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I'm trying to solve the question below:

"An $8$ year old child usually has $24$ teeth. A dental researcher has estimated that on an average, $8$ year old kids in a geographical region have $1.8$ decayed teeth each. Give LP formulations for the problems of finding the minimum and maximum possible values for the fraction of $8$ year old kids in that region who have $2$ or more decayed teeth."

I know that finding the min would just be $-$(max), so that's not a problem, but I'm stuck on how to use the "$1.8$ decayed teeth on average" as a constraint. The only thing I can think of is letting $x$ be the number of decayed teeth and making one of the constraints $\frac{x}{24}=1.8$, but then wouldn't that just make the objective function always give the answer that the fraction of kids with $2$+ decayed teeth be $0$??

Aiden Chow
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plzhelpme
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1 Answers1

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Let $x_i$, $0 \leq i \leq 24$, be the fraction of 8 year old kids in the geographical region having $i$ decayed teeth. Notice we have the constraint $$ \sum_{i=0}^{24} x_i = 1 \text{.} $$ Now to represent your constraint, temporarily let $N$ be the number of 8 year old kids in the geographical region. For each $i$, $x_i N$ of them have $i$ decayed teeth. So the average number of decayed teeth, $A$, is $$ A = \frac{1}{N} \sum_{i=0}^{24} i x_i N = \sum_{i=0}^{24} i x_i = 1.8 \text{,} $$ the rightmost equality being a plain linear constraint.

Eric Towers
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