Exam 19: Linear Programming

When plotting constraints,it is best to ignore the inequality aspect of the equation.

A(n)________ is a value that the decision maker cannot control and that does not change when the solution is implemented.

Scenario D.1 Lisa lives out in the country with her seven cats and avoids driving into the big city as much as possible.She has decided to make her own cat food and has the following nutritional guidelines.Each four ounce portion must contain 20 units of protein,15 units of vitamin A,and 10 units of vitamin B.She has eggs,tomatoes,and chicken meat as possible inputs to her cat food.Each ounce of eggs contains 5 units of protein,4 units of Vitamin A,and 3 units of Vitamin B.Each ounce of tomatoes contains 1 unit of protein,5 units of Vitamin A,and 14 units of Vitamin B.Each ounce of chicken contains 22 units of protein,14 units of Vitamin A,and 5 units of Vitamin B.Chicken costs 40 cents per ounce,tomatoes cost 8 cents per ounce,and eggs cost 12 cents per ounce. -Referring to Scenario D.1,what is an appropriate objective function for this scenario?

Use the graphical technique to find the optimal solution for this objective function and associated constraints. Maximize: Z = 8A + 5B Subject To: Constraint 1 4A + 5B < 80 Constraint 2 7A + 4B < 120 A,B > 0 a.Graph the problem fully in the following space.Label the axes carefully,plot the constraints,shade the feasibility region,identify all candidate corner points,and indicate which one yields the optimal answer.

A binding constraint has slack but does not have surplus.

D.5 The Transportation Method Table D.1 Tanfastic,Inc. ,a manufacturer of swimwear,is in the process of developing a production plan for the coming spring break.The ending inventory for January is 30 units.Undertime is paid,at a rate of $5.00 per unit.Details are shown in the following POM for Windows table. -Use the information in Table D.1.Given the information in the optimal tableau,what is the overtime cost in dollars per unit?

A small oil company has a refining budget of $200,000 and would like to determine the optimal production plan for profitability.The following table lists the costs associated with its three products. Marketing has a budget of $50,000,and the company has 750,000 gallons of crude oil available.Each gallon of gasoline contributes 14 cents of profits,heating oil provides 10 cents,and plastic resin 30 cents per unit.The refining process results in a ratio of two units of heating oil for each unit of gasoline produced.This problem has been modeled as a linear programming problem and solved on the computer.The set up and output follows: a.Give a linear programming formulation for this problem.Make the variable definitions and constraints line up with the computer output. b.What product mix maximizes the profit for the company using its limited resources? c.How much plastic resin is produced if profits are maximized? d.Give a full explanation of the meaning of the three numbers listed following. First Number: Slack or surplus of 42500 for the #2 Marketing Budget constraint. Second Number: Shadow price of 0 for the #1 Refining Budget constraint. Third Number: An upper limit of "infinity" for the right-hand-side value for the #1 Refining Budget constraint.

Scenario D.1 Lisa lives out in the country with her seven cats and avoids driving into the big city as much as possible.She has decided to make her own cat food and has the following nutritional guidelines.Each four ounce portion must contain 20 units of protein,15 units of vitamin A,and 10 units of vitamin B.She has eggs,tomatoes,and chicken meat as possible inputs to her cat food.Each ounce of eggs contains 5 units of protein,4 units of Vitamin A,and 3 units of Vitamin B.Each ounce of tomatoes contains 1 unit of protein,5 units of Vitamin A,and 14 units of Vitamin B.Each ounce of chicken contains 22 units of protein,14 units of Vitamin A,and 5 units of Vitamin B.Chicken costs 40 cents per ounce,tomatoes cost 8 cents per ounce,and eggs cost 12 cents per ounce. -Referring to Scenario D.1,which of the following statements is BEST?

An = constraint is often used for certain mandatory relationships.

Which of the following statements regarding linear programming is NOT true?

The Really Big Shoe Company is a manufacturer of basketball shoes and football shoes.Ed Sullivan,the manager of marketing,must decide the best way to spend advertising resources.Each football team sponsored requires 120 pairs of shoes.Each basketball team requires 32 pairs of shoes.Football coaches receive $300,000 for shoe sponsorship and basketball coaches receive $1,000,000.Ed's promotional budget is $30,000,000.The Really Big Shoe Company has a very limited supply (4 liters or 4,000cc)of flubber,a rare and costly raw material used only in promotional athletic shoes.Each pair of basketball shoes requires 3cc of flubber,and each pair of football shoes requires 1cc of flubber.Ed desires to sponsor as many basketball and football teams as resources allow.However,he has already committed to sponsoring 19 football teams and wants to keep his promises. a.Give a linear programming formulation for Ed.Make the variable definitions and constraints line up with the computer output appended to this exam. b.Solve the problem graphically,showing constraints,feasible region,and isoprofit lines.Circle the optimal solution,making sure that the isoprofit lines drawn make clear why you chose this point.(Show all your calculations for plotting the constraints and isoprofit line on the left to get credit. ) c.Solve algebraically for the corner point on the feasible region. d.Part of Ed's computer output is shown following.Give a full explanation of the meaning of the three numbers listed below.Based on your graphical and algebraic analysis,explain why these numbers make sense.(Hint: He formulated the budget constraint in terms of $000. )See the computer printout that follows. First Number: The shadow price of 0.0104 for the "Flubber" constraint. Second Number: The slack or surplus of 6383.334 for the "Budget" constraint. Third Number: The lower limit of 12.2807 for the "Commitment" constraint.

Scenario D.1 Lisa lives out in the country with her seven cats and avoids driving into the big city as much as possible.She has decided to make her own cat food and has the following nutritional guidelines.Each four ounce portion must contain 20 units of protein,15 units of vitamin A,and 10 units of vitamin B.She has eggs,tomatoes,and chicken meat as possible inputs to her cat food.Each ounce of eggs contains 5 units of protein,4 units of Vitamin A,and 3 units of Vitamin B.Each ounce of tomatoes contains 1 unit of protein,5 units of Vitamin A,and 14 units of Vitamin B.Each ounce of chicken contains 22 units of protein,14 units of Vitamin A,and 5 units of Vitamin B.Chicken costs 40 cents per ounce,tomatoes cost 8 cents per ounce,and eggs cost 12 cents per ounce. -Referring to Scenario D.1,assume that an optimal serving contains 0.89 ounces of chicken and 0.52 ounces of tomatoes.Which of the following statements is BEST?

________ are the limitations that restrict the permissible choices for the decision variables.

For the line that has the equation 4X1 + 8X2 = 88,an axis intercept is:

Table D.3 The Harper Company is in the process of production planning for the next four quarters.The company follows a policy of a stable workforce and uses overtime and subcontracting to meet uneven forecasted demand.Anticipation inventory is also allowed,but not backorders.Undertime is paid,at a rate of $5.00 per unit.The beginning (or current)inventory is 25 units.Details are shown in the following POM for Windows table. -Use the information in Table D.3.What is the total cost of the optimal production plan?

Linear programming is useful for allocating scarce resources among competing demands.

Scenario D.1 Lisa lives out in the country with her seven cats and avoids driving into the big city as much as possible.She has decided to make her own cat food and has the following nutritional guidelines.Each four ounce portion must contain 20 units of protein,15 units of vitamin A,and 10 units of vitamin B.She has eggs,tomatoes,and chicken meat as possible inputs to her cat food.Each ounce of eggs contains 5 units of protein,4 units of Vitamin A,and 3 units of Vitamin B.Each ounce of tomatoes contains 1 unit of protein,5 units of Vitamin A,and 14 units of Vitamin B.Each ounce of chicken contains 22 units of protein,14 units of Vitamin A,and 5 units of Vitamin B.Chicken costs 40 cents per ounce,tomatoes cost 8 cents per ounce,and eggs cost 12 cents per ounce. -Referring to Scenario D.1,what is an appropriate constraint for this scenario?

In linear programming,each parameter is assumed to be known with certainty.

The ________ is the upper and lower limit of an objective function coefficient over which the optimal values of the decision variables remain unchanged.

A portfolio manager is trying to balance investments between bonds,stocks and cash.The return on stocks is 12 percent,9 percent on bonds,and 3 percent on cash.The total portfolio is $1 billion,and he or she must keep 10 percent in cash in accordance with company policy.The fund's prospectus promises that stocks cannot exceed 75 percent of the portfolio,and the ratio of stocks to bonds must equal two.Formulate this investment decision as a linear programming problem,defining fully your decision variables and then giving the objective function and constraints.