Problem 1 You would like to make a nutritious meal of eggs, edamame, and elbow macaroni. The meal should provide at least 40 g of carbohydrates, at...

Problem 1 You would like to make a nutritious meal of eggs, edamame, and elbow macaroni. The meal should provide at least 40 g of carbohydrates, at least 20 g of protein, and no more than 50 g of fat. An egg (one serving) 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. (You don’t need to find the optimal solution to the model you formulate.

Biketech manufactures road bikes and mountain bikes. They want to determine the number of each type of bike to produce in order to maximize profit over the next four-week planning horizon. Constraints affecting the production quantities are the production capacities in three departments: welding, assembly, and testing. For the four-week planning period, 260 hours of welding time, 340 hours of assembly time, and 320 hours of testing time are available. The time required to manufacture the two types of bicycles is shown in the following table: Labor (Hours)

                          Road                                    Mountain

Welding              6                                         8

Assembly             4                                      2

Testing               2                                       1

Road bikes provide a profit of $60 per unit and mountain bikes provide a profit of $50 per unit. Formulate a linear optimization model that will maximize Biketech’s profits and satisfy the constraints. (You don’t need to find the optimal solution to the model you formulate.)

Dunder Mifflin supplies paper to corporate clients. An important client needs 1500 reams of paper. Dunder Mifflin can procure the paper from several suppliers. The characteristics of the suppliers are as follows

                              Cost per ream             Delivery time                   Reams available

Supplier A                      3.50                       5 days                       200

Supplier B                     2.00                       10 days                        600

Supplier C                    6.50                          15 days                      600

Supplier D                   5.00                           4 days                         200

Supplier E                 4.00                         6 days                               200

When procuring the paper, Dunder Mifflin needs to ensure the following conditions are satisfied: (a) The client requires that at least 500 of the 1500 reams be delivered within the next 7 days. (b) In order to maintain good relations with Supplier C, Dunder Mifflin must order at least 100 reams from them. (c) Because of a rivalry between supplier B and D, Dunder Mifflin must order at least as many reams from supplier D as they do from supplier B. (d) Each supplier has a limited number of reams of paper available, as shown in the table. Formulate a linear optimization model to help Dunder Mifflin procure the paper at the lowest cost. (You don’t need to find the optimal solution to the model you formulate.)

Consider the linear optimization model

 Maximize x + y

Subject to 2x + y ≤ 8

2x + 3y ≤ 12

x, y ≥ 0

Graph the constraints and identify the feasible region. (b) Draw the isoquant representing all combinations of x and y that make the objective function equal to 2. (c) Determine the optimal values of x and y. (d) Label the optimal solution on you graph. (e) Calculate the optimal value of the objective function.

Consider the linear optimization model

Maximize 3x − 2y

Subject to 5x + 2y ≥ 20

2x + 3y ≤ 12

x, y ≥ 0

(a) Graph the constraints and identify the feasible region. (b) Draw the isoquant representing all combinations of x and y that make the objective function equal to 12. (c) Determine the optimal values of x and y. (d) Label the optimal solution on you graph. (e) Calculate the optimal value of the objective function

Consider the linear optimization model

Maximize 2x + y

Subject to x + 2y ≤ 8

x − y ≤ 5

x, y ≥ 0

(a) Graph the constraints and identify the feasible region.

(b) Choose a value and draw the isoquant representing all combinations of x and y that make the

objective function equal to that value.

(c) Determine the optimal values of x and y.

(d) Label the optimal solution on you graph.