Question 1

A linear programming problem has two constraints 2X + 4Y ≥ 100 and 1X + 8Y ≤ 100, plus nonnegativity constraints on X and Y. Which of the following statements about its feasible region is true? A) There are four corner points including (50, 0) and (0, 12.5). B) The two corner points are (0, 0) and (50, 12.5). C) The graphical origin (0, 0) is in the feasible region. D) The feasible region is triangular in shape, bounded by (50, 0), (33-1/3, 8-1/3), and (100, 0). E) The feasible region cannot be determined without knowing whether the problem is to be minimized or maximized.

Accepted Answer

To find the feasible region, we graph the two constraints on the same coordinate axes. The shaded area where both inequalities are satisfied is the feasible region.

2X + 4Y ≥ 100 can be simplified to X + 2Y ≥ 50, or Y ≥ -0.5X + 25.
1X + 8Y ≤ 100 can be simplified to Y ≤ -0.125X + 12.5.

Graphing these two inequalities, we get a triangular feasible region bounded by (50, 0), (33-1/3, 8-1/3), and (100, 0). Therefore, statement D is true.

Statement A is false since there are only three corner points in this feasible region. Statement B is false since (0, 0) is not a corner point. Statement C is false since the origin is not in the feasible region. Statement E is false since the shape of the feasible region is independent of whether we are minimizing or maximizing.

Question 2

A linear programming problem has two constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100, plus nonnegativity constraints on X and Y. Which of the following statements about its feasible region is true? A) There are four corner points including (50, 0) and (0, 12.5). B) The two corner points are (0, 0) and (50, 12.5). C) The graphical origin (0, 0) is not in the feasible region. D) The feasible region includes all points that satisfy one constraint, the other, or both. E) The feasible region cannot be determined without knowing whether the problem is to be minimized or maximized.

Accepted Answer

The feasible region of a linear programming problem is defined by the intersection of all constraints, including nonnegativity constraints. The given constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100, along with nonnegativity constraints on X and Y, define a polygonal feasible region on the XY-plane. The origin (0,0) always satisfies the given constraints and nonnegativity, making it a part of the feasible region. The points (50, 0) and (0, 12.5) are found by setting Y and X to zero in the respective constraints, showing they are corner points of the feasible region. The feasible region is determined by the intersection of the constraints, not by the objective function (whether to minimize or maximize), making E incorrect. The statement that the feasible region includes all points that satisfy one constraint, the other, or both (D) is misleading because points must satisfy all constraints simultaneously to be in the feasible region. Therefore, the correct statement is that there are four corner points including (50, 0) and (0, 12.5), which is a characteristic of the feasible region defined by the given constraints and nonnegativity conditions.

Question 3

The region which satisfies all of the constraints in graphical linear programming is called the
A) area of optimal solutions
B) area of feasible solutions
C) profit maximization space
D) region of optimality
E) region of non-negativity

Accepted Answer

The region which satisfies all of the constraints in graphical linear programming is called the area of feasible solutions. This region includes all the points that satisfy the constraints and are therefore feasible for the problem. The optimal solution(s) will lie within this region, and may be found by further analysis or optimization techniques. Therefore, option B is the best choice.

Question 4

For the two constraints given below, which point is in the feasible region of this minimization problem? (1) 14x + 6y > 42 $\quad $$\quad $(2) x - y > 3
A) x = -1, y = 1
B) x = 0, y = 4
C) x = 2, y = 1
D) x = 5, y = 1
E) x = 2, y = 0

Accepted Answer

For x = 5, y = 1: (1) 14(5) + 6(1) = 70 + 6 = 76 > 42, (2) 5 - 1 = 4 > 3. Both constraints are satisfied.

Question 5

A linear programming problem has two constraints 2X + 4Y = 100 and 1X + 8Y ≤ 100, plus non-negativity constraints on X and Y. Which of the following statements about its feasible region is true? A) The points (100, 0) and (0, 25) both lie outside the feasible region. B) The two corner points are (33-1/3, 8-1/3) and (50, 0). C) The graphical origin (0, 0) is not in the feasible region. D) The feasible region is a straight line segment, not an area. E) All of the above are true.

Accepted Answer

The feasible region is the shaded area below the line 1X + 8Y ≤ 100 and to the right of the line 2X + 4Y = 100. The points (100, 0) and (0, 25) both lie outside the feasible region. The two corner points are (33-1/3, 8-1/3) and (50, 0). The graphical origin (0, 0) is not in the feasible region. The feasible region is an area, not a straight line segment.

Question 6

What combination of a and b will yield the optimum for this problem?
Maximize $6a + $15b, subject to (1) 4a + 2b < 12 and (2) 5a + 2b < 20 and (3) a, b ≥ 0.
A) a = 0, b = 0
B) a = 3, b = 3
C) a = 0, b = 6
D) a = 6, b = 0
E) cannot solve without values for a and b

Accepted Answer

The answer of What combination of a and b will...

Question 7

Using the iso-profit line solution method to solve a maximization problem requires that we
A) find the value of the objective function at the origin
B) move the iso-profit line away from the origin until it barely touches some part of the feasible region
C) move the iso-cost line to the lowest level that still touches some part of the feasible region
D) test the objective function value of every corner point in the feasible region
E) none of the above

Accepted Answer

The iso-profit line solution method involves drawing iso-profit lines (lines containing points with equal profit) and moving them away from the origin until they just touch the feasible region. The last line to touch the feasible region will be tangent to a corner point of the feasible region, which is an optimal solution. Thus, choice B is the correct answer. None of the other choices are part of this method.

Question 8

For the constraints given below, which point is in the feasible region of this maximization problem?
(1) 14x + 6y < 42$\quad $$\quad $ (2) x - y < 3 $\quad $$\quad $(3) x, y ≥ 0
A) x = 2, y = 1
B) x = 1, y = 5
C) x = -1, y = 1
D) x = 4, y = 4
E) x = 2, y = 8

Accepted Answer

For option A, substituting x = 2 and y = 1 into the constraints: (1) 14(2) + 6(1) = 28 + 6 = 34 < 42, (2) 2 - 1 = 1 < 3, and (3) both x and y are non-negative, hence it satisfies all the given constraints.

Question 9

What combination of x and y will yield the optimum for this problem?
Maximize $3x + $15y, subject to (1) 2x + 4y < 12 and (2) 5x + 2y < 10 and (3) x, y ≥ 0.
A) x = 2, y = 0
B) x = 0, y = 3
C) x = 0, y = 0
D) x = 1, y = 5
E) none of the above

Accepted Answer

To find the optimum, we can graph the constraints and identify the feasible region. The objective function is maximized at a vertex of the feasible region. Testing the vertices (which are the intersections of the constraints), we find that the option B (x = 0, y = 3) satisfies all constraints and maximizes the objective function $3x + $15y, yielding the maximum value of $45.

Question 10

The feasible region in the diagram below is consistent with which one of the following constraints?

A) 8X1 + 4X2 ≥ 160
B) 4X1 + 8X2 ≤ 160
C) 8X1 - 4X2 ≤ 160
D) 8X1 + 4X2 ≤ 160
E) 4X1 - 8X2 ≤ 160

Accepted Answer

We can see from the graph that the feasible region is below the line 8X1 + 4X2 = 160, which means that any point in that region would satisfy the inequality 8X1 + 4X2 ≤ 160. Therefore, the correct choice is D.

Question 11

Which of the following sets of constraints results in an unbounded maximizing problem? A) X + Y > 100 and X + Y < 50 B) X + Y > 15 and X - Y < 10 C) X + Y < 10 and X > 5 D) X < 10 and Y < 20 E) All of the above have a bounded maximum.

Accepted Answer

The constraints X + Y > 15 and X - Y < 10 do not limit the values of X and Y in a way that bounds the feasible region, allowing for infinitely large values of X and Y.

Question 12

If cars sell for $500 profit and trucks sell for $300 profit which of the following represents the objective function?
A) Maximize = 500C+300T
B) Minimize = 500C + 300T
C) Maximize = 500C - 300T
D) Minimize = 300T- 500C
E) None of the above

Accepted Answer

The objective is to maximize the profit, which is given by the formula Profit = Revenue - Cost. Here, the revenue will be the selling price of the cars/trucks, and the cost will be the cost price of the cars/trucks plus any additional expenses.

Let's assume that the cost price of a car is C and the cost price of a truck is T. Then, the revenue from selling a car will be C + 500 (profit). Similarly, the revenue from selling a truck will be T + 300 (profit).

Hence, the objective function will be:

Maximize = (C + 500) + (T + 300)
Maximize = 500C + 300T

Therefore, the correct choice is A.

Question 13

What combination of x and y will yield the optimum for this problem?
Minimize $3x + $15y, subject to (1) 2x + 4y < 12 and (2) 5x + 2y < 10 and (3) x, y ≥ 0.
A) x = 2, y = 0
B) x = 0, y = 3
C) x = 0, y = 0
D) x = 1, y = 5
E) none of the above

Accepted Answer

The answer of What combination of x and y will...

Question 14

The feasible region in the diagram below is consistent with which one of the following constraints?

A) 8X1 + 4X2 ≤ 160
B) 8X1 + 4X2 ≥ 160
C) 4X1 + 8X2 ≤ 160
D) 8X1 - 4X2 ≤ 160
E) 4X1 - 8X2 ≤ 160

Accepted Answer

The answer of The feasible region in the diagram below...

Question 15

Which of the following combinations of constraints has no feasible region? A) X + Y > 15 and X - Y < 10 B) X + Y > 5 and X > 10 C) X > 10 and Y > 20 D) X + Y > 100 and X + Y < 50 E) All of the above have a feasible region.

Accepted Answer

Constraint D implies that the sum of X and Y is both greater than 100 and less than 50, which is impossible. Therefore, there is no feasible region for this set of constraints.

Question 16

The corner-point solution method requires
A) identifying the corner of the feasible region that has the sharpest angle
B) moving the iso-profit line to the highest level that still touches some part of the feasible region
C) moving the iso-profit line to the lowest level that still touches some part of the feasible region
D) finding the coordinates at each corner of the feasible solution space
E) none of the above

Accepted Answer

The corner-point solution method involves finding the coordinates of each corner of the feasible solution space, and then evaluating the objective function at each corner to determine the optimal solution. It does not involve identifying the sharpest angle or moving the iso-profit line.

Question 17

An iso-profit line
A) can be used to help solve a profit maximizing linear programming problem
B) is parallel to all other iso-profit lines in the same problem
C) is a line with the same profit at all points
D) all of the above
E) none of the above

Accepted Answer

An iso-profit line represents all combinations of two products that yield the same profit, making it useful in solving profit maximizing linear programming problems. All iso-profit lines in the same problem are parallel because they represent different levels of profit but have the same slope, which is determined by the ratio of the profits per unit of the products. Since each point on an iso-profit line represents the same total profit, all statements A, B, and C are correct, making D the correct choice.

Question 18

A maximizing linear programming problem has two constraints: 2X + 4Y < 100 and 3X + 10Y < 210, in addition to constraints stating that both X and Y must be nonnegative. The corner points of the feasible region of this problem are
A) (0, 0), (50, 0), (0, 21), and (20, 15)
B) (0, 0), (70, 0), (25, 0), and (15, 20)
C) (20, 15)
D) (0, 0), (0, 100), and (210, 0)
E) none of the above

Accepted Answer

The corner points of the feasible region are determined by the intersection points of the constraints. The constraints 2X + 4Y < 100 and 3X + 10Y < 210, along with X ≥ 0 and Y ≥ 0, form a bounded region. The intersections of these lines with each other and with the axes give the corner points: (0, 0) where the axes intersect, (50, 0) and (0, 21) where each constraint intersects an axis, and (20, 15) where the two constraints intersect each other.

Question 19

In linear programming, a statement such as "maximize contribution" becomes a(n)
A) constraint
B) slack variable
C) objective function
D) violation of linearity
E) decision variable

Accepted Answer

In linear programming, the objective function is the function that needs to be optimized or maximized (in case of a maximization problem) or minimized (in case of a minimization problem). Constraints are the limitations that the problem faces, slack variables are introduced to effectively deal with constraints, decision variables are the variables whose values are to be determined through optimization, and violation of linearity refers to a situation where a linear programming problem violates the linearity condition. Hence, in the given context, the statement "maximize contribution" would be the objective function.

Question 20

Which of the following is not a requirement of a linear programming problem?
A) an objective function, expressed in terms of linear equations
B) constraint equations, expressed as linear equations
C) an objective function, to be maximized or minimized
D) alternative courses of action
E) for each decision variable, there must be one constraint or resource limit

Accepted Answer

Linear programming problems require an objective function (which can be either maximized or minimized), constraint equations (which must be linear), and alternative courses of action. However, there is no requirement that each decision variable must have its own unique constraint or resource limit. Constraints typically apply to combinations of variables, not necessarily on a one-to-one basis with variables.

Quiz 22: Linear Programming

A linear programming problem has two constraints 2X + 4Y ≥ 100 and 1X + 8Y ≤ 100, plus nonnegativity constraints on X and Y. Which of the following statements about its feasible region is true?

A linear programming problem has two constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100, plus nonnegativity constraints on X and Y. Which of the following statements about its feasible region is true?

The region which satisfies all of the constraints in graphical linear programming is called the

For the two constraints given below, which point is in the feasible region of this minimization problem? (1) 14x + 6y > 42 $\quad$ $\quad$ (2) x - y > 3

A linear programming problem has two constraints 2X + 4Y = 100 and 1X + 8Y ≤ 100, plus non-negativity constraints on X and Y. Which of the following statements about its feasible region is true?

What combination of a and b will yield the optimum for this problem? Maximize $6a + $15b, subject to (1) 4a + 2b < 12 and (2) 5a + 2b < 20 and (3) a, b ≥ 0.

Using the iso-profit line solution method to solve a maximization problem requires that we

For the constraints given below, which point is in the feasible region of this maximization problem? (1) 14x + 6y < 42 $\quad$ $\quad$ (2) x - y < 3 $\quad$ $\quad$ (3) x, y ≥ 0

What combination of x and y will yield the optimum for this problem? Maximize $3x + $15y, subject to (1) 2x + 4y < 12 and (2) 5x + 2y < 10 and (3) x, y ≥ 0.

The feasible region in the diagram below is consistent with which one of the following constraints?

Which of the following sets of constraints results in an unbounded maximizing problem?

If cars sell for $500 profit and trucks sell for $300 profit which of the following represents the objective function?

What combination of x and y will yield the optimum for this problem? Minimize $3x + $15y, subject to (1) 2x + 4y < 12 and (2) 5x + 2y < 10 and (3) x, y ≥ 0.

The feasible region in the diagram below is consistent with which one of the following constraints?

Which of the following combinations of constraints has no feasible region?

The corner-point solution method requires

An iso-profit line

A maximizing linear programming problem has two constraints: 2X + 4Y < 100 and 3X + 10Y < 210, in addition to constraints stating that both X and Y must be nonnegative. The corner points of the feasible region of this problem are

In linear programming, a statement such as "maximize contribution" becomes a(n)

Which of the following is not a requirement of a linear programming problem?

Quiz 22: Linear Programming

A linear programming problem has two constraints 2X + 4Y ≥ 100 and 1X + 8Y ≤ 100, plus nonnegativity constraints on X and Y. Which of the following statements about its feasible region is true?

A linear programming problem has two constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100, plus nonnegativity constraints on X and Y. Which of the following statements about its feasible region is true?

The region which satisfies all of the constraints in graphical linear programming is called the

For the two constraints given below, which point is in the feasible region of this minimization problem? (1) 14x + 6y > 42 \quad \quad (2) x - y > 3

A linear programming problem has two constraints 2X + 4Y = 100 and 1X + 8Y ≤ 100, plus non-negativity constraints on X and Y. Which of the following statements about its feasible region is true?

What combination of a and b will yield the optimum for this problem? Maximize $6a + $15b, subject to (1) 4a + 2b < 12 and (2) 5a + 2b < 20 and (3) a, b ≥ 0.

Using the iso-profit line solution method to solve a maximization problem requires that we

For the constraints given below, which point is in the feasible region of this maximization problem? (1) 14x + 6y < 42 \quad \quad (2) x - y < 3 \quad \quad (3) x, y ≥ 0

What combination of x and y will yield the optimum for this problem? Maximize $3x + $15y, subject to (1) 2x + 4y < 12 and (2) 5x + 2y < 10 and (3) x, y ≥ 0.

The feasible region in the diagram below is consistent with which one of the following constraints?

Which of the following sets of constraints results in an unbounded maximizing problem?

If cars sell for $500 profit and trucks sell for $300 profit which of the following represents the objective function?

What combination of x and y will yield the optimum for this problem? Minimize $3x + $15y, subject to (1) 2x + 4y < 12 and (2) 5x + 2y < 10 and (3) x, y ≥ 0.

The feasible region in the diagram below is consistent with which one of the following constraints?

Which of the following combinations of constraints has no feasible region?

The corner-point solution method requires

An iso-profit line

A maximizing linear programming problem has two constraints: 2X + 4Y < 100 and 3X + 10Y < 210, in addition to constraints stating that both X and Y must be nonnegative. The corner points of the feasible region of this problem are

In linear programming, a statement such as "maximize contribution" becomes a(n)

Which of the following is not a requirement of a linear programming problem?

For the two constraints given below, which point is in the feasible region of this minimization problem? (1) 14x + 6y > 42 $\quad$ $\quad$ (2) x - y > 3

For the constraints given below, which point is in the feasible region of this maximization problem? (1) 14x + 6y < 42 $\quad$ $\quad$ (2) x - y < 3 $\quad$ $\quad$ (3) x, y ≥ 0