Random Demand: Fixed Order Quantity Dr. Ron Lembke

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

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

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.

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

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

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

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

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

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

Fixed-Time Period Model S = Order-Up-To Level S = 40*(7+2) + 1.2 * Sqrt(7+2)*10 S = 40*9 + 1.2 * 3 * 10 S = 360 + 36 = 396 Cycle stock = 360 Safety Stock = 36 = 40 = 2 days = 10 = 1.20 = 7 days Service level = 88.5%

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

Two Types of Service Cycle Demand 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 40 0 Sum 1,450 55 Type I: 8 of 10 periods 80% service Type II: 1,395 / 1,450 = 96%

Summary Fixed Order Quantity – always order same Fixed Time Period 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