Question

A cell company offers calls for day hours and night hours. Calls can be availed 8 hours during the day and 4 hours at night and atmost 10 hours a day. The profit on the day calls is \(\mathrm{Rs} 60\) per hour and on night calls, is Rs 50 per hour. How many hours during the day and at night a customer must use to fetch a maximum profit to the company? (1) \(6 \mathrm{hrs}\) during day and 4 hours at night (2) 5 hrs during day and 5 hours at night (3) 8 hours during day and 2 hours at night (4) 6 hours during day and 2 hours at night

Short answer

Answer: The customer should use 8 hours during the day and 2 hours at night to maximize the cell company's profit.

Step-by-step solution

Identify the decision variables

Let x be the number of hours for day calls and y for night calls.

Write down the constraints

The constraints based on the given problem are: 1. Day hours available (x): \(0 \leq x \leq 8\) 2. Night hours available (y): \(0 \leq y \leq 4\) 3. Total hours per day (x+y): \( x + y \leq 10\)

Write down the profit function to maximize

The profit function, P(x,y), is given by: \(P(x,y) = 60x + 50y\)

Analyze each option provided

Now, we will calculate the profit for each option provided and compare which option yields the maximum profit: 1. 6 hrs during day and 4 hours at night: \(P(6,4) = 60(6) + 50(4) = 360 + 200 = 560\) 2. 5 hrs during day and 5 hours at night: \(P(5,5) = 60(5) + 50(5) = 300 + 250 = 550\) 3. 8 hours during day and 2 hours at night: \(P(8,2) = 60(8) + 50(2) = 480 + 100 = 580\) 4. 6 hours during day and 2 hours at night: \(P(6,2) = 60(6) + 50(2) = 360 + 100 = 460\)

Choose the best option with maximum profit

Comparing the profits from all options, we find that option 3 yields the maximum profit of Rs 580. Therefore, a customer must use 8 hours during the day and 2 hours at night to fetch a maximum profit to the cell company.

Key concepts

Decision Variables

When solving a linear programming problem, we begin by identifying the decision variables. These are the variables that we can control and that will ultimately impact the outcome of our objective, which, in the context of our exercise, is to maximize the company's profit from day and night calls. Here, we have two decision variables, denoted as x and y. x represents the number of day hours a customer uses, and y represents the night hours used. Understanding these variables is crucial because they form the backbone of our optimization model and dictate the feasibility and optimality of the solutions.

Constraint Formulation

After setting our decision variables, we need to establish the constraints that limit the range of our solutions. Constraining factors create boundaries within which the decision variables can operate. In our cell company example, there are three clear constraints:

• x must be between 0 and 8 (inclusive), representing the available day hours.
• y must be between 0 and 4 (inclusive), standing for the available night hours.
• The total of day and night hours (x + y) must not exceed 10 hours, capturing the maximum allowable usage per day.

These constraints are formulated using linear inequalities, which means they'll plot as straight lines on a graph, creating a 'feasibility region' where possible solutions exist. When constructing these constraints, special attention must be given to ensuring they represent the real-life limitations as accurately and completely as possible in order to find a valid and optimal solution.

Profit Maximization

The ultimate goal in our problem is to maximize the company's profit, making this a profit maximization model. We accomplish this by forming a profit function that is dependent on our decision variables. In this case, the profit function is P(x, y) = 60x + 50y, where 60 and 50 are the profit per hour from day and night calls, respectively. Our task then is to find the values of x and y within the feasible region (determined by our constraints) that will give us the highest possible value of P(x, y). This often involves evaluating the profit function at the vertices of the feasibility region or using mathematical techniques like the Simplex Method for more complex problems.

Linear Inequalities

Linear inequalities are mathematical expressions that utilize inequality symbols (such as ≤, ≥) to relate linear functions. In the context of linear programming, they are used to define the constraints of the problem. Each constraint in the cell company example can be expressed as a linear inequality. For instance, the constraint that 'day hours (x) available must be at most 8' can be written as 0 ≤ x ≤ 8. These inequalities play a vital role in determining the feasible region, which is the area containing all the potential solutions satisfying all the imposed constraints. When combined, these inequalities shape the geometric boundaries for our decision variables and help visualize the solution space where we search for the optimum.