Question

The Magnetron Company manufactures and markets microwave ovens. Currently, the company produces two models: full-size and compact. Production is limited by the amount of labor available in the general assembly and electronic assembly departments, as well as by the demand for each model. Each full-size oven requires 2 hours of general assembly and 2 hours of electronic assembly, whereas each compact oven requires 1 hour of general assembly and 3 hours of electronic assembly. In the current production period, there are 500 hours of general assembly labor available and 800 hours of electronic assembly labor available.
In addition, the company estimates that it can sell at most 220 full-size ovens and 180 compact ovens in the current production period. The earnings contribution per oven is $120 for a full-size oven and $130 for a compact oven. The company would like to find an earnings-maximizing production plan for the current production period.

Required:
Formulate the above problem as a linear optimization model

Answer 1

Answer:

Max: $Z = 120x + 130y$

Subject to:

$2x + y \le 500$

$2x + 3y \le 800$

$x \le 220$

$y \le 180$

$x,y \ge 0$

Explanation:

Given

Let:

$x \to Units\ of\ full\ size$

$y \to Units\ of\ compact\ size$

Required

Formulate a linear optimization model

Constraints for time:

For the general assembly (hours), we have the following parameters:

$x \to 2$

$y \to 1$

So, the expression is:

$2x + y$ --- (1)

For the electronic assembly (hours), we have the following parameters:

$x\to 2$

$y \to 3$

So, the expression is:

$2x + 3y$ --- (2)

Solving further [Time available]:

$General\ Assembly \to 500$

$Electronic\ Assembly \to 800$

So, (1) and (2) becomes:

$2x + y \le 500$

$2x + 3y \le 800$

Constraints for selling:

$Full\ Size \to 220$ --- at most

$Compact \to 180$ -- at most

The above can be represented as:

$x \le 220$

$y \le 180$

Earnings contribution:

$Total\ Full\ Size \to 120$

$Total\ Compact \to 130$

The objective function to be maximized can then be modelled as:

$Z = 120x + 130y$