Random Demand: Fixed Order Quantity

1. Random Demand:Fixed Order Quantity Dr. Ron Lembke

2. Random Demand • Don’t know how many we will sell • Sales will differ by period • Average always remains the same • Standard deviation remains constant How would our policies change? • How would our order quantity change? • EOQ balances ordering vs holding, and is unchanged • How would our reorder point change? • That’s a good question

3. Constant Demand vs Random • Steady demand • Always buy Q • Reorder at R=dL • Sell dL during LT • Inv = Q after arrival • Random demand • Always buy Q • Reorder at R=dL + ? • Sell ? during LT • Inv = ? after arrival Inv Q Q Q Q R R L L

4. Random Demand • Reorder when on-hand inventory is equal to the amount you expect to sell during LT, plus an extra amount of safety stock • Assume daily demand has a normal distribution • If we want to satisfy all of the demand 95% of the time, how many standard deviations above the mean should the inventory level be? • Just considers a probability of running out, not the number of units we’ll be short.

5. Demand over the Lead Time R = Expected Demand over LT + Safety Stock = Average demand per day = Lead Time in days = st deviation of demand over Lead Time z from normal table, e.g. z.95 = 1.65

6. Random Example • What should our reorder point be? • Demand averages 50 units per day, L = 5 days • Total demand over LT has standard deviation of 100 • want to satisfy all demand 90% of the time • To satisfy 90% of the demand, z = 1.28

7. Random Demand • Sometimes use SS • Sometimes don’t • On average use 0 SS • Random demand • Always buy Q • Reorder at R=dL + SS Inv R dL SS L L L L

8. St Dev of Daily Demand • What if we only know the average daily demand, and the standard deviation of daily demand? • Lead time = 4 days, • daily demand = 10, • Daily demand has standard deviation = 5, • What should our reorder point be, if z = 3?

9. St Dev Over LT If the average each day is 10, and the lead time is 4 days, then the average demand over the lead time must be 40. What is the standard deviation of demand over the lead time? Std. Dev. ≠ 5 * 4

10. St Dev Over Lead Time We can add up variances, not standard deviations Standard deviation of demand over LT =

11. Demand Per Day Or, same thing: = Lead time in days = average demand per day = st deviation of demand per day z from normal table, e.g. z.95 = 1.65

12. Random DemandFixed Order Quantity • Demand per day averages 40 with standard deviation 15, lead time is 5 days, service level of 90% = 5 days = 40 = 15 = 1.30

13. Fixed-Time Period Model • Place an order every, say, week. • Time period is fixed, order quantity will vary • Order enough so amount on hand plus on order gets up to a target amount • Q = S – Inv • Order “up to” policies

14. Fixed Order Period • Amount on hand after arrival differs • Order today, next order comes T+L days later • Order every T days • Q = S – Inv • Order comes in L days after being placed T L S Inv On hand On hand + on order T L

15. Service Level Criteria • Type I: specify probability that you do not run out during the lead time • Chance that 100% of customers go home happy • Type II: (Fill Rate) proportion of demands met from stock • 100% chance that this many go home happy, on average

16. Two Types of Service CycleDemand Stock-Outs 1 180 0 2 75 0 3 235 45 4 140 0 5 180 0 6 200 10 7 150 0 8 90 0 9 160 0 10 400 Sum 1,450 55 Type I: 8 of 10 periods 80% service Type II: 1,395 / 1,450 = 96%

17. Summary • Fixed Order Quantity – always order same • Random demand – reorder point needs to change • Standard Deviation over the LT is given • Standard Deviation per day is given • Fixed Time Period • Always order once a month, e.g. • Amount on hand plus on order will add up to S • Different service metrics: Type I, Type II