# Lead time longer than cycle time

I am trying to calculate the reorder point and safety stock for a pharmaceutical product. I am using this formula $$r = \mu_{LTD} + z_\alpha\sigma_{LTD}.$$

$$SS = z_\alpha\sigma_{LTD}.$$

where,

$$\mu_{LTD} = \mu_D\mu_L$$

$$\sigma^2_{LTD} = \mu_L\sigma_D^2 + \mu_D^2\sigma_L^2$$

Let's say I take a time granularity of 1 month. That means my calculations for the mean and standard deviations of demand ($$\mu_D$$ and $$\sigma_D$$) will be done on a monthly basis. I get these values as $$26000$$ and $$6350$$ per month, respectively. The lead time for my product is $$\mu_L = 75$$ days or $$2.5$$ months. The standard deviation of lead time is assumed to be 20% of the mean lead time so that $$\sigma_L = 0.5$$. I also assume a Type 1 service level of 99% to get $$z = 2.33$$. Putting these values directly in the above formula I get the safety stock as $$SS = 38,000$$ and $$r = 103,000$$

The values I got from the formula seem absurdly high to me. The company typically orders about 25000 units of the product with each order. I will have to place 4 orders just to reach the reorder point and once all of those orders arrive, the amount of inventory I would be holding will be tremendous.

My question: Does the formula mandate that the lead time be shorter than the time granularity considered? Another way I thought about it was that since there are $$2$$ complete cycles in each lead time period, placing an order $$2.5$$ months back would be the same as placing it $$0.5$$ months back, which would yield an effective $$\mu_L = 0.5$$. I think the second approach should yield some reasonable numbers but I am not sure if it is correct. Any help is appreciated!

EDIT: I thought about this some more and came up with another approach. Suppose I assumed $$\mu_L$$ to be the "hypothetical lead time" of say $$1.5$$ months. If I do the calculations taking $$\mu_L = 0.5$$ (other parameters remaining same), I get $$SS = 35,000$$ and $$r = 74,000$$. Since this is a "hypothetical reorder point", I would actually place the order 1 month back. So if the reorder point suggests placing an order on 20th May, I would actually place an order on 20th April. The caveats I can see with this approach are that I would need to forecast the demand well to plan placing orders ahead of time. Again, not sure if this approach is any good.

• First, what is your firm policy to calculate the safety stocks? Is it based on the statistical analysis? AFAIK, many of manufacturing company prefer figuring out the SS based on whose experience rather than math analysis. Second, are you sure calculating the SS based on your need? (it has near $30$ percent fluctuations from the demands) – A.Omidi May 14 '20 at 15:43
• @A.Omidi I am not sure what policy they are following, but on looking at their inventory levels, I could make out that the lowest it falls to is about 22,000 units. Also, the levels rarely go beyond 60,000. Could you explain what you mean by 'SS based on need' and the 'fluctuations'? Thanks – Nishant Kumar Gupta May 14 '20 at 16:35
• I mean "by SS based on need" is that, if your company chases customers demands and generating the MPS according to it, you will have to calculate the SS and reorder point regard to real customers demand and choosing the service level by looking on historical data to achieve the minimum fluctuations. As you mentioned about the range of the demand, $[22000, 60000]$, I think what you calculated should not be unrealistic. If the firm does not have any specific policy to figure out the SS and ROP, you could try working on historical data to increase your precision. Is it helpful? – A.Omidi May 14 '20 at 18:14
• Looking at the historical customer demands will certainly help. I think that once I get a hold of the setup costs, holding costs etc, we can refine the "SS" and "r" values further by running some simulations. P.S.-What are your comments on the "Edit" I posted in the question. Is that approach correct? – Nishant Kumar Gupta May 15 '20 at 4:06
• I think your question title is more about, how you can determine the inventory position/level rather than cycle time. – A.Omidi May 15 '20 at 7:38

There is no contradiction or inconsistency in having the lead time be longer than the time unit* for the model. The important thing to remember here is that the reorder point $$r$$ refers to the inventory position (IP), not the inventory level (L). The inventory position equals the on-hand inventory plus the on-order inventory (minus backorders, if any).

So, setting $$r = 103{,}000$$ means you place an order when there are 103,000 on hand and on order (combined). If $$Q = 25{,}000$$ and $$\mu_{LTD} = 26{,}000 \times 2.5 = 65{,}000$$, then you always have a few orders outstanding. So a good portion of the 103,000 units in the IP are really on order, not on hand.

Typically you don't pay holding cost on on-order inventory, so these units don't cost us anything until they arrive.

To take a simpler example, suppose $$\mu=25{,}000$$ and $$\sigma=0$$ (so the demands are deterministic), $$L=2.5$$, $$r=100{,}000$$ (to keep the numbers simpler), and $$Q=25{,}000$$. Then the inventory position (red curve) and inventory level (blue curve) look like this:

(I assumed that at time 0 we have 50,000 units on hand and nothing on order, but you could assume any other numbers and the curves would eventually look the same once we reach steady state.)

The on-hand inventory fluctuates between 37,500 and 62,500, which seems roughly in line with the numbers you are describing in practice.

Another reason you might be seeing a discrepancy is your choice of a 99% type-1 service level. This is a reasonable target but if the company is using something smaller (or, more likely, isn't really using any formula at all and is just winging it), you might see lower inventory levels in practice. For example, if the service level is 95%, then $$r \approx 92,000$$, and all the inventory levels/positions will be about 10% lower than the ones you calculated.

*It's not really a cycle time, unless it's actually a periodic-review model and the firm can only order, say, once per month. But in that case you can get yourself into trouble because you have a lead time that's not an integer multiple of the period length, and things are even worse if the lead time is stochastic. So I'm assuming you're operating in continuous review.

• Assuming that my order quantity Q is always 25,000 my on-hand inventory should at least be greater than 78,000 units for the IP to be upwards of 103,000, right? This is where my confusion arises since the company inventory levels rarely go beyond 60,000. I understand that several orders can be outstanding and the IP will be above "r" in that case. But if those orders were made within a short span of time, I will end up receiving them at almost the same time, leading to large on-hand inventory. P.S-What are your comments on the "Edit" I posted in the question? Is that approach correct? – Nishant Kumar Gupta May 15 '20 at 4:45
• No, you should have 50-75k on order, so the IP fluctuates between 103,000 and 128,000. When the IP jumps up to 128,000, you have 3 orders outstanding so IL = 53,000. When the IP hits 103,000, you have 2 orders outstanding so IL again = 53,000. In between, the IL goes down by ~12,500 and up by ~12,500. So IL fluctuates between ~40,500 and ~65,500. I updated my answer to show a graph and explain the logic a bit more. – LarrySnyder610 May 15 '20 at 12:41
• About your Edit, you might be thinking along the right lines here, but I think what you're getting at is really the inventory position/inventory level dichotomy, and it's cleaner to think about it in those terms, in my opinion. Or at least more consistent with how inventory theory is usually explained. – LarrySnyder610 May 15 '20 at 12:42
• @LarrySnyder610, many thanks for your detailed explanation. would you say please, what is the difference between the IP and the IL in practice? We can divide the LT to the detail parts such as ordering, vendors procurement, etc. Once, we will calculate each item inventory (in the simplest case if we don't need to calculate the mixing inventory) based on you mentioned in the IP section. Also, I know that, in many of the production software, when a production order will be released, inventory position can be flagged and updated/recalculated automatically based on IP. – A.Omidi May 16 '20 at 7:49
• @A.Omidi At any point in time, you have stuff in inventory, and you have stuff that you have ordered but haven't yet received. (Either or both could be 0.) IL = the stuff in inventory (minus backorders, usually), and IP = IL + the stuff you have ordered but haven't received. It doesn't matter how the LT is broken down into various components: anything that has been ordered but not yet received gets counted in IP. – LarrySnyder610 May 16 '20 at 19:06