Question

A furniture manufacturer makes two type of products, chairs and tables. Processing of these products is done on two machines A and B. A chair requires 2 hrs on machine A and 6 hrs on machine B. A table requires 5 hrs on machine A and no time on machine B. There are 16 hrs per day available on machine A and 30 hrs on machine B. Profit gained by manufacturer from 33 a chair and a table is USD 2 and USD 10 respectively. Solve this problem to find the daily production of each of the two products.

Answer 1

We need to find the daily production quantity for two products viz. Chair and Table. Since the objective is not mentioned we will assume that profit maximization will be objective for finding daily production quantity.

Denotating a unit of Table as 'T' and a unit of Chair as 'C' below.

Now we will formulate our equations for objective and given constraints and then will try to find out value of 'T' and 'C' by solving these equations.

Objective: Max Profit = $2 X C + $10 X T

Constraint 1: 2 hrs X C + 5 hrs X T <= 16 hrs (Daily available Machine A time)

Constraint 2: 6 hrs X C + 0 hrs X T <= 30 (Daily available Machine A time)

As per the objective equation, making more tables will give us more profit than chairs and as per the constraints equation we need to focus on Machine A time because it is a scarce resource compared to Machine B time and optimum utilization of Machine A time will finally decide daily production quantity.

Scenario 1: Making only tables viz. 0 Chairs(C=0) hence plugging this value in constraint 1 equation

2hrs X 0 + 5 hrs X T <=16 ; T <=16/5 viz. T <= 3.2 Hence we can make a maximum of 3 Tables and our profit(Objective equation) will be

$2 X 0 + $10 X 3 = $30

Scenario 2: Since we can only maximum of 3 tables so let's find out how much profit we can make if we only make 2 tables and rest as Chairs

2hrs X C + 5 hrs X 2 <=16

2hrs X C + 10 <=16

2 hrs X C <= 16-10

C <=6/2 ; C = 3 . This value of C should also satisfy our constraint 2 equation as well.

Hence we can make 2 Tables and 3 Chairs, hence our profit(Objective equation) will be

$2 X 3 + $10 X 2 = $26. Now we can see that with every reduction in the production of the table, we will get fewer profits.

So, Scenario 1 will be our best-case scenario and hence we should produce 3 tables daily.