Linear Programming Test

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Linear Programming Test - 22

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Question 1
1 / -0

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

A
Linear constraints

B
Bottlenecks in the objective function

C
Non-homogeneity

D
Uncertainty

E
None of the above

Solution

For linear programming, the constraints must be linear.
Question 2
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Apply linear programming to this problem. A firm wants to determine how many units of each of two products (products D and E) they should produce to make the most money. The profit in the manufacture of a unit of product D is $100 and the profit in the manufacture of a unit of product E is $87. The firm is limited by its total available labor hours and total available machine hours. The total labor hours per week are 4,000. Product D takes 5 hours per unit of labor and product E takes 7 hours per unit. The total machine hours are 5,000 per week. Product D takes 9 hours per unit of machine time and product E takes 3 hours per unit. Which of the following is one of the constraints for this linear program?

A
$$5 D + 7 E≤ 5,000$$

B
$$9 D + 3 E ≥4,000$$

C
$$5 D + 7 E = 4,000$$

D
$$5 D + 9 E ≤5,000$$

E
$$9 D + 3 E ≤5,000$$

Solution

Given, product D takes 5 hours per unit of labour, and
product E takes 7 hours per unit of labour.
Therefore, to produce D units of product D takes $$5D$$ hours and
to produce E units of product E takes $$7E$$ hours
Given, total labour hours per week are $$4000$$ hours.
Hence, $$5D+7E\leq 4000$$

Given, product D takes 9 hours per unit of machine time, and
product E takes 3 hours per unit of machine time.
Therefore, to produce D units of product D takes $$9D$$ hours and
to produce E units of product E takes $$3E$$ hours
Given, total machine hours per week are $$5000$$ hours.
Hence, $$9D+3E\leq 5000$$
Question 3
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A constraint in an LP model becomes redundant because:

A
two iso-profit line may be parallel to each other

B
the solution is unbounded

C
this constraint is not satisfied by the solution values

D
none of the above

Solution

A constraint in an LP model becomes redundant when the feasible region doesn't change by the removing the constraint.
For example, $$x+2y\leq 20 $$ and $$2x+4y\leq 40$$ are the constraints.
$$2x+4y\leq 40 \implies 2*(x+2y)\leq 2*20$$
$$\implies x+2y\leq 20$$ which is same as the first constraint.
Therefore $$2x+4y\leq 40$$ can be removed. By removing this constraint feasible region doesn't change.
Question 4
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If two constraints do not intersect in the positive quadrant of the graph, then

A
The problem is infeasible

B
The solution is unbounded

C
One of the constraints is redundant

D
None of the above

Solution

Any linear programming problem must have the following properties:-
1. The relationship between variables and constraints must be linear.
2. The constraints must be non-negative.
3.. objective function must be linear.
Non-negativity conditions are used because the variables cannot take negative values. i.e., it is not possible to have negative resources(land, capital, labour cannot be negative).

Because of the non-negativity condition, the feasible region exists only in I quadrant.
Question 5
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In profit objective function, all lines representing same level of profit are classified as

A
iso-objective lines

B
iso-function lines

C
iso-profit lines

D
iso-cost lines

Solution

An iso-profit line is obtained by equating the objective function with the constant number $$a$$ which results in a linear equation. The graph of the linear equation is a straight line. This straight line represents the $$\text{iso profit line}$$. Each and every point on this iso-profit line has the same objective value $$a$$.

A line parallel to an iso-profit line is also an iso-profit line, but with a different objective function value.
Question 6
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The number of constraints allowed in a linear program is which of the following?

A
Less than 5

B
Less than 72

C
Less than 512

D
Less than 1,024

E
Unlimited

Solution

there is no limit on constraints allowed in linear programming.
so the number of constraints is unlimited.
Question 7
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A feasible solution to an LP problem

A
must satisfy all of the problems constraints simultaneously

B
need not satisfy all of the constraints, only some of them

C
must be a corner point of the feasible region

D
must optimize the value of the objective function

Solution

A feasible solution to an LP problem belongs to the feasible region. Feasible region is the set of all the points that satisfy the problem's constraints including inequalities, equalities and integer constraints.

In the above figure, blue region is the feasible region. All the points in this region are feasible solutions or feasible points which satisfy the given constraints($$4x+3y\leq 480 $$ and $$2x+3y\leq 360$$)
Question 8
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In North west corner rule if the demand in the column is satisfied one must move to the

A
left cell in the next column

B
right cell in the next row

C
right cell in the next column

D
left cell in the next row

Solution

The North West Corner method is used to compute the initial feasible solution. This method starts from the north west (i.e., upper left) cell.

If the supply or demand in the column is satisfied then we should move to the right cell in the next column.

For example, in the above figure, 20 units are assigned to the first cell that satisfies the demand of $$D$$ while the supply is in surplus. Now move to the $$\text{right cell in the next column}$$ and assign 30 units which are available with source $$C$$.
Question 9
1 / -0

The __________ is the method available for solving an L.P.P

A
graphical method

B
least cost method

C
MODI method

D
Hungarian method

Solution

There are different methods to solve an linear programming problem. Such as Graphical method, Simplex method, Ellipsoid method, Interior point methods.
Question 10
1 / -0

In North west corner rule, if the supply in the row is satisfied one must move

A
down in the next row

B
up in the next row

C
right cell in the next column

D
left cell in the next row

Solution

The North West Corner method is used to compute the initial feasible solution. This method starts from the north-west (i.e., upper left) cell.

If the supply or demand in the column is satisfied then we should move to the right cell in the next column.

For example, in the above figure, 20 units are assigned to the first cell that satisfies the demand of $$D$$ while the supply is in surplus.
Now move to the right cell in the next column and assign 30 units which are available with source $$A$$.
Now $$\text{move down in the next row}$$ and assign 40 units which are available with source $$B$$.

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Linear Programming Test - 22

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Linear Programming Test - 22

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Question 1

Linear constraints

Bottlenecks in the objective function

Non-homogeneity

Uncertainty

None of the above

Solution

Question 2

$$5 D + 7 E≤ 5,000$$

$$9 D + 3 E ≥4,000$$

$$5 D + 7 E = 4,000$$

$$5 D + 9 E ≤5,000$$

$$9 D + 3 E ≤5,000$$

Solution

Question 3

two iso-profit line may be parallel to each other

the solution is unbounded

this constraint is not satisfied by the solution values

none of the above

Solution

Question 4

The problem is infeasible

The solution is unbounded

One of the constraints is redundant

None of the above

Solution

Question 5

iso-objective lines

iso-function lines

iso-profit lines

iso-cost lines

Solution

Question 6

Less than 5

Less than 72

Less than 512

Less than 1,024

Unlimited

Solution

Question 7

must satisfy all of the problems constraints simultaneously

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

must be a corner point of the feasible region

must optimize the value of the objective function

Solution

Question 8

left cell in the next column

right cell in the next row

right cell in the next column

left cell in the next row

Solution

Question 9

graphical method

least cost method

MODI method

Hungarian method

Solution

Question 10

down in the next row

up in the next row

right cell in the next column

left cell in the next row

Solution

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