Tamil Nadu Board of Secondary EducationHSC Commerce Class 11th

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

## Chapter 10: Operations Research

Exercise 10.1Exercise 10.2Exercise 10.3Miscellaneous Problems
Exercise 10.1 [Pages 243 - 244]

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

Exercise 10.1 | Q 1 | Page 243

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.

Exercise 10.1 | Q 2 | Page 243

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.

Exercise 10.1 | Q 3 | Page 244

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.

Exercise 10.1 | Q 4. (i) | Page 244

Solve the following linear programming problems by graphical method.

Maximize Z = 6x1 + 8x2 subject to constraints 30x1 + 20x2 ≤ 300; 5x1 + 10x2 ≤ 110; and x1, x2 ≥ 0.

Exercise 10.1 | Q 4. (ii) | Page 244

Solve the following linear programming problems by graphical method.

Maximize Z = 22x1 + 18x2 subject to constraints 960x1 + 640x2 ≤ 15360; x1 + x2 ≤ 20 and x1, x2 ≥ 0.

Exercise 10.1 | Q 4. (iii) | Page 244

Solve the following linear programming problems by graphical method.

Minimize Z = 3x1 + 2x2 subject to the constraints 5x1 + x2 ≥ 10; x1 + x2 ≥ 6; x1 + 4x2 ≥ 12 and x1, x2 ≥ 0.

Exercise 10.1 | Q 4. (iv) | Page 244

Solve the following linear programming problems by graphical method.

Maximize Z = 40x1 + 50x2 subject to constraints 3x1 + x2 ≤ 9; x1 + 2x2 ≤ 8 and x1, x2 ≥ 0.

Exercise 10.1 | Q 4. (v) | Page 244

Solve the following linear programming problems by graphical method.

Maximize Z = 20x1 + 30x2 subject to constraints 3x1 + 3x2 ≤ 36; 5x1 + 2x2 ≤ 50; 2x1 + 6x2 ≤ 60 and x1, x2 ≥ 0.

Exercise 10.1 | Q 4. (vi) | Page 244

Solve the following linear programming problems by graphical method.

Minimize Z = 20x1 + 40x2 subject to the constraints 36x1 + 6x2 ≥ 108; 3x1 + 12x2 ≥ 36; 20x1 + 10x2 ≥ 100 and x1, x2 ≥ 0.

Exercise 10.2 [Pages 249 - 250]

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

Exercise 10.2 | Q 1 | Page 249

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.

Exercise 10.2 | Q 2 | Page 249

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
Exercise 10.2 | Q 3 | Page 249

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

Exercise 10.2 | Q 4 | Page 249

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
Exercise 10.2 | Q 5 | Page 249

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.

Exercise 10.2 | Q 6 | Page 249

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.

Exercise 10.2 | Q 7 | Page 249

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
Exercise 10.2 | Q 8 | Page 249

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.

Exercise 10.2 | Q 9 | Page 249

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.

Exercise 10.2 | Q 10 | Page 250

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.

Exercise 10.3 [Pages 250 - 257]

### 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

Exercise 10.3 | Q 1 | Page 250

The critical path of the following network is

• 1 - 2 - 4 - 5

• 1 - 3 - 5

• 1 - 2 - 3 - 5

• 1 - 2 - 3 - 4 - 5

Exercise 10.3 | Q 2 | Page 250

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 comer point?

• x1 = 18, x2 = 24

• x1 = 15, x2 = 30

• x1 = 2.5, x2 = 35

• x1 = 20.5, x2 = 19

Exercise 10.3 | Q 3 | Page 250

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

• Ej – Ei = Lj – Li = tij

• Ei – Ej = Lj – Li = tij

• Ej – Ei = Li – Lj = tij

• Ej – Ei = Lj – Li ≠ tij

Exercise 10.3 | Q 4 | Page 250

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.

Exercise 10.3 | Q 5 | Page 250

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.

Exercise 10.3 | Q 6 | Page 250

A solution which maximizes or minimizes the given LPP is called

• a solution

• a feasible solution

• an optimal solution

• none of these

Exercise 10.3 | Q 7 | Page 251

In the given graph the coordinates of M1 are

• x1 = 5, x2 = 30

• x1 = 20, x2 = 16

• x1 = 10, x2 = 20

• x1 = 20, x2 = 30

Exercise 10.3 | Q 8 | Page 257

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

Exercise 10.3 | Q 9 | Page 251

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

Exercise 10.3 | Q 10 | Page 251

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.

Exercise 10.3 | Q 11 | Page 251

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

Exercise 10.3 | Q 12 | Page 251

The objective of network analysis is to

• Minimize total project cost

• Minimize total project duration

• Minimize production delays, interruption and conflicts

• All the above

Exercise 10.3 | Q 13 | Page 251

Network problems have the advantage in terms of project

• Scheduling

• Planning

• Controlling

• All the above

Exercise 10.3 | Q 14 | Page 251

In critical path analysis, the word CPM mean

• Critical path method

• Crash project management

• Critical project management

• Critical path management

Exercise 10.3 | Q 15 | Page 251

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

Miscellaneous Problems [Page 252]

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

Miscellaneous Problems | Q 1 | 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 M1 and M2. Product A requires one minute of processing time on M1 and two minutes on M2, While B requires one minute on M1 and one minute on M2. Machine M1 is available for not more than 7 hrs 30 minutes while M2 is available for 10 hrs during any working day. Formulate this problem as a linear programming problem to maximize the profit.

Miscellaneous Problems | Q 2 | Page 252

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.

Miscellaneous Problems | Q 3 | Page 252

Solve the following linear programming problem graphically.

Maximise Z = 4x1 + x2 subject to the constraints x1 + x2 ≤ 50; 3x1 + x2 ≤ 90 and x1 ≥ 0, x2 ≥ 0.

Miscellaneous Problems | Q 4 | Page 252

Solve the following linear programming problem graphically.

Minimize Z = 200x1 + 500x2 subject to the constraints: x1 + 2x2 ≥ 10; 3x1 + 4x2 ≤ 24 and x1 ≥ 0, x2 ≥ 0.

Miscellaneous Problems | Q 5 | Page 252

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.

Miscellaneous Problems | Q 6 | Page 252

Solve the following linear programming problem graphically.

Maximize Z = 60x1 + 15x2 subject to the constraints: x1 + x2 ≤ 50; 3x1 + x2 ≤ 90 and x1, x2 ≥ 0.

Miscellaneous Problems | Q 7 | Page 252

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
Miscellaneous Problems | Q 8 | Page 252

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
Miscellaneous Problems | Q 9 | Page 252

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.

Miscellaneous Problems | Q 10 | Page 252

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

Exercise 10.1Exercise 10.2Exercise 10.3Miscellaneous Problems

## 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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