Deck 20: Linear Programming

Question 1

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

A)A parameter is also known as a decision variable.
B)Linearity assumes proportionality and additivity.
C)The product-mix problem is a one-period type of aggregate planning problem.
D)One reasonable sequence for formulating a model is defining the decision variables,writing out the objective function,and writing out the constraints.

Answer 1

A

Question 2

One assumption of linear programming is that a decision maker cannot use negative quantities of the decision variables.

Answer 2

True

Question 3

The objective function Maximize Z = 3x2 + 4y is appropriate for linear programming.

Answer 3

False

Question 4

The simplex method is an interactive algebraic procedure for solving linear programming problems.

Question 5

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

Question 6

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

A)(0,22).
B)(6,0).
C)(6,22).
D)(0,11).

Question 7

The terms slack and surplus both refer to having too much of a resource.

Question 8

A parameter is a region that represents all permissible combinations of the decision variables in a linear programming model.

Question 9

Decision variables are represented in both the objective function and the constraints while formulating a linear program.

Question 10

Only corner points should be considered for the optimal solution to a linear programming problem.

Question 11

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

A)A linear programming problem can have more than one optimal solution.
B)Most real-world linear programming problems are solved on a computer.
C)If a binding constraint were relaxed,the optimal solution wouldn't change.
D)A surplus variable is added to a > constraint to convert it to an equality.

Question 12

The graphical method is a practical method for solving product mix problems of any size,provided the decision maker has sufficient quantities of graph paper.

Question 13

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

Question 14

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

Question 15

A binding constraint is the amount by which the left-hand side falls short of the right-hand side.

Question 16

A constraint is a limitation that restricts the permissible choices.

Question 17

Degeneracy occurs when the linear program model consists of only an infeasible region.

Question 18

A manager is interested in using linear programming to analyze production for the ensuing week.She knows that it will take exactly 1.5 hours to run a batch of product A and that this batch will consume two tons of sugar.This is an example of the linear programming assumption of:

A)linearity.
B)certainty.
C)continuous variables.
D)whole numbers.

Question 19

A binding constraint has slack but does not have surplus.

Question 20

Sensitivity analysis is not possible with the graphical method of linear programming.

Question 21

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,an optimal four ounce serving contains 3.6 ounces of eggs and .4 ounces of tomatoes.Which of the following statements is best?

A)The serving contains more protein than is necessary.
B)The serving contains less Vitamin B than is necessary.
C)The serving contains more Vitamin A than is necessary.
D)The serving contains less chicken than is necessary.

Question 22

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?

A)4*Eggs + 8*Tomatoes + 14*Chicken ≤ 15
B)4*Eggs + 8*Tomatoes + 14*Chicken ≥ 15
C)4*Eggs + 8*Tomatoes + 14*Chicken = 15
D)15*VitaminA = Eggs + Tomatoes +Chicken

Question 23

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?

A)0 units
B)1 through 5 units
C)6 through 10 units
D)11 units and above

Question 24

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?

A)300 units
B)350 units
C)400 units
D)450 units

Question 25

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 level plan strategy for the six-month planning period.What is the cost for month 6 of their level plan?

A)between $16,200 and $16,600
B)between $16,600 and $17,000
C)between $17,000 and $17,400
D)between $17,400 and $17,800

Question 26

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.Given the information in the optimal tableau,what is the demand forecast for quarter 2?

A)fewer than 30 units
B)greater than 30 units but fewer than or equal to 35 units
C)greater than 35 units but fewer than or equal to 40 units
D)greater than 40 units

Question 27

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?

A)3*Eggs + 14*Tomatoes + 8*Chicken ≥ 8
B)22*Protein + 15*VitaminA + 8*VitaminB ≥ 8
C).12*Egg + .5*Tomato + .4*Chicken ≥ 4
D)22*Protein + 15*VitaminA + 8*VitaminB ≥ 4

Question 28

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 total cost of the optimal production plan?

A)less than or equal to $1,000
B)greater than $1,000 but less than or equal to $2,000
C)greater than $2,000 but less than or equal to $3,000
D)greater than $3,000

Question 29

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?

A)Max Z = .12*Egg + .5*Tomato + .4*Chicken
B)Max Z = 22*Protein + 15*VitaminA + 8*VitaminB
C)Min Z = .12*Egg + .5*Tomato + .4*Chicken
D)Min Z = 22*Protein + 15*VitaminA + 8*VitaminB

Question 30

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:

A)the next worker that offers to work an extra 8 hours should receive at least $80.
B)you can send someone home 6 hours early and still pay them the $60 they would have earned while on the clock.
C)you would be willing pay up to $60 for someone to work another 6 hours.
D)you would lose $80 if one of your workers missed an entire 8 hour shift.

Question 31

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 plant has no limits on the number of units produced by overtime or subcontractors,what is the lowest-cost chase plan that is possible for the six-month period?

A)less than $50,000
B)between $50,000 and $100,000
C)between $100,000 and $150,000
D)between $150,000 and $200,000

Question 32

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,an optimal four ounce serving contains 3.6 ounces of eggs and .4 ounce of tomatoes.Which of the following statements is best?

A)The serving costs about 21 cents.
B)The serving costs about 17 cents
C)The serving costs about 34 cents.
D)The serving costs about 45 cents.

Question 33

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?

A)$8,000
B)$9,852
C)$11,317
D)$12,631

Question 34

A manager is interested in deciding production quantities for products A,B,and C.He has an inventory of 20 tons each of raw materials 1,2,3,and 4 that are used in the production of products A,B,and C.He can further assume that he can sell all of what he makes.Which of the following statements is correct?

A)The manager has four decision variables.
B)The manager has three constraints.
C)The manager has three decision variables.
D)The manager can solve this problem graphically.

Question 35

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.Given the information in the optimal tableau,what is the inventory carrying cost,in dollars per unit per quarter?

A)less than $1
B)greater than $1 but less than or equal to $2
C)greater than $2 but less than or equal to $3
D)greater than $3

Question 36

Consider a corner point to a linear programming problem,which lies at the intersection of the following two constraints: 6X1 + 15X2 < 390
2X1 + X2 < 50
Which of the following statements about the corner point is true?

A)X1 < 21
B)X1 > 25
C)X1 < 10
D)X1 > 17

Question 37

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.

A)none
B)two units
C)three units
D)greater than three units

Question 38

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 level plan for the planning period,what will the regular output be for month 3?

A)between 290 and 330 units
B)between 330 and 370 units
C)between 370 and 410 units
D)between 410 and 450 units

Question 39

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?

A)two units
B)three units
C)four units
D)five units

Question 40

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 level plan for the planning period,what will the total output be for month 3?

A)between 350 and 400 units
B)between 400 and 450 units
C)between 450 and 500 units
D)more than 500 units

Question 41

A(n)________ limits the ability to improve the objective function.

Question 42

________ is useful for allocating scarce resources among competing demands.

Question 43

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

Question 44

________ represent choices the decision maker can control.

Question 45

Each coefficient or given constant is known by the decision maker with ________.

Question 46

The ________ is an expression in linear programming models that states mathematically what is being maximized or minimized.

Question 47

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

Question 48

A modeler is limited to two or fewer decision variables when using the ________.

Question 49

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.Given the information in the optimal tableau,what is the subcontracting cost,in dollars per unit?

A)$5
B)$7
C)$10
D)$12

Question 50

The assumption of ________ allows a decision maker to combine the profit from one product with the profit from another to realize the total profit from a feasible solution.

Question 51

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

Question 52

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

Question 53

The interval over which the right-hand-side parameter of a constraint can vary while its shadow price remains valid is the ________.

Question 54

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.

Question 55

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

Question 56

For an "equal" constraint,only points ________ are feasible solutions.

Question 57

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

Question 58

The ________ problem is a one-period type of aggregate planning problem,the solution of which yields optimal output quantities of a group of products or services,subject to resource capacity and market demand conditions.

Question 59

A(n)________ is the marginal improvement in the objective function value caused by relaxing a constraint by one unit.

Question 60

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

Question 61

________ occurs in a linear programming problem when the number of nonzero variables in the optimal solution is fewer than the number of constraints.

Question 62

A producer has three products,A,B,and C,which are composed from many of the same raw materials and subassemblies by the same skilled workforce.Each unit of product A uses 15 units of raw material X,a single purge system subassembly,a case,a power cord,three labor hours in the assembly department,and one labor hour in the finishing department.Each unit of product B uses 10 units of raw material X,five units of raw material Y,two purge system subassemblies,a case,a power cord,five labor hours in the assembly department,and 90 minutes in the finishing department.Each unit of product C uses five units of raw material X,25 units of raw material Y,two purge system subassemblies,a case,a power cord,seven labor hours in the assembly department,and three labor hours in the finishing department.Labor between the assembly and finishing departments is not transferable,but workers within each department work on any of the three products.There are three full-time (40 hours/week)workers in the assembly department and one full-time and one half-time (20 hours/week)worker in the finishing department.At the start of this week,the company has 300 units of raw material X,400 units of raw material Y,60 purge system subassemblies,40 cases,and 50 power cords in inventory.No additional deliveries of raw materials are expected this week.There is a $90 profit on product A,a $120 profit on product B,and a $150 profit on product C.The operations manager doesn't have any firm orders,but would like to make at least five of each product so he can have the products on the shelf in case a customer wanders in off the street.
Formulate the objective function and all constraints,and clearly identify each constraint by the name of the resource or condition it represents.

Question 63

Provide three examples of operations management decision problems for which linear programming can be useful,and why.

Question 64

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

Question 65

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.

Question 66

A snack food producer runs four different plants that supply product to four different regional distribution centers.The division operations manager is focused on one product,so he creates a table showing each plant's monthly capacity and each distribution center's monthly demand (both amounts in cases)for the product.The division manager supplements this table with the cost data to ship one case from each plant to each distribution center.Formulate an objective function and constraints that will solve this problem using linear programming.

Question 67

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.

Question 68

You observe a linear programming problem that has been solved using the graphical method of linear programming.The feasible region and optimal solution are clearly labeled.How could you identify the slack or surplus amounts in the scenario?

Question 69

A very confused manager is reading a two-page report given to him by his student intern."She told me that she had my problem solved,gave me this,and then said she was off to her production management course," he whined."I gave her my best estimates of my on-hand inventories and requirements to produce,but what if my numbers are slightly off? I recognize the names of our four models W,X,Y,and Z,but that's about it.Can you figure out what I'm supposed to do and why?" You take the report from his hands and note that it is the answer report and the sensitivity report from Excel's solver routine.
Explain each of the highlighted cells in layman's terms and tell the manager what they mean in relation to his problem.

Question 70

What are the limitations of the graphical method of linear programming?

Question 71

What are the assumptions of linear programming? Provide examples of each.

Question 72

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.

Question 73

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.

Question 74

What does the feasible region represent?

Question 75

Briefly describe the meaning of a shadow price.Provide an example of how a manager could use information about shadow prices to improve operations?

Question 76

A manufacturer builds finished items A,B,and C from five different components.They currently have several finished units,a number of items that are partially complete,and many components that have just been delivered from their suppliers.The finished items,incomplete items and raw components can all be assigned some monetary value even though the manufacturer typically does not sell anything except finished items.The manufacturer needs to raise capital quickly so they formulate a linear program to help them decide on the most profitable way ahead.Their linear programming expert forgets to restrict their decision variables to non-negative values and is surprised when the computer output tells them that finished item A and C should be negative? If the company always follows the advice of their linear programming analysis,what should they do and why?

Question 77

What are some potential abuses or misuses of linear programming (beyond violation of basic assumptions)?

Question 78

What is the meaning of a slack or surplus variable?

Question 79

The CZ Jewelry Company produces two products: (1)engagement rings and (2)jeweled watches.The production process for each is similar in that both require a certain number of hours of diamond work and a certain number of labor hours in the gold department.Each ring takes four hours of diamond work and two hours in the gold shop.Each watch requires three hours in diamonds and one hour in the gold department.There are 240 hours of diamond labor available and 100 hours of gold department time available for the next month.Each engagement ring sold yields a profit of $9;each watch produced may be sold for a $10 profit.
a.Give a complete formulation of this problem,including a careful definition of your decision variables.Let the first decision variable, (X1),deal with rings,the second decision variable, (X2),with watches,the first constraint with diamonds,and the second constraint with gold.
b.Graph the problem fully in the following space.Label the axes carefully,plot the constraints,shade the feasibility region,plot at least one isoprofit line that reveals the optimal solution,circle the corner points and highlight the optimal corner point so found,and solve for it algebraically.(Show all your work to get credit. )

Question 80

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.