A manufacturing company of two types of products. Each product must be processed in two departments. Product XX requires three hours per unit in department 1 and two hours per unit in department 2. Product YY requires three hours per unit in department 1 and two hours per unit in department 2. Departments 1 and 2 have , respectively 90 and 100 hours available each week, the profit margins of the two products are, respectively, $30 dollars and $20 dollars per unit. Model the objective function and constraints to obtain maximum utility

To model the objective function and constraints for obtaining maximum utility, we can follow these steps: Define the decision variables: Let’s denote the number of units of Product XX and Product YY produced as x and y, respectively. Define the objective function: The objective function represents the value that we are trying to optimize. In this case, we want to maximize the utility. The utility can be calculated by multiplying the profit margin per unit with the number of units produced for each product and summing them up. Therefore, the objective function can be defined as: Objective Function: Maximize 30x + 20y Define the constraints: Constraints are conditions that must be satisfied. In this case, we have two types of constraints: Department 1 constraint: The total processing time required for Product XX and Product YY in department 1 should not exceed the available hours in department 1 (90 hours). The processing time for Product XX is 3 hours per unit, and for Product YY, it is also 3 hours per unit. Therefore, the department 1 constraint can be defined as: Department 1 Constraint: 3x + 3y ≤ 90 Department 2 constraint: The total processing time required for Product XX and Product YY in department 2 should not exceed the available hours in department 2 (100 hours). The processing time for both products in department 2 is 2 hours per unit. Therefore, the department 2 constraint can be defined as: Department 2 Constraint: 2x + 2y ≤ 100 Non-negativity constraint: The number of units produced cannot be negative. Therefore, we need to add non-negativity constraints for both products: Non-negativity Constraint: x ≥ 0 and y ≥ 0 By defining these decision variables, objective function, and constraints, we can model the problem to obtain maximum utility.