Question

Answer Questions 2 and 3 based on the following LP problem.

Let     P1 = number of Product 1 to be produced

          P2 = number of Product 2 to be produced

          P3 = number of Product 3 to be produced

Maximize 100P1 + 120P2 + 90P3         Total profit

Subject to

        8P1 + 12P2 + 10P3 ≤ 7280       Production budget constraint

            4P1 + 3P2 + 2P3 ≤ 1920       Labor hours constraint

                                   P1 > 200         Minimum quantity needed for Product 1 constraint

                                   P2 > 100        Minimum quantity needed for Product 2 constraint

               And P1, P2, P3 ≥ 0             Non-negativity constraints

The QM for Windows output for this problem is given below.

Linear Programming Results:

            P1        P2        P3                   RHS     Dual

Maximize         100      120      90                               

Constraint 1      8          12        10        <=        7280    0

Constraint 2      4          3          2          <=        1920    45

Constraint 3      1          0          0          >=        200      -80

Constraint 4      0          1          0          >=        100      -15

Solution           200      100      410                  68900

Solution List:

Variable           Status   Value

P1        Basic    200

P2        Basic    100

P3        Basic    410

slack 1 Basic    380

slack 2 NONBasic        0

surplus 3          NONBasic        0

surplus 4          NONBasic        0

Optimal Value (Z)        68900

Ranging Results:

Variable           Value   Reduced Cost   Original Val      Lower Bound   Upper Bound

P1        200      0          100      -Infinity           180

P2        100      0          120      -Infinity           135

P3        410      0          90        80        Infinity

            Dual Value       Slack/Surplus   Original Val      Lower Bound   Upper Bound

Constraint 1      0          380      7280    6900    Infinity

Constraint 2      45        0          1920    1100    1996

Constraint 3      -80       0          200      168.33 405

Constraint 4      -15       0          100      0          373.33

2. (a) Determine the optimal solution and the optimal value and interpret their meanings.

(b) Determine the slack (or surplus) value for each constraint and interpret its meaning.

3. (a) What are the ranges of optimality for the profit of Product 1, Product 2 and Product 3?

(b) Find the dual prices of the four constraints and interpret their meanings. What are the ranges in which each of these dual prices is valid?

(c) If the profit contribution of Product 2 changes from $120 per unit to $128 per unit, what will be the optimal solution? What will be the new total profit? (Note: Answer this question using the ranging results given above. Do not solve the problem again).

(d) Which resource should be obtained in larger quantity to increase the profit most? (Note: Answer this question using the ranging results given above. Do not solve the problem again).

4. An insurance agent plans to sell three types of policies— homeowner’s insurance, auto insurance and life insurance. The average amount of profit returned per year by each type of insurance policy is as follows:

Policy                            Yearly Profit/Policy

Homeowner’s               $50

Auto                 40

Life                  75

Each homeowner’s policy will cost $18.20, each auto policy will cost $14.50 and each life insurance policy will cost $30.50 to sell and maintain. He has projected a budget of $80,000 per year. In addition, the sale of a homeowner’s policy will require 6.5 hours of effort; the sale of an auto policy will require 3.7 hours of effort and the sale of a life insurance policy will require 10.5 hours of effort. There are a total of 28,000 hours of working time available per year from himself and his employees.

He wants to sell at least twice as many auto policies as homeowner’s policies.

Formulate a linear programming model that meets these restrictions and maximizes total yearly profit for the agent.

(a) Define the decision variables.

(b) Determine the objective function. What does it represent?

(c) Determine all the constraints. Briefly describe what each constraint represents.

Note: Do NOT solve the problem after formulating

5. The Delicious Snacks manufactures a snack mix by blending three ingredients: a dried fruit mixture, a nut mixture, and a cereal mixture. Information about the three ingredients (per ounce) is shown below.

Ingredient	Cost	Fat Grams	Protein grams	Calories
Dried Fruit Mixture	1.10	1	1	180
Nut Mixture	1.00	10	8	415
Cereal Mixture	0.70	4	5	130

The company wants to know how many ounces of each mixture to put into the blend. The blend should contain no more than 1250 calories and no more than 20 grams of fat. It should contain at least 15.5 grams of protein. Dried fruit mixture must be at least 25% of the weight of the blend, and nut mixture must be no more than 40% of the weight of the blend.

Formulate a linear programming model that meets these restrictions and minimizes the cost of the blend by determining

(a) The decision variables.

(b) Determine the objective function. What does it represent?

(c) Determine all the constraints. Briefly describe what each constraint represents.

Note: Do NOT solve the problem after formulating

6. A professor has been contacted by a company willing to work with student consulting teams. The Company needs help with four projects. There are four student teams available to work on these projects. The estimated time of completion (in hours) of each project by each team is given in the following table.

                     Project A    Project B     Project C   Project D

_________________________________________________

Team 1           40                25                32              28      

Team 2           40                34                27              40

Team 3           35                30                24              30

Team 4           32                 -                  20              20  

_________________________________________________

Team 4 cannot be assigned to Project B because they do not have enough training to do that project. The professor wants Team 1 to be assigned to Project B or Project D. The objective of this assignment problem is to minimize the total time of completion of all the projects.

                                                                                                                                      

(a) Define the decision variables.

(b) Determine the objective function. What does it represent?

(c) Determine all the constraints. Briefly describe what each constraint represents.

Note: Do NOT solve the problem

Answer 1

(5)

(a)

x = Grams of dried fruit mixture

y = Grams of Nut mixture

z = Grams of Cereal mixture

C = Cost in $

(b)

Minimize C = 1.1x + y + 0.7z. This represents the minimum cost of the blend.

(c)

180x + 415y + 130z ≤ 1250 [Maximum calories requirement]

x + 10y + 4z ≤ 20 [Maximum fat requirement]

x + 8y + 5z ≥ 15.5 [Minimum protein requirement]

x ≥ 0.25(x + y + z), that is 3x – y – z ≥ 0 [Dried fruit mixture weight to total weight ratio]

y ≤ 0.4(x + y + z), that is x – 1.5y + z ≥ 0 [Nut mixture weight to total weight ratio]

x, y, z ≥ 0 [Non-negativity restrictions]

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