Pgcbm/531 Week 4 Linear Programming Approach
Operational Research (OR)
DR. SUPRIYA KUMAR DE
ASSIGNMENT
XLRI-PGCBM-18
NAME: GH. RASOOL WANI
SMS ID: 2217429
Table of Contents
1. Problem: 1
2. Solution: 2
2.1 Manual Approach 2 2.2 Linear programming approach 2 2.2.1 Decision Variables: 3 2.2.2 Objective Function: 3 2.2.3 Constraints: 3 2.3 Excel Solution. 6 2.3.1 Excel Solution Embedded: 6
3. Analysis: 6
3.1 Sensitivity Analysis: Objective Function 6 3.2 Sensitivity Analysis: Right Hand Side of the constraints 7
Problem:
Mr. Ramesh Chandra is a Software Development Project Manager in one of the renowned Indian Software Services Company, namely “ABC Technologies”. The company’s business division has recently won a project from a major …show more content…
European telecom company. This Project is very important for the “ABC Technologies” as it is first time the company is entering in the telecom sector for the software services.
The “ABC Technologies” has been given whole end to end responsibility for this software package from Requirement Analysis till final Deployment.
Customer has communicated recently to the business director of “ABC Technologies” that they would like to see the final Project Plan with a Breakdown of the tasks, Total Cost of the Project and Total duration of the Project by end of next week.
“ABC Technologies” management has decided to assign Ramesh Chandra as the Project Manager for this Project and has been told to come up with the Project Plan so that they can share it with the Customer by end of next week. Ramesh has already been provided with all the required inputs so that he can be prepared for the plan.
Mr. Ramesh worked very hard and came up with the following plan.
[pic]
Table-1: Initial Plan
Durations and Effort are here in Person Days (pdays).
AS per this plan, Activities/Tasks A-B-C-D-E-F-H-I-L-M are on Critical Path and the Project duration comes to 64 days.
The Total Cost of the Project comes to $20,000.
During the plan discussion with the customer, customer commented that the duration of 64 days for this project is not acceptable to them and they would like the project duration be reduced by at least 20 days with a very minimal extra cost impact.
Ramesh agreed during the meeting that he would try to reduce the duration and will come up with the new plan in a couple of days.
Solution:
Being a Project Manager, Ramesh knows that Project time cam be reduced by a technique called Crashing which takes care to reduce the duration on a minimal extra cost.
To start the Crashing, Ramesh needs to know how much time each activity can be crashed to and how much it will cost. He starts this activity and comes with the following analysis for each activity.
Durations and Effort are here in Person Days (pdays).
[pic]
Table-2: Plan with Crash Time and Crash Cost against each activity.
There are two approaches to the problem:
1 Manual Approach
For crashing project time, the first task is to identify the normal critical path and the critical activities. Then it is required to determine the crash cost per time period (cost-time slopes) for various activities. The cost-time slopes can be computed using the following formula:
Crash Cost – Normal Cost Slope = -------------------------------------- Normal Time –Crash Time
The next step is to identify the activity on the critical path with the smallest crash cost per time period. This activity will be crashed to the maximum possible extent or to the point at which management’s desired deadline has been reached.
Then it should be checked that the critical path that were being crashed is still critical. If the path is still critical then crash the activity that has second lowest cost-time slopes and continue this process until the goal has been reached. If, a reduction in a critical activity time causes a non-critical path or paths to become critical, then the crash will be continued along with the new critical path based on the lowest cost time slopes of the new path.
2 Linear programming approach
The linear programming is a tool for decision making under certain situation. So, the basic assumption of this approach is that to know some relevant data with certainty.
The basic data requirements are as follows:
1) To know the project network with activity time, and their precedence sequence.
2) To what extent an activity can be crashed.
3) The crash cost associated with per unit of time for all activities.
These requirements are identified in Table-2.
Now let’s define the variable of the problem.
1 Decision Variables:
For each activity “i”, following two variables can be defined.
Ti = Time at each activity “i”, starts.
Ci = Number of periods (days in this case) by which an activity “i”, is crashed.
2 Objective Function:
The objective function here is to minimize the total cost of crashing the project down to 47 (64-20 = 44) days. Using the crash cost per week computed in Table-2.
The Crash Cost per week per activity is computed using following formula:
Crash Cost – Normal Cost -------------------------------- Normal Time –Crash Time
So the objective function can be expressed as follows.
MIN (Z) = $0Ca+ $500Cb +$222Cc +$500Cd +$545Ce +$500Cf +$500Cg +$500Ch +667Ci+250Cj+ $286Ck + $400Cl +$667Cm
3 Constraints:
This objective function is subject to some constraints. These constraints can be classified in to three categories.
1) Precedence Constraints:
These set of constraints describe the structure of the network. As we know, the activities of a project are interrelated, the starting of some activities is dependent upon the completion of some other activities; we must have to establish work sequence of the activities through constraints. For this problem the precedence constraints are defined as below:
The precedence for each activity is given in below Table-3 in column “Predecessors”
[pic]
Table-3: Precedence Relationship of the Activities
[pic]
Fig-1: Project Network Diagram
|A->B |
|B->C |
|C->D |
|C->K |
|C->J |
|D->E
|
|E->F |
|E->K |
|E->G |
|E->J |
|F->H |
|H->I |
|G->I |
|I->L |
|J->L |
|K->L |
|L->M |
|Tb >= Ta + 4-Ca |i.e. |Tb-Ta+ Ca >=4 |
|Tc >= Tb + 8-Cb |i.e. |Tc-Tb+ Cb >=8 |
|Td>= Tc+9-Cc |i.e. |Td - Tc + Cc >=9 |
|Tk >= Tc+9-Cc |i.e. |Tk - Tc + Cc>=9 |
|Tj >= Tc+9-Cc |i.e. |Tj – Tc + Cc>=9 |
|Te >=Td+6-Cd |i.e. |Te – Td + Cd>=6 |
|Tf >= Te+11-Ce |i.e. |Tf - Te + Ce >=11 |
|Tk >= Te+11-Ce |i.e. |Tk – Te + Ce >=11 |
|Tg>= Te+11-Ce |i.e. |Tg - Te + Ce >=11 |
|Tj >= Te+11-Ce |i.e. |Tj – Te + Ce >=11 |
|Th >= Tf + 4- Cf |i.e. |Th - Tf + Cf >= 4 |
|Ti >= Th + 8-Ch |i.e. |Ti – Th + Ch >=8 |
|Ti >= Tg + 4-Cg |i.e. |Ti - Tg + Cg >=4 |
|Tl >= Ti + 6-Ci |i.e. |Tl – Ti + Ci >=6 |
|Tl >= Tj + 16-Cj |i.e. |Tl – Tj + Cj >=16 |
|Tl >= Tk + 14-Ck |i.e. |Tl – Tk + Ck>=14 |
|Tm >= Tl + 5-Cl |i.e. |Tm - Tl + Cl >=5 |
Table-4: Precedence Constraints
For e.g. consider the precedence relationship between activities A and B. Activity A starts at time Ta and its duration is (4-Ca) days where Ca is the duration activity A can be crashed. Hence activity A finishes at time (Ta+4-Ca).This implies that activity B start time (Tb) can be no earlier than (Ta+4-Ca). It can be expressed mathematically as:
Tb >= Ta + 4-Ca , it can also be written as Tb-Ta+ Ca >=4.
In a similar fashion, precedence relationship of all other activities are expressed in Table-4
2) Crash Time Constraints:
We can reduce the time to complete an activity by simply increasing the resources or by improving the productivity which also require the commitment of additional resources. But it is not possible to reduce the time required to complete an activity after a certain threshold limit. Strive for such intention will result in superfluous resources employment which will be an inefficient approach. That is why the allowable time to crash an activity has a limit.
We need a second set of constraints to restrict the number of periods by which an activity can be crashed using the crash time limits given in Table-2. We can write these constraints as:
|Ca
DR. SUPRIYA KUMAR DE
ASSIGNMENT
XLRI-PGCBM-18
NAME: GH. RASOOL WANI
SMS ID: 2217429
Table of Contents
1. Problem: 1
2. Solution: 2
2.1 Manual Approach 2 2.2 Linear programming approach 2 2.2.1 Decision Variables: 3 2.2.2 Objective Function: 3 2.2.3 Constraints: 3 2.3 Excel Solution. 6 2.3.1 Excel Solution Embedded: 6
3. Analysis: 6
3.1 Sensitivity Analysis: Objective Function 6 3.2 Sensitivity Analysis: Right Hand Side of the constraints 7
Problem:
Mr. Ramesh Chandra is a Software Development Project Manager in one of the renowned Indian Software Services Company, namely “ABC Technologies”. The company’s business division has recently won a project from a major …show more content…
European telecom company. This Project is very important for the “ABC Technologies” as it is first time the company is entering in the telecom sector for the software services.
The “ABC Technologies” has been given whole end to end responsibility for this software package from Requirement Analysis till final Deployment.
Customer has communicated recently to the business director of “ABC Technologies” that they would like to see the final Project Plan with a Breakdown of the tasks, Total Cost of the Project and Total duration of the Project by end of next week.
“ABC Technologies” management has decided to assign Ramesh Chandra as the Project Manager for this Project and has been told to come up with the Project Plan so that they can share it with the Customer by end of next week. Ramesh has already been provided with all the required inputs so that he can be prepared for the plan.
Mr. Ramesh worked very hard and came up with the following plan.
[pic]
Table-1: Initial Plan
Durations and Effort are here in Person Days (pdays).
AS per this plan, Activities/Tasks A-B-C-D-E-F-H-I-L-M are on Critical Path and the Project duration comes to 64 days.
The Total Cost of the Project comes to $20,000.
During the plan discussion with the customer, customer commented that the duration of 64 days for this project is not acceptable to them and they would like the project duration be reduced by at least 20 days with a very minimal extra cost impact.
Ramesh agreed during the meeting that he would try to reduce the duration and will come up with the new plan in a couple of days.
Solution:
Being a Project Manager, Ramesh knows that Project time cam be reduced by a technique called Crashing which takes care to reduce the duration on a minimal extra cost.
To start the Crashing, Ramesh needs to know how much time each activity can be crashed to and how much it will cost. He starts this activity and comes with the following analysis for each activity.
Durations and Effort are here in Person Days (pdays).
[pic]
Table-2: Plan with Crash Time and Crash Cost against each activity.
There are two approaches to the problem:
1 Manual Approach
For crashing project time, the first task is to identify the normal critical path and the critical activities. Then it is required to determine the crash cost per time period (cost-time slopes) for various activities. The cost-time slopes can be computed using the following formula:
Crash Cost – Normal Cost Slope = -------------------------------------- Normal Time –Crash Time
The next step is to identify the activity on the critical path with the smallest crash cost per time period. This activity will be crashed to the maximum possible extent or to the point at which management’s desired deadline has been reached.
Then it should be checked that the critical path that were being crashed is still critical. If the path is still critical then crash the activity that has second lowest cost-time slopes and continue this process until the goal has been reached. If, a reduction in a critical activity time causes a non-critical path or paths to become critical, then the crash will be continued along with the new critical path based on the lowest cost time slopes of the new path.
2 Linear programming approach
The linear programming is a tool for decision making under certain situation. So, the basic assumption of this approach is that to know some relevant data with certainty.
The basic data requirements are as follows:
1) To know the project network with activity time, and their precedence sequence.
2) To what extent an activity can be crashed.
3) The crash cost associated with per unit of time for all activities.
These requirements are identified in Table-2.
Now let’s define the variable of the problem.
1 Decision Variables:
For each activity “i”, following two variables can be defined.
Ti = Time at each activity “i”, starts.
Ci = Number of periods (days in this case) by which an activity “i”, is crashed.
2 Objective Function:
The objective function here is to minimize the total cost of crashing the project down to 47 (64-20 = 44) days. Using the crash cost per week computed in Table-2.
The Crash Cost per week per activity is computed using following formula:
Crash Cost – Normal Cost -------------------------------- Normal Time –Crash Time
So the objective function can be expressed as follows.
MIN (Z) = $0Ca+ $500Cb +$222Cc +$500Cd +$545Ce +$500Cf +$500Cg +$500Ch +667Ci+250Cj+ $286Ck + $400Cl +$667Cm
3 Constraints:
This objective function is subject to some constraints. These constraints can be classified in to three categories.
1) Precedence Constraints:
These set of constraints describe the structure of the network. As we know, the activities of a project are interrelated, the starting of some activities is dependent upon the completion of some other activities; we must have to establish work sequence of the activities through constraints. For this problem the precedence constraints are defined as below:
The precedence for each activity is given in below Table-3 in column “Predecessors”
[pic]
Table-3: Precedence Relationship of the Activities
[pic]
Fig-1: Project Network Diagram
|A->B |
|B->C |
|C->D |
|C->K |
|C->J |
|D->E
|
|E->F |
|E->K |
|E->G |
|E->J |
|F->H |
|H->I |
|G->I |
|I->L |
|J->L |
|K->L |
|L->M |
|Tb >= Ta + 4-Ca |i.e. |Tb-Ta+ Ca >=4 |
|Tc >= Tb + 8-Cb |i.e. |Tc-Tb+ Cb >=8 |
|Td>= Tc+9-Cc |i.e. |Td - Tc + Cc >=9 |
|Tk >= Tc+9-Cc |i.e. |Tk - Tc + Cc>=9 |
|Tj >= Tc+9-Cc |i.e. |Tj – Tc + Cc>=9 |
|Te >=Td+6-Cd |i.e. |Te – Td + Cd>=6 |
|Tf >= Te+11-Ce |i.e. |Tf - Te + Ce >=11 |
|Tk >= Te+11-Ce |i.e. |Tk – Te + Ce >=11 |
|Tg>= Te+11-Ce |i.e. |Tg - Te + Ce >=11 |
|Tj >= Te+11-Ce |i.e. |Tj – Te + Ce >=11 |
|Th >= Tf + 4- Cf |i.e. |Th - Tf + Cf >= 4 |
|Ti >= Th + 8-Ch |i.e. |Ti – Th + Ch >=8 |
|Ti >= Tg + 4-Cg |i.e. |Ti - Tg + Cg >=4 |
|Tl >= Ti + 6-Ci |i.e. |Tl – Ti + Ci >=6 |
|Tl >= Tj + 16-Cj |i.e. |Tl – Tj + Cj >=16 |
|Tl >= Tk + 14-Ck |i.e. |Tl – Tk + Ck>=14 |
|Tm >= Tl + 5-Cl |i.e. |Tm - Tl + Cl >=5 |
Table-4: Precedence Constraints
For e.g. consider the precedence relationship between activities A and B. Activity A starts at time Ta and its duration is (4-Ca) days where Ca is the duration activity A can be crashed. Hence activity A finishes at time (Ta+4-Ca).This implies that activity B start time (Tb) can be no earlier than (Ta+4-Ca). It can be expressed mathematically as:
Tb >= Ta + 4-Ca , it can also be written as Tb-Ta+ Ca >=4.
In a similar fashion, precedence relationship of all other activities are expressed in Table-4
2) Crash Time Constraints:
We can reduce the time to complete an activity by simply increasing the resources or by improving the productivity which also require the commitment of additional resources. But it is not possible to reduce the time required to complete an activity after a certain threshold limit. Strive for such intention will result in superfluous resources employment which will be an inefficient approach. That is why the allowable time to crash an activity has a limit.
We need a second set of constraints to restrict the number of periods by which an activity can be crashed using the crash time limits given in Table-2. We can write these constraints as:
|Ca