The diet problem

I want to limit my carbs, my calories, my fat, and my protein, but I don’t want to starve to death.

One way to look at it is to minimize carbohydrates subject to calories between 1800 and 2200, protein between 77 and 95 grams, and calories from fat less than 220.

That’s a linear program, which I know how to solve. All that’s left to specify is the characteristics of the foods I want to pick from.

Let’s start with Kroger brand CarbWatch yogurt (not really a yogurt). It has 80 calories, 10 from fat, 12 grams of protein and 4 grams of carbs.

A diet of just the low carb yogurt won’t work, it has an infeasible solution to the linear program since to get enough calories from just the one food would greatly exceed my protein limits.

So as a second food I need to add something with very small protein contents. Let’s try a banana. That has 93 calories, 4+ from fat, 1 gram of protein, and 24 grams of carbohydrates..

About 6-7 yogurt containers and 13-14 bananas seems to be a workable mix although I think I’d tire of that diet pretty quickly and I expect that it would cause some deficiencies in vitamins my model isn’t tracking.

To add a little variety let’s think about a spoon full of peanut butter now and then, maybe on a cracker. One cracker with a tablespoon of peanut butter has 99 calories, 65 from fat, 24 grams of protein and 76 of carbs. Adding it still results in a two food solution.

One quirk of linear programming models is that an optimal solution will have no more than m nonzero food quantities if there are m restrictions on the model. My model has restrictions on calories, protein, and calories from fat so the model has room for at most 3 food choices. The three I’ve put into the model aren’t the right 3 to provide a mix though. Let’s try some more.

Kroger store brand has a frozen cauliflower side dish with a low fat cheese sauce., 60 calories, 10 from fat, 2 grams of protein and 10 grams of carbs. Adding that to my mix of food to pick from gives me a 3 food solution, 5 containers of the carbwatch yogurt, 6 bananas and 14 servings of the cauliflower with low fat cheese sauce.

Again, not a daily diet I’m going to look forward to. I like cauliflower, but 14 packages a day is a little much.

To get a 4 food solution I’m going to need a model with at least 4 constraints, in addition to adding more foods to pick from. I'll work on that tomorrow.