#### Chapters

Chapter 2: Algebra

Chapter 3: Analytical Geometry

Chapter 4: Trigonometry

Chapter 5: Differential Calculus

Chapter 6: Applications of Differentiation

Chapter 7: Financial Mathematics

Chapter 8: Descriptive Statistics and Probability

Chapter 9: Correlation and Regression Analysis

Chapter 10: Operations Research

## Chapter 10: Operations Research

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 10 Operations ResearchExercise 10.1 [Pages 243 - 244]

A company produces two types of pens A and B. Pen A is of superior quality and pen B is of lower quality. Profits on pens A and B are ₹ 5 and ₹ 3 per pen respectively. Raw materials required for each pen A is twice as that of pen B. The supply of raw material is sufficient only for 1000 pens per day. Pen A requires a special clip and only 400 such clips are available per day. For pen B, only 700 clips are available per day. Formulate this problem as a linear programming problem.

A company produces two types of products say type A and B. Profits on the two types of product are ₹ 30/- and ₹ 40/- per kg respectively. The data on resources required and availability of resources are given below.

Requirements |
Capacity available per month |
||

Product A |
Product B |
||

Raw material (kgs) | 60 | 120 | 12000 |

Machining hours/piece | 8 | 5 | 600 |

Assembling (man hours) | 3 | 4 | 500 |

Formulate this problem as a linear programming problem to maximize the profit.

A company manufactures two models of voltage stabilizers viz., ordinary and auto-cut. All components of the stabilizers are purchased from outside sources, assembly and testing is carried out at the company’s own works. The assembly and testing time required for the two models are 0.8 hours each for ordinary and 1.20 hours each for auto-cut. Manufacturing capacity 720 hours at present is available per week. The market for the two models has been surveyed which suggests a maximum weekly sale of 600 units of ordinary and 400 units of auto-cut. Profit per unit for ordinary and auto-cut models has been estimated at ₹ 100 and ₹ 150 respectively. Formulate the linear programming problem.

**Solve the following linear programming problems by graphical method.**

Maximize Z = 6x_{1} + 8x_{2} subject to constraints 30x_{1} + 20x_{2 }≤ 300; 5x_{1} + 10x_{2} ≤ 110; and x_{1}, x_{2} ≥ 0.

**Solve the following linear programming problems by graphical method.**

Maximize Z = 22x_{1} + 18x_{2} subject to constraints 960x_{1} + 640x_{2} ≤ 15360; x_{1} + x_{2} ≤ 20 and x_{1}, x_{2} ≥ 0.

**Solve the following linear programming problems by graphical method.**

Minimize Z = 3x_{1} + 2x_{2} subject to the constraints 5x_{1} + x_{2} ≥ 10; x_{1} + x_{2} ≥ 6; x_{1 }+ 4x_{2} ≥ 12 and x_{1}, x_{2} ≥ 0.

**Solve the following linear programming problems by graphical method.**

Maximize Z = 40x_{1} + 50x_{2} subject to constraints 3x_{1} + x_{2} ≤ 9; x_{1} + 2x_{2} ≤ 8 and x_{1}, x_{2} ≥ 0.

**Solve the following linear programming problems by graphical method.**

Maximize Z = 20x_{1} + 30x_{2} subject to constraints 3x_{1} + 3x_{2} ≤ 36; 5x_{1} + 2x_{2} ≤ 50; 2x_{1} + 6x_{2} ≤ 60 and x_{1}, x_{2} ≥ 0.

**Solve the following linear programming problems by graphical method.**

Minimize Z = 20x_{1} + 40x_{2} subject to the constraints 36x_{1} + 6x_{2} ≥ 108; 3x_{1} + 12x_{2} ≥ 36; 20x_{1} + 10x_{2} ≥ 100 and x_{1}, x_{2} ≥ 0.

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 10 Operations ResearchExercise 10.2 [Pages 249 - 250]

**Draw the network for the project whose activities with their relationships are given below:**

Activities A, D, E can start simultaneously; B, C > A; G, F > D, C; H > E, F.

**Draw the event oriented network for the following data:**

Events |
1 | 2 | 3 | 4 | 5 | 6 | 7 |

Immediate Predecessors |
- | 1 | 1 | 2, 3 | 3 | 4, 5 | 5, 6 |

**Construct the network for the projects consisting of various activities and their precedence relationships are as given below:**

A, B, C can start simultaneously A < F, E; B < D, C; E, D < G

Construct the network for each the projects consisting of various activities and their precedence relationships are as given below:

Activity |
A | B | C | D | E | F | G | H | I | J | K |

Immediate Predecessors |
- | - | - | A | B | B | C | D | E | H, I | F, G |

Construct the network for the project whose activities are given below.

Activity |
0 - 1 | 1 - 2 | 1 - 3 | 2 - 4 | 2 - 5 | 3 - 4 | 3 - 6 | 4 - 7 | 5 - 7 | 6 - 7 |

Duration (in week) |
3 | 8 | 12 | 6 | 3 | 3 | 8 | 5 | 3 | 8 |

Calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity. Determine the critical path and the project completion time.

A project schedule has the following characteristics

Activity |
1 - 2 | 1 - 3 | 2 - 4 | 3 - 4 | 3 - 5 | 4 - 9 | 5 - 6 | 5 - 7 | 6 - 8 | 7 - 8 | 8 - 10 | 9 - 10 |

Time |
4 | 1 | 1 | 1 | 6 | 5 | 4 | 8 | 1 | 2 | 5 | 7 |

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.

Draw 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.

Jobs |
1 - 2 | 1 - 3 | 2 - 4 | 3 - 4 | 3 - 5 | 4 - 5 | 4 - 6 | 5 - 6 |

Duration |
6 | 5 | 10 | 3 | 4 | 6 | 2 | 9 |

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.

A Project has the following time schedule

Activity |
1 - 2 | 1 - 6 | 2 - 3 | 2 - 4 | 3 - 5 | 4 - 5 | 6 - 7 | 5 - 8 | 7 - 8 |

Duration (in days) |
7 | 6 | 14 | 5 | 11 | 7 | 11 | 4 | 18 |

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 use the activities in a construction projects and relevant information

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

Duration(in days) |
22 | 27 | 12 | 14 | 6 | 12 |

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.

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 10 Operations ResearchExercise 10.3 [Pages 250 - 257]

#### Choose the correct answer

The critical path of the following network is

1 - 2 - 4 - 5

1 - 3 - 5

1 - 2 - 3 - 5

1 - 2 - 3 - 4 - 5

Maximize: z = 3x_{1} + 4x_{2} subject to 2x_{1} + x_{2} ≤ 40, 2x_{1} + 5x_{2} ≤ 180, x_{1}, x_{2} ≥ 0. In the LPP, which one of the following is feasible comer point?

x

_{1}= 18, x_{2}= 24x

_{1}= 15, x_{2}= 30x

_{1}= 2.5, x_{2}= 35x

_{1}= 20.5, x_{2}= 19

One of the conditions for the activity (i, j) to lie on the critical path is

E

_{j}– E_{i}= L_{j}– L_{i}= t_{ij}E

_{i}– E_{j}= L_{j}– L_{i}= t_{ij}E

_{j}– E_{i}= L_{i}– L_{j}= t_{ij}E

_{j}– E_{i}= L_{j}– L_{i}≠ t_{ij}

In constructing the network which one of the following statements 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 a network while numbering the events which one of the following statements 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.

A solution which maximizes or minimizes the given LPP is called

a solution

a feasible solution

an optimal solution

none of these

In the given graph the coordinates of M_{1} are

x

_{1}= 5, x_{2}= 30x

_{1}= 20, x_{2}= 16x

_{1}= 10, x_{2}= 20x

_{1}= 20, x_{2}= 30

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

6

15

25

31

The minimum value of the objective function Z = x + 3y subject to the constraints 2x + y ≤ 20, x + 2y ≤ 20, x > 0 and y > 0 is

10

20

0

5

Which of the following is not correct?

Objective that we aim to maximize or minimize

Constraints that we need to specify

Decision variables that we need to determine

Decision variables are to be unrestricted.

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

The objective of network analysis is to

Minimize total project cost

Minimize total project duration

Minimize production delays, interruption and conflicts

All the above

Network problems have the advantage in terms of project

Scheduling

Planning

Controlling

All the above

In critical path analysis, the word CPM mean

Critical path method

Crash project management

Critical project management

Critical path management

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

No feasible solution

unique optimum solution

multiple optimum solution

none of these

### Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide Chapter 10 Operations ResearchMiscellaneous Problems [Page 252]

A firm manufactures two products A and B on which the profits earned per unit are ₹ 3 and ₹ 4 respectively. Each product is processed on two machines M_{1} and M_{2}. Product A requires one minute of processing time on M_{1} and two minutes on M_{2}, While B requires one minute on M_{1} and one minute on M_{2}. Machine M_{1} is available for not more than 7 hrs 30 minutes while M_{2} is available for 10 hrs during any working day. Formulate this problem as a linear programming problem to maximize the profit.

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 at least 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.

**Solve the following linear programming problem graphically.**

Maximise Z = 4x_{1} + x_{2} subject to the constraints x_{1} + x_{2} ≤ 50; 3x_{1} + x_{2} ≤ 90 and x_{1} ≥ 0, x_{2} ≥ 0.

**Solve the following linear programming problem graphically.**

Minimize Z = 200x_{1} + 500x_{2} subject to the constraints: x_{1} + 2x_{2} ≥ 10; 3x_{1} + 4x_{2} ≤ 24 and x_{1} ≥ 0, x_{2} ≥ 0.

**Solve the following linear programming problem graphically.**

Maximize Z = 3x_{1} + 5x_{2} subject to the constraints: x_{1} + x_{2} ≤ 6, x_{1} ≤ 4; x_{2} ≤ 5, and x_{1}, x_{2} ≥ 0.

**Solve the following linear programming problem graphically.**

Maximize Z = 60x_{1} + 15x_{2} subject to the constraints: x_{1} + x_{2} ≤ 50; 3x_{1} + x_{2} ≤ 90 and x_{1}, x_{2} ≥ 0.

**Draw a network diagram for the following activities.**

Activity code |
A | B | C | D | E | F | G | H | I | J | K |

Predecessor activity |
- | A | A | A | B | C | C | C, D | E, F | G, H | I, J |

**Draw the network diagram for the following activities.**

Activity code |
A | B | C | D | E | F | G |

Predecessor activity |
- | - | A | A | B | C | D, E |

A Project has the following time schedule

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

Duration(in days) |
6 | 8 | 4 | 9 | 2 | 7 |

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.

The following table gives the characteristics of the project

Activity |
1 - 2 | 1 - 3 | 2 - 3 | 3 - 4 | 3 - 5 | 4 - 6 | 5 - 6 | 6 - 7 |

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

## Chapter 10: Operations Research

## Tamil Nadu Board Samacheer Kalvi solutions for Class 11th Business Mathematics and Statistics Answers Guide chapter 10 - Operations Research

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Concepts covered in Class 11th Business Mathematics and Statistics Answers Guide chapter 10 Operations Research are Linear Programming Problem (L.P.P.), Network Analysis.

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