Module B

The diet problem is known in agricultural applications as the: -fertilizer problem. -feed-mix problem. -crop-rotation problem. -egg-choice problem. -genetic-transformation problem.

feed-mix problem.

In which of the following has LP been applied successfully? -minimizing distance traveled by school buses carrying children -minimizing 911 response time for police patrols -minimizing labor costs for bank tellers while maintaining service levels -determining the distribution system for multiple warehouses to multiple destinations -All of these

All of these

Binary variables can only take on the values of 1 or 2. True/False

False

Constraints are needed to solve linear programming problems by hand, but not by computer. True/False

False

If we wish to ensure that decision variable values in a linear program are integers rather than fractions, the generally accepted practice is to round the solutions to the nearest integer values. True/False

False

In linear programming, if there are three constraints, each representing a resource that can be used up, the optimal solution must use up all of each of the three resources. True/False

False

The graphical method of solving linear programs can handle only maximization problems. True/False

False

A linear programming problem has three constraints, plus nonnegativity constraints on X and Y. The constraints are: 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? -50 -30 -20 -15 -10

30

The feasible region in the diagram below is consistent with which one of the following X2 is 40, X1 is 20. -8X1 + 4X2 ≥ 160 -4X1 + 8X2 ≤ 160 -8X1 - 4X2 ≤ 160 -8X1 + 4X2 ≤ 160 -4X1 - 8X2 ≤ 160

8X1 + 4X2 ≤ 160

For a linear programming problem with the constraints 2X + 4Y ≤ 100 and 1X + 8Y ≤ 100, two of its corner points are (0, 0) and (0, 25). True/False

False

In terms of linear programming, the fact that the solution is infeasible implies that the "profit" can increase without limit. True/False

False

Linear programming is an appropriate problem-solving technique for decisions that have no alternative courses of action. True/False

False

Solving a linear programming problem with the iso-profit line solution method requires that we move the iso-profit line to each corner of the feasible region until the optimum is identified. True/False

False

The region that satisfies all of the constraints in linear programming is called the region of optimality. True/False

False

If cars sell for $500 profit and trucks sell for $300 profit, which of the following represents the objective function? -Maximize profit = 500C + 300T -Minimize profit = 500C + 300T -Maximize profit = 500C - 300T -Minimize profit = 300T - 500C -Maximize profit = 800(T + C)

Maximize profit = 500C + 300T

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 linear programming constraint. -Q + R + S + T ≤ 4 -Q ≥ R + S + T -Q - R - S - T ≤ 0 -Q / (R + S + T) ≤ 0 -Q ≤ R + Q ≤ S + Q ≤ T

Q - R - S - T ≤ 0

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 linear programming constraint. -S / (T + U) ≥ 4 -S - .25T - .25U ≥ 0 -4S ≤ T + U -S ≥ 4T / 4U -S ≥ .25T + S ≥ .25U

S - .25T - .25U ≥ 0

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? -There are four corner points including (50, 0) and (0, 12.5). -The two corner points are (0, 0) and (50, 12.5). -The graphical origin (0, 0) is not in the feasible region. -The feasible region includes all points that satisfy one constraint, the other, or both. -The feasible region cannot be determin

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

Which of the following correctly describes all iso-profit lines for an LP maximization problem? -They all pass through the origin -They are all parallel. -They all pass through the point of maximum profit. -Each line passes through at least 2 corners. -All of these

They are all parallel.

A common form of the product-mix linear programming problem seeks to find that combination of products and the quantity of each that maximizes profit in the presence of limited resources. True/False

True

Computer software provides a simple way to guarantee only integer solutions to linear programming problems. True/False

True

In linear programming, a statement such as "maximize contribution" becomes an objective function when the problem is formulated. True/False

True

One requirement of a linear programming problem is that the objective function must be expressed as a linear equation or inequality. True/False

True

The optimal solution to a linear programming problem lies within the feasible region. True/False

True

Which of the following combinations of constraints has no feasible region? -X + Y ≥ 15 and X - Y ≤ 10 -X + Y ≥ 5 and X ≥ 10 -X ≥ 10 and Y ≥ 20 -X + Y ≥ 100 and X + Y ≤ 50 -X ≤ -5

X + Y ≥ 100 and X + Y ≤ 50

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? -3X ≥ Y -X ≤ 3Y -X + Y ≥ 3 -X - 3Y ≥ 0 -3X ≤ Y

X - 3Y ≥ 0

Capital Co. is considering 5 different projects. Define Xi as a binary variable that equals 1 if project i is undertaken and 0 otherwise, for i = 1, 2, 3, 4, 5. Which of the following represents the constraint(s) stating that projects 2, 3, and 4 cannot all three be undertaken simultaneously? -X2 + X3 + X4 ≤ 3 -X1 + X2 + X3 + X4 + X5 ≤ 3 -X2 + X3 + X4 ≤ 1 -X2 + X3 + X4 ≤ 2 -X2 + X3 ≤ 1 and X3 + X4 ≤ 1

X2 + X3 + X4 ≤ 2

Data Corp. is considering which of 4 different projects to undertake in order to maximize its net present value (NPV). Define Xi as a binary variable that equals 1 if project i is undertaken and 0 otherwise, for i = 1, 2, 3, 4. The NPV and required capital (in millions) for each project are listed below. -X2 + X3 ≤ 1 -X2 - X3 ≤ 0 -X3 - X2 ≤ 0 -X2 - X3 ≤ 1 -X3 - X2 ≤ 1

X3 - X2 ≤ 0

What is the region that satisfies all of the constraints in linear programming called? -area of optimal solutions -area of feasible solutions -profit maximization space -region of optimality -region of non-negativity

area of feasible solutions

The corner-point solution method requires: -identifying the corner of the feasible region that has the sharpest angle. -moving the iso-profit line to the highest level that still touches some part of the feasible region. -moving the iso-profit line to the lowest level that still touches some part of the feasible region. -finding the coordinates at each corner of the feasible solution space. -None of these

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

Using the iso-profit line solution method to solve a maximization problem requires that we: -find the value of the objective function at the origin. -move the iso-profit line away from the origin until it barely touches some part of the feasible region. -move the iso-cost line to the lowest level that still touches some part of the feasible region. -test the objective function value of every corner point in the feasible region. -None of these

move the iso-profit line away from the origin until it barely touches some part of the feasible region.

In a linear programming formulation, a statement such as "maximize contribution" becomes a(n): -constraint. -slack variable. -objective function. -violation of linearity. -decision variable.

objective function.

Which of the following is not a requirement of a linear programming problem? -an objective function, expressed in linear terms -constraints, expressed as linear equations or inequalities -an objective function to be maximized or minimized -alternative courses of action -one constraint or resource limit for each decision variable

one constraint or resource limit for each decision variable

Which of the following is an algorithm for solving linear programming problems of all sizes? -duplex method -multiplex method -shadow price method -simplex method -

simplex method

A shadow price (or dual value) reflects which of the following in a maximization problem? -the marginal gain in the objective realized by subtracting one unit of a resource -the market price that must be paid to obtain additional resources -the increase in profit that would accompany one added unit of a scarce resource -the reduction in cost that would accompany a one unit decrease in the resource -the profit contribution necessary for that item to be included in the optimal solution

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

The main disadvantage of introducing constraints into a linear program that enforce some or all of the decision variables to be either integer or binary is that: -the programs may take longer to solve. -the solutions will no longer be optimal. -Excel can no longer be used to solve the program. -the constraints are difficult to formulate. -we cannot have "yes-or-no" decisions in the linear program.

the programs may take longer to solve.

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. -x = 2, y = 0 -x = 0, y = 3 -x = 0, y = 0 -x = 1, y = 5 -x = 0, y = 5

x = 0, 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. -x = 2, y = 0 -x = 0, y = 3 -x = 0, y = 0 -x = 1, y = 5 -x = 0, y = 5

x = 0, y = 3

For the following constraints, which point is in the feasible region of this minimization problem? (1) 14x + 6y >= 42 (2) x - y >= 3 -x = -1, y = 1 -x = 0, y = 4 -x = 2, y = 1 -x = 5, y = 1 -x = 2, y = 0

x = 5, y = 1