A small-appliance manufacturer has plants in Baltimore and Philadelphia, each of

Question

A small-appliance manufacturer has plants in Baltimore and Philadelphia, each of which produces toasters and blenders. The Baltimore plant can produce at most 800 appliances in one day at a cost of $12 per toaster and $15 per blender. The Philadelphia plant can produce at most 500 appliances one day at a cost of $10 per toaster and $20 per blender. A rush order is received for 600 toasters and 300 blenders. The manufacturer will choose the number of toasters and blenders produced at each plant in order to minimize costs. Name in words the four quantities that must be determined. Express the four quantities in (a) algebraically using only the two variables, x and y. Write the complete system of inequalities (in terms of x and y) needed to solve the problem. Write the objective function in terms of x and y.

Explanation / Answer

(a) The four qualtities that must be determiend are

1. Number of toasters produced at Baltimore plant

2. Number of blenders produced at Baltimore plant

3. Number of toasters produced at Philadelphia plant

4. Number of blenders produced at Philadelphia plant

(b) Let x, and y denote the number of toasters and number of blenders produced at Baltimore plant respectively.

Hence, number of toasters produced at Philadelphia plant = 600 – x; number of blenders produced at Philadelphia plant = 300 – y.

(c) Since, the Baltimore plant can produce at most 800 appliances in one day, the quantity (x + y) should not exceed 800. In other words, (x + y) 800.

Since, the Philadelphia plant can produce at most 500 appliances in one day, the quantity {(600 – x) + (300 – y)} should not exceed 500. In other words, {(600 – x) + (300 – y)} 500 or, (900 – x – y) 500.

Hence, the complete system of inequalities (in terms of x and y) needed to solve the problem is given by

x + y 800

900 – x – y 500

x, y 0 (Non-negativity restriction)

(d) The objective of the manufacturer is to choose the number of toasters and blenders produced at each plant in order to minimize the cost.

Cost of producing x toasters and y blenders at Baltimore plant is $(12x + 15y).

Cost of producing (600 – x) toasters and (300 – y) blenders at Philadelphia plant is

${10(600 – x) + 20(300 – y)} = $(12000 – 10x – 20y).

Hence, the objective function is Minimize z = (12x + 15y) + (12000 – 10x – 20y)

Minimize z = 12000 + 2x – 5y                        (Answer)

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