1. Other Inventory Models 1

2. Continuous Review or Q System 2 The EOQ model is based on several assumptions, one being that there is a constant demand. This may not be realistic. Next we consider some models that allow demand to occur more at random. In the EOQ model once the best order size, Q, was determined, the firm would order at a regular interval that divided the period (year) up into D/Q times. So, the same amount was ordered at the same interval. In the Q system we will study the same amount will be ordered at different intervals of time. Then, in the P system (I say more later) different amounts will be ordered at a fixed time interval. The Q system relies on the normal distribution, so we turn there next.

3. Q System 3 Say daily demand is normal with mean = 200 and standard deviation = 150. Then we might have a question like, “what is the probability daily demand will be 350, or less?” 200 350 daily demand To answer the question we see the normal distribution and we have to find the area under the curve to the left of 350. We resort to a z calculation = (value (like 350) minus mean)/st dev. Our z = (350 – 200)/150 = 1.00 (usually we round z to 2 decimals). Then we go to a z table and see the value with z = 1.00 of.8413 and we say the probability is 84.13.

4. Q System 4 Page 344 has a table that rounds z to 1 decimal and the service level is 84.1 (84.13 rounded to 1 decimal). So this table really has an abbreviated version of the z table. Now, when orders are placed to replenish inventory, it takes some number of days for the order to arrive. If demand during this lead time is higher than our trigger reorder level R (the level such that when our inventory reaches this level an order of Q Will be made), the demand will not be met and we say there is a stockout. If demand is less than this trigger reorder level then demand will be met and we can calculate the fill rate or service level as the percentage of customer demand satisfied by inventory.

5. Q System 5 Before we had an example of daily demand being normal with mean = 200 and standard deviation = 150. Now, if the lead time is 4 days, then the demand during the 4 days of lead time will be normal with mean = 4 times daily mean demand of 200 = 800 and standard deviation = sqrt(4 days)times daily standard deviation of 150 = 300. On the next slide I show a normal distribution with mean of 800 and a standard deviation of 300. We will use this graph and related ideas to help us determine what the trigger reorder amount R should be.

6. Q System 6 800 950 1100 1400 demand over lead time Say in general the mean demand over the lead time period is m, which equals 800 here. For a while I am just going to play a hypothetical game. I am going to ask what would happen if our trigger order amount R is various amounts. In fact I will look at the cases where R = 800, R = 950, R = 1100 and R =1400. Let’s do this next, but refer back to this slide to “see” what is going on.

7. Q System 7 Say we make R = 1400, meaning that if our stock position reaches R we will reorder some amount (I will say how much to order later). Since our lead time here is 4 days this will mean that over the next 4 days if actual demand is 1400 or less than we will have enough inventory to meet the demand. But, if actual demand is over 1400 we will not have enough on hand and there will be a stockout. (Have you ever gone to a store to buy something, maybe it was even advertised, and the store ran out? How do you feel at that point? Are you bummed out, upset or just plain seething with anger? Stores do not what to bum you out, but schtuff happens!)

8. Q System 8 Since demand is random, and here assumed normal, we can calculate what percentage of the time demand will be above or below the trigger amount R, here picked to be 1400. The z for 1400 is (1400 – 800)/300 = 2.0. The table on page 344 tells us in this case we would meet demand 97.7 percent of the time. Thus our service level or fill rate would be that we meet 97.7 percent of customer demand from inventory. Similarly, 100 – 97.7 = 2.3 percent of the time we would have a stockout. If R = 1100, z = (1100 – 800)/300 = 1.0 and the service level will be 84.1% and the stockout % will be 15.9%. If R = 950, z = (950 – 800)/300 =.5 and the service level will be 69.1% and the stockout % will be 30.9%. If R = 800, z =(800 – 800)/300 = 0 and the service level will be 50% and the stockout % will be 50%

9. Q System 9 What I have done here is talk about hypothetical R values, levels of the stock position that would trigger an order be made. With different R values we see different service levels and stockout %’s. We could work in reverse to what I have presented. If demand over lead time has mean = 800 and standard deviation = 300, what should R be to make the service level 97.7? The z there is 2.0. Thus 2.0 = (R – 800)/300 and solving for R we get R = 800 + 2.0(300) = 1400. In general, R = m + zσ = mean over lead time + safety stock. Thus, we need to think about how serious our customers become if there is a stock out. The more serious the higher the service level should be and thus the higher the z we pick.

10. Q System 10 I mentioned before I would say how much should be ordered. Just order the EOQ on a yearly average demand basis. Thus, if average demand is 200 per day, and say we are open 5 days a week for 50 weeks then annual average demand is 250(5)(50) = 50,000 units per year. If S = $20 per order, i = 20% per year and C = $10 per unit, the the EOQ Q = sqrt[{2(20)(50000)}/{.2(10)}] = sqrt(1000000) = 1000. So, 1000 units would be ordered when R is reached. Since annual demand has an average of 50,000 and we order in lots of 1000 we will make an average of 50 orders per year. Since there are 250 working days in the year orders will be made on average every 250/50 = 5 days.

11. Q System 11 What should R be if you want a service level of 95.0? Note 95.0 is not in the table on page 344, but it is in the middle of 94.5 and 95.5 so we take the z in the middle of the associated z’s for a z = 1.65. R = 800 + 1.65(300) = 1295. The order amount R depends of the service level desired. The Q amount to order still is picked by the EOQ method, but using annual average demand.