Exam 20: Linear Programming

In linear programming,a(n)________ is a point that lies at the intersection of two (or possibly more)constraint lines on the boundary of the feasible region.

Table D.5 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.5.What is the anticipation inventory at the end of the first quarter?

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.

While glancing over the sensitivity report,you note that the stitching labor has a shadow price of $10 and a lower limit of 24 hours with an upper limit of 36 hours.If your original right hand value for stitching labor was 30 hours,you know that:

Table D.5 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.5.According to the optimal production plan,what is the total ending inventory in the second quarter? Assume that the plan that uses the least amount of subcontracting is used.

An equality constraint requires that only the points on the line described by the constraint are feasible.

NYNEX must schedule round-the-clock coverage for its telephone operators.To keep the number of different shifts down to a manageable level,it has only four different shifts.Operators work eight-hour shifts and can begin work at either midnight,8 a.m. ,noon,or 4 p.m.Operators are needed according to the following demand pattern,given in four-hour time blocks. Formulate this scheduling decision as a linear programming problem,defining fully your decision variables and then giving the objective function and constraints.

What is degeneracy in the context of linear programming? Why is degeneracy a concern?

Table D.4 Bahouth Enterprises produces a variety of hookahs for clients around the globe.Their small plant has a highly flexible workforce that can switch between products seamlessly.They forecast using a six-month planning period and have a demand forecast as shown in the table.The per-unit costs for each output option the sales and operations planner has at his disposal are indicated in the table.Regular output costs $40 per unit,overtime production is $60 per unit,and subcontracting is $70 per unit.Holding inventory from one month to the next costs $2 per unit per month and a backlog costs $5 per unit per month.Regular plant capacity is 300 units per month. -Use the information in Table D.4.The plant has no limits on the number of units produced by overtime or subcontractors and adopts a chase plan strategy for the six-month planning period.What is the cost for month 6 of their chase plan?

Table D.3 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 22 units of protein,15 units of vitamin A,and 8 units of vitamin B.She has eggs,tomatoes,and chicken meat as possible inputs to her cat food.Each ounce of eggs contains 6 units of protein,4 units of Vitamin A,and 3 units of Vitamin B.Each ounce of tomatoes contains 1 unit of protein,8 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 8 units of Vitamin B.Chicken costs 40 cents per ounce,tomatoes cost 5 cents per ounce,and eggs cost 12 cents per ounce. -Referring to Table D.3,what is an appropriate constraint for this scenario?

Table D.3 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 22 units of protein,15 units of vitamin A,and 8 units of vitamin B.She has eggs,tomatoes,and chicken meat as possible inputs to her cat food.Each ounce of eggs contains 6 units of protein,4 units of Vitamin A,and 3 units of Vitamin B.Each ounce of tomatoes contains 1 unit of protein,8 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 8 units of Vitamin B.Chicken costs 40 cents per ounce,tomatoes cost 5 cents per ounce,and eggs cost 12 cents per ounce. -Referring to Table D.3,what is an appropriate objective function for this scenario?

The ________ represents all permissible combinations of the decision variables in a linear programming model.

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

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.

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 22 units of protein,15 units of vitamin A,and 8 units of vitamin B.She has eggs,tomatoes,and chicken meat as possible inputs to her cat food.Each ounce of eggs contains 6 units of protein,4 units of Vitamin A,and 3 units of Vitamin B.Each ounce of tomatoes contains 1 unit of protein,8 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 8 units of Vitamin B.Chicken costs 40 cents per ounce,tomatoes cost 5 cents per ounce,and eggs cost 12 cents per ounce.To make the production process as easy as possible,she would like to make exactly four ounces of cat food from her recipe.She used POM for Windows and received the following results.Provide an interpretation.

________ is the amount by which the left-hand side falls short of the right-hand side in a linear programming model.

Table D.5 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.5.According to the optimal production plan,what is the overtime production in the second quarter?

________ is an assumption that the decision variables must be either positive or zero.

Table D.4 Bahouth Enterprises produces a variety of hookahs for clients around the globe.Their small plant has a highly flexible workforce that can switch between products seamlessly.They forecast using a six-month planning period and have a demand forecast as shown in the table.The per-unit costs for each output option the sales and operations planner has at his disposal are indicated in the table.Regular output costs $40 per unit,overtime production is $60 per unit,and subcontracting is $70 per unit.Holding inventory from one month to the next costs $2 per unit per month and a backlog costs $5 per unit per month.Regular plant capacity is 300 units per month. -Use the information in Table D.4.If the planner decides to adopt a chase plan for the planning period,what will the regular output be for month 1?

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