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Questions Related to business maths

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?

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

  1. $(30,60)$

  2. $(105,0)$

  3. $(0,210)$

  4. $(100,10)$

  5. None of the above

Show answer
Correct Option: B,D
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.

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

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

  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

Show answer
Correct Option: D
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. 

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?

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

  1. $(50,40)$

  2. $(30,50)$

  3. $(60,30)$

  4. $(90,20)$

  5. None of these

Show answer
Correct Option: C
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

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

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

  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

Show answer
Correct Option: D
Explanation:

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

Mark the wrong statement:

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

  1. The primal and dual have equal number of variables.

  2. The shadow price indicates the change in the value of the objective function, per unit increase in the value of the RHS.

  3. The shadow price of a non-binding constraint is always equal to zero.

  4. The information about shadow price of a constraint is important since it may be possible to purchase or, otherwise, acquire additional units of the concerned resource.

Show answer
Correct Option: A
Explanation:

The number of variables in dual is equal to the number of constraints in the primal and the  number of variables in primal is equal to the number of constraints in the dual.


Therefore, the primal and dual doesn't have equal number of variables.

In linear programming context, sensitivity analysis is a technique to

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

  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.

Show answer
Correct Option: D
Explanation:

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

Choose the wrong statement:

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

  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.

Show answer
Correct Option: A
Explanation:

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

The number of constraints allowed in a linear program is which of the following?

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

  1. Less than 5

  2. Less than 72

  3. Less than 512

  4. Less than 1,024

  5. Unlimited

Show answer
Correct Option: E
Explanation:

there is no limit on constraints allowed in linear programming.
so the number of constraints is unlimited.

Which of the following is an essential condition in a situation for linear programming to be useful?

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

  1. Linear constraints

  2. Bottlenecks in the objective function

  3. Non-homogeneity

  4. Uncertainty

  5. None of the above

Show answer
Correct Option: A
Explanation:

For linear programming, the constraints must be linear.

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

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

  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.

Show answer
Correct Option: D
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.