# ___________________________________________________________________________ Operations Research  Jan Fábry Probabilistic Inventory Models.

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___________________________________________________________________________ Operations Research  Jan Fábry Probabilistic Inventory Models

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

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

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

Time Inventory Level 0 ___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Probabilistic Model with Continuous Demand d r d q Cycle ICycle II Placing Order Shortage

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Probabilistic Model with Continuous Demand Probability distribution of demand μQμQ Demand μ Q + σ Q μ Q – σ Q Mean of demand μ Q Standard deviation σ Q

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Example – Brewery Probabilistic Model with Continuous Demand  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 Inventory Models Example – Brewery Probabilistic Model with Continuous Demand  Mean of annual demand  Standard deviation of annual demand μ Q = 120 000 cases σ Q = 12 000 cases  Annual holding cost  Ordering cost  Lead time c 1 = 20 CZK per case c 2 = 12 000 CZK per order d = 1/2 of month = 1/24 of year

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Example – Brewery  Optimum order quantity Probabilistic Model with Continuous Demand

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Example – Brewery  Mean of demand within the LEAD TIME = = Optimum reorder point Probabilistic Model with Continuous Demand  Standard deviation of demand within the LEAD TIME

5 000 Lead-Time Demand 5 500 6 000 4 500 4 000 ___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Probabilistic Model with Continuous Demand Example – Brewery Mean μ d = 5 000 Standard deviation σ d = 500

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

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Example – Brewery Service level - definition Probabilistic Model with Continuous Demand 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 Inventory Models Example – Brewery  Implemented reorder point (for the given service level p) Probabilistic Model with Continuous Demand Optimum reorder point Safety stock level

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Probabilistic Model with Continuous Demand d Time Inventory Level 0 d r *

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Probabilistic Model with Continuous Demand w 0 Time Inventory Level d r * d r p

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Example – Brewery  Mean of total cost Probabilistic Model with Continuous Demand Holding cost of safety stock  Objective: find such SAFETY STOCK level w that satisfies the given SERVICE LEVEL p and minimizes MEAN of TOTAL COST  TC

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Example – Brewery Determination of optimum SAFETY STOCK level Probabilistic Model with Continuous Demand SERVICE LEVEL Real LEAD-TIME DEMAND Implemented REORDER POINT

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Example – Brewery Determination of optimum SAFETY STOCK level Probabilistic Model with Continuous Demand Real LEAD-TIME DEMAND Q d ~ N (r *, σ d ) ~ N (0, 1) Transformation

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

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

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Example – Brewery  Optimum SAFETY STOCK level Probabilistic Model with Continuous Demand p = 0.95 p = 0.99

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Example – Brewery  Optimum mean of total annual cost Probabilistic Model with Continuous Demand p = 0.95 p = 0.99

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

___________________________________________________________________________ Operations Research  Jan Fábry 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 - surplus - stockout - stockout penalty !!!

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

___________________________________________________________________________ Operations Research  Jan Fábry Inventory Models Example – Happyland  Bakery department – optimize everyday order of rolls  Purchase price = 1 CZK per roll Single-Period Decision Model  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 Inventory Models Example – Happyland  Daily demand – normal probabilistic distribution  = 10 000 rolls Single-Period Decision Model  = 500 rolls  Objective: determine optimum order quantity

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

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

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

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

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

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

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

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

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