Paper/ Subject Code 86001/ Operation Research
TYBMS Semester 6 Operation Research
(Q.P. November 2022 with Solution)
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.
_________________________________________________
Q.1 A) State whether following statements True or False:(Attempt any8) (8)
1. Operation Research, Methodology consists of definition, solution and validation only.
Ans: False. Operation Research methodology typically involves problem formulation, model construction, solution, validation, and implementation phases.
2. A linear programing problem may have more than one set of solutions.
Ans: True. A linear programming problem can have multiple solutions, especially if the feasible region is unbounded.
3. In graphical method of LPP, if the objective function is parallel to a constraints, the constraint is infeasible.
Ans: False. If the objective function is parallel to a constraint line in the graphical method of linear programming, it means the constraint is redundant or there are infinite feasible solutions, not necessarily infeasible.
4. It is possible for an equation in the simplex table to have both a slack and surplus variable at the same time
Ans: False. In the simplex method, an equation in the simplex table cannot have both a slack and a surplus variable at the same time because they represent different types of variables.
5. The Hungarian method for solving an assignment problem can also be used to solve a transportation problem.
Ans: True. The Hungarian method, which is an algorithm for solving the assignment problem, can be adapted to solve transportation problems.
6. Regret matrix is made to convert a maximization problem into minimization problem in assignment.
Ans: False. The regret matrix is used in decision theory to assess the consequences of different decisions, but it doesn't necessarily convert a maximization problem into a minimization problem.
7. In transportation problem, number of basic allotted (or Occupied) cells should be exactly m + n + 1 that it becomes non-degenerate.
Ans: False. The number of basic allotted cells in a transportation problem should be exactly m + n - 1, where m is the number of sources and n is the number of destinations, to ensure non-degeneracy.
8. In PERT, the total project completion time follows a normal probability distribution.
Ans: False. In PERT (Program Evaluation and Review Technique), the total project completion time follows a beta distribution, not a normal distribution.
9. The orders of completion of jobs are dependent on the sequence of jobs.
Ans: True. The order of completion of jobs can be dependent on the sequence of jobs in certain scheduling problems.
10. Different saddle points in the same payoff matrix always have the same payoff.
Ans: False. Different saddle points in the same payoff matrix can have different payoffs.
Q.1. (B) Match the right and closely related answer from Column Y with the text/term given in Column X. (Attempt Any 7 question) (7)
|
Column X |
Column Y |
|
1) Key Column |
a) Method of penalties |
|
2) feasible region |
b) Crash cost per day |
|
3)VAM Method |
c) Region of feasible solution |
|
4) Dummy Row or Column |
d) Incoming variable in Simplex |
|
5) NWCR |
e) LST-EST or LET-EFT |
|
6) Cost Slope in Crashing |
f) Optimistic time |
|
7) Total Float value |
g) Saddle point not available in the game |
|
8) Shortest activity time in PERT |
h) Unbalance problem |
|
9) Mixed strategy game |
i) Time required by a job or any machine |
|
10) Processing time |
j) Top left side corner of the side |
Ans:
Column X | Column Y |
1) Key Column | d) Incoming variable in Simplex |
2) feasible region | c) Region of feasible solution |
3)VAM Method | a) Method of penalties |
4) Dummy Row or Column | h) Unbalance problem |
5) NWCR | j) Top left side corner of the side |
6) Cost Slope in Crashing | b) Crash cost per day |
7) Total Float value | e) LST-EST or LET-EFT |
8) Shortest activity time in PERT | f) Optimistic time |
9) Mixed strategy game | g) Saddle point not available in the game |
10) Processing time | i) Time required by a job or any machine |
Q.2. (A) Use graphical method to solve the following linear programming problem: (8)
Objective Function:
Max Z = 40X + 80Y
Subject to: 2X + 3Y ≤ 48
X ≤ 15
Y ≤ 10
X ≥ 0
Y ≥ 0
Q.2. (B) Max Z = 100 + 80Y (Max Z = 100X + 80Y )
Subject to: 6X + 4Y ≤ 7200
2X + 4Y ≤ 4000
X ≥ 0 Y ≥ 0
Find Optimal Solution by Simplex Method.
OR
Q.2. (C) A Company is transporting its units from three factories F_{1} F_{x} F_{J} to four warehouses W_{1} W_{2} W_{3} and W_{4} The supply and demand of units with transportation cost per unit ( Rs .) are given below with feasible solution (The numbers which are in circle indicates number of units transported from Factory to warehouse)
i) Test the solution for optimality (3)
(ii) If solution is not optimal find optimal solution. (4)
Q.2. (D) Five salesmen are to be assigned to five territories. Based on past performance, the following table shows the annual sales (is Rs.lakh). that can be generated by each salesman in each territory. Find Optimum assignment to maximize sales, (8)
Salesman | Territory | ||||
T1 | T2 | T3 | T4 | T5 | |
S1 | 26 | 14 | 10 | 12 | 9 |
S2 | 31 | 27 | 30 | 14 | 16 |
S3 | 15 | 18 | 16 | 25 | 30 |
S4 | 17 | 12 | 21 | 30 | 25 |
S5 | 20 | 19 | 25 | 16 | 10 |
Salesman | Territory | ||||
T1 | T2 | T3 | T4 | T5 | |
S1 | 5 | 17 | 21 | 19 | 22 |
S2 | 0 | 4 | 1 | 17 | 15 |
S3 | 16 | 13 | 15 | 6 | 1 |
S4 | 14 | 19 | 10 | 1 | 6 |
S5 | 11 | 12 | 6 | 15 | 21 |
Salesman | Territory | ||||
T1 | T2 | T3 | T4 | T5 | |
S1 | 0 | 12 | 16 | 14 | 17 |
S2 | 0 | 4 | 1 | 17 | 15 |
S3 | 15 | 12 | 14 | 5 | 0 |
S4 | 13 | 18 | 9 | 0 | 5 |
S5 | 5 | 6 | 0 | 9 | 15 |
Salesman | Territory | ||||
T1 | T2 | T3 | T4 | T5 | |
S1 | 0 | 12 | 16 | 14 | 17 |
S2 | 0 | 4 | 1 | 17 | 15 |
S3 | 15 | 12 | 14 | 5 | 0 |
S4 | 13 | 18 | 9 | 0 | 5 |
S5 | 5 | 6 | 0 | 9 | 15 |
Salesman | Territory | ||||
T1 | T2 | T3 | T4 | T5 | |
S1 | 0 | 12 | 16 | 14 | 17 |
S2 | 0 | 4 | 1 | 17 | 15 |
S3 | 15 | 12 | 14 | 5 | 0 |
S4 | 13 | 18 | 9 | 0 | 5 |
S5 | 5 | 6 | 0 | 9 | 15 |
Salesman | Territory | ||||
T1 | T2 | T3 | T4 | T5 | |
S1 | 0 | 12 | 16 | 14 | 17 |
S2 | 0 | 0 | 1 | 17 | 15 |
S3 | 15 | 12 | 14 | 5 | 0 |
S4 | 13 | 18 | 9 | 0 | 5 |
S5 | 5 | 6 | 0 | 9 | 15 |
Salesmen | Territory | Sales (Rs. Lacs) |
S1 S2 S3 S4 S5 | T1 T2 T3 T4 T5 | 26 27 30 30 25 |
Optimal Sales = Rs. 138 Lacs |
|
Activities |
Time (Days) |
|
A (1-2) |
3 |
|
B (1-3) |
4 |
|
C(1-4) |
6 |
|
D (2-3) |
5 |
|
E (3-6) |
6 |
|
F (4-7) |
5 |
|
G (5-8) |
4 |
|
H (6-8) |
7 |
|
I (7-8) |
4 |
(i) Construct a network diagram, find critical path and project completion time. (3)
(ii) Tabulate/Calculate Earliest Start and Finish Time, Latest Start and Finish Time and Total Float. (5)
Q.3. (B) From the data given below
(1) Draw a diagram (1)
(2) Find Critical path (1)
3. Crash Systematically the activities any determine optimal project duration: (4)
Activities | 1-2 | 1-2 | 2-4 | 2-5 | 3-4 | 4-5 |
Normal time | 8 | 4 | 2 | 10 | 5 | 3 |
Normal cost | 100 | 150 | 50 | 100 | 100 | 80 |
Crash time | 6 | 2 | 1 | 5 | 1 | 1 |
Crashed cost | 200 | 350 | 90 | 400 | 200 | 100 |
Indirect cost is Rs. 70 per day
Solution:
OR
Activities | Average Expected Time in weeks (te) | Standard deviation |
1-2 | 3 | 4/6 |
1-3 | 4 | 4/6 |
2-5 | 5 | 4/6 |
2-4 | 6 | 2/6 |
5-6 | 7 | 4/6 |
4-6 | 8 | 4/6 |
3-6 | 9 | 4/6 |
6-7 | 3 | 2/6 |
Solution:
_1.jpg)
_2.jpg)
Q.4. (A) You are given the Pay-off (Profit in Rs.) matrix in respect of Two-Person-Zero-sum Game as follows.
(i) Find the Maximin Strategy
(ii) Find the Minimax Strategy
(iii) What is the value of the Game?
Q.4. (B) Six jobs I, II, III, IV and VI are to be processed on two machine
Jobs
Processing Time (Min.)
Machine A
Machine B
I
5
8
II
2
6
III
10
3
Iv
9
4
V
6
3
VI
8
9
Jobs
Processing Time (Min.)
Machine A
Machine B
I
5
8
II
2
6
III
10
3
Iv
9
4
V
6
3
VI
8
9
(i) Find the sequence that minimizes the total elapsed time required to complete the jobs. (2)
(ii) Calculate the total elapsed time (3)
(iii) Idle time on for each Machine (3)
OR
Q.4. (C) 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:
(i) complete the job Find the sequence that minimizes the total elapsed time required to
(ii) Calculate the total elapsed time
(iii) Idle time on Machine A. Machine B and Machine C
Q.4 (D) A company produces three product A, B, fer manufacturing (3) raw material P, Q, R are used and Profit per unit for A = Rs * 0.5 B = Rs. 3 and for C = Rs 4
Resources required per units are given below
Maximum raw material availability is P = 80 units, Q = 100 units R = 150 units. (Formulate linear programming problem
.jpg)
Q.5. (A) Define Operations Research and what are the major application areas for operation research techniques?
Ans: Operations Research (OR) is a field of study that uses mathematical modeling, optimization techniques, and analytical methods to solve complex decision-making problems in various domains. It encompasses a wide range of quantitative methods and tools to help organizations make better decisions, improve efficiency, and optimize resource allocation.
Major application areas for Operations Research techniques include:
1. Supply Chain Management: OR techniques are widely used to optimize supply chain operations, including inventory management, production planning, distribution network design, and transportation logistics. OR helps organizations streamline their supply chain processes, reduce costs, and improve customer service levels.
2. Transportation and Logistics: OR methods are applied to solve transportation and logistics problems, such as vehicle routing, scheduling, facility location, and freight optimization. These techniques help companies improve the efficiency of their transportation systems, minimize transportation costs, and enhance delivery performance.
3. Manufacturing and Production: OR plays a crucial role in optimizing manufacturing and production processes. It helps companies with capacity planning, production scheduling, resource allocation, facility layout design, and quality control. By applying OR techniques, organizations can increase productivity, reduce lead times, and optimize production costs.
4. Finance and Investment: OR techniques are used in financial modeling, portfolio optimization, risk management, and investment decision-making. These methods help investors and financial institutions make informed decisions regarding asset allocation, portfolio diversification, and risk mitigation strategies.
5. Healthcare Management: OR is applied in healthcare to optimize resource allocation, patient scheduling, hospital operations, healthcare facility layout, and healthcare delivery processes. OR techniques help healthcare providers improve patient outcomes, reduce waiting times, and optimize resource utilization.
6. Project Management: OR methods are utilized in project management to optimize project scheduling, resource allocation, budgeting, and risk analysis. These techniques help project managers plan and execute projects more effectively, minimize project delays, and maximize project outcomes.
7. Marketing and Revenue Management: OR techniques are used in marketing analytics, pricing optimization, demand forecasting, and revenue management. These methods help companies optimize pricing strategies, maximize revenue, and improve marketing campaign effectiveness.
8. Environmental Management: OR is applied in environmental modeling, resource allocation, pollution control, and sustainability analysis. OR techniques help policymakers and environmental scientists make informed decisions regarding environmental conservation, natural resource management, and sustainable development.
Operations Research techniques have diverse applications across various industries and sectors, helping organizations optimize processes, make better decisions, and achieve their strategic objectives efficiently.
(B) What are the limitation of operation research techniques?
Ans: Quantitative techniques can be classified in two groups, statistical techniques and programming techniques.
OR
Q.5. (C) Answer Any 3 of the following (15)
(i) Feasible region f solution in LPP graphical method.
Ans: In Linear Programming Problems (LPPs), the feasible region represents the set of all feasible solutions that satisfy the constraints of the problem. When using the graphical method to solve LPPs, the feasible region is visually represented by the area in the graph where all constraints are simultaneously satisfied.
The feasible region is determined in the graphical method:
1. Plot Constraints: Each constraint in the LPP is represented as a linear inequality on the graph. For example, if the constraint is \(ax + by \leq c\), then it represents a line on the graph. If it is an equality, then it represents a boundary line, and if it is an inequality, it represents a boundary line and a half-plane.
2. Identify Feasible Region: The feasible region is the intersection of all half-planes formed by the constraints. It is the region of the graph where all constraints are satisfied simultaneously. This region is typically bounded by the constraint lines or may extend to infinity if the problem is unbounded.
3. Determine Feasible Solutions: Any point within the feasible region represents a feasible solution to the LPP. These solutions satisfy all the constraints of the problem. The objective is to find the optimal solution within this feasible region that maximizes or minimizes the objective function.
4. Optimal Solution: Once the feasible region is identified, the optimal solution is determined by evaluating the objective function at various points within the feasible region. The point that maximizes or minimizes the objective function while lying within the feasible region is considered the optimal solution.
(ii) 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.
(iii) Three time estimates in PERT
Ans: In Program Evaluation and Review Technique (PERT), three time estimates are utilized to estimate the duration of activities within a project. These estimates help in predicting the most likely duration for completing each activity. The three time estimates are:
1. Optimistic Time (a): This is the shortest possible time that an activity can be completed in under ideal conditions. It assumes that everything proceeds as smoothly as possible without any delays or obstacles. The optimistic time estimate represents the best-case scenario.
2. Most Likely Time (M): This estimate represents the duration that the activity is most likely to take under normal circumstances. It considers typical factors such as resources, skills, and potential obstacles that may arise during the activity's execution.
3. Pessimistic Time (b): The pessimistic time estimate reflects the longest duration that an activity might take if everything goes wrong or if unexpected problems occur. It considers worst-case scenarios, such as resource shortages, technical difficulties, or unforeseen delays.
These three time estimates are used to calculate the Expected Time (TE) for each activity using the formula:
Te = a
+ 4m + b / 6
Once the Expected Time for each activity is determined, it is used in conjunction with the project network diagram to calculate the overall project duration and identify the critical path—the sequence of activities with the longest total duration, which determines the minimum time required to complete the project.
(iv) 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.
(v) Objectives of critical path
Ans: 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.
Subject | April Q.P. 2019 |
Operation Research | |
Media Planning & Management | |
International Marketing | |
Brand Management | |
Retail Management | |
International Financial |
){kind=link}
0 Comments