Modular B: Linear Programming

A linear programming problem has three constraints: 2X + 10Y ≤ 100 4X + 6Y ≤ 120 6X + 3Y ≤ 90 What is the largest quantity of X that can be made without violating any of these constraints? a. 50 b. 30 c. 20 d. 15 e. 10

d. 15

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

a. (0, 0), (50, 0), (0, 21), and (20, 15)

A linear programming problem has two constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100. 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.

a. There are four corner points including (50, 0) and (0, 12.5).

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

a. x = 2, y = 1

A linear programming problem has three constraints: 2X + 10Y ≤ 100 4X + 6Y ≤ 120 6X + 3Y ≥ 90 What is the largest quantity of X that can be made without violating any of these constraints? a. 50 b. 30 c. 20 d. 15 e. 10

b. 30

A linear programming problem contains a restriction that reads "the quantity of S must be no less than one-fourth as large as T and U combined." Formulate this as a constraint ready for use in problem solving software. a. S / (T + U) ≥ 4 b. S - .25T -.25U ≥ 0 c. 4S ≤ T + U d. S ≥ 4T / 4U e. none of the above

b. S - .25T -.25U ≥ 0

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

b. area of feasible solutions

Which of the following represents valid constraints in linear programming? a. 2X ≥ 7X*Y b. 2X * 7Y ≥ 500 c. 2X + 7Y ≥ 100 d. 2X2 + 7Y ≥ 50 e. All of the above are valid linear programming constraints.

c. 2X + 7Y ≥ 100

A linear programming problem contains a restriction that reads "the quantity of Q must be no larger than the sum of R, S, and T." Formulate this as a constraint ready for use in problem solving software. a. Q + R + S + T ≤ 4 b. Q ≥ R + S + T c. Q - R - S - T ≤ 0 d. Q / (R + S + T) ≤ 0 e. none of the above

c. Q - R - S - T ≤ 0

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. 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

c. a = 0, b = 6

In sensitivity analysis, a zero shadow price (or dual value) for a resource ordinarily means that a. the resource is scarce b. the resource constraint was redundant c. the resource has not been used up d. something is wrong with the problem formulation e. none of the above

c. the resource has not been used up

A maximizing linear programming problem with variables X and Y and constraints C1, C2, and C3 has been solved. The dual values (not the solution quantities) associated with the problem are X = 0, Y = $10, C1 = $2, C2 = $0.50, and C3 = $0. Which statement below is true? a. One more unit of the resource in C1 would reduce the objective function value by $2. b. One more unit of the resource in C2 would add one-half unit each of X and Y. c. The resources in C1 and C2 have not been used up. d. The optimal solution makes only X; the quantity of Y must be zero. e. All of the above are true.

d. The optimal solution makes only X; the quantity of Y must be zero.

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

d. X + Y > 100 and X + Y < 50

The corner point solution method requires a. finding the value of the objective function at the origin 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

d. finding the coordinates at each corner of the feasible solution space

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

d. x = 5, y = 1

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

e. for each decision variable, there must be one constraint or resource limit

A maximizing linear programming problem with variables X and Y and constraints C1, C2, and C3 has been solved. The dual values (not the solution quantities) associated with the problem are X = 0, Y = 0, C1 = $2, C2 = $0.50, and C3 = $0. Which statement below is false? a. One more unit of the resource in C1 would add $2 to the objective function value. b. One more unit of the resource in C2 would add one more unit each of X and Y. c. The resource in C3 has not been used up d. The resources in C1 and in C2, but not in C3, are scarce. e. All of the above are true.

b. One more unit of the resource in C2 would add one more unit each of X and Y.

Using the graphical 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 to the highest level that still 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. apply the method of simultaneous equations to solve for the intersections of constraints e. none of the above

b. move the iso-profit line to the highest level that still touches some part of the feasible region

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. 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

b. x = 0, y = 3

A firm makes two products, Y and Z. Each unit of Y costs $10 and sells for $40. Each unit of Z costs $5 and sells for $25. If the firm's goal were to maximize sales revenue, the appropriate objective function would be a. maximize $40Y = $25Z b. maximize $40Y + $25Z c. maximize $30Y + $20Z d. maximize 0.25Y + 0.20Z e. none of the above

c. maximize $30Y + $20Z

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

c. objective function

A shadow price (or dual value) reflects which of the following in a maximization problem? a. the marginal gain in the objective realized by subtracting one unit of a resource b. the market price that must be paid to obtain additional resources c. the increase in profit that would accompany one added unit of a scarce resource d. the reduction in cost that would accompany a one unit decrease in the resource e. none of the above

c. the increase in profit that would accompany one added unit of a scarce resource

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. 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

c. x = 0, y = 0

A linear programming problem has two constraints 2X + 4Y ≥ 100 and 1X + 8Y ≤ 100. 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.

d. The feasible region is triangular in shape, bounded by (50, 0), (33-1/3, 8-1/3), and (100, 0).

A linear programming problem contains a restriction that reads "the quantity of X must be at least three times as large as the quantity of Y." Which of the following inequalities is the proper formulation of this constraint? a. 3X ≥ Y b. X ≤ 3Y c. X + Y ≥ 3 d. X - 3Y ≥ 0 e. 3X ≤ Y

d. X - 3Y ≥ 0

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.

e. All of the above are true.

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. none of the above e. all of the above

e. all of the above