Use R Program to solve A company produces interior and exterior paints from three raw materials, M1,M2, and M3. The maximum daily available amounts of M1, M2, and M3 are 26,12 , and 15 tons, respectively. Table 1 shows the quantities of raw materials M1, M2, and M3 needed per ton of each exterior and interior paints. Table 1 The maximum daily demand for interior paint is 5 tons, so the company should produce at most 5 tons of the interior paint per day. Profit per ton of interior paint is $70,000. Profit per ton of the exterior paint depends on its amount. Table 2 shows profit per ton of the exterior paint at different amounts. Table 2 For example, if the production amount for the exterior paint is 7 tons, the profit from the exterior paint is calculated as follows: Profit =60,0004+80,0003=480,000 [Hint: The above profit is equivalent to: 60,000min(4,7)+80,000max(74,0).] The company wants to determine the optimal production quantities (in tons) of interior and exterior paints such that the total daily profit is maximized. a) [5 points] Please define the decision variables, the objective function, and constraints. Then, write the mathematical formulation of your optimization model. Please note that you should use mathematical notation NOT the R proaramming format. For example, you can use X as the production amount of exterior paint and Y as the production amount of interior paint. b) [8 points] Create a function in R that, given the vector of decision variables as the argument, returns the objective value for feasible solutions and penalizes the infeasible solutions. c) [3 points] Use optim() to solve your model. Save the optim()'s solution and its objective value in two variables with the names of your choice. d) [4 points] Try 3 different ways (e.g. changing the initial feasible solution, trying different algorithms, changing the optim's setting) to improve the optim(1)'s solution and discuss the results