Stochastic Inventory Theory Professor Stephen R. Lawrence Leeds School of Business University of Colorado Boulder, CO

Stochastic Inventory Theory Single Period Stochastic Inventory Model  “Newsvendor” model Multi-Period Stochastic Inventory Models  Safety Stock Calculations  Expected Demand & Std Dev Calculations  Continuous Review (CR) models  Periodic Review (PR) models

Single Period Stochastic Inventory “Newsvendor” Model

Single-Period Independent Demand “Newsvendor Model:” One-time buys of seasonal goods, style goods, or perishable items Examples:  Newspapers, Christmas trees;  Supermarket produce;  Fad toys, novelties;  Fashion garments;  Blood bank stocks.

Newsvendor Assumptions Relatively short selling season; Well defined beginning and end; Commit to purchase before season starts; Distribution of demand known or estimated; Significant lost sales costs (e.g. profit); Significant excess inventory costs.

Single-Period Inventory Example A T-shirt silk-screening firm is planning to produce a number of custom T-shirts for the next Bolder Boulder running event. The cost of producing a T-shirt is $6.00, with a selling price of $ After BB concludes, demand for T-shirts falls off, and the manufacturer can only sell remaining shirts for $3.00 each. Based on historical data, the expected demand distribution for BB T-shirts is: How many T-shirts should the firm produce to maximize profits? QuantityProbabilityCumulative

Opportunity Cost of Unmet Demand Define: U = opportunity cost of unmet demand (underproduce - understock) Example: U = sales price - cost of production = = $6 lost profit / unit

Cost of Excess Inventory Define: O = cost of excess inventory (overproduce - overstock) Example: O = cost of production - salvage price = = $3 loss/unit

Solving Single-Period Problems Where Pr(x≤Q*) is the “critical fractile” of the demand distribution. Example U = cost of unmet demand (understock) U = = 6 profit O = cost of excess inventory (overstock) O = = 3 loss Produce/purchase quantity Q* that satisfies the ratio Optimal Solution:

Translation to Textbook Notation LawrenceTextbook Understock costUC us Overstock costOC os Probability of understocking Pr(x  Q) P us Critical fractile Pr(x  Q) Critical fill rate (cfr)

Alternate Solution Where Pr(x>Q*) is the “critical fractile” that represents the probability of a stockout when starting with an inventory of Q* units. Produce/purchase quantity Q* that satisfies the ratio Some textbooks use an alternative representation of the critical fractile: NOTE: to use a standard normal Z-table, you will need Pr(x≤Q*), NOT Pr(x>Q*)

Solving Single-Period Problems Example U = cost of unmet demand (underage) U = = 6 profit O = cost of excess inventory (overage) O = = 3 loss Example: Pr(x ≤ Q) = 6 / ( ) = 0.667

Solving Single-Period Problems D ,222

Inventory Spreadsheet

Multi-Period Stochastic Inventory Models Continuous Review (CR) models Periodic Review (PR) models

Key Assumptions Demand is probabilistic Average demand changes slowly Forecast errors are normally distributed with no bias Lead times are deterministic

Key Questions How often should inventory status be determined? When should a replenishment order be placed? How large should the replenishment be?

Types of Multi-Period Models (CR) continuous review  Reorder when inventory falls to R (fixed)  Order quantity Q (fixed)  Interval between orders is variable (PR) periodic review  Order periodically every T periods (fixed)  Order quantity q (variable)  Inventory position I at time of reorder is variable Many others…

Notation B = stockout cost per item TAC = total annual cost of inv. L = order leadtime D = annual demand d(L) = demand during leadtime h = holding cost percentage H = holding cost per item I = current inventory position Q = order quantity (fixed) Q* = optimal order quantity q = order quantity (variable) T = time between orders R = reorder point (ROP) S = setup or order cost SS = safety stock C = per item cost or value.

Demand Calculations

Demand over Leadtime Multiply known demand rate D by leadtime L  Be sure that both are in the same units! Example  Mean demand is D = 20 per day  Leadtime is L= 40 days  d(L) = D x L = 20 x 40 = 800 units

Demand Std Deviation over Leadtime Multiply demand variance  2 by leadtime L Example  Standard deviation of demand  = 4 units per day  Calculate variance of demand  2 = 16  Variance of demand over leadtime L=40 days is  L 2 = L  2 = 40×16=640  Standard deviation of demand over leadtime L is  L = [L  2 ] ½ = 640 ½ = 25.3 units Remember  Variances add, standard deviations don’t!

Safety Stock Calculations

Safety Stock Analysis The world is uncertain, not deterministic  demand rates and levels have a random component  delivery times from vendors/production can vary  quality problems can affect delivery quantities  Murphy lives safety stock R L time Inventory Level Q 0 stockout! SS

Inventory / Stockout Trade-offs Inventory Level R SS 0 time R SS R Large safety stocks Few stockouts High inventory costs Small safety stocks Frequent stockouts Low inventory costs Balanced safety stock, stockout frequency, inventory costs

Safety Stock Example Service policies are often set by management judgment (e.g., 95% or 99% service level)  Monthly demand is 100 units with a standard deviation of 25 units. If inventory is replenished every month, how much safety stock is need to provide a 95% service level? Assume that demand is normally distributed. Alternatively, optimal service level can be calculated using “Newsvendor” analysis

Continuous Review (CR) Stochastic Inventory Models

Always order the same quantity Q Replenish inventory whenever inventory level falls below reorder quantity R Time between orders varies Replenish level R depends on order lead-time L Requires continuous review of inventory levels R QQ Q Q L time Inventory Level (CR) Continuous Review System

Safety Stock and Reorder Levels Reorder Level = Safety Stock + Mean Demand over Leadtime R = SS + D L L Distribution of demand over leadtime L Stockout! Inventory Level time R SS 0 DLDL

(CR) Order-Point, Order-Quantity  Continuous review system  Useful for class A, B, and C inventories  Replenish when inventory falls to R;  Reorder quantity Q.  Easy to understand, implement  “Two-bin” variation

(CR) Implementation Implementation  Determine Q using EOQ-type model  Determine R using appropriate safety-stock model Practice  Reserve quantity R in second “bin” (i.e. a baggy)  Put order card with second bin  Submit card to purchasing when second bin is opened  Restock second bin to R upon order arrival

(CR) Example Consider a the following product  D = 2,400 units per year  C = $100 cost per unit  h = 0.24 holding fraction per year (H = hC = $24/yr)  L = 1 month leadtime  S = $ 200 cost per setup  B = $ 500 cost for each backorder/stockout   L = 125 units per month variation Management desires to maintain a 95% in-stock service level.

(CR) Example Whenever inventory falls below 406, place another order for 200 units

Total Inventory Costs for CR Policies TAC = Total Annual Costs TAC = Ordering + Holding + Expected Stockout Costs TAC = $10,044 per year (CR policy)

Periodic Review (PR) Stochastic Inventory Models

Multi-Period Fixed-Interval Systems Requires periodic review of inventory levels Replenish inventories every T time units Order quantity q (q varies with each order) TTT LLL    I I I Inventory Level time q q q

Periodic Review Details Order quantity q must be large enough to cover expected demand over lead time L plus reorder period T (less current inventory position I ) Exposed to demand variation over T+L periods TTT LLL    I I I Inventory Level time q q q

(PR) Periodic-Review System Periodic review (often Class B,C inventories) Review inventory level every T time units Determine current inventory level I Order variable quantity q every T periods Allows coordinated replenishment of items Higher inventory levels than continuous review policies

(PR) Implementation Implementation  Determine Q using EOQ-type model;  Set T=Q/D (if possible --T often not in our control)  Calculate q as sum of required safety stock, demand over leadtime and reorder interval, less current inventory level Practice  Interval T is often set by outside constraints  E.g., truck delivery schedules, inventory cycles, …

(PR) Policy Example Consider a product with the following parameters:  D = 2,400 units per year  C = $100  h = 0.24 per year (H = hC = $24/yr)  T = 2 months between replenishments  L = 1 month  S = $200  B = $500 cost for each backorders/stockouts  I = 100 units currently in inventory   L = 125 units per month variation Management desires to maintain a 95% in-stock service level.

(PR) Policy Example Suppose that this is given by circumstances…

Total Inventory Costs for PR Policies TAC = Total Annual Costs TAC = Ordering + Holding + Expected Stockout Costs TAC = $14,718 per year (PR policy)

Further Information American Production and Inventory Control Society (APICS)  Professional organization of production, inventory, and resource managers Offers professional certifications in production, inventory, and resource management

Further Information Institute for Supply Management  (  Previously the National Association of Purchasing Managers (NAPM) Professional organization of supply chain managers Offers certifications in supply chain management