Test bank chap 2 and three
Which of the following is a valid objective function for a linear programming problem? a. Min 8xy b. Min 4x + 3y + (1/2)z c. Min 5x 2 + 6y 2 d. Max (x 1 + x 2 )/x 3
B. Min 4x + 3y + (1/2)z
A linear programming. Problem can be both unbounded an infeasible T/F.
False
Solve the following system of simultaneous equations. 6X + 2Y = 50 2X + 4Y = 20
X = 8, Y =1
42. Use this graph to answer the questions. Max 20X + 10Y s.t. 12X + 15Y ≤ 180 15X + 10Y ≤ 150 3X − 8Y ≤ 0 X, Y ≥ 0 a. Which area (I, II, III, IV, or V) forms the feasible region? b. Which point (A, B, C, D, or E) is optimal? c. Which constraints are binding? d. Which slack variables equal zero?
a. Area III is the feasible region. b. Point D is optimal. c. Constraints 2 and 3 are binding. d. S 2 and S 3 are equal to 0.
And infeasible problem is one in which the objective function can be increased to infinity T/F
False
44. 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
The complete optimal solution is X = 15, Y = 0, Z = 75, S 1 = 0, S 2 = 10, S 3 = 90
For minimization problem, and the solution is considered to be unbounded if the value may be infinitely small tea/F
True
20. The maximization or minimization of a desired quantity is the a. goal of management science. b. decision for decision analysis. c. constraint of operations research. d. objective of linear programming.
Objective of the linear programming
The graphical solution procedure is useful only for linear programs involving a. two decision variables. b. more than two decision variables. c. a single constraint. d. None of these are correct.
A
The constraint 5x 1 − 2x 2 ≤ 0 passes through the point (20, 50). a. True b. False
True
57. Solve the following linear program graphically. Max 4X + 5Y s.t. X + 3Y ≤ 22 −X + Y ≤ 4 Y ≤ 6 2X − 5Y ≤ 0 X, Y ≥ 0
Two extreme points exist (points A and B below). The optimal solution is X = 10, Y = 6, and Z = 2760 (point B).
39. Solve the following system of simultaneous equations. 6X + 4Y = 40 2X + 3Y = 20
X = 4, Y = 4
33. If there is a maximum of 4,000 hours of labor available per month and 300 ping-pong balls (x 1 ) or 125 wiffle balls (x 2 ) can be produced per hour of labor, which of the following constraints reflects this situation? a. 300x 1 + 125x 2 >/= 4,000 b. 300x 1 + 125x 2 </= 4,000 c. 425(x 1 + x 2 ) </= 4,000 d. 300x 1 + 125x 2 = 4,000
b. 300x 1 + 125x 2 </= 4,000
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.
C
The dual price associated with a constraint is the change in the value of the solution per unit decrease in the right-hand side of the constraint.t/f
False
The dual value in the dual price are identical for a minimization problem t/f
False
52. Does the following linear programming problem exhibit infeasibility, unboundedness, or alternative optimal solutions? Explain. Min 1X + 1Y s.t. 5X + 3Y ≤ 30 3X + 4Y ≥ 36 Y ≤ 7 X, Y ≥ 0
The problem is infeasible.
An objective function reflects the relevant cost of labor hours used in production rather than treating them as a sunk cost. The correct interpretation of the dual price associated with the labor hours constraint is the a. maximum premium (say for overtime) over the normal price that the company would be willing to pay. b. upper limit on the total hourly wage the company would pay. c. reduction in hours that could be sustained before the solution would change. d. number of hours by which the right-hand side can change before there is a change in the solution poi
A
27. Which of the following special cases does NOT require reformulation of the problem in order to obtain a solution? a. alternative optimality b. infeasibility c. unboundedness d. Each case requires a reformulation.
Alternate optimality
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.
At the optimal solution, a redundant constraint will have zero slack
Sensitivity analysis information in computer output is based on the assumption that a. no coefficient changes. b. one coefficient changes. c. two coefficients change. d. all coefficients change.
B
When the cost of a resource is sunk, then the dual price can be interpreted as the a. minimum amount the firm should be willing to pay for one additional unit of the resource. b. maximum amount the firm should be willing to pay for one additional unit of the resource. c. minimum amount the firm should be willing to pay for multiple additional units of the resource. d. maximum amount the firm should be willing to pay for multiple additional units of the resource.
B
The amount the objective function coefficient of a decision variable would have to improve before that variable would have a positive value in the solution is the a. dual price. b. surplus variable. c. reduced cost. d. upper limit.
C
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. All of these are correct.
C
Alternative optimal solutions occur when there's no feasible solution to the problem T/F
False
Because surplus variables represent the amount by which the solution exceeds a minimum target, they are given positive coefficient in the objective function T/F
False
For a minimization problem, a positive dual price indicates the value of the objective function will increase t/f
False
I had a problem optimal solution. A redundant constraint will have zero slack T/F.
False
If the range of feasibility for b 1 is between 16 and 37, then if b 1 = 22, the optimal solution will not change from the original optimal solution. a. True b. False
False
Only binding constraints form the shape/boundaries of the feasible region. T/F.
False
The constraint 2x 1 − x 2 = 0 passes through the point (200, 100). a. True b. False
False
37. A variable added to the left-hand side of a less-than-or-equal-to constraint to convert the constraint into an equality is a a. standard variable. b. slack variable. c. surplus variable. d. nonnegative variable.
Slack variable
50. 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. ANSWER: Let X = number of rolls of Grade X carpet to make Y = number of rolls of Grade Y carpet to make Max 200X + 160Y s.t. 50X + 40Y ≤ 3000 25X + 28Y ≥ 1800 20X + 15Y ≤ 1500 X, Y ≥ 0
The complete optimal solution is X = 30, Y = 37.5, Z = 12,000, S 1 = 0, S 2 = 0, S 3 = 337.5
46. 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, Y ≥ 0
The complete optimal solution is X = 4, Y = 2, Z = 18, S 1 = 8, S 2 = 0, S 3 = 0
45. 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, Y ≥ 0
The complete optimal solution is X = 4.304, Y = 6.087, Z = 26.87, S 1 = 0, S 2 = 0, S 3 = 25.043
43. 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, Y ≥ 0
The complete optimal solution is X = 6, Y = 3, Z = 48, S 1 = 6, S 2 = 0, S 3 = 0
51. Does the following linear programming problem exhibit infeasibility, unboundedness, or alternative optimal solutions? Explain. Min 3X + 3Y s.t. 1X + 2Y ≤ 16 1X + 1Y ≤ 10 5X + 3Y ≤ 45 X, Y ≥ 0
The problem has alternative optimal solutions.
A negative dual price indicates that increasing the right-hand side of the associated constraint would be detrimental to the objective. T/f
True
A small change in the objective function coefficient can necessitate modifying the optimal solution t/f
True
An optimal solution to a linear programming. Problem can be found at an extreme point of the feasible region for the problem T/F.
True
And unbounded feasible region might not result in an unbounded solution for a minimization or maximization problem T/F
True
Classical sensitivity analysis provides no information about changes resulting from a change in the coefficient of a variable in a constraint. T/f
True
If a constraint is redundant, it can be removed from the problem without affecting the feasible region. T/F.
True
If the optimal value of a decision variable is zero, and it's reduced cost is zero this indicates that alternative optimal solutions exist T/F
True
In linear programming problem the objective function and the constraints must be linear functions of the decision variables T/F
True
In order to tell the impact of a change in a constraint coefficient, the change must be made and then the model resolved t/f
True
No matter what value it has each objective function line is parallel to every other objection function line no problem T/F
True
The optimal solution to any linear programming. Problem is the same as the optimal solution to the standard form of the problem T/F.
True
The point (3, 2) is feasible for the constraint 2x 1 + 6x 2 ≤ 30. a. True b. False
True
The reduced cost of a variable is the dual value of the corresponding nonnegativity constraint t/f
Trye
When two or more objective function coefficients are changed simultaneously, further analysis is necessary to determine whether the optimal solution will change. T/f
Trye
56. Solve the following linear program graphically. How many extreme points exist for this problem? Min 150X + 210Y s.t. 3.8X + 1.2Y ≥ 22.8 Y ≥ 6 Y ≤ 15 45X + 30Y = 630 X, Y ≥ 0
Two extreme points exist (points A and B below). The optimal solution is X = 10, Y = 6, and Z = 2760 (point B).
41. For the following linear programming problem, determine the optimal solution using the graphical solution method. Max −X + 2Y s.t. 6X − 2Y ≤ 3 −2X + 3Y ≤ 6 X + Y ≤ 3 X, Y ≥ 0
X = 0.6 and Y = 2.4
47. For the following linear programming problem, determine the optimal solution using the graphical solution method. Are any of the constraints redundant? If yes, identify the constraint that is redundant. Max X + 2Y s.t. X + Y ≤ 3 X − 2Y ≥ 0 Y ≤ 1 X, Y ≥ 0
X = 2 and Y = 1 Yes, there is a redundant constraint; Y ≤ 1
40. Consider the following linear programming problem: Max 8X + 7Y s.t. 15X + 5Y ≤ 75 10X + 6Y ≤ 60 X + Y ≤ 8 X, Y ≥ 0 a. Use a graph to show each constraint and the feasible region. b. Identify the optimal solution point on your graph. What are the values of X and Y at the optimal solution? c. What is the optimal value of the objective function?
a. b. The optimal solution occurs at the intersection of constraints 2 and 3. The point is X = 3, Y = 5. c. The value of the objective function is 59.
41. The binding constraints for this problem are the first and second. Min x 1 + 2x 2 s.t. x 1 + x 2 ≥ 300 2x 1 + x 2 ≥ 400 2x 1 + 5x 2 ≤ 750 x 1 , x 2 ≥ 0 a. Keeping c 2 fixed at 2, over what range can c 1 vary before there is a change in the optimal solution point? b. Keeping c 1 fixed at 1, over what range can c 2 vary before there is a change in the optimal solution point? c. If the objective function becomes Min 1.5x 1 + 2x 2 , what will be the optimal values of x 1 , x 2 , and the objective function? d. If the objective function becomes Min 7x 1 + 6x 2 , what constraints will be binding? e. Find the dual price for each constraint in the original problem.
a. 0.8 ≤ c 1 ≤ 2 b. 1 ≤ c 2 ≤ 2.5 c. x 1 = 250, x 2 = 50, z = 475 d. Constraints 1 and 2 will be binding. e. Dual prices are 0.33, 0, and 0.33. (The first and third values are negative.)
40. The optimal solution of this linear programming problem is at the intersection of constraints 1 and 2. Max 2x 1 + x 2 s.t. 4x 1 + 1x 2 ≤ 400 4x 1 + 3x 2 ≤ 600 1x 1 + 2x 2 ≤ 300 x 1 , x 2 ≥ 0 a. Over what range can the coefficient of x 1 vary before the current solution is no longer optimal? b. Over what range can the coefficient of x 2 vary before the current solution is no longer optimal? c. Compute the dual prices for the three constraints.
a. 1.33 ≤ c 1 ≤ 4 b. 0.5 ≤ c 2 ≤ 1.5 c. Dual prices are 0.25, 0.25, and 0.
53. A businessman is considering opening a small specialized trucking firm. To make the firm profitable, it must have a daily trucking capacity of at least 84,000 cubic feet. Two types of trucks are appropriate for the specialized operation. Their characteristics and costs are summarized in the table below. Note that truck two requires three 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. Capacity Drivers Truck Cost (cu. ft.) Needed Small $18,000 2,400 1 Large $45,000 6,000 3 Solve the problem graphically and note that there are alternative optimal solutions. a. Which optimal solution uses only one type of truck? b. Which optimal solution utilizes the minimum total number of trucks? c. Which optimal solution uses the same number of small and large trucks?
a. 35 small, 0 large b. 5 small, 12 large c. 10 small, 10 large
49. 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. Type A Type B Shade tolerance 1 1 Traffic resistance 2 1 Drought resistance 2 5 a. 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? b. Which targets will be exceeded? c. How much will the blend cost?
a. Let A = pounds of Type A seed in the blend B = 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. b. Constraint 2 has a surplus value of 150. c. The cost is 350.
48. Maxwell Manufacturing makes two models of felt-tip marking pens. Requirements for each lot of pens are given below. Fliptop Model Tiptop Model Available Plastic 3 4 36 Ink assembly 5 4 40 Molding time 5 2 30 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?
a. Let F = number of lots of Fliptop pens to produce T = number of lots of Tiptop pens to produce Max 1000F + 1000T s.t. 3F + 4T ≤ 36 5F + 4T ≤ 40 5F + 2T ≤ 30 F, T ≥ 0 b. The complete optimal solution is F = 2, T = 7.5, Z = 9500, S 1 = 0, S 2 = 0, S 3 = 5 c. There is an excess of 5 units of molding time available.
42. Excel Solver tool has been used in the spreadsheet below to solve a linear programming problem with a maximization objective function and all ≤ constraints. Input Section Objective Function Coefficients X Y 4 6 Constraints Avail. #1 3 5 60 #2 3 2 48 #3 1 1 20 Output Section Variables 13.333 4 Profit 53.333 24 77.333333 Constraint Usage Slack #1 60 1.789E-11 #2 48 -2.69E-11 #3 17.333 2.6666667 a. Give the original linear programming problem. b. Give the complete optimal solution.
a. Max 4X + 6Y s.t. 3X + 5Y ≤ 60 3X + 2Y ≤ 48 1X + 1Y ≤ 20 X, Y ≥ 0 b. The complete optimal solution is X = 13.333, Y = 4, Z = 73.333, S 1 = 0, S 2 = 0, S 3 =2.667
54. Consider the following linear program: Max 60X + 43Y s.t. X + 3Y ≥ 9 6X − 2Y = 12 X + 2Y ≤ 10 X, Y ≥ 0 a. Write the problem in standard form. b. What is the feasible region for the problem? c. 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.
a. Max 60X + 43Y s.t. X + 3Y − S 1 = 9 6X − 2Y = 12 X + 2Y + S 3 = 10 X, Y, S 1 , S 3 ≥ 0 b. Line segment of 6X − 2Y = 12 between (22/7, 24/7) and (27/10, 21/10). c. Extreme points: (22/7, 24/7) and (27/10, 21/10). First one is optimal, giving Z = 336.
39. In a linear programming problem, the binding constraints for the optimal solution are: 5X + 3Y ≤ 30 2X + 5Y ≤ 20 a. Fill in the blanks in the following sentence: As long as the slope of the objective function stays between _______ and _______, the current optimal solution point will remain optimal. b. Which of these objective functions will lead to the same optimal solution? (1) 2X + 1Y (2) 7X + 8Y (3) 80X + 60Y (4) 25X + 35Y
a. −5/3 and −2/5 b. objective functions (2), (3), and (4)
25. The amount by which the left side of a less-than-or-equal-to constraint is smaller than the right side a. is known as a surplus. b. is known as slack. c. is optimized for the linear programming problem. d. exists for each variable in a linear programming problem.
is known as slack
35. The three assumptions necessary for a linear programming model to be appropriate include all of the following EXCEPT a. proportionality. b. additivity. c. divisibility. d. normality.
normality
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.
An infeasible solution violates all constraints
The cost that varies depending on the values of the decision variables is a a. reduced cost. b. relevant cost. c. sunk cost. d. dual cost.
B
. 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.
C
30. 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.
standard form
If the dual price for the right-hand side of a ≤ constraint is zero, there is no upper limit on its range of feasibility. a. True b. False
true
If two or more objective function coefficients are changed simultaneously, further analysis is necessary to determine whether the optimal solution will change. t/f
true
36. A redundant constraint results in a. no change in the optimal solution(s). b. an unbounded solution. c. no feasible solution. d. alternative optimal solutions.
No changes in the optimal solutions
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 these are correct.
None of these is correct
In which part(s) of a linear programming formulation would the decision variables be stated? a. objective function and the left-hand side of each constraint b. objective function and the right-hand side of each constraint c. the left-hand side of each constraint only d. the objective function only
Objective function in the left-hand side of each constraint
When the right-hand sides of two constraints are each increased by one unit, the objective function value will be adjusted by the sum of the constraints' dual prices. T/f
False
55. Solve the following linear program graphically. Max 5X + 7Y s.t. X ≤ 6 2X + 3Y ≤ 19 X + Y ≤ 8 X, Y ≥ 0
From the graph below, we see that the optimal solution occurs at X = 5, Y = 3, and Z = 46.
When no solution to the linear programming, problem satisfies all the constraints, including the non-negativity conditions it is considered
Infeasible
Relevant cause should be reflected in the objective function, but sunk cost should not T/F
True
Constraints limit the degree to which the objective in a linear programming problem is satisfied. T/F.
Trye
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.
A
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.
B
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.
A
28. An improvement in the value of the objective function per unit increase in a right-hand side is the a. sensitivity value. b. constraint coefficient. c. slack value. d. None of these are correct.
D
Based on the per-unit increase in the right-hand side of the constraint, the dual price measures the a. increase in the value of the optimal solution. b. decrease in the value of the optimal solution. c. improvement in the value of the optimal solution. d. change in the value of the optimal solution.
D
Sensitivity analysis is sometimes referred to as a. feasibility testing. b. duality analysis. c. alternative analysis. d. postoptimality analysis.
D
It is not possible to have more than one optimal solution to a linear programming problem T/F
False
The amount of a sunk cost will vary, depending on the values of the decision variable T/F
False
29. A constraint that does NOT affect the feasible region of the solution is a a. nonnegativity constraint. b. redundant constraint. c. standard constraint. d. slack constraint.
Redundant constraint
If the range of feasibility indicates that the original amount of a resource, which was 20, can increase by 5, then the amount of the resource can increase to 25. T/f
True
There is a dual price for every decision variable in a model t/f
False
28. 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.
0
Are values that are used to determine how much are how many of something to produce invest, etc.
Decisión variables
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.
alternative optimal solutions
33. Which of the following is NOT a question answered by standard sensitivity analysis information? a. If the right-hand-side value of a constraint changes, will the objective function value change? b. Over what range can a constraint right-hand-side value change without the constraint dual price possibly changing? c. By how much will the objective function value change if the right-hand-side value of a constraint changes beyond the range of feasibility? d. By how much will the objective function value change if a decision variable coefficient in the objective function changes within the range of optimality?
C
A cost that is incurred no matter what values the decision variables assume is a(n) a. reduced cost. b. optimal cost. c. sunk cost. d. dual cost.
C
Sensitivity analysis is concerned with how certain changes affect the a. feasible solution. b. unconstrained solution. c. optimal solution. d. degenerative solution.
C
A redundant constraint cannot be removed from the problem without affecting the feasible region t/F
False
38. The dual price for a </= (less than or = to) constraint will a. always be greater than 0. b. always be less than 0. c. equal 0
always be greater than 0
Increasing the right-hand side of a nonbinding constraint will not cause a change in the optimal solution. a. True b. False
false
