**Stochastic Inventory TheoryProfessor Stephen R.** LawrenceLeeds School of BusinessUniversity of ColoradoBoulder, CO 80309-0419

**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 $12.00. 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?

**Opportunity Cost of Unmet Demand** • Define: • U = opportunity cost of unmet demand (underproduce - understock) • Example: • U = sales price - cost of production • = 12 - 6 • = $6 lost profit / unit

**Cost of Excess Inventory** • Define: • O = cost of excess inventory • (overproduce - overstock) • Example: • O = cost of production - salvage price • = 6 - 3 • = $3 loss/unit

**Solving Single-Period Problems** Example U = cost of unmet demand (understock) U = 12 - 6 = 6 profit O = cost of excess inventory (overstock)O = 6 - 3 = 3 loss Optimal Solution: Where Pr(x≤Q*) is the “critical fractile” of the demand distribution. Produce/purchase quantity Q* that satisfies the ratio

**Translation to Textbook Notation**

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

**Solving Single-Period Problems** • Example • U = cost of unmet demand (underage) U = 12 - 6 = 6 profit • O = cost of excess inventory (overage)O = 6 - 3 = 3 loss • Example: • Pr(x ≤ Q) = 6 / ( 3 + 6 ) = 0.667

**Solving Single-Period Problems** 0.667 4,222 D

**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…

**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. Notation

**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 variances 2 by leadtime L • Example • Standard deviation of demand s = 4 units per day • Calculate variance of demand s2 = 16 • Variance of demand over leadtime L=40 daysis sL2 = Ls2 = 40×16=640 • Standard deviation of demand over leadtime L issL = [Ls2]½ = 640½ = 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 Inventory Level Q L R SS safety stock 0 stockout! time

**Inventory** Level R R SS R SS SS 0 time Inventory / Stockout Trade-offs Small safety stocks Frequent stockouts Low inventory costs Large safety stocks Few stockouts High 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**

**Inventory** Level Q L R Q Q Q time (CR) Continuous Review System • 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

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

**(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 • sL = 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**

**q** I q I L L L I Inventory Level q time T T T Multi-Period Fixed-Interval Systems • Requires periodicreview of inventory levels • Replenish inventories every T time units • Order quantity q (q varies with each order)

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

**(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 • sL= 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) • www.APICS.org • Professional organization of production, inventory, and resource managers • Offers professional certifications in production, inventory, and resource management

**Further Information** • Institute for Supply Management • (www.ISM.ws) • Previously the National Association of Purchasing Managers (NAPM) • Professional organization of supply chain managers • Offers certifications in supply chain management