A Small Town Wants to Build Some New Recreational Facilities

question 64

Essay

A small town wants to build some new recreational facilities. The proposed facilities include a swimming pool, recreation center, basketball court and baseball field. The town council wants to provide the facilities which will be used by the most people, but faces budget and land limitations. The town has $400,000 and 14 acres of land. The pool requires locker facilities which would be in the recreation center, so if the swimming pool is built the recreation center must also be built. Also the council has only enough flat land to build the basketball court or the baseball field. The daily usage and cost of the facilities (in $1,000) are shown below.
$$\begin{array} { c l c c c } \text { Variable } & \text { Facility } & \text { Usage } & \text { Cost ( C1,000) } & \text { Land } \\\hline \mathbf { X } _ { 1 } & \text { Swimming pool } & 400 & 100 & 2 \\\mathbf { X } _ { \mathbf { 2 } } & \text { Recreation center } & 500 & 200 & 3 \\\mathbf { X } _ { 3 } & \text { Basketball caut } & 300 & 150 & 4 \\\mathbf { X } _ { 4 } & \text { Baseball Eield } & 200 & 100 & 5\end{array}$$ Based on this ILP formulation of the problem and the indicated optimal solution what values should go in cells B5:G12 of the following Excel spreadsheet?
MAX: $\quad 400 x _ { 1 } + 500 x _ { 2 } + 300 x _ { 3 } + 200 x _ { 4 }$
$\text {Subject to:}$
$\begin{array} { l l } 100 \mathbf { x } _ { 1 } + 200 \mathbf { x } _ { \mathbf { 2 } } + 150 \mathbf { x } _ { \mathbf { 3 } } + 100 \mathbf { x } _ { 4 } \leq 400 & \text { budget } \\2 \mathbf { x } _ { 1 } + 3 \mathbf { x } _ { \mathbf { 2 } } + 4 \mathbf { x } _ { \mathbf { 3 } } + 5 \mathbf { x } _ { 4 } \leq 14 & \text { land } \\\mathbf { x } _ { 1 } - \mathbf { x } _ { \mathbf { 2 } } \leq 0 & \text { pool and recreation center } \\\mathbf { x } _ { \mathbf { 3 } } + \mathbf { x } _ { 4 } \leq 1 & \text { basketball and baseball } \\\mathbf { x } _ { \mathrm { i } } = 0.1 &\end{array}$
$\text {Solution: \(\left( x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } \right) = ( 1,1,0,1 )\)}$ $\begin{array}{|c|c|c|c|c|c|c|c|} \hline & \text { A } & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} & \mathrm{F} & \mathrm{G} \\\hline 1 & & & & & & & \\\hline 2 & & & & & & & \\\hline 3 & & & \text { Facilities } & & & & \\\hline 4 & & \text { Pool } & \text { Rec center } & \text { Basketball } & \text { Baseball } & \text { Total usage: } & \\\hline 5 & \text { Select (0=no, 1=yes) } & & & & & & \\\hline 6 & \text { Usage } & & & & & & \\\hline 7 & & & & & & & \\\hline 8 & \text { Resources } & & & & & \text { Used } & \text { Available } \\\hline 9 & \text { Cost } & & & & & & \\\hline 10 & \text { Land } & & & & & & \\\hline 11 & \text { Pool \& Rec center } & & & & & & \\\hline 12 & \text { Basket or Baseball } & & & & & &\\\hline \end{array}$

---

Definitions:

Manufacturing Overhead Account

An accounting category that records the indirect costs associated with producing goods, such as utilities for the factory or salaries for the maintenance staff.

Overapplied Manufacturing Overhead

A situation where the amount of manufacturing overhead allocated to products is more than the actual overhead incurred.

Work in Process

Inventory that includes all materials, labor, and overhead costs incurred for products that are in the production process but not yet completed.

Debit Balance

An amount that reflects a surplus of debits over credits in an account, often indicating assets or expenses.