11th Business Maths Weekly Test - 1 ( Operations Research )

MABS Institution

11th Business Maths Weekly Test - 1 ( Operations Research )-Aug 2020

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

Given an L.P.P maximize Z=2x₁+3x₂ subject to the constrains x₁+x₂≤1, 5x₁+5x₂≥0 and x₁≥0, x₂≥0 using graphical method, we observe

No feasible solution
unique optimum solution
multiple optimum solution
none of these

In the context of network, which of the following is not correct

A network is a graphical representation
A project network cannot have multiple initial and final nodes
An arrow diagram is essentially a closed network
An arrow representing an activity may not have a length and shape

Maximize: z=3x₁+4x₂ subject to 2x₁+x2≤40, 2x₁+5x₂≤180, x₁,x₂≥0 in the LPP, which one of the following is feasible corner point?

x₁=18, x₂=24
x₁=15, x₂=30
x₁=2.5, x₂=35
x₁=20, x₂=19

In constructing the network which one of the following statement is false?

Each activity is represented by one and only one arrow. (i.e) only one activity can connect any two nodes
Two activities can be identified by the same head and tail events
Nodes are numbered to identify an activity uniquely. Tail node (starting point) should be lower than the head node (end point) of an activity
Arrows should not cross each other

In critical path analysis, the word CPM mean

Critical path method
Crash project management
Critical project management
Critical path management

A solution which maximizes or minimizes the given LPP is called

a solution
a feasible solution
an optimal solution
none of these

In a network while numbering the events which one of the following statement is false?

Event numbers should be unique
Event numbering should be carried out on a sequential basis from left to right
The initial event is numbered 0 or 1
The head of an arrow should always bear a number lesser than the one assigned at the tail of the arrow

Network problems have advantage in terms of project

Scheduling
Planning
Controlling
All the above

The critical path of the following network is

1 – 2 – 4 – 5
1– 3– 5
1 – 2 – 3 – 5
1 – 2 – 3 – 4 – 5

The maximum value of the objective function Z = 3x + 5y subject to the constraints x > 0 , y > 0 and 2x + 5y ≤10 is

Part - B

Generic 2 - Answer All Question

Construct a network diagram for the following situtation A < B; B < D, E; C <A and D < G.

A dealer whises to purchase a number of fans and sewing machines. He has only Rs.5760 to invest and has a space for atmost 20 items. A fan costs him Rs.360 and a sewing machine Rs. 240. His expectation is he can sell a fan at a profit of Rs.22 and a sewing machine at a profit of ns. Formulate this as an LPP to maximize his profit?

A producer has 30 and 17 units of labour and capital respectively which he can use to produce two types of goods X and Y. To produce one unit of X, 2 unit of labour and 3 units of capital are required. Similarly, 3 units of labour and 1 unit of capital is required to produce one unit of Y. If X and Yare priced at HOO and H20 per unit respectively, how should the producer use his resources to maximize the total revenue? Formulate the LPP for the above.
Draw a network diagram for the project whose activities and their predecessor relationships are given below

Activity A B C D E F

Predecessor activity - - D A B C

Activity	A	B	C	D	E	F
Predecessor activity	-	-	D	A	B	C

Construct a network diagram for the following situation:
A<D,E; B, D<F; C<G and B<H.

Part - C

Generic 3 - Answer All Question

Solve the following LPP graphically. Maximize \(Z={ x }_{ 1 }+{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }-{ x }_{ 2 }\le -1,{ -x }_{ 1 }+{ x }_{ 2 }\le 0\quad and\quad { x }_{ 1 }+{ x }_{ 2 }\ge 0\)

Solve the following LPP graphically.\(Maximize\quad Z=-{ x }_{ 1 }+2{ x }_{ 2 }\)
Subject to the constraints \(-{ x }_{ 1 }+3{ x }_{ 2 }\le 10,\quad { x }_{ 1 }+{ x }_{ 2 }\le 6,\quad { x }_{ 1 }{ -x }_{ 2 }\le 2\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0\)

Solve the following LPP graphically. Minimize \(Z={ x }_{ 1 }-5{ x }_{ 2 }+20\)
Subject to the constraints \({ x }_{ 1 }-{ x }_{ 2 }\ge 0,\quad { -x }_{ 1 }+2{ x }_{ 2 }\ge 2,\quad { x }_{ 1 }\ge 3,{ x }_{ 2 }\le 4\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0\)
Solve the following LPP graphically.
Maximize \(Z=6{ x }_{ 1 }+5{ x }_{ 2 }\) Subject to the constraints \(3{ x }_{ 1 }+5{ x }_{ 2 }\le 15,\quad 5{ x }_{ 1 }+2{ x }_{ 2 }\le 10\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0\)

Solve the following LPP graphically. Minimize\(Z=-3{ x }_{ 1 }+4{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }+2{ x }_{ 2 }\le 8\quad ,{ 3x }_{ 1 }+{ 2x }_{ 2 }\le 12\quad and\quad \quad { x }_{ 1 }\ge 0,{ x }_{ 2 }\ge 2.\)

Part - D

Generic 5 - Answer All Question

A dietician wishes to mix two types of food F₁ and F₂ in such a way that the vitamin contents of the mixture contains atleast 6units of vitamin A and 9 units of vitamin B. Food F₁ costs Rs.50 per kg and F₂ costs Rs 70 per kg. Food F₁ contains 4 units per kg of vitamin A and 6 units per kg of vitamin B while food F₂ contains 5 units per kg of vitamin A and b units per kg of vitamin B. Formulate the above problem as a linear programming problem to minimize the cost of mixture.

Solve the following linear programming problem graphically.
Maximize Z = 3x₁ + 5x₂ subject to the constraints x₁+x₂≤6, x₁≤4; x₂≤5, and x₁, x₂ ≥0

The following table gives the activities of a project and their duration in days

Activity 1-2 1-3 2-3 2-4 3-4 3-5 4-5

Duration 5 8 6 7 5 4 8

Construct the network and calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and determine the Critical path of the project and duration to complete the project.

The following table gives the characteristics of project

Activity 1-2 1-3 2-3 3-4 3-5 4-6 5-6

Duration (in days) 5 10 3 4 6 6 5

Draw the network for the project, calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and find the critical path. Compute the project duration.
A firm manufactures pills in two sizes A and B. Size A contains 2 mgs of aspirin, 5 mgs of bicarbonate and 1 mg of codeine. Size B contains 1 mg. of aspirin, 8 mgs. of bicarbonate and 6 mgs. of codeine. It is found by users that it requires atleast 12 mgs. of aspirin, 74 mgs.of bicarbonate and 24 mgs. of codeine for providing immediate relief. It is required to determine the least number of pills a patient should take to get immediate relief. Formulate the problem as a standard LLP.

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11th Business Maths Weekly Test - 1 ( Operations Research )

MABS Institution

11th Business Maths Weekly Test - 1 ( Operations Research )-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

Given an L.P.P maximize Z=2x1+3x2 subject to the constrains x1+x2≤1, 5x1+5x2≥0 and x1≥0, x2≥0 using graphical method, we observe

In the context of network, which of the following is not correct

Maximize: z=3x1+4x2 subject to 2x1+x2≤40, 2x1+5x2≤180, x1,x2≥0 in the LPP, which one of the following is feasible corner point?

In constructing the network which one of the following statement is false?

In critical path analysis, the word CPM mean

A solution which maximizes or minimizes the given LPP is called

In a network while numbering the events which one of the following statement is false?

Network problems have advantage in terms of project

The critical path of the following network is

The maximum value of the objective function Z = 3x + 5y subject to the constraints x > 0 , y > 0 and 2x + 5y ≤10 is

Part - B

Generic 2 - Answer All Question

Construct a network diagram for the following situtation A < B; B < D, E; C <A and D < G.

Draw a network diagram for the project whose activities and their predecessor relationships are given below Activity A B C D E F Predecessor activity - - D A B C

Construct a network diagram for the following situation: A<D,E; B, D<F; C<G and B<H.

Part - C

Generic 3 - Answer All Question

Solve the following LPP graphically. Maximize \(Z={ x }_{ 1 }+{ x }_{ 2 }\) Subject to the constraints \({ x }_{ 1 }-{ x }_{ 2 }\le -1,{ -x }_{ 1 }+{ x }_{ 2 }\le 0\quad and\quad { x }_{ 1 }+{ x }_{ 2 }\ge 0\)

Solve the following LPP graphically.\(Maximize\quad Z=-{ x }_{ 1 }+2{ x }_{ 2 }\) Subject to the constraints \(-{ x }_{ 1 }+3{ x }_{ 2 }\le 10,\quad { x }_{ 1 }+{ x }_{ 2 }\le 6,\quad { x }_{ 1 }{ -x }_{ 2 }\le 2\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0\)

Solve the following LPP graphically. Minimize \(Z={ x }_{ 1 }-5{ x }_{ 2 }+20\) Subject to the constraints \({ x }_{ 1 }-{ x }_{ 2 }\ge 0,\quad { -x }_{ 1 }+2{ x }_{ 2 }\ge 2,\quad { x }_{ 1 }\ge 3,{ x }_{ 2 }\le 4\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0\)

Solve the following LPP graphically. Maximize \(Z=6{ x }_{ 1 }+5{ x }_{ 2 }\) Subject to the constraints \(3{ x }_{ 1 }+5{ x }_{ 2 }\le 15,\quad 5{ x }_{ 1 }+2{ x }_{ 2 }\le 10\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0\)

Solve the following LPP graphically. Minimize\(Z=-3{ x }_{ 1 }+4{ x }_{ 2 }\) Subject to the constraints \({ x }_{ 1 }+2{ x }_{ 2 }\le 8\quad ,{ 3x }_{ 1 }+{ 2x }_{ 2 }\le 12\quad and\quad \quad { x }_{ 1 }\ge 0,{ x }_{ 2 }\ge 2.\)

Part - D

Generic 5 - Answer All Question

Solve the following linear programming problem graphically. Maximize Z = 3x1 + 5x2 subject to the constraints x1+x2≤6, x1≤4; x2≤5, and x1, x2 ≥0

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Given an L.P.P maximize Z=2x₁+3x₂ subject to the constrains x₁+x₂≤1, 5x₁+5x₂≥0 and x₁≥0, x₂≥0 using graphical method, we observe

Maximize: z=3x₁+4x₂ subject to 2x₁+x2≤40, 2x₁+5x₂≤180, x₁,x₂≥0 in the LPP, which one of the following is feasible corner point?

Draw a network diagram for the project whose activities and their predecessor relationships are given below

Activity A B C D E F

Predecessor activity - - D A B C

Construct a network diagram for the following situation:
A<D,E; B, D<F; C<G and B<H.

Solve the following LPP graphically. Maximize \(Z={ x }_{ 1 }+{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }-{ x }_{ 2 }\le -1,{ -x }_{ 1 }+{ x }_{ 2 }\le 0\quad and\quad { x }_{ 1 }+{ x }_{ 2 }\ge 0\)

Solve the following LPP graphically.\(Maximize\quad Z=-{ x }_{ 1 }+2{ x }_{ 2 }\)
Subject to the constraints \(-{ x }_{ 1 }+3{ x }_{ 2 }\le 10,\quad { x }_{ 1 }+{ x }_{ 2 }\le 6,\quad { x }_{ 1 }{ -x }_{ 2 }\le 2\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0\)

Solve the following LPP graphically. Minimize \(Z={ x }_{ 1 }-5{ x }_{ 2 }+20\)
Subject to the constraints \({ x }_{ 1 }-{ x }_{ 2 }\ge 0,\quad { -x }_{ 1 }+2{ x }_{ 2 }\ge 2,\quad { x }_{ 1 }\ge 3,{ x }_{ 2 }\le 4\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0\)

Solve the following LPP graphically.
Maximize \(Z=6{ x }_{ 1 }+5{ x }_{ 2 }\) Subject to the constraints \(3{ x }_{ 1 }+5{ x }_{ 2 }\le 15,\quad 5{ x }_{ 1 }+2{ x }_{ 2 }\le 10\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0\)

Solve the following LPP graphically. Minimize\(Z=-3{ x }_{ 1 }+4{ x }_{ 2 }\)
Subject to the constraints \({ x }_{ 1 }+2{ x }_{ 2 }\le 8\quad ,{ 3x }_{ 1 }+{ 2x }_{ 2 }\le 12\quad and\quad \quad { x }_{ 1 }\ge 0,{ x }_{ 2 }\ge 2.\)

Solve the following linear programming problem graphically.
Maximize Z = 3x₁ + 5x₂ subject to the constraints x₁+x₂≤6, x₁≤4; x₂≤5, and x₁, x₂ ≥0