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Math of R&D, Part One


Mister Death
RJ: McFlono McFloninoo

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Joined: Feb 6, 2012
Posted on Apr 1, 2012
The purpose of this post, and hopefully a follow-up, is to examine how and where to invest in research. Because I sell supermarket products, my supply chain and my spreadsheet are completely geared toward the supermarket supply chain, so unless you too are running a complete supermarket supply chain (and are making the same production decisions as me), do not use my numbers except as examples and guides. The formulas, on the other hand, you can plug into your own supply chain and work out your own results. Also, the model as it stands has some big holes in it which I haven't fixed yet, but what comes out of it already is big enough that I felt I had to post as quickly as I could.

In this first post, I'll examine cost-effectiveness - if you spend a dollar on research, how much will it affect your overall quality (and thus your profit)?

Here's how I break things down. For every base price dollar (BP$) of an item that goes on the shelves, I give that product an e-rating of 1 (e for effectiveness). To simplify matters I assume that all items sell at the same rate, so I've given each shelf product a starting e-rating of 1; but it would be easy enough to modify the starting e-ratings, for instance if you wanted to factor in demand levels.

The e-rating of every product is split up according to what contributes to its quality. Apple pie for instance contributes 0.3 to apple, 0.2 to pastry dough, 0.1 to sugar, and 0.4 to apple pie research. The amounts that go to the ingredients continue down the chain until they contribute to a product's research.

Since nothing else contributes to apple pie research, it winds up with an e-rating of 0.4. Other researches get multiple contributions. As a simple example, ham research gets 0.3 from ham itself and 0.06 from ham sandwiches, and that's it, so the e-rating of ham is 0.36. (Ham and ham sandwiches can both get different e-ratings if you make them in different factories. I'm making ham in a smokehouse and the sandwiches in a bakery, but if you made both of them in food processing, the e-rating of your ham would be 0.46.) Bananas get used in 6 different products, 3 of which I make; so my bananas have an e-rating of 1 (banana) + 0.35 (banana chips) + 0.30 (banana cream pie) + 0.16 (banana juice concentrate) = 1.81. Now 75% of that goes to banana research, and the other 25% trickles down to water; so my banana research e-rating is 0.75 * 1.81 = 1.3575. This would go up if I decided to open a food court and sell banana splits, for instance.

What the e-value represents is essentially the amount that quality goes up overall for your products if you do one quality level's worth of research. There are other factors involved, such as lag time as you clear older crappier inventory, but I didn't want to spend forever on the model. If you do research on something with an e-value of 2, your overall quality eventually goes up twice as much as if you research something with an e-value of 1.

A supermarket sells 156 products, and in order to make those I also produce 43 ingredient-only products for a total of 199 (Edit: these numbers were prior to the introduction of black licorice...). My research e-values don't add up to 156 like they should (currently at 155.89), because I'm an imperfect human and missed a chain or two, but I'm looking for them. The e-values I have range from a low of 0.026 for natural gas to 15.383165 for water, the mega-ingredient that contributes to about 85 of the products I make.

So relatively speaking, I'd much rather research water - but only up to a point, because the prices go up for each quality level. It turns out that, because research prices grow exponentially, for any two products there is a set difference in levels where the benefits per research dollar are the same for both products; and it's even easy to calculate, if you're not afraid of logarithms, which I'm not. Just work out q-values for each product as I outline next, and calibrate your favourite product to 0 by adding the right value. I want to set water to zero, so I calculate

Q = 14.0687896813239 + (log(e-value) - log(base)) / log(1.2)

for each product. The e-value you already know about, and "base" is the research price of each item (for my products this ranges from $200 to $2000). The number at the front is the fudge factor that makes water have Q = 0 in my supply chain.

I was expecting there to be some spread in these numbers, maybe a dozen or two levels between the best and worst. Maybe even a couple real dogs because I do have some ingredients that I only need sparingly (bauxite, for instance). Well, it turns out that with current prices and my supply chain, and not pricing R&D buildings in (which is tricky!), the next-best thing to research after water is milk, whose Q is -4.7; it benefits from a magic combination of low research price (base = $200), high research % (90) and high product e-value (7.17). It turns out that milk is the only other product that even starts with a single digit! Third is wheat, almost exactly 10 levels below water, and around -12 you start seeing the better fruits and crops - the 75% research level to quality helps a lot - and some of the power ingredients come next, starting with malt vinegar(!) at -16.1. The mean across all my products is -27.6. There is a huge cluster of products around the -31 to -32 range, and at the very bottom there are three below -40, one of which (neapolitan ice cream, Q = -40.25) is a shelf product! Q = -40.25 means that for every dollar I spend on neapolitan ice cream research, all other things being equal, I'd want to spend 1.2^40 = $1,538 on water research; that I'm not even motivated to research it at all until I Q28 water coming out of my beverage factory. That sort of a gap shouldn't be happening IMHO.

I haven't completed part 2 of my formula yet - factoring in the prices of R&D facilities - because then I lose the exponential niceness and I have to actually do some real work, but I think this is enough to show that there are serious issues with R&D pricing as the situation stands. The good news for you, Scott, is that you have some really big knobs to fiddle with to bring some of these values into more reasonable synch. The base price, naturally, is the first knob, but the second is the 1.2. Have fun...


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