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)

 

****************************************

Please check whether you have got the right question paper.

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.

__________________________________________

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

Click Here

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)

Click here for Solution  

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

Click here for Solution  

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)

Click Here for Solution

           

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

Click Here for Solution

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)

Click Here for Solution

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)

Click Here for Solution

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)

Click Here for Solution

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

Click Here for Solution

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)

Click Here for Solution

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)

Click Here for Solution

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)

Click Here for Solution

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

A non-degenerate solution is a prerequisite for the optimality test in a transportation problem because the optimality test relies on the existence of a complete and unique set of basic variables that satisfy the feasibility conditions without redundancy.

Reasons for Non-Degenerate Solution is Necessary:

1. Ensures Uniqueness of the Solution

   • A transportation tableau with m supply points and n demand points should have exactly (m + n - 1) basic variables to form a feasible solution.
   • If there are fewer than (m + n - 1) allocations (degeneracy), the potential method (used for optimality testing) may fail to assign unique dual variables (ui and vj), leading to incorrect or ambiguous conclusions.

2. Avoids Singular Systems in the U-V Method

   • The Modified Distribution (MODI) method or Stepping Stone method requires solving a set of equations for dual variables (ui and vj).
   • A degenerate solution may result in a singular system, where some equations become dependent, making it impossible to determine all ui and vj values uniquely.

3. Prevents Cycling or Indeterminate Loops

   • Degeneracy can lead to loop formation during the improvement step in the transportation algorithm, causing the method to enter a cycle without improving the solution.
   • A non-degenerate solution ensures that there is a clear pivoting process to move toward optimality.

4. Maintains the Validity of Opportunity Cost Calculation

   • The opportunity cost (ΔCij) is calculated using the dual variables.
   • If degeneracy exists, some opportunity costs might not be computed correctly, leading to an incorrect assessment of optimality.

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

Definition of Operations Research (OR):

Operations Research (OR) is a scientific approach to decision-making that uses mathematical models, statistical analysis, and optimization techniques to solve complex problems and improve efficiency in organizations. It involves formulating, analyzing, and optimizing problems related to resource allocation, logistics, scheduling, and decision-making under constraints.

Major Application Areas of Operations Research Techniques:

1. Manufacturing & Production Management

   • Inventory control (EOQ models, Just-in-Time)
   • Production scheduling (Job shop scheduling, Assembly line balancing)
   • Facility layout and design

2. Transportation & Logistics

   • Routing and scheduling (Vehicle routing problem, Traveling salesman problem)
   • Supply chain management
   • Warehouse and distribution management

3. Healthcare & Medical Decision-Making

   • Hospital resource allocation (Beds, staff scheduling)
   • Epidemic modeling
   • Optimizing patient flow and appointment scheduling

4. Finance & Banking

   • Portfolio optimization (Risk analysis, Investment strategies)
   • Credit risk management
   • Fraud detection using data analysis

5. Military & Defense

   • War game simulations
   • Logistics and resource planning
   • Strategic defense planning

6. Telecommunications & Network Optimization

   • Optimal network design
   • Bandwidth allocation
   • Traffic management in communication networks

7. Energy & Utilities

   • Power generation and distribution optimization
   • Renewable energy resource allocation
   • Smart grid management

8. Marketing & Revenue Management

   • Demand forecasting
   • Pricing and revenue optimization
   • Customer segmentation and targeted advertising

9. Human Resource Management

   • Workforce planning and scheduling
   • Performance evaluation models
   • Recruitment and selection optimization

10. Sports & Entertainment

• Tournament scheduling
• Performance analysis using data science
• Seating and ticket pricing strategies

OR

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

1. Assumption on LPP

Ans: 

Linear Programming (LP) is a mathematical method used to optimize a linear objective function, subject to linear constraints. The following key assumptions underlie LP models:

1. Linearity: The relationships in the objective function and constraints must be linear. This means that the effect of decision variables is proportional, and there are no powers or nonlinear terms.

2. Additivity: The total effect of decision variables is the sum of their individual effects. Contributions from variables in the objective function and constraints combine additively.

3. Divisibility: Decision variables can take any fractional values (continuous variables). This assumes resources can be divided into smaller parts without loss.

4. Certainty: All coefficients in the objective function and constraints are known with certainty and remain constant. This assumes no variability in parameters like costs, resources, or returns.

5. Non-Negativity: Decision variables cannot take negative values, reflecting real-world constraints where resources or activities cannot have negative quantities

2 Objective of Network analysis

The objective of network analysis is to understand, model, and optimize the structure and behavior of networks to achieve specific goals. Networks can represent various systems, such as communication, social, transportation, electrical, and biological systems. The analysis provides insights into the relationships, dynamics, and efficiency within these systems.

Primary Objectives of Network Analysis:

1. Understand the Structure:

   • Identify nodes (individual entities) and edges (connections).
   • Understand the topology (e.g., centralized, decentralized, hierarchical).
   • Classify networks (e.g., random, scale-free, small-world).

2. Measure Network Properties:

   • Analyze metrics like degree distribution, centrality, clustering coefficient, and path length.
   • Evaluate network density, connectivity, and robustness.

3. Identify Key Nodes and Connections:

   • Detect influential nodes using metrics like degree centrality, betweenness centrality, or eigenvector centrality.
   • Find critical connections that maintain the network's integrity or performance.

4. Detect Communities and Subgroups:

   • Identify clusters or communities within the network.
   • Understand group dynamics and interactions.

5. Optimize Network Performance:

   • Enhance network efficiency by optimizing routing, load distribution, or resource allocation.
   • Minimize bottlenecks and vulnerabilities.

6. Predict and Model Behavior:

   • Forecast the spread of information, diseases, or influence within the network.
   • Simulate network dynamics under different scenarios (e.g., failures, attacks).

7. Enhance Security and Resilience:

   • Identify vulnerabilities and mitigate risks, such as node or link failures.
   • Design robust networks resistant to disruptions.

8. Support Decision-Making:

• Provide actionable insights for network design, expansion, or intervention.
• Assist in resource planning and operational strategies.

3. Project crashing

Ans: 

Project crashing is a project management technique used to reduce the duration of a project by accelerating certain activities. It involves allocating additional resources, such as labor, equipment, or financial investment, to critical path tasks to achieve faster completion. The goal of crashing is to minimize project time while keeping costs and resource use within acceptable limits.

Features of Project Crashing:

1. Focus on Critical Path: Only activities on the critical path (tasks that directly affect the project duration) are considered for crashing.
2. Trade-off Between Time and Cost: Crashing usually increases costs as additional resources are deployed, but it is justified if the time savings bring higher value (e.g., meeting deadlines or avoiding penalties).
3. Cost-Slope Analysis: The cost of reducing the duration of an activity is analyzed to determine the most cost-effective tasks to crash.

Common Methods of Crashing:

• Adding extra workforce or overtime.
• Using more advanced technology or equipment.
• Simplifying or overlapping tasks (fast-tracking).

Project crashing must be carefully planned to avoid diminishing returns, overburdening resources, or impacting project quality.

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)
Year	Month	Link	Link
IMP Q.			Solution
2019	April	Download	Solution
2019	November	Download	Solution
2022	November	Download	Solution
2023	April	Download	Solution
2023	November	Download	Solution
2024	April	Download	Solution
2024	November	Download	Solution

Click Here to Download Question Papers