OM3020 - Chapter 3

For the constraints given below, which point is in the feasible region of this minimization problem?

14x + 6y ≤ 42 x + 3y ≥ 6 A) x = 2; y = 5 B) x = 1; y = 2 C) x = 2; y = 1 D) x = 2; y = 3

Use the constraints given below and determine which of the following points is feasible.

14x + 6y ≤ 42 x - y ≤ 3 A) x = 1; y = 4 B) x = 2; y = 8 C) x = 2; y = 4 D) x = 3; y = 0.5

Which of the following could not be a linear programming problem constraint?

A) A + B ≤ -3 B) A - B ≤ -3 C) A - B ≤ 3 D) -A + B ≤ -3

For a profit maximization problem, if the allowable increase for a coefficient in the objective function is infinite, then profits are unbounded.

False

Most computer linear programming packages readily accept constraints entered in fractional form, such as X1/X3.

False

Sensitivity analysis can be used to determine the effect on the solution for changing several parameters at once.

False

Sensitivity ranges can be computed only for the right-hand sides of constraints.

False

The sensitivity range for a constraint quantity value is the range over which the optimal values of the decision variables do not change.

False

The sensitivity range for an objective function coefficient is the range of values over which the profit does not change.

False

To determine the change in the objective function for any increase in right-hand side, simply multiple the shadow price by the increase in the right-hand side.

False

When the right-hand sides of two constraints are both increased by one unit, the value of the objective function will be adjusted by the sum of the constraints' prices.

False

Given the following linear program that maximizes revenue:

Max Z = 15x + 20y s.t. 8x + 5y ≤ 40 4x + y ≥ 4 What is the maximum revenue at the optimal solution? A) $120 B) $160 C) $185 D) $200

Given the following linear programming problem:

Max Z = 15x + 20y s.t. 8x + 5y ≤ 40 4x + y ≥ 4 What would be the values of x and y that will maximize revenue? A) x = 5; y = 0 B) x = 0; y = 8 C) x = 0; y = 1 D) x = 1; y = 0

What combination of x and y is a feasible solution that minimizes the value of the objective function?

Min Z = 3x + 15y 2x + 4y ≥ 12 5x + 2y ≥ 10 A) x = 0; y = 3 B) x = 0; y = 5 C) x = 5; y = 0 D) x = 6; y = 0

If we change the constraint quantity to a value outside the sensitivity range for that constraint quantity, the shadow price will change.

True

One type of sensitivity analysis is to add a new constraint to the model.

True

Sensitivity analysis allows the modeler to relax the certainty assumption.

True

Sensitivity analysis determines how a change in a parameter affects the optimal solution.

True

The Allowable Increase for Calimari in the sensitivity report was 25 with a final value of 13, a shadow price of $16, and an objective coefficient of 10. Luigi added 5 units of Calimari and increased his profit by $80.

True

The marginal value of any scarce resource is the dollar amount one should be willing to pay for one additional unit of that scarce resource.

True

The sensitivity range for a constraint quantity value is the range over which the shadow price is valid.

True

The sensitivity range for an objective function coefficient is the range of values over which the current optimal solution point (product mix) will remain optimal.

True

The shadow price for a positive decision variable is 0.

True

The terms "shadow price" and "dual price" mean the same thing.

True

Captain Stubing of The Pacific Princess seeks to maximize the return for their scheduled 14 day tour of Europe and has a number of options available to him. He can ply his guests with alcohol, upsell them on fancier restaurant fare or include more expensive excursion options. These alternatives are not without tradeoffs, since different guests prefer different options, depending largely on their age and wherewithal. Among the limitations Captain Stubing must consider is the number of excursions; they must offer at least five alternatives per day for each the ten days they will reach port. In addition, the restaurant choices must exceed 12 major styles of cuisine and the bar themes down in The Grotto should rotate every other day for the 14 days. It's possible to rotate them twice a day, but any more than that and poor Isaac spends more time tearing down and setting up than he does mixing libations. Ideally, there should be at least one different bar theme for every cuisine type. The total budget for excursions, restaurants and bar has been set by the parent company at $150,000. It costs $1,500 to stock supplies for a major cuisine category, it costs $5,000 to include each different excursion, and it costs $900 to set up with a different bar theme. Based on historical data, Captain Stubing believes that each new bar setup will generate $1,500 profit, each new cuisine type will bring in $5,000, and each excursion type will generate $17,000 for the ship.

What should Captain Stubing's objective function be? A) Max Z = 1500 Bar + 5000 Food + 17000 Excursion B) Max Z = Bar + Food + Excursion C) Max Z = 900 Bar + 1500 Food + 5000 Excursion D) Max Z = 5 Excursions + 12 Cuisine + 7 Bar + 150000 Budget What is the appropriate constraint for the budget? A) 1 Bar + 1 Food + 1 Excursion ≤ 150,000 B) 900 Bar + 1500 Food + 5,000 Excursion ≤ 150,000 C) 1,500 Bar + 5,000 Food + 17,000 Excursion ≤ 150,000 D) 7 Bar + 12 Food + 5 Excursion ≤ 150,000 What is the appropriate constraint for the requirement that there should be at least one different bar setup for every different type of food? A) Bar - Food ≤ 0 B) Bar + Food ≤ 0 C) Food - Bar ≤ 0 D) Food + Bar ≥ 0

A plant manager is attempting to determine the production schedule of various products to maximize profit. Assume that a machine hour constraint is binding. If the original amount of machine hours available is 200 minutes., and the range of feasibility is from 130 minutes to 300 minutes, providing two additional machine hours will result in:

a different product mix, different total profit

For a maximization problem, the shadow price measures the ________ in the value of the optimal solution, per unit increase for a given ________.

improvement, resource

The sensitivity range for a(n) ________ coefficient is the range of values over which the current optimal solution point (product mix) will remain optimal.

objective function

Sensitivity analysis is the analysis of the effect of ________ changes on the ________.

parameter, optimal solution

For a maximization problem, assume that a constraint is binding. If the original amount of a resource is 4 lbs. and the range of feasibility (sensitivity range) for this constraint is from 3 lbs. to 6 lbs., increasing the amount of this resource by 1 lb. will result in the:

same product mix, different total profit

________ is the analysis of the effect of parameter changes on the optimal solution.

sensitivity

The sensitivity range for a constraint quantity value is also the range over which the ________ is valid.

shadow price

A shadow price reflects which of the following in a maximization problem?

the marginal gain in the objective that would be realized by adding one unit of a resource

For a linear programming problem, assume that a given resource has not been fully used. We can conclude that the shadow price associated with that constraint:

will have a value of zero.

The reduced cost (shadow price) for a positive decision variable is ________.

zero

For a resource constraint, either its slack value must be ________ or its shadow price must be ________.

zero, zero