Mathematics · Economics

Optimization and Mathematical Programming

1,582 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

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

A stockist wishes to optimize the number of perishable items he needs to stock in any month in his store. The demand distribution for this perishable item is given in the table:

The stockist pays Rs. 70 for each item and he sells each at Rs. 90. If the stock is left unsold in any month, he can sell the item at Rs. 50 each. There is no penalty for unfulfilled demand. To maximize the expected profit, the optimal stock level is

  1. 5 units

  2. 4 units

  3. 3 units

  4. 2 units
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

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

Linear programming used to optimize mathematical procedure and is

  1. subset of mathematical programming

  2. dimension of mathematical programming

  3. linear mathematical programming

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

Linear programming is an extremely powerful tool for addressing a wide range of applied optimization problems. A short list of application areas is resource allocation, production scheduling, warehousing, layout, transportation scheduling, facility location, flight crew scheduling, portfolio optimization, parameter estimation.
So ,linear programming is used to subset mathematical programming.

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

In linear programming, oil companies used to implement resources available is classified as

  1. implementation modeling

  2. transportation models

  3. oil model

  4. resources modeling
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In linear programming, $\text{transportation model}$ are applied to problems related to the study of efficient transportation routes.

For oil companies, how effectively the available resources are transported to different destinations with minimum cost.

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

Which of the following is a property of all linear programming problems?

  1. alternate courses of action to choose from

  2. minimization of some objective

  3. a computer program

  4. usage of graphs in the solution

  5. usage of linear and nonlinear equations and inequalities
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
According to Robbins, the resources(capital, land, labour, materials, ...) are always limited. Every resource have multiple uses.

The problem before any organisation or manager is to choose the best alternatives which can maximize the profit or minimize the cost of production. Linear programming is the method which is used to select the best possible alternatives from the all alternatives.

According to William M. Fox, "Linear programming is a planning technique that permits some objective function to be maximized or minimized within the framework of given situational restrictions"

Therefore, the linear programming is the process of selecting best courses of action to choose from various alternatives.