Billy Penny is trying to determine how many units of two types of lawn mowers to

Question

Billy Penny is trying to determine how many units of two types of lawn mowers to produce each day. One of these is the Standard model, while the other is the Deluxe model. The profit per unit on the Standard model is $60, while the profit per unit on the Deluxe model is $40. The Standard model requires 20 minutes of assembly time, while the Deluxe model requires 35 minutes of assembly time. The Standard model requires 10 minutes of inspection time, while the Deluxe model requires 15 minutes of inspection time. The company must fill an order for 6 Deluxe models. There are 450 minutes of assembly time and 180 minutes of inspection time available each day. How many units of each product should be manufactured to maximize profits?

Explanation / Answer

Assume that manufacturer produce x units of standard Model and y units of Deluxe Model therefore for profit maximization we have

Maximize Z = 60 *x +40 *y

                                Subject to         

                                                20x + 35y < 450 (assembly time constraint)……(I)

                                                10x + 15y < 180 (Inspection time constraint)……(II)

                                                x > 0, y > 6         (Non-negative constraint)

To solve the equations, we have two equations

20 x + 35 y = 450 …………………. (1)

10 x + 15 y = 180………………………… (2)

Multiply Eq (2) by 2 and subtract from eq (1), now we will get

20 x + 35 y = 450

20 x + 30 y = 360

Or 5 y = 90 y = 18

Putting the value of y = 18 in eq (1) we can get

20 x + 35 * 18 = 450 or 20 x = 450 – 630 = -180 therefore x = -9 but x is a non negative number

There is no optimum solution for this question and even production of 18 units of y is not possible as it needs 18* 15 = 270 units of assembly time but we have only 180 which can produce only 10 units of y

Now suppose if 10 units of y and 0 units of x produced then profit

Z = 60* 0 + 40 * 10 = $ 400

Suppose if 6 units of y ( minimum requirement of deluxe production is 6 ) and 9 units of x ( putting y=6 in equation 1 & 2) produced then profit

Z = 60* 9 + 40 * 6 = $ 780

Now optimum combination of standard and deluxe to maximize the profit is 9 units of x and 6 units of y should be produced. And profit will be $ 780 in that case.

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