Tuesday, 14 July 2009

Two sides of the "Diet Problem"

Yesterday I contributed a comment to Laura McLay's blog about Operations Research in the USA. She was considering the way that food manufacturers mark the nutritional content of their products. Here are her comments.

I wrote:
There is a similar debate going on in the UK. Our Food Standards Agency has proposed a “traffic light” system showing that certain ingredients are low, medium or high. Some manufacturers have adopted the FSA system, others have refused to use it. Some supermarkets (and in the UK, the food retailing sector is dominated by a few large supermarket chains) have chosen their own systems.

It doesn’t look as if the FSA has used any OR in their research; I would have thought that OR could have helped answer the question that doesn’t really get tackled “What information will people use, and how will they use it?”.

It is possible to see some of the research reports that led to the FSA recommendations at http://www.food.gov.uk/foodlabelling/signposting/siognpostlabelresearch/alt
I was surprised at one response that said 25% of consumers always read nutritional labels. The question which led to this response was badly phrased and it looks as if the response was badly understood. Just watch the shoppers in your local shop; do 25% of them read every label?

Historians of O.R. reckon that Stigler's diet problem was one of the catalysts of the development of linear programming at the end of the 1940's ane early 1950's. I, like many academics, have used nutrition as a simple example of a medium-sized linear programming problem in my classes. Nutrition is additive, and there are one or two interesting constraints on maxima and minima of nutrients in the human diet. Some of them are straightforward, others are expressed as percentages of other nutrients. Many people (myself included) have used data from McDonald's to see if one can find a minimal cost, "Healthy" diet from that chain. [If you have never encountered this problem, then there are two twists in the modelling. The first is the obvious one that the problem really is an integer programming problem. The second is that sachets of sauce are free and contain nutrients; without constraining the number of sachets that you use in a day, the LP solution uses over 40.] The model has a variety of extensions and lessons for the class, for example, concerning shadow prices. [Apologies to those who do not know what a shadow price is; in this case, I used it as a tool to tell you what the maximum price should be for an item that is not in the diet.]

My former colleague, Alan Munford created an integrated database and optimisation tool for mixing feed for animals, who are less choosy about their diets than humans.(Incidentally, Munford's theorem states that for any random variable X, with mean \mu and variance \sigma^2, then for any value k
Probabilty (abs(X-\mu) \ge k^2\sigma^2) \le minimum of((1/k^2),1) [footnote])

My title was "Two sides of the ...". The second side is the one I alluded to in my comment on Laura's blog. People need information. There is an immense amount that you could list about any item of food. What ought to be put on the packaging of processed food? Those with the commonest allergies need to have simple, clear warnings. That is almost straightforward, though there are numerous unusual allergies whose sufferers need to peruse the whole list of ingredients. So those observations give the essentials: common allergies, list of everything. But what about the extra, general information? As I said, the FSA does not appear to have really though this one through, and there is a case for using some O.R. in answering it.

[footnote] Munford's theorem is a joke. Alan introduced it when he was teaching a class of probability, and proved Chebyshev's theorem, which has the inequality
\le (1/k^2). However all probabilities must be less than 1. A few years after this spoof was introduced, a firstyear student told Alan how excited he was to be taught by someone who had a theorem named after them; this student had been taught by one of our graduates who had swallowed the story that this result carried Alan's name.

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