Operations Research MCQ Quiz - Objective Question with Answer for Operations Research - Download Free PDF

```html

Explanation:

Flow Process Chart and Symbol for 'Inspection'

A flow process chart is a graphical representation used in work study to analyze and improve processes or workflows. It is a tool for mapping out the sequence of operations, inspections, delays, storages, and movements involved in a particular process. Each activity in the flow process chart is represented using standard symbols, making it easier to identify areas for improvement. One of the key activities often depicted in these charts is 'Inspection'.

The symbol used to represent 'Inspection' in a flow process chart is a square. This symbol is standardized and universally accepted in work study practices. The square represents the activity where the quality, quantity, or condition of a product, component, or process is checked or verified. This inspection could be manual or automated, and its purpose is to ensure that the item meets predefined standards or specifications.

Correct Option Analysis:

The correct option is:

Option 1: Square

This option correctly identifies the standard symbol used to represent 'Inspection' in a flow process chart. The square is chosen because it provides a clear visual cue that an inspection is occurring at that particular point in the process. This symbol helps analysts and engineers focus on quality control aspects and locate potential bottlenecks or inefficiencies related to inspections.

Additional Information

Analysis of Other Options:

Option 2: Arrow

The arrow symbol is used in flow process charts to represent movement or transportation of materials, components, or people. It indicates the transfer of items from one location to another within the process. While important in process mapping, it is not related to inspection, making this option incorrect for the given question.

Option 3: Dae

'Dae' does not correspond to any standard symbol used in flow process charts. This option is invalid and does not represent any recognized activity or function in work study practices.

Option 4: Triangle

The triangle symbol in flow process charts represents storage. It is used to depict a point where materials or components are stored temporarily during the process. This symbol is unrelated to inspection activities, making it an incorrect choice for the question.

Option 5: Not Applicable

This option is irrelevant as it does not correspond to any standard symbol or activity in a flow process chart. It does not provide any meaningful information related to inspection.

Conclusion:

The correct answer is Option 1: Square, as it accurately represents 'Inspection' in a flow process chart used in work study. Understanding these symbols is crucial for effectively analyzing and optimizing workflows, ensuring quality control, and improving productivity.

```

Explanation:

Least Cost Method for Solving Transportation Problems

The Least Cost Method (LCM) is a systematic approach for solving transportation problems in operations research. It aims to minimize the cost of transporting goods from multiple supply sources to multiple demand destinations while adhering to the constraints of supply and demand.

Step-by-Step Process:

The first step in the Least Cost Method is to identify the cell with the lowest cost in the transportation matrix and allocate as many units as possible to that cell. This step is essential as it ensures the solution starts with the least cost allocation, thereby paving the way for an optimal solution.

Step 1: Identify the Cell with the Lowest Cost

In a transportation matrix, each cell represents a transportation route between a supply source and a demand destination, along with its associated cost. The first step is to scan the matrix and locate the cell with the lowest transportation cost. This cell is the most economical route for transporting goods, and allocating units to it minimizes the overall cost.

Step 2: Allocate Units to the Selected Cell

Once the cell with the lowest cost is identified, allocate as many units as possible to this cell. The allocation is determined by the minimum value of supply or demand associated with that cell. For example:

• If the supply at the source is 50 units and the demand at the destination is 30 units, allocate 30 units to the cell. The supply at the source will reduce by 30 units, leaving 20 units, while the demand at the destination will become zero.

Step 3: Adjust Supply and Demand Values

After allocating units to the cell, adjust the supply and demand values accordingly. If the supply or demand becomes zero, cross out the respective row or column in the matrix, as it has been completely satisfied. If both supply and demand become zero simultaneously, cross out either the row or column, but ensure a dummy allocation of zero in the remaining cell.

Step 4: Repeat the Process

Repeat the process by identifying the next cell with the lowest cost in the remaining transportation matrix, and allocate units to it. Continue adjusting supply and demand values until all supply and demand requirements are satisfied.

Step 5: Final Allocation

The process ends when all rows and columns in the transportation matrix are crossed out, indicating that all supply and demand constraints have been met. The total transportation cost is calculated by multiplying the allocated units in each cell by their respective transportation costs and summing them up.

Correct Option Analysis:

Option 1: Identify the cell with the lowest cost and allocate as many units as possible.

This option correctly describes the first step in the Least Cost Method. Identifying the cell with the lowest cost ensures that the allocation starts with the most economical route, minimizing transportation costs. Allocating as many units as possible to this cell adheres to the supply and demand constraints while progressing towards an optimal solution.

Additional Information

Analysis of Other Options:

Option 2: Allocate units to the cell with the highest cost. 

This option is incorrect because allocating units to the cell with the highest cost does not minimize the overall transportation cost. The Least Cost Method specifically focuses on minimizing costs by starting with the cell with the lowest transportation cost.

Option 3: Adjust the supply and demand values by adding a dummy row or column.

This option is partially correct but does not represent the first step in the Least Cost Method. Adding a dummy row or column is a step performed when the supply and demand are not balanced, ensuring the problem is solvable. However, the initial step is identifying the cell with the lowest cost and allocating units to it.

Option 4: Cross out rows or columns with zero supply or demand.

This option describes a step performed later in the process, after allocating units to the lowest cost cell and adjusting supply and demand values. It is not the first step in the Least Cost Method.

Conclusion:

Option 1 is the correct answer as it accurately describes the first step in the Least Cost Method for solving transportation problems. Starting with the cell with the lowest cost ensures the solution progresses towards minimizing the overall transportation cost.

```html

Explanation:

Critical Path Method (CPM):

The Critical Path Method (CPM) is a project management technique used to plan, schedule, and control complex projects. It focuses on identifying the longest sequence of dependent tasks required to complete a project, ensuring timely delivery. The critical path is essential because it determines the shortest possible project duration and highlights tasks that cannot be delayed without affecting the overall project timeline.

Primary Purpose of Identifying the Critical Path:

The correct option is:

Option 3: To identify the longest sequence of dependent tasks that dictates the project’s completion time.

Identifying the critical path is crucial for project managers as it provides a clear understanding of the tasks that directly impact the project’s duration. These tasks, known as "critical tasks," require close monitoring and efficient execution because any delays in these tasks will cause delays in the entire project.

Detailed Explanation:

1. Definition: The critical path is the sequence of dependent tasks that forms the longest duration in a project. It determines the minimum time required for project completion.

2. Calculation: To calculate the critical path, project managers use tools like network diagrams, where tasks are represented as nodes, and dependencies are depicted as arrows. The critical path is identified by summing up the durations of tasks along different paths and selecting the one with the longest duration.

3. Importance: The critical path allows project managers to:

• Focus on tasks that must be completed on time to avoid project delays.
• Allocate resources efficiently to critical tasks.
• Identify float (slack) for non-critical tasks, enabling flexibility in resource management.
• Plan realistic timelines and manage stakeholder expectations.

Applications:

The Critical Path Method is widely used across industries, such as construction, software development, event planning, manufacturing, and more. It is particularly valuable for large-scale projects with multiple interdependent tasks.

Important Information

Analysis of Other Options:

Option 1: To calculate the average time required for project completion.

This option is incorrect because the critical path method does not focus on calculating the average time for project completion. Instead, it identifies the sequence of tasks that dictates the shortest project duration.

Option 2: To determine the least expensive way to complete the project.

This option is incorrect because CPM is primarily concerned with time management, not cost optimization. While cost considerations can be integrated into project planning, they are not the primary focus of CPM.

Option 4: To find the most efficient allocation of resources across tasks.

This option is incorrect because CPM does not directly address resource allocation efficiency. Resource leveling or resource optimization techniques are used for that purpose.

Option 5: No relevant content provided for this option.

```

```html

Explanation:

The Purpose of a Slack Variable in a Linear Programming Problem:

A slack variable is a critical concept in Linear Programming (LP) used to transform inequality constraints (≤ type) into equality constraints. The primary purpose of introducing slack variables is to facilitate the application of mathematical techniques, such as the Simplex Method, which requires all constraints to be expressed as equations.

Let us delve deeper into the details of how slack variables work and why they are essential:

1. Understanding the Problem:

Linear Programming problems often involve constraints that are inequality conditions, typically of the form:

• Ax + By ≤ C

Here, 'x' and 'y' are the decision variables, and 'C' is the constant representing the resource or capacity limit.

Such inequality constraints are challenging to handle directly in optimization methods like the Simplex Method, which operates on equations rather than inequalities. This is where slack variables come into play.

2. Definition of a Slack Variable:

A slack variable is a non-negative variable that is added to an inequality constraint (≤ type) to convert it into an equality. Mathematically, if the original constraint is:

• Ax + By ≤ C

We introduce a slack variable 'S' such that:

• Ax + By + S = C

Here, 'S' is the slack variable, and it represents the unused portion of the resource or capacity. For the constraint to hold true, 'S' must always be ≥ 0.

3. Role of Slack Variables in Linear Programming:

The inclusion of slack variables serves several purposes:

• Converting Inequalities to Equations: As mentioned earlier, optimization techniques like the Simplex Method require all constraints to be expressed as equations. Slack variables make this conversion possible.
• Maintaining Feasibility: By ensuring that the slack variables are non-negative, the feasibility of the solution is preserved. This means that the constraints are always satisfied.
• Interpreting Results: The values of the slack variables in the optimal solution provide insights into the utilization of resources. For example, if 'S = 0', it indicates that the corresponding resource is fully utilized. If 'S > 0', it signifies that some portion of the resource remains unused.

4. Example:

Consider a simple LP problem:

• Maximize Z = 3x + 2y
• Subject to:
• 2x + y ≤ 10
• x + 3y ≤ 15
• x, y ≥ 0

To convert the inequalities into equations, we introduce slack variables 'S1' and 'S2':

• 2x + y + S1 = 10
• x + 3y + S2 = 15
• S1, S2 ≥ 0

Now, the constraints are in a form suitable for applying the Simplex Method.

5. Advantages of Using Slack Variables:

• Simplifies Computations: By converting inequalities into equations, slack variables simplify the mathematical computations involved in solving LP problems.
• Enables Graphical and Algebraic Solutions: Slack variables make it possible to use graphical methods for two-variable problems and algebraic methods like the Simplex Method for larger problems.
• Provides Insights: The values of slack variables in the optimal solution help in understanding resource utilization and identifying bottlenecks.

Correct Option Analysis:

The correct option is:

Option 1: To convert a constraint into an equation when the constraint is a ≤ type.

This option accurately describes the purpose of a slack variable. Slack variables are specifically introduced to convert ≤ type constraints into equalities, ensuring compatibility with optimization techniques like the Simplex Method.

Important Information:

Analysis of Other Options:

• Option 2: To reduce the number of variables in the problem.

This option is incorrect. In fact, introducing slack variables increases the number of variables in the problem. However, this increase is necessary to convert inequalities into equations and make the problem solvable using standard optimization techniques.

• Option 3: To convert a constraint into an equation when the constraint is a ≥ type.

This option is incorrect. For ≥ type constraints, surplus variables (not slack variables) are introduced to convert the inequality into an equation. Additionally, artificial variables may also be introduced depending on the method used.

• Option 4: To represent the objective function.

This option is incorrect. Slack variables are not used to represent the objective function. Instead, they are used to modify the constraints. The objective function remains as it is and is maximized or minimized based on the problem requirements.

```

Explanation:

The Two-Phase Method in Linear Programming

The Two-Phase Method is a systematic approach used in linear programming to solve problems that involve artificial variables. It is particularly useful when the initial problem lacks a feasible solution or when the constraints require the introduction of artificial variables to start the simplex algorithm. The method is divided into two phases:

Phase I: The objective of Phase I is to find a feasible solution to the problem. This involves minimizing the sum of all artificial variables. If the minimum value of this sum is zero, then a feasible solution exists, and the algorithm proceeds to Phase II. Otherwise, if the minimum value is greater than zero, it implies that the problem has no feasible solution.

Phase II: Once a feasible solution is obtained in Phase I, the algorithm moves to Phase II. Here, the goal is to solve the original linear programming problem using the original objective function. The feasible solution found in Phase I serves as the starting point for this phase.

Correct Option Analysis:

The correct option is:

Option 1: Phase II involves solving the linear programming problem with the original objective function, starting from the feasible solution obtained in Phase I.

This statement accurately describes the function of Phase II in the Two-Phase Method. Once a feasible solution is identified in Phase I, the focus shifts to optimizing the original objective function. The feasible solution obtained serves as the initial basic feasible solution, and the simplex algorithm proceeds to find the optimal solution.

Additional Information

Important Information:

Let us analyze why the other options are incorrect:

Option 2: Phase II is used to minimize the sum of artificial variables added during Phase I.

This statement is incorrect because minimizing the sum of artificial variables is the primary goal of Phase I, not Phase II. Phase II focuses on solving the original problem with the actual objective function.

Option 3: Phase II is used to determine the feasibility of the solution found in Phase I.

This statement is misleading. The feasibility of the solution is determined in Phase I. Phase II assumes that a feasible solution has already been found and works towards optimizing the original objective function.

Option 4: Phase II involves adding more artificial variables to refine the feasible solution found in Phase I.

This statement is incorrect because no additional artificial variables are added in Phase II. The artificial variables introduced in Phase I are either removed or ignored once a feasible solution is found. Phase II focuses solely on optimizing the original objective function using the feasible solution obtained from Phase I.

Concept:

In CPM:

The standard deviation of critical path:

σ_cp = 

σ_cp = 

Where, σ₁, σ₂, ...., σ₈, σ₉ are the standard deviation of each activity on the critical path   

Calculation:

Given:

σ1, σ2, ...., σ8, σ9 = 3

σcp = 

σ_cp = 

σ_cp =  = 9

∴ the standard deviation of the critical path is 9.

Explanation:

A project may be defined as a combination of interrelated activities which must be executed in a certain order before the entire task can be completed.

The aim of planning is to develop a sequence of activities of the project so that the project completion time and cost are properly balanced.

To meet the objective of systematic planning, the management has evolved several techniques applying network strategy.

PERT (Programme Evaluation and Review Technique) and CPM (Critical Path Method) are network techniques which have been widely used for planning, scheduling and controlling the large and complex projects.

• PERT (Project Evaluation and Review Technique) approach takes account of the uncertainties. In this approach, 3-time values are associated which each activity. So it is probabilistic.

• CPM (Critical Path Method) involves the critical path which is the largest path in the network from starting to ending event and defines the minimum time required to complete the project. So it is deterministic.

Difference between PERT and CPM (Critical Path Method)

Concept:

Project Evaluation and Review Technique (PERT) is probabilistic in nature and is based upon three-time estimates to complete an activity.

Optimistic Time (to): It is the minimum time that will be taken to complete an activity if everything goes according to the plan.

Pessimistic Time (tp): It is the maximum time that will be taken to complete an activity when everything goes against the plan.

Most likely time (tm): It is the time required to complete a project when an activity is executed under normal conditions.

Average or most expected time is given by 

The variance gives the measure of uncertainty of activity completion. The variance of the activity is given by 

Variance, 

Standard duration, 

Calculation:

Given:

t_p = 10 days, t_o = 4 days

The variance of the activity is 1.

Concept:

In a transportation problem with m supply points and n demand points

Number of constraints = m + n

Number of variables = m × n

Number of equations = m + n - 1

Calculation:

Given:

m = 4, n = 5

Number of constraints = m + n = 4 + 5 = 9

The correct answer is Kolkata.

Key Points

• Indian Railways is divided into 18 zones and 73 divisions.
• A Divisional Railway Manager (DRM) heads the division and he/she reports to General Manager (GM).
• A Railway Division is the smallest administrative unit of Railways.
• North Zone is the largest zone.

Given below is the list of all railway zones and their headquarters:

Explanation

Slack or Event Float

• Slack corresponds to the event in PERT.
• Float corresponds to activity in CPM.

Slack

• It is defined as the amount of time by which an event can be delayed without delaying the project schedule.
• Slack of an event = Latest Start Time – Earliest Start Time OR Latest Finish Time – Earliest Finish Time

There are three types of floats.

Concept:

For a system of equation, the number of possible basic solution is calculated by - 

n = number of variables.

m = number of equations.

Inequalities must be converted into equalities.

Calculation:

Given:

X1 + X2 + X3 ≤ 5

X₁ + X₂ + X₃ + S₁ + 0S₂ = 5     (1)

2X1 + X2 + 3X3 ≤ 10

2X1 + X2 + 3X₃ + 0S₁ + S₂ = 10      (2)

n = number of variables = 5

m = number of equations = 2

∴ number of basic solution = 

∴ 

Explanation:

PERT stands for "Program Evaluation and Review Technique". This network model is used for project scheduling.

Difference between PERT and CPM (Critical Path Method)

PERT does take uncertainties involved in the estimation of times, therefore three-time estimates have been taken for the calculation project duration. They are optimistic (t_o), pessimistic (t_p), and most likely (t_m).

Therefore, the option 2 is the incorrect statement among the given options.

Explanation:

If   is the number of units shipped from i^th source to j^th destination, then the equivalent LPP model will be

Minimize 

Subjected to:

If total supply = total demand then it is a balanced transportation problem otherwise it is called an unbalanced transportation problem.

There will be (m + n - 1) basic independent variables out of (m x n) variables.

Explanation:

PERT stands for Program Evaluation and Review Technique and was developed to address the needs of projects for which the time and cost estimates tend to be quite uncertain.

It has a probabilistic approach and hence suitable for the projects which are to be conducted for the first time or projects related to research and development.

PERT uses 3 cases:

• Optimistic time ⇒ estimates the shortest possible time required for the completion of the activity.
• Most likely time ⇒ estimates the time required for the completion of activity under normal circumstances.
• Pessimistic time ⇒ estimates the longest possible time required for the completion of the activity.