Indycar Case

We may illustrate more realistic forms of knapsack problems by considering the (fictitious) dilemma of mechanics in the Indy Car racing team. Six different features might still be added to this year's car to improve its top speed. Table 11.1 lists their estimated costs and speed enhancements.

Suppose first that Indy Car wants to maximize the performance gain without exceeding a budget of $35,000. Using decision variables

\(x_{j} \begin{cases}1 & \text { if feature } j \text { is added } \\ 0 & \text { otherwise }\end{cases}\)

we can formulate the problem as the knapsack model

That is, we maximize total performance subject to a budget constraint. An optimal solution chooses features 1,4 , and 5 for a gain of 25 miles per hour.

Suppose now that the Indy Car team decides they simply must increase speed by 30 miles per hour to have any chance of winning the next race. Ignoring the budget, they wish to find the minimum cost way to achieve at least that much performance.

This scenario leads to an alternative, minimize knapsack form. With variables (11.3), we obtain

\(\begin{array}{lll}

\min & 10.2 x_{1}+6.0 x_{2}+23.0 x_{3}+11.1 x_{4}+9.8 x_{5}+31.6 x_{6} & (\text { cost }) \\

\text { s.t. } & 8 x_{1}+3 x_{2}+15 x_{3}+7 x_{4}+10 x_{5}+12 x_{6} \geq 30 & \text { (mph required) } \\

& x_{1}, \ldots, x_{6}=0 \text { or } 1 &

\end{array}\)

This model minimizes cost subject to a performance requirement. An optimal solution now chooses features 1,3 , and 5 at cost $43,000.

Price: $11.15

Solution: The downloadable solution consists of 7 pages, 415 words and 5 charts.
Deliverable: Word Document