Mission College Linear Optimization Exercises

1. Decide whether each of the following statements is true or false. You do not need to explain your answer.
   1. There can be only one optimal solution.
   2. The set of solution points that satisfies all of a linear programming problem’s constraints simultaneously is defined as the feasible region in graphical linear programming.
   3. An objective function is necessary in a maximization problem but is not required in a minimization problem.
   4. The objective function shown is a legitimate expression in linear programming: Max Profit = $7X + $8Y + $5XY.
2. Optimization in Daily Life: Think about your favorite activity, hobbies or any kind of activity from your daily life. Find a problem which can be modeled with optimization. Now, think about the formulation steps for this activity using optimization and answer the formulation step questions below. (Note that you do not need to write a mathematical model.)
   1. What is the decision to make?
   2. What is the objective criterion?
   3. What are the constraints for the decision?
3. Optimization for Business Decision Making: Think about a job/internship you have had in the past or you would like to have in the future. Consider business decisions for this work and demonstrate how you could use optimization to improve your decision making. Note that you need to write a question and formulate the problem using a mathematical model for this question. In your formulation make sure to
   1. Clearly define your variables.
   2. Develop an objective function for the proposed problem.
   3. Write the constraints and label them clearly.
4. You would like to make a nutritious meal of eggs, edamame, and elbow macaroni. The meal should provide at least 30 g of carbohydrates, at least 20 g of protein, and no more than 60 g of fat. An egg contains 2 g of carbohydrates, 17 g of protein, and 14 g of fat. A serving of edamame contains 12 g of carbohydrates, 12 g of protein, and 6 g of fat. A serving of elbow macaroni contains 43 g of carbohydrates, 8 g of protein, and 1 g of fat. An egg costs $2, a serving of edamame costs $5, and a serving of elbow macaroni costs $3. Formulate a linear optimization model that could be used to determine the number of servings of egg, edamame, and elbow macaroni that should be in the meal in order to meet the nutrition requirements at minimal cost. In your formulation make sure to: a.Clearly define your variables b.Develop an objective function for the proposed problem c.Write the constraints and label them clearly.

5. Sunny Painting produces both interior and exterior paints from two raw materials, 𝑀1􏰀 and 𝑀2􏰁. The usage of raw material 𝑀1􏰀 by exterior paint is 6 tons/day and the usage of raw material𝑀1􏰀 by interior paint is 4 tons/day. Similarly, the Usage of raw material 𝑀2􏰁 by exterior paint and interior paint are one and two tons per day, respectively. The daily availabilities of raw materials 𝑀1􏰀 and𝑀2􏰁 are limited to 24 and 6 tons, respectively. In order to understand the demand for each product, you developed a market survey. The survey indicates that the daily demand for interior paint cannot exceed that for exterior paint by more than 1 ton. Also, the maximum daily demand for interior paint is 2 tons. The profit per ton is 5 for the exterior and 4 for the interior paint. As the analyst of the Sunny Painting, you want to determine the optimum (best) product mix of interior and exterior paints that maximizes the total daily profit of the company. Formulate this problem: a.Clearly define your variables b.Develop an objective function for the proposed problem c.Write the constraints and label them.