Chapter 11.8, Problem 24AYU

Production Scheduling In a factory, machine 1 produces 8-inch (in.) pliers at the rate of 60 units per hour (h) and 6-in. pliers at the rate of 70 units/h. Machine 2 produces 8-in. pliers at the rate of 40 units/h and 6-in. pliers at the rate of 20 units/h. It costs $\$50/\text{h}$ to operate machine 1, and machine 2 costs $\$30/\text{h}$ to operate. The production schedule requires that at least 240 units of 8-in. pliers and at least 140 units of 6-in. pliers be produced during each 10-h day. Which combination of machines will cost the least money to operate?

To determine

To solve: The given linear programming problem.

Answer to Problem 24AYU

Solution:

The minimum expense is when Machine 1 is operated for 30 minutes and machine 2 is operated for 5 hours and 15 minutes.

Explanation of Solution

Given:

Machine	8 inch pliers	6 inch pliers	Cost of operation
1	60 units/hour	70 units/hour	$\$50/\text{hour}$
2	40 units/hour	20 units/hour	$\$30/\text{hour}$

In 10 hours of production schedule requires atleast, 240 units of 8 inch pliers and 140 units of 6 inch pliers.

Calculation:

Begin by assigning symbols for the two variables.

$x=$ Number of hours Machine 1 operates.

$y=$ Number of hours Machine 2 operates.

a. If $C$ is the total cost of operating the two machines, then

$C=50x+30y$

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

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

From the given data we get

$60x+40y\ge 240$

$70x+20y\ge 140$

$x\le 10$

$y\le 10$

Therefore, the linear programming problem may be stated as

Minimize, $C=50x+30y$

Subject to

$x\ge 0$

$y\ge 0$

$60x+40y\ge 240 =>3x+2y\ge 12$

$70x+20y\ge 140 =>7x+2y\ge 14$

$x\le 10$

$y\le 10$

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

Corner points are	Value of objective function
$\left(0,10\right)$	300
$\left(0,7\right)$	210
$\left(1/2,21/4\right)$	$182.5$
$\left(4,0\right)$	200
$\left(10,0\right)$	500
$\left(10,10\right)$	800

The minimum expense is when Machine 1 is operated for 30 minutes and machine 2 is operated for 5 hours and 15 minutes.