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

In transportation models designed in linear programming, points of demand is classified as

  1. ordination

  2. transportation

  3. destinations

  4. origins
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In linear programming, transportation modeltransportation model are applied to problems related to the study of efficient transportation routes. i.e., how effectively the available resources are transported to different destinations with minimum cost.

Therefore, the points of demand is classified as destinations.

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

Consider the following linear programming problem:

Maximize $12X + 10Y$
Subject to: $4X + 3Y ≤ 480$
  $2X + 3Y ≤ 360$
  all variables $ ≥0$

Which of the following points $(X,Y)$ could be a feasible corner point?

  1. $(40,48)$
  2. $(120,0)$
  3. $(180,120)$
  4. $(30,36)$

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

Given constraints, $4x+3y\leq 480$ and $2x+3y\leq 360$


first, draw the graph for equations $4x+3y= 480$ and $2x+3y= 360$

for $4x+3y= 480$
substitute y=0 we get, $4x=480 \implies x=120$
substitute x=0 we get, $3y=480 \implies y=160$
therefore, $4x+3y= 480$ line passes through (120,0) and (0,160) as shown in fig.
Hence, $4x+3y\leq 480$ includes the region below the line.


for $2x+3y= 360$
substitute y=0 we get, $2x=360 \implies x=180$
substitute x=0 we get, $3y=360 \implies y=120$
therefore, $4x+3y= 480$ line passes through $(180,0)$ and $(0,120)$ as shown in fig.
Hence, $2x+3y\leq 360$ includes the region below the line.

the shaded region as shown in figure is intersection region. From the figure, $(0,120)$ and $(120,0)$ are feasible corner points.

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

Consider the following linear programming problem:

Maximize $12X + 10Y$
Subject to: $4X + 3Y ≤ 480$
  $2X + 3Y ≤ 360$
all variables $ ≥0$

Which of the following points $(X,Y)$ is feasible?

  1. $(10,120)$
  2. $(120,10)$
  3. $(30,100)$
  4. $(60,90)$

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

Consider point $(10,120)$ , It doesnt satisfy the inequality $2X+3Y \le 360$

Consider point $(120,10)$ , it doesnt satisfy the inequality $4X+3Y \le 480$
Consider point $(30,100)$ , it satisfies all the inequalities . So this point is feasible
Consider point $(60,90)$ , it doesnt satisfies the inequalities $2X+3Y \le 360$ and $4X+3Y \le 480$
Therefore the correct option is $C$

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

Unboundedness is usually a sign that the LP problem.

  1. has finite multiple solutions.

  2. is degenerate.

  3. contains too many redundant constraints.

  4. has been formulated improperly.

  5. none of the above.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A linear programming problem is said to have unbounded solution if it has infinite number of solutions. I.e., the problem has been formulated improperly

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

The first step in formulating an LP problem is

  1. graph the problem.

  2. perform a sensitivity analysis.

  3. identify the objective and the constraints.

  4. define the decision variables.

  5. understand the managerial problem being faced.
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

The first step in formulating an linear programming problem is to understand the managerial problem being faced i.e., determine the quantities that are needed to solve the problem.

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

Consider the following linear programming problem:

Maximize $5X + 6Y$
Subject to: $4X + 2Y ≤ 420$
  $1X + 2Y ≤ 120$
  all variables  $≥0$

Which of the following points $(X,Y)$ is in the feasible region?

  1. $(30,60)$
  2. $(105,0)$
  3. $(0,210)$
  4. $(100,10)$

  5. None of the above
Reveal answer Fill a bubble to check yourself
B,D Correct answer
Explanation
Feasible points are the points that satisfy the constraints.
Therefore, substitute the options in the constraint equations and verify. 

$\mathrm A.$ substituting  $(30,60)$ in $4x+2y\leq 420$
we get $4\times 30+2\times 60\leq 420$
$\implies 120+120 \leq 420 \implies 240\leq 420$ True
substituting  $(30,60)$ in $1x+2y\leq 120$
we get $1\times 30+2\times 60\leq 120$
$\implies 30+120 \leq 120 \implies 150\leq 120$ False

$\mathrm B.$ substituting  $(105,0)$ in $4x+2y\leq 420$
we get $4\times 105+2\times 0\leq 420$
$\implies 420+0 \leq 420 \implies 420\leq 420$ True
substituting  $(105,0)$ in $1x+2y\leq 120$
we get $1\times 105+2\times 0\leq 120$
$\implies 105+0 \leq 120 \implies 105\leq 120$ True

$\mathrm C.$ substituting  $(0,210)$ in $4x+2y\leq 420$
we get $4\times 0+2\times 210\leq 420$
$\implies 0+420 \leq 420 \implies 420\leq 420$ True
substituting  $(0,210)$ in $1x+2y\leq 120$
we get $1\times 0+2\times 210\leq 120$
$\implies 0+240 \leq 120 \implies 240\leq 120$ False


$\mathrm D.$ substituting  $(100,10)$ in $4x+2y\leq 420$
we get $4\times 100+2\times 10\leq 420$
$\implies 400+20 \leq 420 \implies 420\leq 420$ True
substituting  $(100,10)$ in $1x+2y\leq 120$
we get $1\times 100+2\times 10\leq 120$
$\implies 100+20 \leq 120 \implies 120\leq 120$ True

Therefore option B and D are the points in the feasible region.

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

In order for a linear programming problem to have a unique solution, the solution must exist

  1. at the intersection of the nonnegativity constraints.

  2. at the intersection of a nonnegativity constraint and a resource constraint.

  3. at the intersection of the objective function and a constraint.

  4. at the intersection of two or more constraints.

  5. none of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In order for a linear programming problem to have a unique solution, the solution must exist at the intersection of two or more constraints. Then the problem becomes convex and has a single optimum(maximum or minimum) solution. 

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

Consider the following linear programming problem:

Maximize $5X + 6Y$
Subject to: $4X + 2Y ≤ 420$
  $1X + 2Y ≤ 120$
  all variables $≥ 0$

Which of the following points $(X,Y)$ is feasible?

  1. $(50,40)$
  2. $(30,50)$
  3. $(60,30)$
  4. $(90,20)$

  5. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Feasible points are the points that satisfy the constraints.
Therefore, substitute the options in the constraint equations and verify. 

$\mathrm A.$ substituting  $(50,40)$ in $4x+2y\leq 420$
we get $4\times 50+2\times 40\leq 420$
$\implies 200+80 \leq 420 \implies 280\leq 420$ True
substituting  $(50,40)$ in $1x+2y\leq 120$
we get $1\times 50+2\times 40\leq 120$
$\implies 50+80 \leq 120 \implies 130\leq 120$ False

$\mathrm B.$ substituting  $(30,50)$ in $4x+2y\leq 420$
we get $4\times 30+2\times 50\leq 420$
$\implies 120+100 \leq 420 \implies 220\leq 420$ True
substituting  $(30,50)$ in $1x+2y\leq 120$
we get $1\times 30+2\times 50\leq 120$
$\implies 30+100 \leq 120 \implies 130\leq 120$ False

$\mathrm C.$ substituting  $(60,30)$ in $4x+2y\leq 420$
we get $4\times 60+2\times 30\leq 420$
$\implies 240+60 \leq 420 \implies 300\leq 420$ True
substituting  $(60,30)$ in $1x+2y\leq 120$
we get $1\times 60+2\times 30\leq 120$
$\implies 60+60 \leq 120 \implies 120\leq 120$ True

$\mathrm A.$ substituting  $(90,20)$ in $4x+2y\leq 420$
we get $4\times 90+2\times 20\leq 420$
$\implies 360+40 \leq 420 \implies 400\leq 420$ True
substituting  $(90,20)$ in $1x+2y\leq 120$
we get $1\times 90+2\times 20\leq 120$
$\implies 90+40 \leq 120 \implies 130\leq 120$ False

Therefore option C (60,30) is the feasible point

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

Which of the following statements about an LP problem and its dual is false?

  1. If the primal and the dual both have optimal solutions, the objective function values for both problems are equal at the optimum

  2. If one of the variables in the primal has unrestricted sign, the corresponding constraint in the dual is satisfied with equality

  3. If the primal has an optimal solution, so has the dual

  4. The dual problem might have an optimal solution, even though the primal has no (bounded) optimum
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

if one of the problems(primal, dual) is infeasible then the other problem is infeasible. Hence, the option D is the false statement.

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

In linear programming context, sensitivity analysis is a technique to

  1. Allocate resources optimally.

  2. Minimize cost of operations.

  3. Spell out relation between primal and dual.

  4. Determine how optimal solution to LPP changes in response to problem inputs.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A sensitivity analysis is performed to determine the sensitivity of the solution to changes in parameters.
Option D is correct.

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

Choose the wrong statement:

  1. In order that dual to an LPP may be written, it is necessary that it has at least as many constraints as the number of variables.

  2. The dual represents an alternate formulation of LPP with decision variables being implicit values.

  3. The optimal values of the dual variables can be obtained by inspecting the optimal tableau of the primal problem as well.

  4. Sensitivity analysis is carried out having reference to the optimal tableau alone.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In order to write LPP, it is not necessary that it has at least as many constraints as the number of variables.

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

Choose the most correct of the following statements relating to primal-dual linear programming problems:

  1. Shadow prices of resources in the primal are optimal values of the dual variables.

  2. The optimal values of the objective functions of primal and dual are the same.

  3. If the primal problem has unbounded solution, the dual problem would have infeasibility.

  4. All of the above.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

From the primal-dual relationship,

The shadow prices of resources in the primal are optimal values of the dual variables.

If one of the problems has an optimal feasible solution then the other problem also has an optimal feasible solution. The optimal objective function value is same for both primal and dual problems.

If one problem has an unbounded solution then the other problem is infeasible.

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

To write the dual; it should be ensured that  
I. All the primal variables are non-negative.
II. All the bi values are non-negative.
III. All the constraints are $≤$ type if it is maximization problem and $≥$ type if it is a minimization problem.

  1. I and II

  2. II and III

  3. I and III

  4. I, II and III
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To write the dual, then all the primal variables must be non-negative.

All the constraints are $\leq$ type if it ia maximization problem and $\geq$ type if it is a minimization problem.