Chapter 1—Introduction MULTIPLE CHOICE 1. The field of management science a. concentrates on the use of quantitative methods to assist in decision making. b. approaches decision making rationally, with techniques based on the scientific method. c. is another name for decision science and for operations research. d. each of these choices are true. ANS: D
PTS: 1
TOP: Introduction
2. Identification and definition of a problem a. cannot be done until alternatives are proposed. b. is the first step of decision making. c. is the final step of problem solving. d. requires consideration of multiple criteria. ANS: B
PTS: 1
TOP: Problem solving and decision making
3. Decision alternatives a. should be identified before decision criteria are established. b. are limited to quantitative solutions c. are evaluated as a part of the problem definition stage. d. are best generated by brain-storming. ANS: A
PTS: 1
TOP: Problem solving and decision making
4. Decision criteria a. are the choices faced by the decision maker. b. are the problems faced by the decision maker. c. are the ways to evaluate the choices faced by the decision maker. d. must be unique for a problem. ANS: C
PTS: 1
TOP: Problem solving and decision making
5. In a multicriteria decision problem a. it is impossible to select a single decision alternative. b. the decision maker must evaluate each alternative with respect to each criterion. c. successive decisions must be made over time. d. each of these choices are true. ANS: B
PTS: 1
TOP: Problem solving and decision making
6. The quantitative analysis approach requires a. the manager's prior experience with a similar problem. b. a relatively uncomplicated problem. c. mathematical expressions for the relationships. ANS: C
PTS: 1
TOP: Quantitative analysis and decision making
7. A physical model that does not have the same physical appearance as the object being modeled is a. an analog model. b. an iconic model.
c. a mathematical model. d. a qualitative model. ANS: A
PTS: 1
TOP: Model development
8. Inputs to a quantitative model a. are a trivial part of the problem solving process. b. are uncertain for a stochastic model. c. are uncontrollable for the decision variables. d. must all be deterministic if the problem is to have a solution. ANS: B
PTS: 1
TOP: Model development
9. When the value of the output cannot be determined even if the value of the controllable input is known, the model is a. analog. b. digital. c. stochastic. d. deterministic. ANS: C
PTS: 1
TOP: Model development
10. The volume that results in total revenue being equal to total cost is the a. break-even point. b. marginal volume. c. marginal cost. d. profit mix. ANS: A
PTS: 1
TOP: Break-even analysis
11. Management science and operations research both involve a. qualitative managerial skills. b. quantitative approaches to decision making. c. operational management skills. d. scientific research as opposed to applications. ANS: B
PTS: 1
TOP: Introduction
12. George Dantzig is important in the history of management science because he developed a. the scientific management revolution. b. World War II operations research teams. c. the simplex method for linear programming. d. powerful digital computers. ANS: C
PTS: 1
TOP: Introduction
13. The first step in problem solving is a. determination of the correct analytical solution procedure. b. definition of decision variables. c. the identification of a difference between the actual and desired state of affairs. d. implementation. ANS: C
PTS: 1
TOP: Problem solving and decision making
14. Problem definition a. includes specific objectives and operating constraints.
b. must occur prior to the quantitative analysis process. c. must involve the analyst and the user of the results. d. each of these choices are true. ANS: D
PTS: 1
TOP: Quantitative analysis
15. A model that uses a system of symbols to represent a problem is called a. mathematical. b. iconic. c. analog. d. constrained. ANS: A
PTS: 1
TOP: Model development
TRUE/FALSE 1. The process of decision making is more limited than that of problem solving. ANS: T
PTS: 1
TOP: Problem solving and decision making
2. The terms 'stochastic' and 'deterministic' have the same meaning in quantitative analysis. ANS: F
PTS: 1
TOP: Model development
3. The volume that results in marginal revenue equaling marginal cost is called the break-even point. ANS: F
PTS: 1
TOP: Problem solving and decision making
4. Problem solving encompasses both the identification of a problem and the action to resolve it. ANS: T
PTS: 1
TOP: Problem solving and decision making
5. The decision making process includes implementation and evaluation of the decision. ANS: F
PTS: 1
TOP: Problem solving and decision making
6. The most successful quantitative analysis will separate the analyst from the managerial team until after the problem is fully structured. ANS: F
PTS: 1
TOP: Quantitative analysis
7. The value of any model is that it enables the user to make inferences about the real situation. ANS: T
PTS: 1
TOP: Model development
8. Uncontrollable inputs are the decision variables for a model. ANS: F
PTS: 1
TOP: Model development
9. The feasible solution is the best solution possible for a mathematical model. ANS: F
PTS: 1
TOP: Model solution
10. A company seeks to maximize profit subject to limited availability of man-hours. Man-hours is a controllable input. ANS: F
PTS: 1
TOP: Model development
11. Frederick Taylor is credited with forming the first MS/OR interdisciplinary teams in the 1940's. ANS: F
PTS: 1
TOP: Introduction
12. To find the choice that provides the highest profit and the fewest employees, apply a single-criterion decision process. ANS: F
PTS: 1
TOP: Problem solving and decision making
13. The most critical component in determining the success or failure of any quantitative approach to decision making is problem definition. ANS: T
PTS: 1
TOP: Quantitative analysis
14. The first step in the decision making process is to identify the problem. ANS: T
PTS: 1
TOP: Introduction
15. All uncontrollable inputs or data must be specified before we can analyze the model and recommend a decision or solution for the problem. ANS: T
PTS: 1
TOP: Quantitative analysis
16. In quantitative analysis, the optimal solution is the mathematically-best solution. ANS: T
PTS: 1
TOP: Quantitative analysis
17. If you are deciding to buy either machine A, B, or C with the objective of minimizing the sum of labor, material and utility costs, you are dealing with a single-criterion decision. ANS: T
PTS: 1
TOP: Problem solving and decision making
18. Model development should be left to quantitative analysts; the model user's involvement should begin at the implementation stage. ANS: F
PTS: 1
TOP: Problem solving and decision making
19. A feasible solution is one that satisfies at least one of the constraints in the problem. ANS: F
PTS: 1
TOP: Model solution
20. A toy train layout designed to represent an actual railyard is an example of an analog model. ANS: F
PTS: 1
TOP: Model development
SHORT ANSWER 1. Should the problem solving process be applied to all problems?
ANS: Answer not provided. PTS: 1
TOP: Problem solving and decision making
2. Explain the difference between quantitative and qualitative analysis from the manager's point of view. ANS: Answer not provided. PTS: 1
TOP: Quantitative analysis and decision making
3. Explain the relationship among model development, model accuracy, and the ability to obtain a solution from a model. ANS: Answer not provided. PTS: 1
TOP: Model solution
4. What are three of the management science techniques that practitioners use most frequently? How can the effectiveness of these applications be increased? ANS: Answer not provided. PTS: 1
TOP: Methods used most frequently
5. What steps of the problem solving process are involved in decision making? ANS: Answer not provided. PTS: 1
TOP: Introduction
6. Give three benefits of model development and an example of each. ANS: Answer not provided. PTS: 1
TOP: Model development
7. Explain the relationship between information systems specialists and quantitative analysts in the solution of large mathematical problems. ANS: Answer not provided. PTS: 1
TOP: Data preparation
8. Define and contrast the terms feasible solution, infeasible solution and optimal solution. ANS:
Answer not provided. PTS: 1
TOP: Model solution
9. Define three forms of models and provide an example of each. ANS: Answer not provided. PTS: 1
TOP: Model development
10. Explain the difference between controllable and uncontrollable inputs to a mathematical model and provide an example of each. ANS: Answer not provided. PTS: 1
TOP: Model development
PROBLEM 1. A snack food manufacturer buys corn for tortilla chips from two cooperatives, one in Iowa and one in Illinois. The price per unit of the Iowa corn is $6.00 and the price per unit of the Illinois corn is $5.50. a. Define variables that would tell how many units to purchase from each source. b. Develop an objective function that would minimize the total cost. c. The manufacturer needs at least 12000 units of corn. The Iowa cooperative can supply up to 8000 units, and the Illinois cooperative can supply at least 6000 units. Develop constraints for these conditions. ANS: a. b. c.
Let x1 = the number of units from Iowa Let x2 = the number of units from Illinois Min 6x1 + 5.5x2 x1 + x 2 12000 x1 8000 x1 6000
PTS: 1
TOP: Model development
2. The relationship d = 5000 − 25p describes what happens to demand (d) as price (p) varies. Here, price can vary between $10 and $50. a. How many units can be sold at the $10 price? How many can be sold at the $50 price? b. Model the expression for total revenue. c. Consider prices of $20, $30, and $40. Which of these three price alternative will maximize total revenue? What are the values for demand and revenue at this price? ANS: a. b.
For p = 10, d = 4750 For p = 50, d = 3750 TR = p(5000 − 25p)
c.
For p = 20, d = 4500, TR = $90,000 For p = 30, d = 4250, TR = $127,500 For p = 40, d = 4000, TR = $160,000 (maximum total revenue)
PTS: 1
TOP: Model development
3. There is a fixed cost of $50,000 to start a production process. Once the process has begun, the variable cost per unit is $25. The revenue per unit is projected to be $45. a. Write an expression for total cost. b. Write an expression for total revenue. c. Write an expression for total profit. d. Find the break-even point. ANS: a. b. c. d.
C(x) = 50000 + 25x R(x) = 45x P(x) = 45x − (50000 + 25x) x = 2500
PTS: 1
TOP: Break-even analysis
4. An author has received an advance against royalties of $10,000. The royalty rate is $1.00 for every book sold in the United States, and $1.35 for every book sold outside the United States. Define variables for this problem and write an expression that could be used to calculate the number of books to be sold to cover the advance. ANS: Let x1 = the number of books sold in the U.S. Let x2 = the number of books sold outside the U.S. 10000 = 1x1 + 1.35x2 PTS: 1
TOP: Break-even analysis
5. A university schedules summer school courses based on anticipated enrollment. The cost for faculty compensation, laboratories, student services, and allocated overhead for a computer class is $8500. If students pay $920 to enroll in the course, how large would enrollment have to be for the university to break even? ANS: Enrollment would need to be 10 students. PTS: 1
TOP: Break-even analysis
6. As part of their application for a loan to buy Lakeside Farm, a property they hope to develop as a bed-and-breakfast operation, the prospective owners have projected: Monthly fixed cost (loan payment, taxes, insurance, maintenance) Variable cost per occupied room per night Revenue per occupied room per night a.
Write the expression for total cost per month. Assume 30 days per month.
$ $
$6000 20 75
b. c.
Write the expression for total revenue per month. If there are 12 guest rooms available, can they break even? What percentage of rooms would need to be occupied, on average, to break even?
ANS: a. b. c.
C(x) = 6000 + 20(30)x (monthly) R(x) = 75(30)x (monthly) Break-even occupancy = 3.64 or 4 occupied rooms per night, so they have enough rooms to break even. This would be a 33% occupancy rate.
PTS: 1
TOP: Break-even analysis
7. Organizers of an Internet training session will charge participants $150 to attend. It costs $3000 to reserve the room, hire the instructor, bring in the equipment, and advertise. Assume it costs $25 per student for the organizers to provide the course materials. a. How many students would have to attend for the company to break even? b. If the trainers think, realistically, that 20 people will attend, then what price should be charged per person for the organization to break even? ANS: a.
b.
C(x) = 3000 + 25x R(x) = 150x Break-even students = 24 Cost = 3000 + 25(20) Revenue = 20p Break-even price = 175
PTS: 1
TOP: Break-even analysis
8. In this portion of an Excel spreadsheet, the user has given values for selling price, the costs, and a sample volume. Give the cell formula for a. cell E12, break-even volume. b. cell E16, total revenue. c. cell E17, total cost. d. cell E19, profit (loss). A 1 2 3 4 5 6 7 8 9 10 11 12 13
B
C
D
E
Break-even calculation Selling price per unit
10
Costs Fix cost Variable cost per unit
8400 4.5
Break-even volume
14 15 16 17 18 19
Sample calculation Volume Total revenue Total cost
2000
Profit (loss)
ANS: a. b. c. d.
=E9/(E6-E10) =E15*E6 =E9+E10*E15 =E16-E17
PTS: 1
TOP: Spreadsheets for management science
9. A furniture store has set aside 800 square feet to display its sofas and chairs. Each sofa utilizes 50 sq. ft. and each chair utilizes 30 sq. ft. At least five sofas and at least five chairs are to be displayed. a. Write a mathematical model representing the store's constraints. b. Suppose the profit on sofas is $200 and on chairs is $100. On a given day, the probability that a displayed sofa will be sold is .03 and that a displayed chair will be sold is .05. Mathematically model each of the following objectives: 1. Maximize the total pieces of furniture displayed. 2. Maximize the total expected number of daily sales. 3. Maximize the total expected daily profit. ANS: a.
b.
50s + 30c 800 s5 c5 (1) Max s + c (2) Max .03s + .05c (3) Max 6s + 5c
PTS: 1
TOP: Model development
10. A manufacturer makes two products, doors and windows. Each must be processed through two work areas. Work area #1 has 60 hours of available production time per week. Work area #2 has 48 hours of available production time per week. Manufacturing of a door requires 4 hours in work area #1 and 2 hours in work area #2. Manufacturing of a window requires 2 hours in work area #1 and 4 hours in work area #2. Profit is $8 per door and $6 per window. a. Define decision variables that will tell how many units to build (doors and windows) per week. b. Develop an objective function that will maximize total profit per week. c. Develop production constraints for work area #1 and #2. ANS: a. b.
Let D = the number of doors to build per week Let N = the number of windows to build per week Weekly Profit = 8D + 6W
c.
4D + 2W 60 2D + 4W 48
PTS: 1
TOP: Model development
11. A small firm builds galvanized swing sets. The investment in plant and equipment is $200,000. The variable cost per swing set is $500. The selling price of the swing set is $1000. How many swing sets would have to be sold for the firm to break even? ANS: 400 swing sets PTS: 1
TOP: Break-even analysis
12. A computer rework center has the capacity to rework 300 computers per day. The expected number of computers needing to be reworked per day is 225. The center is paid $26 for each computer reworked. The fixed cost of renting the reworking equipment is $250 per day. Work space rents for $150 per day. The cost of material is $18 per computer and labor costs $3 per computer. What is the break-even number of computers reworked per day? ANS: 80 computers PTS: 1
TOP: Break-even analysis
13. To establish a driver education school, organizers must decide how many cars, instructors, and students to have. Costs are estimated as follows. Annual fixed costs to operate the school are $30,000. The annual cost per car is $3000. The annual cost per instructor is $11,000 and one instructor is needed for each car. Tuition for each student is $350. Let x be the number of cars and y be the number of students. a. Write an expression for total cost. b. Write an expression for total revenue. c. Write an expression for total profit. d. The school offers the course eight times each year. Each time the course is offered, there are two sessions. If they decide to operate five cars, and if four students can be assigned to each car, will they break even? ANS: a. b. c. d.
C(x) = 30000 + 14000x R(y) = 350y P(x,y) = 350y − (30000 + 14000x) Each car/instructor can serve up to (4 students/session)(2 sessions/course)(8 courses/year) = 64 students annually. Five cars can serve 320 students. If the classes are filled, then profit for five cars is 350(320) − (30000 + 14000(5)) = 12000 so the school can reach the break-even point.
PTS: 1
TOP: Break-even analysis
14. Zipco Printing operates a shop that has five printing machines. The machines differ in their capacities to perform various printing operations due to differences in the machines' designs and operator skill levels. At the start of the workday there are five printing jobs to schedule. The manager must decide what the job-machine assignments should be. a. How could a quantitative approach to decision making be used to solve this problem? b. What would be the uncontrollable inputs for which data must be collected? c. Define the decision variables, objective function, and constraints to appear in the mathematical model. d. Is the model deterministic or stochastic? e. Suggest some simplifying assumptions for this problem. ANS: a. b. c.
d. e.
A quantitative approach to decision making can provide a systematic way for deciding the job-machine pairings so that total job processing time is minimized. How long it takes to process each job on each machine, and any job-machine pairings that are unacceptable. Decision variables: one for each job-machine pairing, taking on a value of 1 if the pairing is used and 0 otherwise. Objective function: minimize total job processing time. Constraints: each job is assigned to exactly one machine, and each machine be assigned no more than one job. Stochastic: job processing times vary due to varying machine set-up times, variable operator performance, and more. Assume that processing times are deterministic (known/fixed).
PTS: 1
TOP: Model development
15. Consider a department store that must make weekly shipments of a certain product from two different warehouses to four different stores. a. How could a quantitative approach to decision making be used to solve this problem? b. What would be the uncontrollable inputs for which data must be gathered? c. What would be the decision variables of the mathematical model? the objective function? the constraints? d. Is the model deterministic or stochastic? e. Suggest assumptions that could be made to simplify the model. ANS: a. b. c.
d. e.
A quantitative approach to decision making can provide a systematic way to determine a minimum shipping cost from the warehouses to the stores. Fixed costs and variable shipping costs; the demand each week at each store; the supplies each week at each warehouse. Decision variables--how much to ship from each warehouse to each store; objective function--minimize total shipping costs; constraints--meet the demand at the stores without exceeding the supplies at the warehouses. Stochastic--weekly demands fluctuate as do weekly supplies; transportation costs could vary depending upon the amount shipped, other goods sent with a shipment, etc. Make the model deterministic by assuming fixed shipping costs per item, demand is constant at each store each week, and weekly supplies in the warehouses are constant.
PTS: 1
TOP: Model development
Chapter 2—An Introduction to Linear Programming MULTIPLE CHOICE 1. The maximization or minimization of a quantity is the a. goal of management science. b. decision for decision analysis. c. constraint of operations research. d. objective of linear programming. ANS: D
PTS: 1
TOP: Introduction
2. Decision variables a. tell how much or how many of something to produce, invest, purchase, hire, etc. b. represent the values of the constraints. c. measure the objective function. d. must exist for each constraint. ANS: A
PTS: 1
TOP: Objective function
3. Which of the following is a valid objective function for a linear programming problem? a. Max 5xy b. Min 4x + 3y + (2/3)z c. Max 5x2 + 6y2 d. Min (x1 + x2)/x3 ANS: B
PTS: 1
TOP: Objective function
4. Which of the following statements is NOT true? a. A feasible solution satisfies all constraints. b. An optimal solution satisfies all constraints. c. An infeasible solution violates all constraints. d. A feasible solution point does not have to lie on the boundary of the feasible region. ANS: C
PTS: 1
TOP: Graphical solution
5. A solution that satisfies all the constraints of a linear programming problem except the nonnegativity constraints is called a. optimal. b. feasible. c. infeasible. d. semi-feasible. ANS: C
PTS: 1
TOP: Graphical solution
6. Slack a. is the difference between the left and right sides of a constraint. b. is the amount by which the left side of a constraint is smaller than the right side. c. is the amount by which the left side of a constraint is larger than the right side. d. exists for each variable in a linear programming problem. ANS: B
PTS: 1
TOP: Slack variables
7. To find the optimal solution to a linear programming problem using the graphical method
a. b. c. d.
find the feasible point that is the farthest away from the origin. find the feasible point that is at the highest location. find the feasible point that is closest to the origin. None of the alternatives is correct.
ANS: D
PTS: 1
TOP: Extreme points
8. Which of the following special cases does not require reformulation of the problem in order to obtain a solution? a. alternate optimality b. infeasibility c. unboundedness d. each case requires a reformulation. ANS: A
PTS: 1
TOP: Special cases
9. The improvement in the value of the objective function per unit increase in a right-hand side is the a. sensitivity value. b. dual price. c. constraint coefficient. d. slack value. ANS: B
PTS: 1
TOP: Right-hand sides
10. As long as the slope of the objective function stays between the slopes of the binding constraints a. the value of the objective function won't change. b. there will be alternative optimal solutions. c. the values of the dual variables won't change. d. there will be no slack in the solution. ANS: C
PTS: 1
TOP: Objective function
11. Infeasibility means that the number of solutions to the linear programming models that satisfies all constraints is a. at least 1. b. 0. c. an infinite number. d. at least 2. ANS: B
PTS: 1
TOP: Alternate optimal solutions
12. A constraint that does not affect the feasible region is a a. non-negativity constraint. b. redundant constraint. c. standard constraint. d. slack constraint. ANS: B
PTS: 1
TOP: Feasible regions
13. Whenever all the constraints in a linear program are expressed as equalities, the linear program is said to be written in a. standard form. b. bounded form. c. feasible form. d. alternative form.
ANS: A
PTS: 1
TOP: Slack variables
14. All of the following statements about a redundant constraint are correct EXCEPT a. A redundant constraint does not affect the optimal solution. b. A redundant constraint does not affect the feasible region. c. Recognizing a redundant constraint is easy with the graphical solution method. d. At the optimal solution, a redundant constraint will have zero slack. ANS: D
PTS: 1
TOP: Slack variables
15. All linear programming problems have all of the following properties EXCEPT a. a linear objective function that is to be maximized or minimized. b. a set of linear constraints. c. alternative optimal solutions. d. variables that are all restricted to nonnegative values. ANS: C
PTS: 1
TOP: Problem formulation
TRUE/FALSE 1. Increasing the right-hand side of a nonbinding constraint will not cause a change in the optimal solution. ANS: F
PTS: 1
TOP: Introduction
2. In a linear programming problem, the objective function and the constraints must be linear functions of the decision variables. ANS: T
PTS: 1
TOP: Mathematical statement of the RMC Problem
3. In a feasible problem, an equal-to constraint cannot be nonbinding. ANS: T
PTS: 1
TOP: Graphical solution
4. Only binding constraints form the shape (boundaries) of the feasible region. ANS: F
PTS: 1
TOP: Graphical solution
5. The constraint 5x1 − 2x2 0 passes through the point (20, 50). ANS: T
PTS: 1
TOP: Graphing lines
6. A redundant constraint is a binding constraint. ANS: F
PTS: 1
TOP: Slack variables
7. Because surplus variables represent the amount by which the solution exceeds a minimum target, they are given positive coefficients in the objective function. ANS: F
PTS: 1
TOP: Slack variables
8. Alternative optimal solutions occur when there is no feasible solution to the problem. ANS: F
PTS: 1
TOP: Alternative optimal solutions
9. A range of optimality is applicable only if the other coefficient remains at its original value. ANS: T
PTS: 1
TOP: Simultaneous changes
10. Because the dual price represents the improvement in the value of the optimal solution per unit increase in right-hand-side, a dual price cannot be negative. ANS: F
PTS: 1
TOP: Right-hand sides
11. Decision variables limit the degree to which the objective in a linear programming problem is satisfied. ANS: F
PTS: 1
TOP: Introduction
12. No matter what value it has, each objective function line is parallel to every other objective function line in a problem. ANS: T
PTS: 1
TOP: Graphical solution
13. The point (3, 2) is feasible for the constraint 2x1 + 6x2 30. ANS: T
PTS: 1
TOP: Graphical solution
14. The constraint 2x1 − x2 = 0 passes through the point (200,100). ANS: F
PTS: 1
TOP: A note on graphing lines
15. The standard form of a linear programming problem will have the same solution as the original problem. ANS: T
PTS: 1
TOP: Surplus variables
16. An optimal solution to a linear programming problem can be found at an extreme point of the feasible region for the problem. ANS: T
PTS: 1
TOP: Extreme points
17. An unbounded feasible region might not result in an unbounded solution for a minimization or maximization problem. ANS: T
PTS: 1
TOP: Special cases: unbounded
18. An infeasible problem is one in which the objective function can be increased to infinity. ANS: F
PTS: 1
TOP: Special cases: infeasibility
19. A linear programming problem can be both unbounded and infeasible. ANS: F
PTS: 1
TOP: Special cases: infeasibility and unbounded
20. It is possible to have exactly two optimal solutions to a linear programming problem.
ANS: F
PTS: 1
TOP: Special cases: alternative optimal solutions
SHORT ANSWER 1. Explain the difference between profit and contribution in an objective function. Why is it important for the decision maker to know which of these the objective function coefficients represent? ANS: Answer not provided. PTS: 1
TOP: Objective function
2. Explain how to graph the line x1 − 2x2 0. ANS: Answer not provided. PTS: 1
TOP: Graphing lines
3. Create a linear programming problem with two decision variables and three constraints that will include both a slack and a surplus variable in standard form. Write your problem in standard form. ANS: Answer not provided. PTS: 1
TOP: Standard form
4. Explain what to look for in problems that are infeasible or unbounded. ANS: Answer not provided. PTS: 1
TOP: Special cases
5. Use a graph to illustrate why a change in an objective function coefficient does not necessarily lead to a change in the optimal values of the decision variables, but a change in the right-hand sides of a binding constraint does lead to new values. ANS: Answer not provided. PTS: 1
TOP: Graphical sensitivity analysis
6. Explain the concepts of proportionality, additivity, and divisibility. ANS: Answer not provided. PTS: 1
TOP: Notes and comments
7. Explain the steps necessary to put a linear program in standard form. ANS:
Answer not provided. PTS: 1
TOP: Surplus variables
8. Explain the steps of the graphical solution procedure for a minimization problem. ANS: Answer not provided. PTS: 1
TOP: Graphical solution procedure for minimization problems
PROBLEM 1. Solve the following system of simultaneous equations. 6X + 2Y = 50 2X + 4Y = 20 ANS: X = 8, Y =1 PTS: 1
TOP: Simultaneous equations
2. Solve the following system of simultaneous equations. 6X + 4Y = 40 2X + 3Y = 20 ANS: X = 4, Y = 4 PTS: 1
TOP: Simultaneous equations
3. Consider the following linear programming problem Max
8X + 7Y
s.t.
15X + 5Y 75 10X + 6Y 60 X+ Y8 X, Y 0
a. b. c.
Use a graph to show each constraint and the feasible region. Identify the optimal solution point on your graph. What are the values of X and Y at the optimal solution? What is the optimal value of the objective function?
ANS:
a.
b. c.
The optimal solution occurs at the intersection of constraints 2 and 3. The point is X = 3, Y = 5. The value of the objective function is 59.
PTS: 1
TOP: Graphical solution
4. For the following linear programming problem, determine the optimal solution by the graphical solution method Max
−X + 2Y
s.t.
6X − 2Y 3 −2X + 3Y 6 X+ Y3 X, Y 0
ANS: X = 0.6 and Y = 2.4
PTS: 1
TOP: Graphical solution
5. Use this graph to answer the questions.
Max
20X + 10Y
s.t.
12X + 15Y 180 15X + 10Y 150 3X − 8Y 0 X,Y0
a. b. c. d.
Which area (I, II, III, IV, or V) forms the feasible region? Which point (A, B, C, D, or E) is optimal? Which constraints are binding? Which slack variables are zero?
ANS: a. b. c. d.
Area III is the feasible region Point D is optimal Constraints 2 and 3 are binding S2 and S3 are equal to 0
PTS: 1
TOP: Graphical solution
6. Find the complete optimal solution to this linear programming problem. Min
5X + 6Y
s.t.
3X + Y 15 X + 2Y 12 3X + 2Y 24 X,Y0
ANS:
The complete optimal solution is PTS: 1
X = 6, Y = 3, Z = 48, S1 = 6, S2 = 0, S3 = 0
TOP: Graphical solution
7. Find the complete optimal solution to this linear programming problem. Max
5X + 3Y
s.t.
2X + 3Y 30 2X + 5Y 40 6X − 5Y 0 X,Y 0
ANS:
The complete optimal solution is PTS: 1
X = 15, Y = 0, Z = 75, S1 = 0, S2 = 10, S3 = 90
TOP: Graphical solution
8. Find the complete optimal solution to this linear programming problem. Max
2X + 3Y
s.t.
4X + 9Y 72 10X + 11Y 110 17X + 9Y 153 X,Y0
ANS:
The complete optimal solution is PTS: 1
X = 4.304, Y = 6.087, Z = 26.87, S1 = 0, S2 = 0, S3 = 25.043
TOP: Graphical solution
9. Find the complete optimal solution to this linear programming problem. Min
3X + 3Y
s.t.
12X + 4Y 48 10X + 5Y 50 4X + 8Y 32 X,Y0
ANS:
The complete optimal solution is PTS: 1
X = 4, Y = 2, Z = 18, S1 = 8, S2 = 0, S3 = 0
TOP: Graphical solution
10. For the following linear programming problem, determine the optimal solution by the graphical solution method. Are any of the constraints redundant? If yes, then identify the constraint that is redundant. Max
X + 2Y
s.t.
X+ Y3 X − 2Y 0 Y1 X, Y 0
ANS: X = 2, and Y = 1 Yes, there is a redundant constraint; Y 1
PTS: 1
TOP: Graphical solution
11. Maxwell Manufacturing makes two models of felt tip marking pens. Requirements for each lot of pens are given below.
Plastic Ink Assembly Molding Time
Fliptop Model 3 5 5
Tiptop Model 4 4 2
The profit for either model is $1000 per lot. a. What is the linear programming model for this problem? b. Find the optimal solution. c. Will there be excess capacity in any resource? ANS: a.
Let F = the number of lots of Fliptop pens to produce Let T = the number of lots of Tiptop pens to produce Max
1000F + 1000T
s.t.
3F + 4T 36 5F + 4T 40 5F + 2T 30 F,T0
Available 36 40 30
b.
c.
The complete optimal solution is F = 2, T = 7.5, Z = 9500, S1 = 0, S2 = 0, S3 = 5 There is an excess of 5 units of molding time available.
PTS: 1
TOP: Modeling and graphical solution
12. The Sanders Garden Shop mixes two types of grass seed into a blend. Each type of grass has been rated (per pound) according to its shade tolerance, ability to stand up to traffic, and drought resistance, as shown in the table. Type A seed costs $1 and Type B seed costs $2. If the blend needs to score at least 300 points for shade tolerance, 400 points for traffic resistance, and 750 points for drought resistance, how many pounds of each seed should be in the blend? Which targets will be exceeded? How much will the blend cost?
Shade Tolerance Traffic Resistance Drought Resistance
Type A 1 2 2
Type B 1 1 5
ANS: Let A = the pounds of Type A seed in the blend Let B = the pounds of Type B seed in the blend Min
1A + 2B
s.t.
1A + 1B 300 2A + 1B 400 2A + 5B 750 A, B 0
The optimal solution is at A = 250, B = 50. Constraint 2 has a surplus value of 150. The cost is 350. PTS: 1
TOP: Modeling and graphical solution
13. Muir Manufacturing produces two popular grades of commercial carpeting among its many other products. In the coming production period, Muir needs to decide how many rolls of each grade should be produced in order to maximize profit. Each roll of Grade X carpet uses 50 units of synthetic fiber, requires 25 hours of production time, and needs 20 units of foam backing. Each roll of Grade Y carpet uses 40 units of synthetic fiber, requires 28 hours of production time, and needs 15 units of foam backing. The profit per roll of Grade X carpet is $200 and the profit per roll of Grade Y carpet is $160. In the coming production period, Muir has 3000 units of synthetic fiber available for use. Workers have been scheduled to provide at least 1800 hours of production time (overtime is a possibility). The company has 1500 units of foam backing available for use. Develop and solve a linear programming model for this problem. ANS: Let X = the number of rolls of Grade X carpet to make Let Y = the number of rolls of Grade Y carpet to make Max
200X + 160Y
s.t.
50X + 40Y 3000 25X + 28Y 1800 20X + 15Y 1500 X,Y0
The complete optimal solution is X = 30, Y = 37.5, Z = 12000, S1 = 0, S2 = 0, S3 = 337.5 PTS: 1
TOP: Modeling and graphical solution
14. Does the following linear programming problem exhibit infeasibility, unboundedness, or alternate optimal solutions? Explain. Min
1X + 1Y
s.t.
5X + 3Y 30 3X + 4Y 36 Y7 X,Y0
ANS: The problem is infeasible.
PTS: 1
TOP: Special cases
15. Does the following linear programming problem exhibit infeasibility, unboundedness, or alternate optimal solutions? Explain. Min
3X + 3Y
s.t.
1X + 2Y 16 1X + 1Y 10 5X + 3Y 45 X,Y0
ANS: The problem has alternate optimal solutions.
PTS: 1
TOP: Special cases
16. A businessman is considering opening a small specialized trucking firm. To make the firm profitable, it is estimated that it must have a daily trucking capacity of at least 84,000 cu. ft. Two types of trucks are appropriate for the specialized operation. Their characteristics and costs are summarized in the table below. Note that truck 2 requires 3 drivers for long haul trips. There are 41 potential drivers available and there are facilities for at most 40 trucks. The businessman's objective is to minimize the total cost outlay for trucks.
Truck Small Large
Cost $18,000 $45,000
Capacity (Cu. Ft.) 2,400 6,000
Drivers Needed 1 3
Solve the problem graphically and note there are alternate optimal solutions. Which optimal solution: a. uses only one type of truck? b. utilizes the minimum total number of trucks? c. uses the same number of small and large trucks? ANS:
a. b. c.
35 small, 0 large 5 small, 12 large 10 small, 10 large
PTS: 1
TOP: Alternative optimal solutions
17. Consider the following linear program: MAX
60X + 43Y
s.t.
X + 3Y 9 6X − 2Y = 12 X + 2Y 10 X, Y 0
a. b. c.
Write the problem in standard form. What is the feasible region for the problem? Show that regardless of the values of the actual objective function coefficients, the optimal solution will occur at one of two points. Solve for these points and then determine which one maximizes the current objective function.
ANS: a.
MAX
60X + 43Y
X + 3Y − S1 = 9 6X − 2Y = 12 X + 2Y + S3 = 10 X, Y, S1, S3 0 Line segment of 6X − 2Y = 12 between (22/7,24/7) and (27/10,21/10). Extreme points: (22/7,24/7) and (27/10,21/10). First one is optimal, giving Z = 336. S.T.
b. c.
PTS: 1
TOP: Standard form and extreme points
18. Solve the following linear program graphically. MAX
5X + 7Y
s.t.
X 6 2X + 3Y 19 X+ Y8 X, Y 0
ANS: From the graph below we see that the optimal solution occurs at X = 5, Y = 3, and Z = 46.
PTS: 1
TOP: Graphical solution procedure
19. Given the following linear program: MIN
150X + 210Y
s.t.
3.8X + 1.2Y 22.8 Y6 Y 15 45X + 30Y = 630 X, Y 0
Solve the problem graphically. How many extreme points exist for this problem? ANS: Two extreme points exist (Points A and B below). The optimal solution is X = 10, Y = 6, and Z = 2760 (Point B).
PTS: 1
TOP: Graphical solution procedure
20. Solve the following linear program by the graphical method. MAX
4X + 5Y
s.t.
X + 3Y 22 −X + Y 4 Y6 2X − 5Y 0 X, Y 0
ANS: Two extreme points exist (Points A and B below). The optimal solution is X = 10, Y = 6, and Z = 2760 (Point B).
PTS: 1
TOP: Graphical solution procedure
Chapter 3—Linear Programming: Sensitivity Analysis and Interpretation of Solution MULTIPLE CHOICE 1. To solve a linear programming problem with thousands of variables and constraints a. a personal computer can be used. b. a mainframe computer is required. c. the problem must be partitioned into subparts. d. unique software would need to be developed. ANS: A
PTS: 1
TOP: Computer solution
2. A negative dual price for a constraint in a minimization problem means a. as the right-hand side increases, the objective function value will increase. b. as the right-hand side decreases, the objective function value will increase. c. as the right-hand side increases, the objective function value will decrease. d. as the right-hand side decreases, the objective function value will decrease. ANS: A
PTS: 1
TOP: Dual price
3. If a decision variable is not positive in the optimal solution, its reduced cost is a. what its objective function value would need to be before it could become positive. b. the amount its objective function value would need to improve before it could become positive. c. zero. d. its dual price. ANS: B
PTS: 1
TOP: Reduced cost
4. A constraint with a positive slack value a. will have a positive dual price. b. will have a negative dual price. c. will have a dual price of zero. d. has no restrictions for its dual price. ANS: C
PTS: 1
TOP: Slack and dual price
5. The amount by which an objective function coefficient can change before a different set of values for the decision variables becomes optimal is the a. optimal solution. b. dual solution. c. range of optimality. d. range of feasibility. ANS: C
PTS: 1
TOP: Range of optimality
6. The range of feasibility measures a. the right-hand-side values for which the objective function value will not change. b. the right-hand-side values for which the values of the decision variables will not change. c. the right-hand-side values for which the dual prices will not change. d. each of these choices are true. ANS: C
PTS: 1
TOP: Range of feasibility