø Annual holding cost per case = 20 CZK Ø Ordering cost – transportation = 11 000 CZK per order – other = 1 000 CZK per order Ø Lead time = ½ of month Ø Objective: minimize total annual cost ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Ø Mean of annual demand μQ = 120 000 cases Ø Standard deviation of annual demand Ø Annual holding cost Ø Ordering cost Ø Lead time σQ = 12 000 cases c 1 = 20 CZK per case c 2 = 12 000 CZK per order d = 1/2 of month = 1/24 of year ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Ø Optimum order quantity ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Ø Mean of demand within the LEAD TIME = = Optimum reorder point Ø Standard deviation of demand within the LEAD TIME ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Standard deviation σd = 500 Mean μd = 5 000 4 500 5 000 5 500 6 000 Lead-Time Demand ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Deterministic model – planned shortages Probabilistic model – random occurance of shortages building of SAFETY STOCK ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous 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. ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Ø Implemented reorder point (for the given service level p) Safety stock level Optimum reorder point ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Inventory Level r* 0 d d Time ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Inventory Level rp r* w 0 d d Time ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Ø Mean of total cost Holding cost of safety stock Ø Objective: find such SAFETY STOCK level w that satisfies the given SERVICE LEVEL p and minimizes MEAN of TOTAL COST TC ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Determination of optimum SAFETY STOCK level SERVICE LEVEL Real LEAD-TIME DEMAND Implemented REORDER POINT ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Determination of optimum SAFETY STOCK level Real LEAD-TIME DEMAND Qd ~ N (r*, σd) ~ N (0, 1) Transformation ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Determination of optimum SAFETY STOCK level ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Ø Optimum SAFETY STOCK level p = 0. 95 p = 0. 99 ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Ø Optimum mean of total annual cost p = 0. 95 p = 0. 99 ______________________________________ Quantitative Methods of Management Jan Fábry

Probabilistic Inventory Models Single-Period Decision Model ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Assumptions Ø Only one order in time period Ø Probabilistic distribution of demand (continuous or discrete) Ø End of time period - surplus - stockout penalty !!! ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Seasonal or perishable items Ø Newspapers – „Newsboy problem“ Ø Bread Ø Flowers Ø Fruits Ø Seasonal clothing Ø Christmas trees Ø Halloween pumpkins ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Example – Happyland Ø Bakery department – optimize everyday order of rolls Ø Purchase price = 1 CZK per roll Ø Selling price = 2 CZK per roll Ø Remaining rolls are changed into crumbs 20 rolls in 1 sack of crumbs Selling price of crumbs = 12 CZK per sack ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Example – Happyland Ø Daily demand – normal probabilistic distribution = 10 000 rolls = 500 rolls Ø Objective: determine optimum order quantity ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Example – Happyland Ø Real daily demand for rolls – Q Ø Daily quantity of ordered rolls - q Q < q Evening Q > q Q = q ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Example – Happyland Q < q ( q – Q ) rolls remain crumbs Ø Marginal loss per 1 roll ML = purchase price – salvage value ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Example – Happyland shortage of ( Q – q ) rolls Q > q Ø Marginal profit loss per 1 roll MPL = selling price – purchase price Q = q Ø No loss ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Example – Happyland No stockout probability p Expected ML = p(ML) Stockout probability (1 – p) Expected MPL = (1 -p)MPL ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Example – Happyland Ø Optimum expected cost Ø Probability with which no stockout occurs (optimum service level) ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Example – Happyland Determination of optimum order quantity SERVICE LEVEL REAL DEMAND ORDER QUANTITY ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Example – Happyland Determination of optimum order quantity Real demand Q ~ N ( , ) ~ N (0, 1) Transformation ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Example – Happyland Determination of optimum order quantity ______________________________________ Quantitative Methods of Management Jan Fábry

Inventory Models Single-Period Decision Model Example – Happyland Ø Optimum order quantity ______________________________________ Quantitative Methods of Management Jan Fábry