CPM MCQ Quiz - Objective Question with Answer for CPM - Download Free PDF

Explanation:

Activity A → B: Solid arrow means B can only start after A ends ⇒ B is dependent on A.

Activity C → D: Solid arrow means D can only start after C ends ⇒ D is dependent on C only.

Dummy C → B: Dummy arrow shows logical dependency ⇒ B is also dependent on C.

(i) Activity B is controlled by activity A – Correct

(ii) Activity D is controlled by both A and C – Incorrect (only C controls D)

(iii) Activity B is independent of activity C – Incorrect (dummy from C to B shows B depends on C)

Explanation:

Given Data:

• Normal duration: 15 days

• Normal cost: Rs. 30,000

• Crashed duration: 12 days

• Crashed cost: Rs. 54,000

• Indirect cost: Rs. 8,500 per day (for P) and Rs. 10,000 per day (for Q)

Cost slope (Crashing cost per day) is given by:

\(\mathrm{C} / \mathrm{s}=\frac{54000-30000}{15-12}=8000 \mathrm{Rs} / \text { day }\)

Crashing is economically beneficial if the savings in indirect cost per day exceed the direct cost of crashing per day.

• If the indirect cost per day is Rs. 8,500 (Statement P):
  Since Rs. 8,500 > Rs. 8,000, it is economically advisable to crash. So, P is true.

• If the indirect cost per day is Rs. 10,000 (Statement Q):
  Since Rs. 10,000 > Rs. 8,000, it is also economically advisable to crash. So, Q is true.

Explanation:

In project management, normal time refers to the planned or standard duration needed to complete a project or an activity without expediting or crashing. During this period, the project operates under normal conditions, without any additional resources or acceleration.

The direct cost of a project typically includes labor, materials, and equipment costs. When a project is completed within the normal time, these costs are usually minimized because:

• There is no need for overtime or additional shifts.

• Standard resources are used without any premium charges.

• There is no need to expedite materials or equipment, which could increase costs.

However, if the project timeline is shortened (crashed), the direct costs increase due to additional expenses to speed up the work. Therefore, the direct cost of a project is at its minimum when completed within normal time.

Explanation:

Direct Project cost:

(i) It is the cost that is directly dependent on the amount of resources involved for the completion of activities. It includes labour, materials, plants, and machining.

(ii) To get the same work done in less time, we have to increases the amount of labor, equipment, and time-saving material that too at extra charges which simply means increases indirect cost.

(iii) The project has the highest cost corresponding to the crash duration and has normal cost corresponding to the normal duration.

Normal time:

(i) Normal time is the standard time that an estimator would usually allow for an activity.

(ii) It is the time with which direct cost does not reduce with the increase in time.

Crash time:

(i) Crash time is the minimum possible time in which an activity can be completed, by employing extra resources.

(ii) Crash time is that time, beyond which the activity cannot be shortened by an amount of increases in the resources.

Concept:

Float:

• Float indicates the time by which, starting or finishing of an activity can be delayed without completion time.
• Float is associated with an activity and is analogous to the term slack.

Independent Float:

• It is the amount of time by which an activity can be delayed when all the preceding activities are completed as late as possible and all succeeding activities started as early as possible.

• Independent float can also be defined as the excess of minimum available time over the required activity duration.

• Consider a partial network as below

• If activity c-d is under consideration, and activity b-c finishes as its latest possible time T^c_L and succeeding activity j-k start at its earliest time Tⁱ_E.

• Then activity i-j can take up any duration from t" to ( TT) without affecting any activity of the project.

• The difference between (T^j_E-Tⁱ_L) and t is known as Independent float for activity i-j Independent float for i-j,

F_ID = (T^j_E - Tⁱ_L) - t^ij

F_ID = (T^j_E - t^ij ) - Tⁱ_L                       [ F_F = T^j_E - Tⁱ_E - t^ij ]

F_ID = (F_F + Tⁱ_E) -Tⁱ_L

F_ID = F_F - (Tⁱ_L - Tⁱ_E)                       [S_i = Tⁱ_L - Tⁱ_E]

F_ID = F_F - S_i

​Solution:

FID = (TjE - TiL) - tij

F_ID = Ej – Li - D (i, j)

Additional Information

• The difference between maximum time available and the actual time required for the completion of the activity
• F_T = (Maximum available Time) - Actual time required.
• Maximum available time we can get when the activity starts at Earliest start time and finish by latest finish time i.e.
• Maximum available time = LFT - EST

The actual time required for performing an activity is the estimated activity time 

F_T = LFT - EST - t^ij

Free Float (F_F):

• It is defined as the amount of time by which an activity can be delayed without affecting the EST of the succeeding activity.
• In other words, it is that portion of total float that can be used by an activity without delaying any succeeding activity. 
• Consider an activity i-j and successor activity j - k.
• EST of i-j = Tⁱ_E, therefore EFT of i-j =Tⁱ_E+t^ij
• EST of j-k =T^j_E, if T^j_E > (Tⁱ_E+t^ij) then activity j-k can not start immediately after completion of activity i-j and the difference between T^j_E and (Tⁱ_E+t^ij) is the free float for activity i-j.
• Free Float for i-j     F_F = T^j_E - (Tⁱ_E + t^ij)

Interfering Float (FIT): 

• It is defined as the difference between the total float and the free float of an activity.
• It is also equal to the slack of head event of head activity as shown below

F_IT = F_T - F_F

Earliest occurrence time (T_E):

• It is the earliest time at which an event can occur i.e. the time by which all the activities leading to an event under consideration are complete. It is also called as Earliest Event time.
• The earliest event time (T_E) is analogous to the term earliest expected time used in PERT, except that the degree of uncertainty involved in the word expected is not there.
• Calculation of Earliest event time is done same as PERT analysis by forwarding pass rule i.e.

T^j_E = Maximum (Tⁱ_E + t^ij)

Latest allowable occurrence time (T_L):

• It is the latest (delayed) time by which an event must be completed to completion time is not affected.
• The latest allowable occurrence time is calculated the same as in PERT by the Backward pass rule. i.e. Tⁱ_L= Minimum (T^j_L - t^ij)
• If the scheduled completion time (T_S) of the project is given, then the latest event time of the final/end event will be equal to T_S 
• If the scheduled completion time (T_S) is not specified, then T_L, is taken equal to the earliest event time T_E of the end/final event.

Latest Finish Time (LFT):

• It is the latest (or delayed) time which the activity can be finished, without delaying the project.
• It is equal to the latest occurrence time T_L of the event at which the activity terminates.

Earliest Finish Time (EFT):

• It is the earliest time by which the activity can be completed. It is equal to the earliest start time

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

• Fulkerson’s rule is used for numbering events involved in the project scheduling network.
• Types of network diagram
  • Event on the node ( EON )
  • Activity on the node ( AON )

Important Points

• Hungarian  method – Assignment problems
• Johnson's  rule – scheduling jobs 
• Simplex method -  linear programming

Early Start Time:

• The earliest point in the schedule at which a task can begin.
• EST = \(T_E^i\)

Early Finish Time:

• The earliest point in the schedule at which a task can finish.
• EFT = \(T_E^i\) + t_ij

Latest Start Time:

• The latest point in the schedule at which a task can start without causing a delay.
• LST = \(T_L^j\) - t_ij

Latest Finish Time:

• The latest point in the schedule at which a task can finish without causing a delay.
• LFT = \(T_L^j\)

Types of float:

1) Total float:

• It is the difference between the maximum time available and the actual time required for the completion of the activity. or It is the difference between the time available for an activity performance and the duration of the activity.
• F_T = LST - EST = LFT - EFT = \(T_L^j - T_E^i - t_e^{ij}\) 

2) Free float:

• It is the amount of time by which an activity can be delayed without affecting the EST of the succeeding activity.
• F_T = \(T_E^j - T_E^i - t_e^{ij}\)

3) Independent float:

• It is the excess of minimum available time over the required activity duration
• F_I = \(T_E^j - T_L^i - t_e^{ij}\) 

4) Interfering float

• It is the difference between the total float and the free float of an activity.
• F_IN = F_T - F_F

Explanation:

Direct Project cost:

(i) It is the cost that is directly dependent on the amount of resources involved for the completion of activities. It includes labour, materials, plants, and machining.

(ii) To get the same work done in less time, we have to increases the amount of labor, equipment, and time-saving material that too at extra charges which simply means increases indirect cost.

(iii) The project has the highest cost corresponding to the crash duration and has normal cost corresponding to the normal duration.

Normal time:

(i) Normal time is the standard time that an estimator would usually allow for an activity.

(ii) It is the time with which direct cost does not reduce with the increase in time.

Crash time:

(i) Crash time is the minimum possible time in which an activity can be completed, by employing extra resources.

(ii) Crash time is that time, beyond which the activity cannot be shortened by an amount of increases in the resources.

Concept

Dummy Activity: An activity that is used to maintain the predefined precedence relationship only during the construction of the project network is called a dummy activity.

Estimate: An approximate calculation or judgment of value

PERT is preferred when there are some uncertainties in the methodology, availability of materials, and the final answer is not clearly known.

Whereas CPM is applied when things are almost certain

For example in the case of a construction project, once the project has been finalized there is little doubt of men, material, and equipment required, time, etc. then PERT is used. 

∴ CPM has limitations in the early stage of the project.

Explanation:

Float:

(i) It is associated with activity times.

(ii) It is analogous to the slack of events in PERT.

(iii) It is the range within which the start or finish time of activity may fluctuate without affecting the project completion time.

(iv) Floats are of the following types:

1. Total float: The time span by which starting or finishing of an activity can be delayed without delaying the completion of the project. Total float of activity affects total float of succeeding as well as preceding activities. 

2. Free float: The delay can be made without delaying succeeding activities. It affects only preceding activities.

3. Independent Float: It is the minimum excess available time which exists without affecting any of succeeding or preceding activities.

4. Interfering Float: It is similar to head event slack.

Gantt charts are mainly used to allocate resources to activities. The resources allocated to activities include staff, hardware, and software. Gantt charts are useful for resource planning. A Gantt chart is a special type of bar chart where each bar represents an activity. The bars are drawn along a timeline. The length of each bar is proportional to the duration of time planned for the corresponding activity.

Concept:

Total float: It is the amount of time that the completion time of an activity can be delayed without affecting project completion time.

Mathematically, Total Float = (L_j - E_i) - t_ij

Calculation:

∴ Total Float = (L_j - E_i) - t_ij = 16 - 10 - 4 = 2 week

Additional Information

Free float: It is the amount of time that the activity completion time can be delayed without affecting the earliest start time of the immediate successor activities in the network.

Mathematically, Free Float = (Ej - Ei) - tij

Independent Float: Amount by which an activity can be delayed without delaying the project; even if all predecessors are at Late Finish and all Successors are at Early Start.

Mathematically, Independent Float = (Ej - Li) - tij

Bar Chart / Gantt Chart

First introduced by Hennery Gantt in 1910. Each bar represents a specific job or activity and length of the bar represent the time of completion of that activity. The starting and ending of the bar represent the starting and beginning of that activity.

Concept:

• Critical Path Methods (CPM) have been used for planning scheduling and control in construction project management.
• It is activity oriented method.

CPM Network is updated whenever there is a difference in the planned and actual performance.