# Estimating optimal inventory in times of high uncertainty due to coronavirus

I'm trying to help out a relative of mine who runs a small highly seasonal business in the clothing/textiles industry. 80% of sales are in Q1 of every year, and the purchasing and manufacturing decisions must usually be placed around May-July given lead time.

Their usual process is to assume a 5% yearly demand increase, so they calculate per-SKU inventory on April and ensure they're ordering enough to reach 105% inventory compared to previous year. Given the variability (and lot's of SKUs), it's not optimal, but simple enough and usually works well for them.

Given the high uncertainty of coronavirus, demand is highly uncertain as many of their customers are affected and would cut orders.

I'm trying to dust-off my Operations Research background from engineering undergrad to provide help that is slightly better than a hand-wavy guess. Right now the guess it to roughly cut all order by 20-40%. Which is obviously a very broad range.

This seems like a newsvendor-type problem, with uncertain demand. I do have access to 5yr historical sales & inventory levels, so can run to estimate the demand distribution. But I have no idea on how to go about estimating future demand, and obtaining and a better Q (order quantity). I also have access to the price & cost of SKUs. I assume a simple newsvendor isn't the right path forward, as D (demand) is highly uncertain (mean decreasing and std likely increasing).

Any recommendations of good pointers and/or practical examples to apply? I hope this is't too practical for this stackexchange.

• Welcome to OR.SE! This question is not at all too practical for this site. Glad you asked it. – LarrySnyder610 May 11 at 3:42
• Not directly relevant, but somewhat: I just saw this call for papers for a session on The Impact of Covid on Inventories as part of the Int'l Symposium on Inventory Research. – LarrySnyder610 May 12 at 15:18

This is indeed a newsvendor problem. The fact that D is very uncertain only makes it more so.

If we were in normal times, the standard approach would be:

• Use your historical data to calculate $$\hat{\mu}$$, an estimate for the mean demand. (Sounds like $$\hat{\mu} = 1.05[\text{last year's demand}]$$ is the go-to estimate for your relative.)
• If your historical data contains both forecasts and actuals, then calculate $$\hat{\sigma}_e$$, an estimate for the standard deviation of the forecast error of the demand. If your historical data doesn't contain forecasts, then approximate $$\hat{\sigma}_e$$ as $$\hat{\sigma}$$, the standard deviation of the observed demands.
• (The reason for using $$\hat{\sigma}_e$$ instead of $$\hat{\sigma}$$ is, roughly speaking, that it's not the volatility in the demand that requires us to hold safety stock, it's our inaccuracy in forecasting that volatility.)
• Set $$Q = \hat{\mu} + z_\alpha\hat{\sigma}_e,$$ where $$\alpha = p/(p+h)$$, $$p$$ is the stockout cost per unit, and $$h$$ is the holding cost per unit. (Alternately, choose a service level $$\alpha$$ and use it directly.)

But of course, these are not normal times. Your historical data will be unreliable at best, and either of your estimates $$\hat{\mu}$$ and $$\hat{\sigma}_e$$ will be subject to lots of inaccuracies.

I guess I've basically just restated your question. But maybe thinking about the problem in these terms—especially the point about $$\hat{\sigma}_e$$—will help structure your thoughts about it.

What's very hard here is that in normal years, your relatively probably has $$\mu \gg \sigma$$, which means the newsvendor logic—basically the $$z_\alpha\hat{\sigma}_e$$ term—adds a lot of value. Next year your relative's (and everybody else's) ability to estimate $$\mu$$ is going to have inaccuracies that are larger than $$\sigma$$. So newsvendor logic might not be too useful.

If I had to choose between (a) being given a solid estimate for $$\mu$$, but I had to order exactly $$\mu$$, or (b) using a rule-of-thumb to estimate $$\mu$$, but I was allowed to use newsvendor logic, I'd probably choose (a). It hurts me to say it, because I'm an inventory-theory afficionado, but it's hard to get around it this year.

• Thanks. Fun to confirm that this is a hard problem. And here I was trying to apply the newsvendor textbook solution. I'm finding some success by extending the problem a bit further to incorporate different D and Q scenarios into the operating financials of this company. To see if there is a better solution when the optimization problem is more about "don't run out of money" and not just "optimize inventory" – leonsas May 12 at 13:25
• Stochastic optimization (optimizing under scenarios) seems like a reasonable way to go in this case. – LarrySnyder610 May 12 at 14:35

There are quite a few wrinkles here, specific to the current sources of uncertainty. Will the suppliers be giving discounts (to get orders after coming off a shutdown)? Discounts would reduce the risk of over-ordering. What normally happens with excess inventory (hold it, sell it at a discount, scrap or donate it, ...)? If they sell off the excess to a secondary market, will that market be there this coming year? Is this a luxury/optional purchase (swimwear) or a necessity (baby clothes ... I'm guessing not that, given the seasonality, but it's just an example). The answer relates to the issue of how much discretionary spending customers will do given the projected deep recession. How are stock-out costs calculated? If loss of customers to competitors is part of that calculation, will all the competitors be back and have inventory this time around?

On the demand forecasting side, it might be possible to find industry-wide sales figures for similar goods (swimsuits, school uniforms, whatever this is) in the aftermath of previous economic downturns (the "Great Depression" here, the 2008 recession, ...). That data won't translate directly into forecasts, but it might provide an idea of how quickly sales of like items have bounced back in the past.

Given all the uncertainty, and the unprecedented nature of the situation, I'm with Larry in doubting that the data is there to let a newsvendor model (let alone anything more complex) work well.