Question

LINEAR PROGRAMMING PROBLEM

2. A production manager wants to determine how many units of each product to produce weekly to maximize weekly profits. Production requirements for the products are shown in the following table.

Product          Material 1 (lbs.)           Material 2 (lbs.)         Labor (hours)

A                         3                                 2                            4

B                         1                                 4                            2

C                         5                                 none                       3.5

Material 1 costs $7 a pound, material 2 costs $5 a pound, and labor costs $15 per hour. Product A sells for $101 a unit, product B sells for $67 a unit, and product C sells for $97.50 a unit. Each week there are 300 pounds of material 1; 400 pounds of material 2; and 200 hours of labor. Also, there is a weekly demand of 10 units of product C each week.

A spreadsheet should be included and a the show formula page and the result page. Please show the weekly demand 10 units working and the section in which the material cost $7 a pound will be placed etc and the amount the product sells for. Example Product A sells for $101 a unit etc.

Answer 1

Let,

x1 = Number of units of Product A to be produced
x2 = Number of units of Product B to be produced
x3 = Number of units of Product C to be produced

Profit = Selling price - Cost price

For A:

Selling price per unit = $101

Cost price per unit = Material 1 cost+material 2 cost + labour cost = 3*7+2*5+4*15 = 91

So profit for A per unit = 101-91 = 20

Like wise for B: profit per unit = 67-(1*7+4*5+2*15) = 10

For C: profit per unit = 97.5-(5*7+3.5*15) = 10

Objective is to maximize profit so objective function = max 20x1+10x2+10x3

Subject to,

3x1+x2+5x3 <= 300 (material 1)
2x1+4x2 <= 400 (material 2)
4x1+2x2+3.5x3<=200 (labour)
x3>= 10 (Weekly demand of C)

x1,x2,x3 >= 0 (non-negativity constraint)

Solving in solver we get,

x1 = Number of units of Product A to be produced = 41.5
x2 = Number of units of Product B to be produced = 0
x3 = Number of units of Product C to be produced = 10

maximized weekly profit = $925

Solver screenshot

Excel formula