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

Explanation:

PERT (Program Evaluation and Review Technique)

• PERT is a project management tool used to schedule, organize, and coordinate tasks within a project. It is an event-oriented technique designed to analyze and represent the tasks involved in completing a project. By focusing on events or milestones, PERT helps in planning and managing complex projects with numerous interdependent activities.
• PERT charts are created by identifying all the activities required to complete a project and the sequence in which they need to be executed. Each activity is represented as an arrow, and the events or milestones are represented as nodes. The technique estimates the shortest, longest, and most likely time required to complete each task, allowing for better planning and control over project timelines.

Applications: PERT is widely used in industries such as construction, software development, engineering, and research and development, where complex projects require detailed planning and coordination.

Explanation:

Dummy Activity

• A dummy activity is an artificial activity introduced in project management and network analysis to maintain the logical sequence and dependencies of activities. It does not represent any actual task or consume any resources or time. Dummy activities are typically represented by dotted lines in network diagrams.

Purpose and Importance:

• Maintaining logical relationships: Dummy activities are introduced to maintain the logical dependencies between tasks where no actual activity exists.
• Clarifying network paths: They help clarify the sequencing and flow of activities in a network diagram, ensuring accurate representation of the project plan.
• Preventing ambiguity: Dummy activities eliminate ambiguity in situations where two or more activities depend on the completion of a common predecessor but are not directly related to each other.

Representation:

• Dummy activities are represented by dotted lines in network diagrams.
• The dotted lines show that the activity is artificial and does not represent any real work or resource consumption.

Characteristics:

• Does not consume time: Dummy activities do not have any duration or consume any time in the project schedule.
• Does not consume resources: Since dummy activities are artificial, they do not require resources for completion.
• Introduced artificially: Dummy activities are created artificially to establish logical relationships and dependencies in the project network diagram.

Explanation:

Critical Path Method (CPM):

• The Critical Path Method (CPM) is a project management technique used to determine the sequence of activities that directly affect the project completion time. It identifies the longest path of dependent activities in the project schedule, known as the critical path. The duration of this path determines the shortest possible project completion time.

Critical Path and Slack:

• The critical path in a project is defined as the sequence of tasks where any delay in one task would directly result in a delay in the overall project completion. The slack (or float) is the amount of time an activity can be delayed without affecting the project’s completion date. For activities on the critical path, the slack is zero because there is no flexibility to delay these activities without impacting the entire project.

Why Slack on the Critical Path is Zero:

• The critical path represents the longest duration path in the project network. If any activity on this path is delayed, the entire project completion time will also be delayed.
• Slack is calculated as the difference between the latest allowable finish time (LF) and the earliest finish time (EF) of an activity:
  Slack = LF - EF
• For activities on the critical path, the LF equals EF because they determine the project’s end date. Therefore, their slack is zero.

Explanation:

Critical Path Method (CPM):

Definition: The Critical Path Method (CPM) is a project management technique used to analyze and schedule tasks within a project. It identifies the longest sequence of dependent tasks (known as the critical path) that determines the shortest possible project duration. Any delay in the tasks on the critical path will directly impact the overall project completion time.

How to Identify the Critical Path:

• List all the tasks required to complete the project.
• Define dependencies between tasks (i.e., which tasks must be completed before others can start).
• Determine the duration of each task.
• Calculate the earliest start (ES) and finish (EF) times for each task by performing a forward pass through the network diagram.
• Calculate the latest start (LS) and finish (LF) times for each task by performing a backward pass through the network diagram.
• Identify the tasks with zero slack (i.e., tasks where ES = LS and EF = LF). These tasks form the critical path.

To identify the critical path, we analyze the given network diagram and follow these steps:

• Step 1: List all paths in the network diagram and calculate the total duration for each path.
• Step 2: Identify the path with the longest duration. This is the critical path because it governs the minimum time required to complete the project.

Based on the given data:

• Path 1: 1-2-3-7
  • Total duration = Sum of durations of tasks along this path.
• Path 2: 1-2-4-5-6-7
  • Total duration = Sum of durations of tasks along this path.
• Path 3: 1-2-4-5-6
  • Total duration = Sum of durations of tasks along this path.
• Path 4: 1-2-4-7
  • Total duration = Sum of durations of tasks along this path.

From the calculations, Path 2 (1-2-4-5-6-7) has the longest duration. Therefore, it is the critical path.

Concept:

This problem involves a queuing system with Poisson arrivals and exponential service times (M/M/1 queue). The probability that an arrival does not have to wait for service is equal to the probability that the system is idle (i.e., no customers are being served).

Given:

• Average time between arrivals (\(\lambda^{-1}\)) = 12 min → Arrival rate, \(\lambda = \frac{1}{12}\) customers per minute
• Average service time (\(\mu^{-1}\)) = 5 min → Service rate, \(\mu = \frac{1}{5}\) customers per minute

Step 1: Calculate the Traffic Intensity (ρ)

The traffic intensity is the ratio of arrival rate to service rate:

\[ \rho = \frac{\lambda}{\mu} = \frac{\frac{1}{12}}{\frac{1}{5}} = \frac{5}{12} ≈ 0.4167 \]

Step 2: Determine the Probability of System Being Idle (P₀)

For an M/M/1 queue, the probability that the system is idle (no customers in the system) is:

\[ P_0 = 1 - \rho = 1 - \frac{5}{12} = \frac{7}{12} ≈ 0.5833 \]

The probability that an arrival does not have to wait for service is equal to the probability that the system is idle, which is P₀ ≈ 0.5833

Concept:

In CPM:

The standard deviation of critical path:

σ_cp = \(\sqrt {Sum\;of\;variance\;along\;critical\;path} \)

σ_cp = \(\sqrt {σ _1^2 + σ _2^2 + \ldots + σ _8^2 + σ _9^2} \)

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

Calculation:

Given:

σ1, σ2, ...., σ8, σ9 = 3

σcp = \(\sqrt {σ _1^2 + σ _2^2 + \ldots + σ _8^2 + σ _9^2} \)

σ_cp = \(\sqrt {3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2} \)

σ_cp = \(\sqrt {9 \times 9} \) = 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 \({t_E} = \left( {\frac{{{t_p}\; + \;4{t_m}\; + {t_o}}}{6}} \right)\)

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

Variance, \(V = {\left( {\frac{{{t_p} - {t_0}}}{6}} \right)^2}\)

Standard duration, \(\sigma = \sqrt {variance} \)

Calculation:

Given:

t_p = 10 days, t_o = 4 days

\({\rm{V}} = {\left( {\frac{{{{\rm{t}}_{\rm{p}}} - {{\rm{t}}_{\rm{o}}}}}{6}} \right)^2} = {\left( {\frac{{10 - 4}}{6}} \right)^2} = 1\)

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 - \({n_C}_m\)

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 = \({n_C}_m ⇒ {5_C}_2\)

∴ \(\frac{5!}{2!\;\times\;(5-2)!}\Rightarrow10\)

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).

\(T_e=\frac{t_o\;+\;4t_m\;+\;t_p}{6}\)

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

Explanation:

If \(x_{ij}\ge 0, \)  is the number of units shipped from i^th source to j^th destination, then the equivalent LPP model will be

Minimize \(Z = \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {{c_{ij}}} } {x_{ij}}\)

Subjected to:

\(\begin{array}{l} \sum\limits_{i = 1}^m {{x_{ij}}} \le {b_i}\,\,(demand)\\ \sum\limits_{j = 1}^n {{x_{ij}}} \le {a_i}\,\,(\sup ply) \end{array}\)

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.