A City Wants to Locate 2 New Fire Fighting Ladder $\quad$

question 8

Essay

A city wants to locate 2 new fire fighting ladder trucks to maximize the number of tall buildings which they can cover within a 3 minute response time. The city is divided into 4 zones. The fire chief wants to locate no more than one of the trucks in either Zone 1 or Zone 2. The number of tall buildings in each zone and the travel time between zones is listed below.
$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\underline{\text { To zone }}$
$\begin{array}{l}\begin{array} { c c c c c c } \text { No. tall building } & \text { From zone } & 1 & 2 & 3 & 4 \\\hline 50 & 1 & 0 & 2 & 1 & 6 \\90 & 2 & 2 & 0 & 4 & 5 \\60 & 3 & 1 & 4 & 0 & 1 \\70 & 4 & 6 & 5 & 1 & 0\end{array}\end{array}$
Based on this ILP formulation of the problem what formulas should go in cells B13:B16, B20:E20, F20, and F23:F24 of the following Excel spreadsheet?
Let X_i = 1 if truck located in zone i, 0 otherwise
$$\begin{array} { c l c } \text { Zone } & \text { Covers these zones } & \text { With this many buldings} \\\hline 1 & 1,2,3 & 200 \\2 & 1,2 & 140 \\3 & 1,3,4 & 180 \\4 & 3,4 & 130\end{array}$$
$\text { MAX: } \quad 200 \mathrm{X}_{1+}+140 \mathrm{X}_{2}+180 \mathrm{X}_{3}+130 \mathrm{X}_{4}$

$\begin{array}{ll}\text { Subject to: } & \mathrm{X}_{1}+\mathrm{X}_{2}+\mathrm{X}_{3}+\mathrm{X}_{4}=2 \\& \mathrm{X}_{1}+\mathrm{X}_{2} \leq 1 \\& \mathrm{X}_{\mathrm{i}}=0,1\end{array}$

---

Definitions:

Gymnosperms

A group of seed-producing plants that include conifers and cycads, characterized by having seeds that are not enclosed in an ovary (fruit).

Large Dinosaurs

Large dinosaurs were prehistoric reptiles, including species like the Tyrannosaurus rex and Brachiosaurus, known for their massive size and diversity during the Mesozoic era.

Amino Acids

Organic compounds that serve as the building blocks of proteins, containing both amine and carboxyl functional groups.

Electrical Discharge

The release and flow of electric charge through a medium, often observed as a spark or lightning in nature.