Mathematics · Economics

Optimization and Mathematical Programming

1,802 Questions

Mathematical programming involves selecting the best element from a set of alternatives based on specific criteria. These concepts are tested in various competitive exams, especially those focusing on decision making and resource allocation. The collection includes problems on linear programming, structural optimization, and computational complexity.

Linear programmingDynamic programmingConvex optimizationInteger programmingStructural optimization methodsMathematical modeling

Optimization and Mathematical Programming Questions

Multiple choice

The cryptarithmetic puzzle is an example for

  1. problem reduction

  2. constraint satisfaction

  3. best-first-search

  4. all of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Cryptarithmetic puzzles require assigning digits to letters such that the arithmetic holds true, which is a classic example of a constraint satisfaction problem where each letter is a variable with constraints on its possible values.

Multiple choice

__________ strategy reasons both in forward and backward directions.

  1. Constraint satisfaction

  2. Hill-climbing search

  3. Generate & test

  4. Means-ends analysis
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Means-ends analysis is a problem-solving strategy that compares the current state to the goal state and identifies the differences, working both forward from the current state and backward from the goal to bridge the gap.

Multiple choice

Consider the linear programme (LP) Max 4x + 6y Subject to 3x + 2y $\leq$ 6 2x + 3y $\leq$ 6 x, y $\leq$ 0 After introducing slack variables s and t, the initial basic feasible solution is represented by the table below (basic variables are s = 6 and t = 6, and the objective function value is 0. ||||||| |---|---|---|---|---|---| | | -4 | -6 | 0 | 0 | 0 | | s | 3 | 2 | 1 | 0 | 6 | | t | 2 | 3 | 0 | 1 | 6 | | | x | y | s | t | RHS |

After some simplex iterations, the following table is obtained ||||||| |---|---|---|---|---|---| | | 0 | 0 | 0 | 2 | 12 | | s | 5/3 | 0 | 1 | -1/3 | 2 | | y | 2/3 | 1 | 0 | 1/3 | 2 | | | x | y | s | t | RHS |

From this, one can conclude that

  1. The LP has unique optimal solution.

  2. The LP has an optimal solution that is not unique.

  3. The LP is infeasible.

  4. The LP is unbounded.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\text{The LP has an optimal solution that is not unique, because zero has appeared in}$ $\text{the non-basic variable (x and y) column, in optimal solution}$

Multiple choice

Consider the linear programme (LP) Max 4x + 6y Subject to 3x + 2y $\leq$ 6 2x + 3y $\leq$ 6 x, y $\leq$ 0 After introducing slack variables s and t, the initial basic feasible solution is represented by the table below (basic variables are s = 6 and t = 6, and the objective function value is 0. ||||||| |---|---|---|---|---|---| | | -4 | -6 | 0 | 0 | 0 | | s | 3 | 2 | 1 | 0 | 6 | | t | 2 | 3 | 0 | 1 | 6 | | | x | y | s | t | RHS |

After some simplex iterations, the following table is obtained ||||||| |---|---|---|---|---|---| | | 0 | 0 | 0 | 2 | 12 | | s | 5/3 | 0 | 1 | -1/3 | 2 | | y | 2/3 | 1 | 0 | 1/3 | 2 | | | x | y | s | t | RHS |

From this, one can conclude thatConsider the linear programme (LP) Max 4x + 6y Subject to 3x + 2y $\leq$ 6 2x + 3y $\leq$ 6 x, y $\leq$ 0 The dual for the LP is
  1. $\text{min 6u+6v}\\ \text{Subject to} \\ 3u+2v\geq4 \\ 2u+3v\geq6 \\ u.v\geq0$
  2. $\text{min 6u+6v}\\ \text{Subject to} \\ 3u+2v\geq4 \\ 2u+3v\leq6 \\ u.v\geq0$
  3. $\text{min 4u+6v}\\ \text{Subject to} \\ 3u+2v\geq4 \\ 2u+3v\geq6 \\ u.v\geq0$
  4. $\text{min 4u+6v}\\ \text{Subject to} \\ 3u+2v\geq4 \\ 2u+3v\geq6 \\ u.v\geq0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Multiple choice

Consider a linear programming problem with two variables and two constraints. The objective function is : Maximize X1 + X2 . The corner points of the feasible region are (0, 0), (0, 2), (2, 0) and (4/3, 4/3). If an additional constraint X1 + X2 $\leq$ 5 is added, the optimal solution is
  1. $\bigg(\frac{5}{3},\frac{5}{3}\bigg)$
  2. $\bigg(\frac{4}{3},\frac{4}{3}\bigg)$
  3. $\bigg(\frac{5}{2},\frac{5}{2}\bigg)$

  4. (5, 0)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Multiple choice

Consider a linear programming problem with two variables and two constraints. The objective function is : Maximize X1 + X2 . The corner points of the feasible region are (0, 0), (0, 2), (2, 0) and (4/3, 4/3). Let Y1 and Y2 be the decision variables of the dual and v1 and v2 be the slack variables of the dual of the given linear programming problem. The optimum dual variables are

  1. Y1 and Y2

  2. Y1 and v1

  3. Y1 and v2

  4. v1 and v2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\text{We know,}\\ \text{The inequality constraints are changed to equality constraints by adding or subtracting a non-negative variable from the left-hand sides of such constraints.}\\ \text{These variable is called slack variables or simply slacks.}\\ \text{They are added if the constraints are $(\leq)$ and subtracted if the constraints are $(\geq).These variables can remain positive throughout the process of solution and their values in the optimal solution given useful information about the problem.$}$ Hence,Optimum dual variables are $v_1$ and $v_2$.

Multiple choice business maths linear programming problems structure of linear programming model linear programming problem operations research

Solving an integer programming problem by rounding off answers obtained by solving it as a linear programming problem (using simplex), we find that

  1. The values of decision variables obtained by rounding off are always very close to the optimal values.

  2. The value of the objective function for a maximization problem will likely be less than that for the simplex solution.

  3. The value of the objective function for a minimization problem will likely be less than that for the simplex solution.

  4. All constraints are satisfied exactly.

  5. None of the above.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Solving an integer programming problem by rounding off answers obtained by solving it as a linear programming problem, we find that the value of the objective function for a maximization problem will likely be less than that for the simplex solution.

Multiple choice business maths linear programming problems structure of linear programming model linear programming problem operations research

If the constraints in linear programming problem are changed

  1. the problem is to be re-evaluated

  2. solution is not defined

  3. the objective function has to be modified

  4. the change in constraints is ignored
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The above question asks for the impact of change in constraints on the Linear programming problem. In this scenario, when there is a change in constraint, the solution will change definitely. Whether the solution exists or not, we can only find once the problem is re-evaluated. 

In an LPP, the objective function is related to the main objective of any problem, either we have to maximize or minimize the function based on the situation whereas the constraints is related to physical restrictions in achieving the defined objective function. In real life problems, there might be situations when the constraints change, but objective function does not changes to accommodate the change in constraints.
Thus, if constraints in linear programming problem is changed, the problem has to be re-evaluated for the same objective function and after solving we can find whether the solution exists or not.

Multiple choice business maths linear programming problems structure of linear programming model linear programming problem operations research

The feasible solution of an LP problem, is ________

  1. must satisfies all of the problem's constraints simultaneously

  2. must be a corner point of the feasible region

  3. need not satisfy all of the constraints, only some of them

  4. must optimize the value of the objective function
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

the feasibe solution of a inear programming probem(LP) is a solution that must satisfy all of the problem's constraints simultaniously