Deck 21: Linear Programming

Question 1

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

Answer 1

True

Question 2

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.

Answer 2

C

Question 3

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.

Answer 3

False

Question 4

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 5

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

Question 6

A site manager has three day laborers available for eight hours each and a burning desire to maximize his return on their wages.The site manager uses linear programming to assign them to two tasks and notes that he has enough work to occupy 21 labor hours.The linear program that the site manager has constructed has

A) slack.
B) surplus.
C) a positive shadow price for labor.
D) no feasible solution.

Question 7

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

Question 8

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 9

A binding constraint has slack but does not have surplus.

Question 10

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

A) X? < 21
B) X? > 25
C) X? < 10
D) X? > 17

Question 11

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

Question 12

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

Question 13

Suppose that the optimal values of the decision variables to a two-variable linear programming problem remain the same as long as the slope of the objective function lies between the slopes of the following two constraints:
2X? + 3X? < 26
2X? + 2X? < 20
The current objective function is:
8X? + 9X? = Z
Which of the following statements about the range of optimality on c? is TRUE?

A) 0 < c? < 2
B) 2 < c? < 6
C) 6 < c? < 9
D) 9 < c? < 12

Question 14

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

Question 15

For the line that has the equation 4X? + 8X? = 88,an axis intercept is

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

Question 16

A constraint is a limitation that restricts the permissible choices.

Question 17

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

Question 18

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.

Question 19

The objective function Maximize Z = 3x² = 4y is appropriate.

Question 20

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

Question 21

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

Question 22

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

Question 23

You are faced with a linear programming objective function of:
Max P = $20X + $30Y
And constraints of:
3X + 4Y = 24 (Constraint A)
5X -Y = 18 (Constraint B)
You discover that the shadow price for Constraint A is 7.5 and the shadow price for Constraint B is 0.Which of these statements is TRUE?

A) You can change quantities of X and Y at no cost for Constraint B.
B) For every additional unit of the objective function you create, you lose 0 units of B.
C) For every additional unit of the objective function you create, the price of A rises by $7.50.
D) The most you would want to pay for an additional unit of A would be $7.50.

Question 24

A(n)________ forms the optimal corner and limits the ability to improve the objective function.

Question 25

________ represent choices the decision maker can control.

Question 26

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

Question 27

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

Question 28

For an = constraint,only points ________ are feasible solutions.

Question 29

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 30

________ is useful for allocating scarce resources among competing demands.

Question 31

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

Question 32

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

Question 33

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 34

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

Question 35

If merely rounding up or rounding down a result for a decision variable is not sufficient when they must be expressed in whole units,then a decision maker might instead use ________ to analyze the situation.

Question 36

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

Question 37

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

Question 38

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

Question 39

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

Question 40

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 41

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

Question 42

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 43

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 44

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

Question 45

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

Question 46

What is the meaning of a slack or surplus variable?

Question 47

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

Question 48

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 49

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 50

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 51

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 output follows:
Solution

Sensitivity Analysis and Ranges
Objective Function Coefficients

Right-Hand-Side Values

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 gasoline 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 constraint 2.
Second Number: Shadow price of 0 for constraint 1.
Third Number: An upper limit of "no limit" for the right-hand-side value constraint 1.

Question 52

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 53

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 54

Case: 1A + 1B + 1C ? 40
Cord: 1A + 1B + 1C ? 50
Assembly Department Labor: 3A + 5B + 7C ? 120
Finish Department Labor: 1A + 1.5B + 3C ? 60
Minimum Production for A: 1A + 0B + 0C ? 5
Minimum Production for B: 0A + 1B + 0C ? 5
Minimum Production for C: 0A + 0B + 1C ? 5
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.
Microsoft Excel 10.0 Answer Report
Worksheet: Supplement D
Report Created: 1/26/2004 11:26:50 AM


Microsoft Excel 10.0 Sensitivity Report
Worksheet: Supplement D
Report Created: 1/26/2004 11:26:50 AM

Question 55

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

Question 56

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 at the end.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.
Solver-Linear Programming
Solution

Sensitivity Analysis and Ranges
Objective Function Coefficients

First Number: The shadow price of 0.0104 for the "Const3" constraint.
Second Number: The slack or surplus of 6383 for the "Const1" constraint.
Third Number: The lower limit of 12.2807 for the "Const1" constraint.

Question 57

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