1 Inventory Control with Stochastic Demand

2  Week 1Introduction to Production Planning and Inventory Control  Week 2Inventory Control – Deterministic Demand  Week 3Inventory Control – Stochastic Demand  Week 4Inventory Control – Stochastic Demand  Week 5Inventory Control – Stochastic Demand  Week 6Inventory Control – Time Varying Demand  Week 7Inventory Control – Multiple Echelons Lecture Topics

3  Week 8Production Planning and Scheduling  Week 9Production Planning and Scheduling  Week 12Managing Manufacturing Operations  Week 13Managing Manufacturing Operations  Week 14 Managing Manufacturing Operations  Week 10Demand Forecasting  Week 11Demand Forecasting  Week 15Project Presentations Lecture Topics (Continued…)

4  Demand per unit time is a random variable X with mean E ( X ) and standard deviation   Possibility of overstocking (excess inventory) or understocking (shortages)  There are overage costs for overstocking and shortage costs for understocking

5  Single period models  Fashion goods, perishable goods, goods with short lifecycles, seasonal goods  One time decision (how much to order)  Multiple period models  Goods with recurring demand but whose demand varies from period to period  Inventory systems with periodic review  Periodic decisions (how much to order in each period) Types of Stochastic Models

6  Continuous time models  Goods with recurring demand but with variable inter-arrival times between customer orders  Inventory system with continuous review  Continuous decisions (continuously deciding on how much to order) Types of Stochastic Models (continued…)

7 Example  If l is the order replenishment lead time, D is demand per unit time, and r is the reorder point (in a continuous review system), then Probability of stockout = P(demand during lead time  r)  If demand during lead time is normally distributed with mean E(D) l, then choosing r = E(D) l leads to Probability of stockout = 0.5

8 The Newsvendor Model

9 Assumptions of the Basic Model  A single period  Random demand with known distribution  Cost per unit of leftover inventory (overage cost)  Cost per unit of unsatisfied demand (shortage cost)

10  Objective: Minimize the sum of expected shortage and overage costs  Tradeoff: If we order too little, we incur a shortage cost; if we order too much we incur a an overage cost

11 Notation

12 The Cost Function

13 The Cost Function (Continued…)

14 Leibnitz’s Rule

15 The Optimal Order Quantity The optimal solution satisfies

16 The Exponential Distribution The Exponential distribution with parameters

17 The Exponential Distribution (Continued…)

18 Example Scenario:  Demand for T-shirts has the exponential distribution with mean 1000 (i.e., G ( x ) = P ( X  x ) = 1- e - x /1000 )  Cost of shirts is $10.  Selling price is $15.  Unsold shirts can be sold off at $8. Model Parameters:  c s = 15 – 10 = $5  c o = 10 – 8 = $2

19 Example (Continued…) Solution: Sensitivity: If c o = $10 (i.e., shirts must be discarded) then

20 The Normal Distribution The Normal distribution with parameters  and , N ( ,  ) If X has the normal distribution N ( ,  ), then ( X -  )/  has the standard normal distribution N (0, 1). The cumulative distributive function of the Standard normal distribution is denoted by .

21 The Normal Distribution (Continued…) G ( Q* )=   Pr( X  Q *)=   Pr[( X -  )/   ( Q* -  )/  ] =   Let Y = ( X -  )/  then Y has the the standard Normal distribution Pr[( Y  ( Q* -  )/  ] =  [( Q* -  )/  ] = 

22 The Normal Distribution (Continued…)  (( Q* -  )/  =   Define  z   such that  z  )   Q* =  + z  

23 The Optimal Cost for Normally Distributed Demand

24 The Optimal Cost for Normally Distributed Demand Both the optimal order quantity and the optimal cost increase linearly in the standard deviation of demand.

25 Example  Demand has the Normal distribution with mean  = 10,000 and standard deviation  = 1,000  c s = 1  c o = 0.5 

26 Example  Demand has the Normal distribution with mean  = 10,000 and standard deviation  = 1,000  c s = 1  c o = 0.5  Q* =  + z   From a standard normal table, we find that z 0.67 = 0.44 Q* =  +  z  

27 Service Levels  Probability of no stockout  Fill rate

28 Service Levels  Probability of no stockout  Fill rate Fill rate can be significantly higher than the probability of no stockout

29 Discrete Demand X is a discrete random variable

30 Discrete Demand (Continued) The optimal value of Q is the smallest integer that satisfies This is equivalent to choosing the smallest integer Q that satisfies or equivalently

31 The Geometric Distribution The geometric distribution with parameter , 0    1

32 The Geometric Distribution The optimal order quantity Q * is the smallest integer that satisfies

33 Extension to Multiple Periods The news-vendor model can be used to a solve a multi-period problem, when :  We face periodic demands that are independent and identically distributed (iid) with distribution G ( x )  All orders are either backordered (i.e., met eventually) or lost  There is no setup cost associated with producing an order

34 Extension to Multiple Periods (continued…) In this case  c o is the cost to hold one unit of inventory in stock for one period  c s is either the cost of backordering one unit for one period or the cost of a lost sale

35 Handling Starting Inventory/backorders

36 Handling Starting Inventory/backorders