TYBMS SEM 6 Operation Research (Q.P. November 2023 with Solution)

 Paper/ Subject Code 86001/ Operation Research

 TYBMS SEM 6

 Operation Research 

(Q.P. November 2023 with Solution)

 

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Note:

1. All questions are compulsory. (Subject to Internal Choice)

2. Figures to the right indicate full marks

3. Use of non-programmable calculator is allowed and mobile phones are not allowed.

4. Normal distribution table is printed on the last page for reference. Support your answers with diagrams / illustrations, wherever necessary

6. Graph papers will be supplied on request.

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Q.1 Multiple Choice Questions: (Attempt Any 8)                (8)

1. Operation research, which is very powerful tool for ________.

a) Research

b) Decision Making

c) Operations

d) None of the above

2. Constraints in LPP model represent ________. 

a) Limitations

b) Requirement

c) Balancing limitations and requirements

d) All of the above

3. While solving a LP model graphically, the area bounded by the constraints is called ________.

a) Feasible region

b) Infeasible region

c) Unbounded solution

d) None of the above

4. For solving an assignment problem, which method is used ?

a) British

b) American

c) German

d) None of the above

5. The solution to a transportation problem with m' rows (supplies) & n columns (destination) is feasible if number of positive allocations are

a) m*n

b) m-n

c) m-n-1

d) m-n-l

6. When total supply doesn't match with total demand ________. case exists a transportation problem.

a) unbalanced

b) alternate optimal solution

c) degeneracy

d) none of the above

7. Floats for critical activities will be always ________.

a) zero

b) backward

c) > 1

d) none of the above

8.The two types of costs involved in the process of crashing in a project are costs.

a) direct and indirect

b) partial and full

c) measurable and non-measurable

d) none of these

9. The various courses of actions available to each player are called as

a) Saddle points

b) strategies

c) Pay-offs

d) n player game

10. The total time required to complete all the job-sequencing problem is known as

a) idle time

b) processing time

c) elapsed time

d) processing time

Q.1 B) Match the following (Any Seven)                (7)

Group A	Group B
1 Network scheduling	a) Marketing. Finance
2. Application of OR	b) PERT and CPM
3. Linear programming	c. Worker's allocation
4 Assignment problem	d) Quantitative in nature
5 OR techniques	e. Maximizing minimizing objective function
6. MODI method	f) No of columns to of rows
7. Add dummy row	g) At the start
8. Crash Cost per day	h) Feasible solution
9. Pure strategy game	1) Cost slope
10. A job with smallest processing time on machine A is placed	j) Saddle point exits

 Ans: 

Group A	Group B
1 Network scheduling	b) PERT and CPM
2. Application of OR	a) Marketing. Finance
3. Linear programming	e) Maximizing minimizing objective function
4 Assignment problem	c) Worker's allocation
5 OR techniques	d) Quantitative in nature
6. MODI method	h) Feasible solution
7. Add dummy row	f) No of columns to of rows
8. Crash Cost per day	i) Cost slope
9. Pure strategy game	j) Saddle point exits
10. A job with smallest processing time on machine A is placed	g) At the start

 

Q.2 A) There are four machines M1, M2, M3, M4. There are five jobs P, Q, R, S and T. Cost of doing each job on each machine is given below (in Rs. Hundreds) machine M2 cannot process job R and machine 3 cannot process job P. Find optional assignment of Machines and jobs to minimize total cost.

Jobs Machine	P	Q	R	S	T
M1	9	11	15	10	11
M2	12	9	-	10	9
M3	-	11	14	11	7
M4	14	8	12	7	8

Find optional assignment of Machines and jobs to minimize total cost.            (8)

Q.2 B) Use Graphical method to solve the following Linear programming problem: (7)

Objective Function:-

Max Z = 3X₁ + 4X₂

Subject to constraints:

3X₁ + 2X₂ ≤ 18

X₁ ≤ 5

X₂ ≤ 6

 X_{1 ,} X₂  ≥ 0

OR

Q.2 C)Two nutrients N₁ and N2 are recommended for athletes which are available in two products P₁ and P2 Minimum daily intake for N₁ and N2 is 30and 40 units respectively. cost for 1 unit for P₁ and P2 is Rs. 100 and Rs. 150 respectively 1 unit of P₁ contains 3 units of N₁ and 5 units of N2 1 unit of P2 contains 5 units of N₁ and 7 units of Ne N2

Formulate LPP                                                     (8)

               

Q.2 D) Find Optimal Solution by Simplex Method.            (7)

Max. Z = 3X₁ + 5X2

Subject to:-

X₁ + X3 _{= 4}

X2 + X4 = 6

3X₁ + 2X2 + X5 = 12

X₁  X2  X3  X4  X5 ≥ 0

Q.3A) small project consists of following activities:

Activity	Preceding Activities	Time (Days)
A		4
B		5
C		7
D	A	6
E	B	7
F	C	6
G	D	5
H	E	8
I	F	5

i) Draw network diagram and find critical path and project completion time.        (4)

ii) Find earliest and latest starting and finishing time of all activities. (EST, EFT. LFT. LST) (4)

Q. 3 B) The following is an intermediate table in the solution of a transportation problem.

Figures in the top right corner of every cell represent the cost of transporting (in Rs.) one unit from plants to distribution centers. Allocations are circled

i) Is the solution optimal? If not. find the optimal solution.            (5)

ii) Does the problem have an alternate solution? If so, show alternate optimal solution.    (2)

OR

Q. 3 C) M/S Motilal Limited have taken up a special project consisting of 10 activities whose three point time estimates are listed in the table below. Activities are marked with their node numbers.

Activity	Time Estimates in Week
	Optimistic	Most Likely	Pessimistic
1-2	1	2	3
1-3	1	2	3
1-4	1	2	3
2-5	2	9	20
3-5	4	5	14
3-7	3	6	15
5-7	1	2	9
4-6	2	4	6
6-7	3	3	3
7-8	4	4	4

i, Draw network diagram and find expected completion time of project.            (2)

ii. Identity critical paths.                                (2)

iii. Find the probability that the project is completed in 17 weeks.                (3)

iv. What is the probability that the project will not be completed in 20 weeks?        (2)

v. If the project includes a penalty clause of Rs. 1000 per week for any delay beyond 19 weeks. What is the probability of paying a penalty of more than Rs. 5000.                                (2)

vi. What project duration will have 95" confidence of completion?            (2)

vii. If the project manager completes the project 95% confidence in 21 weeks, by how much time should he crash the average project completion time?             (2)

Q. 4 A) Five jobs are to be processed on 2 machines A and B. Find (i) Total elapsed time. (ii) Idle time for each machine A and B for the following data giving processing time in hours.    (8)

Jobs	I	II	III	IV	V
Machine A	18	14	11	14	15
Machine B	13	12	15	16	17

Q. 4 B) you are given the Pay-off (profit in Rs.) matrix in respect of a Two-person-Zero 

		Player B
		B1	B2	B3	B4
Player A	A1	13	14	-4	-12
	A2	8	9	0	5
	A3	7	-5	-2	-8
	A4	-9	-5	0	-2

Sum game as follows:

i) Find the Maximum strategy                    (3)

ii) Find the Minimax strategy.                    (2)

iii) What is Value of game?                        (2)

OR

Q. 4 C) The time and cost details of a project are as below:

Activities	Predecessor Activity	Normal Time (Days)	Crash Time (Days)	Normal Cost (Rs.)	Crash Cost
A	-	4	3	60	90
B	-	6	4	150	250
C	-	2	1	38	60
D	A	5	3	150	250
E	C	2	2	100	100
F	A	7	5	115	175
G	D,E,B	4	2	100	240

Determine the project duration, which will return in minimum total project cost.    (5)

Q.4 D) There are five jobs namely 1,2,3,4 and 5 each of which must go through machines A, B and C in the order ABC Processing time (in hours) are given below:

Jobs	1	2	3	4	5
Machine A	20	25	23	22	24
Machine B	21	22	19	20	19
Machine C	23	24	22	25	20

(i) Find the sequence that minimizes the total elapsed time required to complete the job        (2)

(ii) Calculate the total elapsed time            (2)

(iii) Idle time on Machine A. Machine B and Machine C.            (3)

Q.5 A) Why is a non-degenerate solution a prerequisite for optimality test of a transportation solution? (8)

B) Define Operations Research and what are the major application areas for operation research techniques? (7)

OR

Q.5 Answer Any 3 of the following              (15)

1. Assumption on LPP

Ans: 

(1) Available quantities of resources and consumption per unit from resources is known exactly and with certainty.

(2) Production of finished products is possible in any fractions, so is consumption of resources.

(3) All external factors are constant.

(4) The problem involves only one major objective.

2 Objective of Network analysis

3, Project crashing

Ans:  In that case, the critical path will have to be shortened or reduced. This can be done by reducing completion time of some or all of the critical activities. To achieve this, we will need to employ extra resources. This process of shortening the critical path to achieve earlier completion of the project is called project crashing.

4. Basis and non-basis variable in simplex table

Ans: 

In the simplex method, which is an algorithm for solving linear programming problems, the terms "basis" and "non-basis" variables are used to describe the variables involved in the current solution of the linear program. 

1. Basis Variables: Basis variables are those variables that are explicitly set to non-zero values in the current feasible solution. These variables correspond to the basic feasible solution, which is a solution where a subset of the variables is set to non-zero values, while the rest are set to zero. The number of basis variables is equal to the number of constraints in the linear programming problem. The values of basis variables are determined by solving a system of linear equations derived from the constraints.

2. Non-Basis Variables: Non-basis variables are those variables that are not explicitly set to non-zero values in the current feasible solution. These variables correspond to the non-basic variables, which are usually set to zero in the current solution. Non-basis variables are typically represented as variables that can potentially enter the basis to improve the objective function value. During each iteration of the simplex method, a non-basis variable may become a basis variable by entering the basis, while an existing basis variable may exit the basis and become a non-basis variable.

In the simplex table, basis variables are usually represented as columns, while non-basis variables are represented as rows. The table contains coefficients representing the contributions of each variable to the objective function and the constraints. The simplex method iteratively modifies this table to improve the objective function value until an optimal solution is found.

Understanding basis and non-basis variables is crucial for implementing the simplex method and solving linear programming problems efficiently. By iteratively adjusting the basis and non-basis variables, the simplex method converges to the optimal solution of the linear programming problem.

5. Objectives of critical path

The critical path method (CPM) is a project management technique used to identify the sequence of tasks that determine the minimum duration required to complete a project. The critical path represents the longest path through a project network diagram and determines the shortest possible project duration. The primary objectives of identifying and analyzing the critical path include:

1. Determining Project Duration: By identifying the critical path, project managers can determine the shortest possible duration required to complete the project. This information is crucial for planning and scheduling resources effectively.

2. Resource Allocation: Understanding the critical path helps in allocating resources efficiently. Tasks on the critical path cannot be delayed without extending the project's overall duration. Thus, it is essential to ensure that sufficient resources are allocated to critical tasks to prevent delays in the project timeline.

3. Task Prioritization: Tasks on the critical path are critical to the project's success. Therefore, they require special attention and priority in terms of monitoring, management, and resource allocation. By identifying the critical path, project managers can focus their efforts on managing these critical tasks to keep the project on track.

4. Risk Management: The critical path highlights tasks that are most susceptible to delays. By focusing on these critical tasks, project managers can proactively identify potential risks and develop strategies to mitigate them. This helps in minimizing the likelihood of delays and ensuring successful project completion.

5. Schedule Compression: Knowing the critical path allows project managers to identify opportunities for schedule compression. By focusing on critical tasks or adding resources to critical activities, project managers can accelerate the project timeline without affecting its overall duration.

Elective: Operation Research (CBCGS)
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