1. 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.
2. 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
3. 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.
4. To find the optimal solution to a linear programming problem using the graphical method
a. find the feasible point that is the farthest away from the origin.
b. find the feasible point that is at the highest location.
c. find the feasible point that is closest to the origin.
d. None of the alternatives is correct.
5. 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.
6. Infeasibility means that the number of solutions to the linear programming models that satisfies all constraints is
a. at least 1.
c. an infinite number.
d. at least 2.
7. A constraint that does not affect the feasible region is a
a. non-negativity constraint.
b. redundant constraint.
c. standard constraint.
d. slack constraint.
8. 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.
1. (50 points)
An ad campaign for a new snack chip will be conducted in a limited geographical area and can use TV time, radio time, and newspaper ads. Information about each medium is shown below.
Medium Cost Per Ad # Reached Exposure Quality
TV 500 10000 30
Radio 200 3000 40
Newspaper 400 5000 25
If the number of TV ads cannot exceed the number of radio ads by more than 4, and if the advertising budget is $10000, develop the model that will maximize the number reached and achieve an exposure quality of at least 1000. Find the optimal solution using Management Scientist or Excel Solver
2. (50 points)
Tots Toys makes a plastic tricycle that is composed of three major components: a handlebar-front wheel-pedal assembly, a seat and frame unit, and rear wheels. The company has orders for 12,000 of these trikes.
As indicated in the table below, the company obviously does not have the resources available to manufacture everything needed for the completion of 12000 tricycles, so it has arranged to purchase additional components, as necessary. Develop a linear programming model to tell the company how many of each component should be manufactured and how many should be purchased in order to provide 12000 fully completed tricycles at the minimum cost. Find the optimal solution using Management Scientist or Excel Solver
Component Plastic Time Space Cost to Manufacture Cost to Purchase
Front 3 10 2 8 12
Seat/Frame 4 6 2 6 9
Each rear wheel .5 2 .1 1 3
Available 50000 160000 30000
This posting contains solutions to following questions on Linear programming.
Linear Programming Study Questions
1. A furniture maker produces tables and chairs. Each product must go through a three stage manufacturing process assembly, finishing, and inspection. Each table requires 3 hours of assembly, 2 hours of finishing, and 1 hour of inspection. Each chair requires 2 hours of assembly, 2 hours or finishing, and 1 hour of inspection. The profit per table is $120; the profit per chair is $80. Each week there are 200 hours of assembly time available, 180 hours of finishing time, and 40 hours of inspection time. Linear programming is to be used to develop a production schedule. Define the variables as follows:
T = number of tables produced each week
C = number of chairs produced each week
According to the above Exhibit, which describes a production problem, which of the following would be necessary constraint in the problem?
a. T + C < 40
b. T + C < 200
c. T + C < 180
d. 120T + 80C > 1000
2. A furniture maker produces tables and chairs. Each product must go through a three stage manufacturing process assembly, finishing, and inspection. Each table requires 3 hours of assembly, 2 hours of finishing, and 1 hour of inspection. Each chair requires 2 hours of assembly, 2 hours or finishing, and 1 hour of inspection. The profit per table is $120; the profit per chair is $80. Each week there are 200 hours of assembly time available, 180 hours of finishing time, and 40 hours of inspection time. Linear programming is to be used to develop a production schedule. Define the variables as follows:
T = number of tables produced each week
C = number of chairs produced each week
According to the above Exhibit, which describes a production problem, what is the objective function?
a. Maximize T + C
b. Maximize 120T + 80C
c. Maximize 200T + 200 C
d. Minimize 6T + 5C
3. Considering the following linear programming problem:
Maximize 40 X + 30 X2 + 60X2
Subject to: X1 + X2 + X3 > 90
12X1 + 8X2 + 10X3 < 1500
X1, X2, X3>0
How many slack, surplus, and artificial variables would be necessary if the simplex were used to solve the problem?
a. 3 slack, 3 surplus, and 3 artificial
b. 1 slack, 2 surplus, and 2 artificial
c. 1 slack, 4 surplus, and 4 artificial
d. 1 slack, 1 surplus, and 1 artificial
4. A feasible solution to a linear programming problem
a. Must satisfy all of the problem's constraints simultaneously
b. Need not satisfy all of the constraints, only the non-negativity constraints
c. Must be a corner point of the feasible region
d. Must give the maximum possible profit
5. Production scheduling is amenable to solution by linear programming because
a. The optimal product combination will minimize production risk
b. Linear programming will allow investment losses to be minimized
c. Scheduling requires specific, narrowly defined constraints
d. Objective functions and constraints can be readily developed and are relatively stable over time
6. The Cj - Zj of a simplex tableau gives
a. The number of units of each basic variable that must be removed from the solution if a new variable is entered
b. The gross profit loss given up by adding one unit of a variable into the solution
c. The next profit or loss that will result from introducing one unit of the variable indicated in that column into the solution
d. The maximal value a variable can take on and still have all constraints satisfied
7. Which of the following is NOT a part of every linear programming problem formulation?
a. An objective function
b. A set of constraints
c. Non-negativity constraints
d. A redundant constraint
8. The number -2 in the X2 column and X1 row of a simplex tableau implies that
a. If 1 unit of X2 is added to the solution, X1 will decrease by 2
b. If 1 unit of X1 is added to the solution, X2 will decrease by 2
c. If 1 unit of X2 is added to the solution, X1 will increase by 2
d. If 1 unit of X1 is added to the solution, X2 will increase by 2
9. What is the maximum possible value for the objective function in the linear programming problem?
Maximize 12X + 10Y
Subject to: 4X + 3Y < 480
2X + 3Y < 360
all variables >0
10. Which of the following is NOT true about slack variables in a simplex tableau?
a. They are used to convert ,,T constraint inequalities to equations
b. They represent unused resources
c. They require the addition of an artificial variable
d. They yield no profit
11. Using linear programming to maximize audience exposure in an advertising campaign is an example of the type of linear programming application known as:
a. Media selection
b. Marketing research
c. Portfolio assessment
d. Media budgeting
12. The substitution rate give
a. The Number of units of each basic variable that must be removed from the solution if a new variable is entered
b. The gross profit or loss given up by adding one unit of a variable into the solution
c. The net profit or loss that will result from introducing one unit of the variable indicted in that column into the solution
d. The maximal value a variable can take on and still have all the constraints satisfied
13. Which of the following is NOT a property of all linear programming problems?
a. The presence of restrictions
b. Optimization of some objective
c. The need for a computer program
d. Alternate courses of action to choose from
14. The following does not represent a factor a manager might consider when employing linear programming from a production scheduling:
a. Labor capacity
b. Space limitations
c. Product demand
d. Risk assessment
15. The graphical solution to a linear programming problem
a. Includes the corner point method and the isoprofit line solution method
b. Is useful for four or fewer decision variables
c. Is inappropriate for more than two constraints
d. Is the most difficult approach, but is useful as a learning tool
16. The selection of specific investments from among a wide variety of alternatives is the type of LP problem known as
a. The product mix problem
b. The investment banker problem
c. The Wall Street problem
d. The portfolio selection problem
17. Consider the following general form of a linear programming problem:
Subject to: Amount of resource A used < 100 units
Amount of resource B used < 240 units
Amount of resource B used < 150 units
The shadow price for S1 is 25, for S2 is 0, and for S3 is 40. If the right-hand side of the constraint 1 is changed from 100 to 101, what happens to maximum possible profit?
a. It would increase by 25
b. It would decrease by 25
c. It would increase by 40
d. It would decrease by 40
18. Determining the most efficient allocation of people, machines, equipment, etc., is characteristic of the LP problem type know as
a. Production scheduling
b. Labor planning
19. The corner point solution method
a. Will yield different results from the isoprofit line solution method
b. Requires that the profit all corners of the feasible region be compared
c. Will provide one, and only one, optimum
d. Requires that all corners created by all constraints be compared
20. In a maximization problem, when one or more of the solution variables and the profit can be made infinitely large without violating any constraints, then the linear program has
a. An infeasible solution
b. An unbounded solution
c. A redundant constraint
d. Alternate optimal solutions