Question Video: Forming the Set of Inequalities and the Objective Function of a Linear Programming Real-World Problem Mathematics

In a workshop, two workers produce two types of iron desks: type 𝐴 and type 𝐵. One worker builds the desks and the other sprays them. It takes the first worker 4 hours to build one desk of type 𝐴 and 3 hours to build one desk of type 𝐵. It takes the second worker 3 hours to spray one desk of type 𝐴 and 4 hours to spray one desk of type 𝐵. The first person works at least 5 hours a day, and the other works a maximum of 7 hours a day. If the workshop earns a profit of 60 LE from each desk (of either type), determine the objective function and inequalities required for calculating the number of desks of each type to be produced every day to maximize the profit 𝑃.

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Video Transcript

In a workshop, two workers produce two types of iron desks: type 𝐴 and type 𝐵. One worker builds the desks and the other sprays them. It takes the first worker four hours to build one desk of type 𝐴 and three hours to build one desk of type 𝐵. It takes the second worker three hours to spray one desk of type 𝐴 and four hours to spray one desk of type 𝐵. The first person works at least five hours a day, and the other works a maximum of seven hours a day. If the workshop earns a profit of 60 Egyptian pounds from each desk of either type, determine the objective function and inequalities required for calculating the number of desks of each type to be produced every day to maximize the profit 𝑃.

This is an example of a linear programming problem. We are asked to calculate the objective function. This is a linear function of the variables and can be used to optimize the solution. In this question, it will be used to maximize the profit 𝑃.

In any real-life problem, there will be restrictions or constraints, such as time and money. We write these as a set of inequalities. We will begin by letting 𝑥 be the number of desks of type 𝐴 that are produced and 𝑦 be the number of desks of type 𝐵 that are produced each day. Since the number of desks of each type cannot be negative, our first two inequalities are 𝑥 is greater than or equal to zero and 𝑦 is greater than or equal to zero.

We are told that the first worker who builds the desks takes four hours to build a desk of type 𝐴 and three hours to build a desk of type 𝐵. The total time that he spends building desks each day can therefore be written as the expression four 𝑥 plus three 𝑦. We are told that he works for at least five hours a day. This means that four 𝑥 plus three 𝑦 must be greater than or equal to five. The second worker spends three hours to spray a desk of type 𝐴 and four hours to spray a desk of type 𝐵. The total time that he spends working each day can be written as three 𝑥 plus four 𝑦. As he spends a maximum of seven hours working a day, this must be less than or equal to seven.

We now have a set of four inequalities or constraints. 𝑥 is greater than or equal to zero. 𝑦 is greater than or equal to zero. Four 𝑥 plus three 𝑦 is greater than or equal to five. And three 𝑥 plus four 𝑦 is less than or equal to seven. Our final step is to work out the objective function which will help us maximize the profit 𝑃.

We are told that the workshop earns a profit of 60 Egyptian pounds from each desk. Therefore, the profit 𝑃 is equal to 60𝑥 plus 60𝑦. Whilst it is not required in this question, we could work out the maximum profit by firstly sketching the four inequalities on the 𝑥𝑦-coordinate plane. This would create a feasible region. And we know that the optimal solution occurs at one of the vertices of this feasible region.

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