Final Multiple Choice
Which of the following vectors or matrices must be examined during the optimality test?
(c_B)(B^-1)(A)-c and (c_B)(B^-1)
The standard form for a linear program includes
1. 'Less than or equal to' constraints 2. 'Nonnegativity' constraints 3. 'Maximize' objective function
When the objective function has cross-product terms, e.g., x1 * x2, which of the four assumptions of linear programming is most likely violated?
Additivity
The simplex method is a procedure requiring what level of mathematics to operate?
Algebra
When revising the final tableau in order to determine the impact of a change in parameters, which of the following portions of the final tableau do you leave as-is in order to execute the assessment?
B^-1 and c_B*B^-1
When evaluating a change to the RHS of a functional constraint given the final tableau of a linear programming problem, which of the following computations is necessary to determine if the current solution is still optimal?
B^-1*b
Sensitivity analysis is designed to account for violations of which assumption(s) of linear programming?
Certainty
Sensitivity analysis is typically used to mitigate concerns about violations of which of the assumptions of linear programming?
Certainty
Which of the four assumptions is seldom satisfied precisely?
Certainty
Which of the following sensitivity analysis operations is the most difficult to assess?
Changes in the coefficient of a basic variable
Requiring decision variables to be integer values violates which of the assumptions of linear programming?
Divisibility
What type of constraints are always in the augmented form of a linear programming model? (excluding non-negativity constraints)
Equality constraints
T/F Every set of n constraint boundary equations chosen from n + m constraints yields a CPF solution.
False
T/F Given a feasible linear programming problem, there are an infinite number of CPF solutions.
False
T/F If a linear program has an infinite number of feasible solutions, then it has an infinite number of alternate optima.
False
T/F The Simplex Method in matrix form does not require a min-ratio test.
False
T/F The amounts shipped from a dummy source represent a surplus.
False
T/F The left-hand side of a benefit constraint represents the minimum acceptable level of the benefit.
False
T/F For a linear programming problem with n decision variables and m functional constraints, two CPF solutions are adjacent to each other if they share n constraint boundaries.
False
T/F The cost of distribution is directly proportional to the number of sources distributing.
False
After considering all constraints for a linear program, the resulting region of permissible values of the decision variables is called the
Feasible region
What is always true about a bounded feasible region with feasible solutions?
If there are multiple optimal solutions, then at least two must be BF solutions.
A mathematical function of the decision variables that provides a measure of performance is called the
Objective Function
When a constraint has a squared term, i.e., x2, on the left-hand side of the inequality, which assumption of linear programming is most likely violated?
Proportionality
Suppose you've found an optimal solution to a linear-programming problem using the simplex method only to discover the original right-hand side values are different. Using the fundamental insight to recompute the correct variable values, how do you know if reoptimization is necessary?
Reoptimization is necessary only if any of the new variable values are negative
Given a simplex tableau for a linear programming problem in standard form, the variable values for the associated dual problem are found where?
Row 0 where (c subscript B B to the power of negative 1 end exponent space A minus c) and (c subscript B B to the power of negative 1 end exponent) are computed
What are potential applications of the fundamental insight?
Sensitivity analysis Revised simplex method Shadow prices
_______ variables are used to convert functional inequality constraints to equivalent equality constraints.
Slack
The column vector b has m elements because ________
There are m functional constraints
T/F If an optimal solution exists for a linear programming problem, it must be a Corner Point Feasible solution.
True
T/F Linear programming models should be placed in standard form prior to converting to augmented form.
True
T/F The Simplex Method in matrix form requires an optimality test?
True
T/F Branching on a variable can decrease the initial LP relaxation objective.
True
T/F For any feasible linear programming problem, there are an infinite number of solutions and only a finite number of Corner Point Feasible solutions.
True
T/F If there is a single optimal solution, that solution always occurs at a corner-point feasible solution.
True
Which of the following is false regarding shortest path and maximum flow LP formulations? a. Decision variables are binary for both problems b. There can be multiple optimal solutions for both problems c. Flow in must equal flow out for non-terminal nodes d. Both have source and destination nodes
a
Which of the following are properties of a basic solution? a. The values of the basic variables are obtained as the simultaneous solution of the system of equations. b. The number of nonbasic variables equals the total number of variables minus the number of functional constraints. c. If the basic variables satisfy all constraints except for the non-negativity constraints, the basic solution is a BF solution. d. Basic variables are set equal to zero. e. The number of basic variables equals the number of equality constraints found in the augmented form of the linear programming problem. f. Each variable is designated as either a nonbasic or a basic variable.
a, c, e, f
Which of the following statements is true regarding a maximization IP? a. The initial LP relaxation provides an upper bound on the optimal IP objective. b. maximization IPs violate the additivity assumption c. We fathom a branch when the current z for a non-integer solution is less than that of the initial relaxed LP z* value d. Branching on a variable can increase the initial relaxed LP z* value
a.
Each time the simplex method performs an iteration to move from the current corner point feasible solution to a better one, it ______ chooses a CPF solution that is _________ to the current one.
always; adjacent
The basis matrix, B, has which of the following characteristics? a. It must be singular b. Its determinant must be non-zero c. Usually is an m X m identity matrix upon initialization of the simplex method. d. It must contain the columns from [A, I] corresponding to the non-basic variables. e. It must be a square matrix f. It must be invertible
b, c, e, f
When solving a linear program, which of the following solution scenarios are possible? a. Infinite optimal corner point feasible (CPF) solutions b. An optimal solution c. Unbounded objective d. Infinite corner point feasible (CPF) solutions e. No feasible solutions g. infinite optimal solutions
b. e. g.
The objective for a ______ problem is to find the best ______ of ingredients into final products to meet certain specifications.
blending; blend
Shadow prices are calculated using which formula?
c_B*B^-1
The fundamental insight assures us that we can compute any number in the final simplex tableau if we know the parameters of the model in the initial tableau and which of the following values?
c_B*B^-1 and B^-1
Sensitivity analysis can be used to analyze changes in what parameters?
coefficients within functional constraint objective function coefficients right-hand side values
Introducing a new variable in the primal problem is equivalent to introducing a new _________ in the dual problem.
constraint
Variables in the dual problem map to ___________ in the primal problem.
constraints
If a Basic Feasible Solution (BFS) has at least one Basic Variable with a value of zero, the solution is __________
degenerate
The purpose of the minimum ratio test is to ______.
determine which basic variable will become a nonbasic variable
Fixed-requirement constraints are typically represented mathematically with ______ constraints.
equality
The objective of the minimum ratio test is to
find the leaving basic variable by increasing the entering basic variable from 0
Suppose you are provided the optimal solution to a standard-form maximization problem. Which of the following parameter changes cannot increase the optimal objective function value?
increasing the RHS value of a nonbinding constraint
Given a suboptimal basic feasible solution to the primal problem of a linear programming problem, the complementary solution to the dual problem must be ______________
infeasible
Changing the coefficients of a nonbasic variable always effects the problem in the following ways:
it depends...the modified problem may or may not be feasible and/or optimal
The simplex method chooses an edge to move along by determining which edge has the _________ rate of improvement.
largest
The book introduces new notation on page 183 to distinguish between information after any iteration of the simplex method and information after the _________ iteration of the simplex method.
last
Consider a linear programming problem with n decision variables and m constraints. How many basic variables will a basic solution have?
m
What type of solution will result from an LP with an objective function that is parallel to one of the constraints?
multiple solutions
If a corner point feasible solution has no adjacent corner point feasible solutions that are better, then it ______ be an optimal solution.
must
For any LP problem with n decision variables, how many constraint boundaries must two adjacent corner feasible solutions share?
n-1
The generic equation below represents what type of linear program construct? Amount shipped out - amount shipped in = required amount
net flow constraints
Reduced costs, i.e., the values c subscript B B to the power of negative 1 end exponent space A minus c, tell us the amount an objective function coefficient for a _________ must change prior to that variable's activity becoming profitable to start.
non-basic variable
Right-hand-side values in the dual problem are ___________ in the primal problem.
objective function coefficients
The "SOB" method stands for the ________-_______-_______ method.
sensible-odd-bizarre
If the RHS of a functional constraint changes within its allowable range, which of the following values can be used to compute the impact to the optimal objective function value?
shadow prices
If the sum of the percentage changes of multiple RHS values does not exceed 100 percent, then the __________ are valid for computing the overall impact to the objective function value.
shadow prices
When every coefficient of the entering Basic Variable is either negative or zero, this means _____________
that Z is unbounded
When selecting the entering Basic Variable from a simplex tableau, I should select the variable whose coefficient in the objective function row, i.e., row (0), is _______________
the most negative
Which aspect of a linear programming problem is least likely to be estimated?
the number of decision variables
When the final tableau of the simplex method has a Nonbasic Variable with a zero coefficient in row (0), that means __________
there are multiple optimal solutions
Constraints in the primal problem map to ___________ in the dual problem.
variables
Suppose x and y are complementary solutions to the primal and dual problems, respectively. If x is a non-optimal, BF solution and the primal problem is bounded, then
y is feasible and the dual problem is bounded