Chapter 11.8, Problem 21AYU

Banquet Seating A banquet hall offers two types of tables for rent: 6-person rectangular tables at a cost of $\$28$ each and 10-person round tables at a cost of $\$52$ each. Kathleen would like to rent the hall for a wedding banquet and needs tables for 250 people. The hall can have a maximum of 35 tables, and the hall has only 15 rectangular tables available. How many of each type of table should be rented to minimize cost and what is the minimum cost?

Source: facilities.princeton.edu

To determine

To solve: The given linear programming problem.

Answer to Problem 21AYU

Solution:

The minimum cost is 1252, when renting 15 rectangular tables and 16 round tables.

Explanation of Solution

Given:

• Cost of renting 6 person rectangular table - $\$28$.
• Cost of renting 10 person round table - $\$52$.
• Maximum number of tables that can be placed in the hall - 35.
• Number of guests - 250.
• Number of available rectangular tables - 15.

Calculation:

Step 1:

Begin by assigning symbols for the two variables.

$x=$ Number of 6 person rectangular table.

$x=$ Number of 10 person round table.

If $C$ is the total cost of renting the tables, then

$C=28x+52y$

Step 2:

The goal is to minimize $C$ subject to certain constraints on $x$ and $y$. Because $x$ and $y$ represents the number of tables, the only meaningful values of $x$ and $y$ are non-negative.

Therefore, $x\ge 0,y\ge 0$.

From the given data we get

$x+y\le 35$

$x\le 15$

$6x+10y\ge 250$

Therefore, the linear programming problem may be stated as

Minimize $C=28x+52y$

Subject to

$x\ge 0$

$y\ge 0$

$x+y\le 35$

$x\le 15$

$6x+10y\ge 250$

Step 3:

The graph of the constraints is illustrated in the figure below.

Corner points are	Value of objective function
$\left(0,35\right)$	1820
$\left(0,25\right)$	1300
$\left(15,16\right)$	1252
$\left(15,20\right)$	1460

The minimum cost is 1252, when renting 15 rectangular tables and 16 round tables.