The following linear programing questions...

1. (a) What are the two different types of sensitivity ranges? Describe each type briefly and give a real world example for each type.

(b) What is a marketing problem in applications of linear programming? Briefly discuss the decision variables, the objective function and constraint requirements in a marketing problem. Give a real world example of a marketing problem.

(c) What is the required format of a linear programming problem to be solved by QM for Windows? What results are available from QM for Windows after solving a linear programming problem? Discuss briefly.

(d) What is a transportation problem? Briefly discuss the decision variables, the objective function and constraint requirements in a transportation problem. Give a real world example of the transportation problem.


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
P4 = number of Product 4 to be produced

Maximize 80P1 + 100P2 + 120P3 + 70P4 Total profit
Subject to
8P1 + 12P2 + 10P3 + 8P4 ≤ 6000 Production budget constraint
4P1 + 3P2 + 2P3 + 3P4 ≤ 2000 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, P4 ≥ 0 Non-negativity constraints

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

Linear Programming Results:
Variable Status Value
P1 Basic 200
P2 Basic 100
P3 Basic 320
P4 NONBasic 0
slack 1 NONBasic 0
slack 2 Basic 260
surplus 3 NONBasic 0
surplus 4 NONBasic 0
Optimal Value (Z) 64400

Original problem w/answers:
P1 P2 P3 P4 RHS Dual
Maximize 80 100 120 70
Constraint 1 8 12 10 8 <= 6000 12
Constraint 2 4 3 2 3 <= 2000 0
Constraint 3 1 0 0 0 >= 200 -16
Constraint 4 0 1 0 0 >= 100 -44
Solution-> 200 100 320 0 Optimal Z-> 64400

Ranging Results:
Variable Value Reduced Cost Original Val Lower Bound Upper Bound
P1 200 0 80 -Infinity 96
P2 100 0 100 -Infinity 144
P3 320 0 120 100 Infinity
P4 0 26 70 -Infinity 96

Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound
Constraint 1 12 0 6000 2800 7300
Constraint 2 0 260 2000 1740 Infinity
Constraint 3 -16 0 200 0 308.3333
Constraint 4 -44 0 100 0 366.6667


2. (a) Determine the optimal solution and 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, Product 3, and Product 4?
(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 $100 per unit to $125 per unit, what will be the optimal solution? What will be the new total profit? (Note: Answer this question by using the ranging results given above).
(d) Which resource should be obtained in larger quantity to increase the profit most? (Note: Answer this question using the ranging results given above.).


4. A professional football player is retiring, and he is thinking about going into the insurance business. He 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 $35
Auto 24
Life 60

Each homeowner's policy will cost $15, each auto policy will cost $12.50 and each life insurance policy will cost $32 to sell and maintain. He has projected a budget of $72,000 per year. In addition, the sale of a homeowner's policy will require 6 hours of effort; the sale of an auto policy will require 3.2 hours of effort and the sale of a life insurance policy will require 10 hours of effort. There are a total of 30,000 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.

How many of each type of insurance policy he would have to sell each year in order to maximize profit?

(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 Charm City 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 0.90 1 1 175
Nut Mixture 0.80 12 7 410
Cereal Mixture 0.48 5 4 128

The company wants to know how many ounces of each mixture to put into the blend. The blend should contain no more than 1200 calories and no more than 22 grams of fat. It should contain at least 15 grams of protein. Dried fruit mixture must be at least 20% of the weight of the blend, and nut mixture must be no more than 50% 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.
A) Decision variables:

X1= grams of dried fruit mixture

X2 = grams of nut mixture

X3= grams of cereal mixture

B) The objective function is c = 0.90x1 + 0.80x2 + 0.48x3


C) The contraints are:

175x1 + 410x2 + 128x3 ≤ 1200

1x1 + 12x2 + 5x3 ≤ 22

1x1 + 7x2 + 4x3 ≥ 15

3x - y - z ≥ 0

-x + y - z ≤ 0

X1, X2, X3 ≥ 0



The complete LP problem is:

Minimize c = 0.75x + 0.65y + 0.35z
Subject to:
180x + 400y + 120z ≤ 1000

x + 10y + 3z ≤ 25

x + 6y + 2z ≥ 12

3x - y - z ≥ 0

-x + y - z ≤ 0

with x ≥ 0, y ≥ 0, z ≥ 0

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 1 Project 2 Project 3 Project 4
_________________________________________________
Team A 32 25 22 24
Team B 45 34 27 45
Team C 40 - 24 28
Team D 30 25 20 15
_________________________________________________

Team C cannot be assigned to Project 2 because they do not have enough training to do that project. The professor wants Team B to be assigned to Project 2 or Project 3. 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 after formulating.