Paper/ Subject Code 86001/ Operation Research
TYBMS SEM 6 :
Operation Research
(Q.P. April 2024 with Solution)
Important Write a short Notes
1) April 2019 Q.P. with Solution (PDF)
2) November 2019 Q.P. with Solution (PDF)
4) April 2023 Q.P. with Solution (PDF)
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.
______________________________________
Q1 (A) Multiple choice questions (Attempt Any 8): (8)
1. A BFS of a LPP is said to be _______ if at least one of the basic variables is zero
a) Degenerated
b) Non-degenerated
c) Infeasible
d) Feasible
2 . For solving an assignment problem, which method is used?
a) Hungarian
b) American
c) German
d) Italian
3. A feasible solution is called a basic feasible solution if the number of non-negative allocations is equal to _______.
a) m-n+1
b) m-n-1
c) m-n-1
d) m+n-1
4. Which method is an iterative procedure for solving LPP in a finite number of steps
a) Simplex algorithm
b) Simplex method
c) slack method
d) M-method
5. An objective function is maximized when it is a ________ function.
a) Profit
b) passive
c) cost
d) time
6. In an assignment problem involving 5 workers & 5 jobs, the total number of assignments possible is
a) 15
b) 10
c) 5
d) 20
7. is known as the time by which activity completion time can be delayed without affecting the start of succeeding activities,
a) Total float
b) interfering float
c) independent float
d) Free float
8. The probability of a normal curve is ________
a) 60%
b) 70%
c) 50%
d) 55%
9. What is the probability of project completing in 55 days if the expected project completion time is 47 days & table value is :(+0.4890)
a) 98.90%
b) 99%
c) 90%
d)98%
10. What is the total idle time if jobs are processed on 2 machines and idle time on machine A is 11 and on machine B is 15.
a) 25
b) 26
c) 23
d) 24
Q1 (B) True or false (Attempt Any 7) operation: (7)
i. concerned with using scientific approach i.e. logical reasoning to solve problems for the management by ensuring optimum utilization of resources
Ans: TRUE Linear programming is indeed concerned with using a scientific approach (logical reasoning) to solve problems for management by ensuring optimal utilization of resources. It's a mathematical method to find the best outcome (maximum profit, minimum cost, etc.) considering constraints.
ii. Any change in the constraint inequalities will have a proportional change in the objective function
Ans: FALSE Changes in constraint inequalities won't necessarily result in a proportional change in the objective function. The relationship between constraints and the objective function is linear, but the impact of a change depends on the specific constraints and objective function being optimized.
iii. In graphical method, infeasibility happens we cannot find feasible region.
Ans: TRUE. In the graphical method of LP, infeasibility occurs when there's no intersection between the constraint lines, indicating no feasible region that satisfies all constraints simultaneously.
iv. Graphical method can be used when the number of decision variable at two
Ans: FALSE The graphical method can be used for problems with more than two decision variables, although it becomes cumbersome to visualize as the number of variables increases. For higher dimensions, the simplex method is a more efficient solution technique.
V An artificial variable is a fictitious variable in LPP problems.
Ans: TRUE Artificial variables are fictitious variables introduced in the simplex method (a common LP technique) to convert an infeasible LP problem into a feasible one for initial calculations. They are eventually removed from the optimal solution.
vii. Surplus variables represent an excess amount of resources utilize
Ans: TRUE Surplus variables represent the unused amount of resources (slack) in a constraint. They are non-negative in the optimal solution and indicate how much a constraint can be relaxed without affecting the objective function.
vii . When the number of lines is not equal to size of matrix the solution is optimum.
Ans: FALSE The number of lines (constraints) not being equal to the size of the matrix doesn't guarantee an optimal solution. There are specific conditions for optimality in LP, and this statement doesn't provide a sufficient criterion.
viii. There are two types of techniques available to find the initial basic feasible solution.
Ans: TRUE There are two main methods to find the initial basic feasible solution in the simplex method: the Northwest corner method and the least-cost method (also known as Vogel's approximation method). Both methods provide a starting point for the iterative optimization process.
ix. The network can have one or more start node and end node.
Ans: TRUE A network in LP (network simplex method) can have one or more source nodes (representing starting points with available resources) and one or more destination nodes (representing goals to be achieved).
X. Pessimistic time is the shortest time period expected to complete the activity.
Ans: FALSE Pessimistic time refers to the longest expected time to complete an activity, not the shortest. It's a concept used in project management, particularly in PERT (Program Evaluation and Review Technique), to estimate project durations considering potential delays.
Q.2 A) A Company manufactures two products A and B. Te manufacture one unit of A. 1.5 mackane hours and 2.5 labour hours are requited. To manufacture product B, 2.5 machine hours and 1.5 labour hours are required. In a month, 300 machine hours and 240 labour hours are available Profit per unt, for A is Rs. 50 and for B is Rs. 40 Formulate as LPP (8)
Q.2 B) Solve following LPP by Simplex method. (7)
Maximize Z = 50X1 + 20X2
Subject to Constraints
20X1 + 10X2 ≤ 500
40X1 + 50X2 ≤ 100
X1, X2 ≥ 0
OR
Q.2 C) Solve following LPP by Graphical method. (7)
Maximize Z = 2X1 + 10X2
Subject to Constraints
2X1 + 5X2 ≤ 16
6X1 ≤ 30
X1, X2 ≥ 0
Q.2 D) A Sales manager has to assign salesmen to four territories. He has four candidates of varying experience and capabilities. The manager assesses the possible profit for each salesman in each territory as given below
|
Salesman |
Territory |
|||
|
|
T1 |
T2 |
T3 |
T4 |
|
S1 |
35 |
27 |
28 |
37 |
|
S2 |
28 |
34 |
29 |
40 |
|
S3 |
35 |
24 |
32 |
33 |
|
S4 |
24 |
32 |
25 |
28 |
Find the assignment of salesmen to the territories so that total profit is Maximum. (8)
Q.3 A) From the Following details of the project
i. Draw the network diagram and identify critical path (3)
ii. Find out Earliest Start and Finish Time. Latest Start and Finish Time of Each activity
|
Activity |
Node |
Duration (Days) |
|
A |
1-2 |
4 |
|
B |
1-3 |
6 |
|
C |
1-5 |
13 |
|
D |
2-3 |
5 |
|
E |
2-4 |
20 |
|
F |
4-6 |
10 |
|
G |
3-6 |
6 |
|
h |
5-6 |
16 |
Q.3 B) A company is transporting it units from three factories F1, F2, F3 with the production capacities of 11,13 and 19 units (in thousands). It has four warehouses W1. W2, and W3. With demands of 6, 10, 12 and 15 units (in thousands).
units cost of transportation is given from each factory to each warehouse.
|
|
W1 |
W2 |
W3 |
W4 |
|
F1 |
42 |
32 |
50 |
26 |
|
F2 |
34 |
36 |
28 |
46 |
|
F3 |
64 |
54 |
36 |
82 |
Construct transportation table and Find Initial feasible solution by Least Cost Method (LCM) (7)
OR
Q.3 C) From the data given below
i. Draw a diagram (2)
ii. Find Critical path (2)
iii. Crash systematically the activities and determine optimal project duration (4)
|
Activity |
Normal
Duration (Days) |
Crash
Cost per day(Rs) |
Maximum
possible Crash Time |
|
1-2 |
6 |
80 |
2 |
|
1-3 |
8 |
90 |
4 |
|
1-4 |
5 |
30 |
2 |
|
2-4 |
3 |
- |
0 |
|
2-5 |
5 |
40 |
2 |
|
3-6 |
12 |
200 |
4 |
|
4-6 |
8 |
50 |
3 |
|
5-6 |
6 |
- |
0 |
Cost of completing eight activities in normal time is Rs. 6500 indirect cost Rs. 160 per day.
Q. 3 D) Five jobs I,II,III,IV and V are to be processed on two machine A and B in order AB
|
Jobs |
Processing Time (Min) |
|
|
|
Machine A |
Machine B |
|
I |
90 |
70 |
|
II |
40 |
80 |
|
III |
40 |
50 |
|
IV |
30 |
10 |
|
V |
25 |
35 |
1) Find the sequence that minimizes the total elapsed time (2)
2) Calculate the total elapsed time (3)
3) Idle time on for each Machine (3)
Q.4 A) There are Six jobs (namely 1,2,3,4,5 and 6), each of which must go through machines. A, and C in the order ABC. Processing Time (in hours) are given below:
|
Jobs |
1 |
2 |
3 |
4 |
5 |
6 |
|
Machine A |
12 |
8 |
7 |
11 |
10 |
5 |
|
Machine B |
3 |
4 |
2 |
5 |
2 |
4 |
|
Machine C |
7 |
10 |
9 |
6 |
11 |
4 |
|
Player A |
Player B |
||||
|
|
B1 |
B2 |
B3 |
B4 |
|
|
A1 |
500 |
260 |
200 |
210 |
|
|
A2 |
-50 |
-100 |
-40 |
240 |
|
|
A3 |
200 |
400 |
160 |
-20 |
|
|
A4 |
250 |
300 |
100 |
50 |
|
Q.4 D) A small project consist of seven activities. Optimistic, most likely and pessimistic time estimated in days are given below
i) Construct the network diagram of PERT network and find expected completion time of the project. (3)
ii) Determine the probability of completing the project in 21 days. (4)
Q.5 A) Define operation Research and What are the Characteristics of Operation research techniques? (8)
Ans:
Operations Research (OR) is a scientific approach to problem-solving and decision-making in complex systems. It involves the application of mathematical models, statistical analysis, and optimization techniques to find the most effective solutions to decision problems, often involving the allocation of limited resources such as time, money, or materials.
Characteristics of Operations Research Techniques
Scientific and Systematic Approach:
- OR applies scientific methods to analyze problems systematically, ensuring logical and objective decision-making.
Interdisciplinary Nature:
- Combines mathematics, statistics, economics, engineering, and other disciplines to address complex problems.
Optimization Focus:
- Aims to find the best possible solution (maximum profit, minimum cost, or optimal resource utilization) under given constraints.
Use of Mathematical Models:
- Represents real-world problems using mathematical models to simulate, analyze, and optimize systems.
Data-Driven Analysis:
- Relies on accurate and quantitative data to formulate problems and validate solutions.
Focus on Complex Systems:
- Deals with multi-variable and multi-objective problems involving interdependent components.
Decision Support:
- Provides quantitative solutions and insights to support managerial and operational decisions.
Iterative Process:
- OR techniques often involve iterative refinements, using simulations and feedback to enhance results.
Computer-Based Solutions:
- Leverages computational tools and software for solving large and complex problems efficiently.
B) Explain Objective of Project Crashing of Network analysis.
Ans:
The primary objective of project crashing in network analysis is to reduce the overall project duration in the most cost-effective manner. It is particularly useful in situations where completing the project earlier is crucial, such as meeting deadlines, avoiding penalties, or taking advantage of market opportunities.
Objectives:
Minimize Project Duration:
- To shorten the project timeline by accelerating critical path activities.
- This ensures the project is completed earlier than originally planned.
Cost-Time Trade-Off Optimization:
- Crashing evaluates the trade-off between additional costs incurred by expediting tasks (e.g., overtime, additional resources) and the benefits of early project completion.
- The goal is to achieve the least additional cost for the desired time reduction.
Meet Deadlines or Critical Milestones:
- To ensure that critical project milestones or external deadlines are met, such as contractual obligations or regulatory requirements.
Respond to External Pressures:
- To address external factors like competitor actions, seasonal market demands, or client requests that demand early completion.
OR
Q.5 C) Write a Short notes on Any Three
1) Degeneracy in transportation
Degeneracy in transportation problems occurs when the number of basic variables in a feasible solution is less than , where is the number of supply points and is the number of demand points. This violates the requirement for a basic feasible solution in linear programming.
Causes of Degeneracy:
- Insufficient Allocations: It arises when not enough independent allocations are made due to certain constraints or when supply and demand are perfectly balanced.
- Tied Values: During the allocation process, if there are multiple equally optimal choices, degeneracy may result.
Implications of Degeneracy:
- It can make solving the transportation problem more complex, as the usual optimization methods (like the stepping stone method or MODI method) may fail without adjustments.
- Degeneracy may lead to cycling or incorrect evaluations in the solution process.
Resolving Degeneracy:
To resolve degeneracy, artificial allocations of zero units (dummy allocations) are introduced in empty cells of the transportation table to satisfy the condition while maintaining the feasibility of the solution.
ii) Dummy activity in network analysis
A dummy activity is an artificial or hypothetical activity used in network diagrams, particularly in project management techniques like CPM (Critical Path Method) and PERT (Program Evaluation and Review Technique). It does not consume time, resources, or cost but is introduced to ensure the logical representation of project dependencies.
Purposes of Dummy Activities:
- Logical Dependencies: To represent relationships between tasks where one task's start or completion depends on another, without directly linking them.
- Avoiding Ambiguity: To differentiate between two activities that share the same start and end points.
- Network Clarity: To maintain the correct sequencing of activities without altering the diagram's logic.
Representation:
- Dummy activities are represented by dashed arrows in the network diagram.
For example, if Task B depends on Task A, and Task C depends on both A and B, a dummy activity may be introduced to clarify these dependencies.
iii) Three time estimates in PERT
In the Program Evaluation and Review Technique (PERT), three-time estimates are used to account for uncertainty in project activities. These estimates help calculate the expected time for each task, considering variability and risk. The three-time estimates are:
Optimistic Time (O): The shortest time an activity can take, assuming everything goes perfectly without any delays.
Most Likely Time (M): The time an activity is expected to take under normal conditions, with standard challenges and delays.
Pessimistic Time (P): The longest time an activity might take, assuming significant delays or challenges.
Expected Time Formula:
The expected time (TE) is calculated using a weighted average:
This formula gives more weight to the most likely estimate while considering variability.
Using these estimates, PERT helps in:
- Predicting project timelines with greater accuracy.
- Identifying potential risks in task durations.
- Supporting probabilistic scheduling to enhance project planning.
iv) Project crashing
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:
- Focus on Critical Path: Only activities on the critical path (tasks that directly affect the project duration) are considered for crashing.
- 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).
- 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.
v) Assumption in LPP
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:
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.
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.
Divisibility: Decision variables can take any fractional values (continuous variables). This assumes resources can be divided into smaller parts without loss.
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.
Non-Negativity: Decision variables cannot take negative values, reflecting real-world constraints where resources or activities cannot have negative quantities.
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Elective: Operation Research (CBCGS) | |||
Year | Month | Question Papers | Link |
IMP Q. |
|
| |
2019 | April | ||
2019 | November | ||
2022 | November | Download | |
2023 | April | ||
2024 | April | ||
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