Description
Project Assignment- XLRI
OPERATIONAL RESEARCH (OR)
INDIVIDUAL ASSIGNMENT- Take a problem from your own organization and apply the concepts taught in the class
NAME: RAJESH KUMAR VERMA SMS ID: 110387 SID NO: RB12044 BATCH: PGCBM-21 FACULTY: DR. SUPRIYA KUMAR DE STUDY CENTER: DAKC, NAVI MUMBAI
RAJESH KUMAR VERMA
PGCBM-21
SMS ID: 110387
PAGE:
1
Table of Contents
1. PROJECT PROBLEM: ............................................................................................................................................................3 2. PROJECT SOLUTION:............................................................................................................................................................4 2.1 MANUAL APPROACH ...........................................................................................................................................................5 2.2 LINEAR PROGRAMMING APPROACH.........................................................................................................................................5 2.2.1 Decision Variables:.................................................................................................................................................6 2.2.2 Objective Function:.................................................................................................................................................6 2.2.3 Constraints:.............................................................................................................................................................6 2.3 EXCEL SOLUTION................................................................................................................................................................9 2.3.1 Excel Solution by Solver- Answer Report: .........................................................................................................10 3. ANALYSIS OF PROJECT SOLUTION:..............................................................................................................................14 3.1 SENSITIVITY ANALYSIS: PROJECT OBJECTIVE........................................................................................................................14 3.2 SENSITIVITY ANALYSIS: RIGHT HAND SIDE CONSTRAINTS.......................................................................................................14
RAJESH KUMAR VERMA
PGCBM-21
SMS ID: 110387
PAGE:
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PROJECT PROBLEM: A Reliance technical Software (RTS) Development Project team has got a assignment to develop software for Wireless business of Reliance. This Project is very important for the RTS division because it is very crucial software which will save 10% of running cost in wireless business. RTS division has been given whole end to end responsibility for this software package from Requirement Analysis till final Deployment and installation. Wireless business has communicated recently to the project manager of RTS 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. RTS division management has decided to assign this project to Mr Iyer as the Project Manager for this Project and given target to plan to present within a week time. Mr. Iyer has come up with the following plan.
Tas k ID A B C D E F Name Effor ts 4 16 9 6 33 4 No of Resour ces utilized 1 2 1 1 3 1 Durati on 4 8 9 6 11 4 Time Savin gs 0 4 4.5 4 5.5 2 A B C D E Predeces sors Cost $1,500.0 0 $3,000.0 0 $1,500.0 0 $1,500.0 0 $4,500.0 0 $1,500.0 0 $1,500.0 0 $3,000.0 0 $3,000.0 0 $3,000.0 0 $3,000.0 0 $1,500.0 0 $1,500.0 0 $30,000. 00 Critic al Yes Yes Yes Yes Yes Yes
1.
Scope of work Software Requirements Design-High Level Design-Low Level Development/Codi ng Unit Testing Specification and Plan Integration Testing Specification and Plan Unit analysis Testing Integration Testing Training Material User Documentation Deployment Post Deployment Review
G H I J K L M
4 16 12 32 42 5
1 2 2 2 2 1
4 8 6 16 14 5 3 64
2 4 3 8 7 2.5 1.5
E F H,G C,E C,E I,J,K L Total Cost
No Yes Yes No No Yes Yes
3 1 Total Project Duration
Table-1: PROJECT PLAN- START WITH Durations and Effort of work are here is in Man Days. RAJESH KUMAR VERMA PGCBM-21 SMS ID: 110387 PAGE: 3
Plan Activities/Tasks A-B-C-D-E-F-H-I-L-M are on Critical Path and the Project duration comes to 64 man days. Total project Cost comes to $30,000. On the finalization of project plan with wireless business, it has been decided that the project is very crucial and duration of 64 days for the project is very high and not acceptable. The wireless business requested that to reduce the project duration by at least 20 days with no extra or minimal extra cost impact. RTS management (Mr Iyer) agreed during the meeting that he would try to reduce the duration and will come up with the new plan within 3 days time.
2.
PROJECT SOLUTION:
Mr Iyer has planned and reduced the project time by a technique called Crashing, which takes care to reduce the duration on a minimal extra cost. Crashing activity started, Mr Iyer 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 Man Days.
No of Resour ces utilized 1 2 1 1 3 1 Pr ed ec es so rs Cra sh Cos t 1,5 00 6,0 00 3,0 00 4,5 00 9,0 00 3,0 00 3,0 00 6,0 00 6,0 00 Extr a Cos t 3,0 00 1,5 00 3,0 00 4,5 00 1,5 00 1,5 00 3,0 00 3,0 00 Cras h Cost/ Day $0 $750 $333 $750 $818 $750
Ta sk ID A B C D E F
Name
Effort s
Durati on
Crash. Durati on 4 4 4.5 2 5.5 2
Time Savin gs 0 4 4.5 4 5.5 2
Cost
Crit ical
Scope of work Software Requirements Design-High Level Design-Low Level Development/Co ding Unit Testing Specification and Plan Integration Testing Specification and Plan Unit analysis Testing Integration Testing
4 16 9 6 33 4
4 8 9 6 11 4
A B C D E
1,50 0 3,00 0 1,50 0 1,50 0 4,50 0 1,50 0 1,50 0 3,00 0 3,00 0
Yes Yes Yes Yes Yes Yes
G H I
4 16 12
1 2 2
4 8 6
2 4 3
2 4 3
E F H, G
$750 $750 $1,00 0
No Yes Yes
RAJESH KUMAR VERMA
PGCBM-21
SMS ID: 110387
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J K L M
Training Material User Documentation Deployment Post Deployment Review
32 42 5 3
2 2 1 1
16 14 5 3 64
8 7 2.5 1.5
8 7 2.5 1.5 Total Cost
C, E C, E I,J, K L
Total Project Duration
3,00 0 3,00 0 1,50 0 1,50 0 30,00 0
6,0 00 9,0 00 3,0 00 3,0 00
3,0 00 6,0 00 1,5 00 1,5 00
$375 $857 $600 $1,00 0
No No Yes Yes
Table-2: Plan with Crash Time and Crash Cost against each activity. Attempted for two approaches to the problem:
2.1
Manual Approach
As per the demand and requirement of wireless group Mr Iyer has decided to crashing project time, 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 for various activities. The cost-time can be computed using the following formula: Crash Cost – Normal Cost Slope = -------------------------------------Normal Time –Crash Time Then next step is to identify that activity on the critical path with the smallest crash cost per time period. This activity will be crashed to the point at which RTS division and wireless management’s desired deadline. 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 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.
2.2
Linear programming approach
Linear programming is a decision making tool under certain situation. The basic assumption of this approach is that to know some relevant data with certainty. Basic data requirements are as follows: 1) To know the project activity time and the precedence sequence. 2) To know what extent an activity can be crashed. 3) Crash cost per unit of time for all activities. The above requirements are explained and calculated in Table-2.
RAJESH KUMAR VERMA
PGCBM-21
SMS ID: 110387
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Now variable of the problem are:
2.2.1 Decision Variables:
To decide variable for each activity “i”, the following two variables can be defined. Ti = Time at each activity “i”, starts. Ci = Number of periods in days by which an activity “i”, is crashed.
2.2.2 Objective Function:
Objective function here is to minimize the total cost of crashing the project down to 44 (64-20 = 44) days. Crash Cost per activity is computed using following formula: Crash Cost – Normal Cost -------------------------------Normal Time –Crash Time Objective function can be expressed as follows. MIN (Z) = $0Ca+ $750Cb +$333Cc +$750Cd +$818Ce +1000Ci+375Cj+ $857Ck + $600Cl +$1000Cm +$750Cf +$750Cg +$750Ch
2.2.3 Constraints:
These constraints can be classified in to three categories. 1) Precedence Constraints: The activities of a project are interrelated and the starting of some activities is dependent upon the completion of some other activities; so we need to establish work sequence of the activities through constraints. For this problem the precedence constraints are defined as below:
RAJESH KUMAR VERMA
PGCBM-21
SMS ID: 110387
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Table-3: Precedence Relationship of the Activities
Task ID A B C D E F G H I J K L M Name Scope of work Software Requirements Design-High Level Design-Low Level Development/Coding Unit Testing Specification and Plan Integration Testing Specification and Plan Unit analysis Testing Integration Testing Training Material User Documentation Deployment Post Deployment Review Predeces sors
A B C D E E F H,G C,E C,E I,J,K L
Fig-1: FLOW DIAGRAM
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SMS ID: 110387
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Table-4: Precedence Constraints
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 Tc >= Tb + 8-Cb Td>= Tc+9-Cc Tk >= Tc+9-Cc Tj >= Tc+9-Cc Te >=Td+6-Cd Tf >= Te+11-Ce Tk >= Te+11-Ce Tg>= Te+11-Ce Tj >= Te+11-Ce Th >= Tf + 4- Cf Ti >= Th + 8-Ch Ti >= Tg + 4-Cg Tl >= Ti + 6-Ci Tl >= Tj + 16-Cj Tl >= Tk + 14-Ck Tm >= Tl + 5-Cl i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. Tl – Tk + Ck>=14 Tm - Tl + Cl >=5 Tb-Ta+ Ca >=4 Tc-Tb+ Cb >=8 Td - Tc + Cc >=9 Tk - Tc + Cc>=9 Tj – Tc + Cc>=9 Te – Td + Cd>=6 Tf - Te + Ce >=11 Tk – Te + Ce >=11 Tg - Te + Ce >=11 Tj – Te + Ce >=11 Th - Tf + Cf >= 4 Ti – Th + Ch >=8 Ti - Tg + Cg >=4 Tl – Ti + Ci >=6 Tl – Tj + Cj >=16
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). Mathematically IT CAN BE EXPRESSED AS: Tb >= Ta + 4-Ca OR IT CAN BE Tb-Ta+ Ca >=4 In the similar fashion, precedence relationship is shown in all other activities are explained in Table-4 2) Crash Time Constraints: Now we can reduce the time to complete an activity by increasing the resources or by improving the productivity. But here it is not possible to reduce the time required to complete an activity after a certain threshold limit. Such intention will result in superfluous resources employment which will be an inefficient approach and because of this allowable time to crash an activity has a limit. As required now 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:
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SMS ID: 110387
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Table-4: Crash Time Constraints
Ca <= 0 Cb <= 4 Cc <= 4.5 Cd <= 4 Ce < = 5.5 Cf <= 2 Cg <= 2 Ch <= 4 Ci <= 3 Cj <= 8 Ck <= 7 Cl <= 2.5 Cm <= 1.5
3) Project Completion Constraints: The project completion constraint will recognize completion of last activities) must take place before the project deadline date. In this project, activity M is the last activity in the project and starts at Tm. The normal duration for activity M is 3 days, and Cm denotes the number of days by which its duration can be crashed. Therefore the actual duration of activity M is (3-Cm), and its completion time is (Tm+3Cm). This can be explained as: Tm+3-Cm <= 44 i.e. Tm-Cm <= 41 (The normal project duration is 64. As per Wireless business, it should be reduced by 20 days, hence the last activity M completion time should be <= 64-20 = 44 days.] 4) Non-Negativity Constraints: All Ti and Ci >= 0
2.3
Excel Solution
Tale-5: Excel Solution
TA TB TC TD TE TF TG TH Ti Tj Tk Tl Tm CA CB CC CD CE CF CG CH Ci Cj Ck Cl Cm
Start Start Start Start Start Start Start Start Start Start Start Start Start Crash C rash Crash C rash Crash C rash Crash C rash Crash Crash Crash Crash Crash A B C D E F G H I J K L M A B C D E F G H I J K L M 0.0 4.0 8.0 1 2.5 1 4.5 20.0 20.0 22.0 30.0 20.0 20.0 36.0 41 .0 0.0 $0 4.0 $750 4.5 $333 4.0 $750 5.5 $81 8 2.0 $750 0.0 $750 0.0 0.0 0.0 0.0 0.0 0.0
$750 $1 ,000 $375
$857 $600 $1 ,000 1 3500
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SMS ID: 110387
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In the above table-5 the activities B, C, D, F and K are required to be crashed by 4.0, 4.5, 4.0, 0.5 and 7.0 days respectively to complete the project within stipulated 44 days. The total cost for crashing will be $13500 2.3.1 Excel Solution by Solver- Answer Report:
M ro o E el 1 .0 A s er R o ic s ft xc 1 nw ep rt W rk h o s eet: [exc s eet.xls o er E tries el h ]S lv n R o C ep rt reated 0 -0 -2 1 1 :3 :5 : 2 6 02 6 0 2
T rg t C ll (M ) a e e in C ell $ B6 A $ C sh co ra st
N e am
O in V e rig al alu 150 3 0 .0
F al V e in alu 150 3 0 .0
A ju b C lls d sta le e C ell N e am $ $ B5 S lu n v lu S rt A o tio a e ta $ $ C5 S lu n v lu S rt B o tio a e ta $ $ D5 S lu n v lu S rt C o tio a e ta $ $ E5 S lu n v lu S rt D o tio a e ta $$ F5 S lu n v lu S rt E o tio a e ta $ $ G5 S lu n v lu S rt F o tio a e ta $ $ H5 S lu n v lu S rt G o tio a e ta $ 5 I$ S lu n v lu S rt H o tio a e ta $$ J5 S lu n v lu S rt I o tio a e ta $ $ K5 S lu n v lu S rt J o tio a e ta $$ L5 S lu n v lu S rt K o tio a e ta $ $ M5 S lu n v lu S rt L o tio a e ta $ $ N5 S lu n v lu S rt M o tio a e ta $ $ O5 S lu n v lu C sh A o tio a e ra $ $ P5 S lu n v lu C sh B o tio a e ra $ $ Q5 S lu n v lu C sh C o tio a e ra $ $ R5 S lu n v lu C sh D o tio a e ra $ $ S5 S lu n v lu C sh E o tio a e ra $$ T5 S lu n v lu C sh F o tio a e ra $ $ U5 S lu n v lu C sh G o tio a e ra $ $ V5 S lu n v lu C sh H o tio a e ra $ $ W5 S lu n v lu C sh I o tio a e ra $$ X5 S lu n v lu C sh J o tio a e ra $ $ Y5 S lu n v lu C sh K o tio a e ra $$ Z5 S lu n v lu C sh L o tio a e ra $ A5 A $ S lu n v lu C sh M o tio a e ra
O in V e rig al alu 0 .0 4 .0 8 .0 1 .5 2 1 .5 4 2 .0 0 2 .0 6 2 .0 2 3 .0 0 2 .0 0 2 .0 0 3 .0 6 4 .0 1 0 .0 4 .0 4 .5 4 .0 5 .5 2 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0
F al V e in alu 0 .0 4 .0 8 .0 1 .5 2 1 .5 4 2 .0 0 2 .0 6 2 .0 2 3 .0 0 2 .0 0 2 .0 2 3 .0 6 4 .0 1 0 .0 4 .0 4 .5 4 .0 5 .5 2 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0
RAJESH KUMAR VERMA
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SMS ID: 110387
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Constraints Cell $AB$8 $AB$9 $AB$10 $AB$11 $AB$12 $AB$13 $AB$14 $AB$15 $AB$16 $AB$17 $AB$18 $AB$19 $AB$20 $AB$21 $AB$22 $AB$23 $AB$24 $AB$25 $AB$26 $AB$27 $AB$28 $AB$29 $AB$30 $AB$31 $AB$32 $AB$33 $AB$34 $AB$35 $AB$36 $AB$37 $AB$38
Name 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 Crash limit A Crash limit B Crash limit C Crash limit D Crash limit E Crash limit F Crash limit G Crash limit H Crash limit I Crash limit J Crash limit K Crash limit L Crash limit M Project finish
Cell Value 4.0 8.0 9.0 18.5 16.5 6.0 11.0 13.0 17.0 11.0 4.0 8.0 4.0 6.0 16.0 14.0 5.0 0.0 4.0 4.5 4.0 5.5 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 41.0
Formula $AB$8>=$AD$8 $AB$9>=$AD$9 $AB$10>=$AD$10 $AB$11>=$AD$11 $AB$12>=$AD$12 $AB$13>=$AD$13 $AB$14>=$AD$14 $AB$15>=$AD$15 $AB$16>=$AD$16 $AB$17>=$AD$17 $AB$18>=$AD$18 $AB$19>=$AD$19 $AB$20>=$AD$20 $AB$21>=$AD$21 $AB$22>=$AD$22 $AB$23>=$AD$23 $AB$24>=$AD$24 $AB$25<=$AD$25 $AB$26<=$AD$26 $AB$27<=$AD$27 $AB$28<=$AD$28 $AB$29<=$AD$29 $AB$30<=$AD$30 $AB$31<=$AD$31 $AB$32<=$AD$32 $AB$33<=$AD$33 $AB$34<=$AD$34 $AB$35<=$AD$35 $AB$36<=$AD$36 $AB$37<=$AD$37 $AB$38<=$AD$38
Status Binding Binding Binding Not Binding Not Binding Binding Binding Not Binding Not Binding Binding Binding Binding Binding Binding Binding Binding Binding Binding Binding Binding Binding Binding Binding Not Binding Not Binding Not Binding Not Binding Not Binding Not Binding Not Binding Binding
Slack 0.0 0.0 0.0 9.5 7.5 0.0 0.0 2.0 6.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 0 0 0 0 0 2 4 3 8 7 2.5 1.5 0
RAJESH KUMAR VERMA
PGCBM-21
SMS ID: 110387
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2.3.2 Excel Solution by Solver- Sensitivity Report
Microsoft Excel 11.0 Sensitivity Report Worksheet: [excel sheet.xls]Solver Entries Report Created: 02-06-2012 16:31:48
Adjustable Cells Cell $B$5 $C$5 $D$5 $E$5 $F$5 $G$5 $H$5 $I$5 $J$5 $K$5 $L$5 $M$5 $N$5 $O$5 $P$5 $Q$5 $R$5 $S$5 $T$5 $U$5 $V$5 $W$5 $X$5 $Y$5 $Z$5 $AA$5 Name Solution value Start A Solution value Start B Solution value Start C Solution value Start D Solution value Start E Solution value Start F Solution value Start G Solution value Start H Solution value Start I Solution value Start J Solution value Start K Solution value Start L Solution value Start M Solution value Crash A Solution value Crash B Solution value Crash C Solution value Crash D Solution value Crash E Solution value Crash F Solution value Crash G Solution value Crash H Solution value Crash I Solution value Crash J Solution value Crash K Solution value Crash L Solution value Crash M Final Reduced Value Cost 0.0 818.2 4.0 0.0 8.0 0.0 12.5 0.0 14.5 0.0 20.0 0.0 20.0 0.0 22.0 0.0 30.0 0.0 20.0 0.0 20.0 0.0 36.0 0.0 41.0 0.0 0.0 0.0 4.0 0.0 4.5 0.0 4.0 0.0 5.5 0.0 2.0 0.0 0.0 750.0 0.0 0.0 0.0 250.0 0.0 306.8 0.0 39.0 0.0 600.0 0.0 181.8 Objective Coefficient 0 0 0 0 0 0 0 0 0 0 0 0 0 0 750 333.3333333 750 818.1818182 750 750 750 1000 375 857.1428571 599.9999999 1000 Allowable Increase 1E+30 1E+30 1E+30 1E+30 1E+30 68.18181818 68.18181818 68.18181818 750 68.18181818 68.18181818 818.1818182 818.1818182 818.1818182 68.18181818 484.8484849 68.18181818 38.96103891 0 1E+30 68.18181818 1E+30 1E+30 1E+30 1E+30 1E+30 Allowable Decrease 818.1818182 818.1818182 68.18181818 68.18181818 68.18181818 38.96103891 0 0 38.96103891 38.96103891 0 38.96103891 181.8181815 1E+30 1E+30 1E+30 1E+30 68.18181818 1E+30 750 0 250.0000001 306.8181818 38.96103891 599.9999999 181.8181816
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SMS ID: 110387
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Constraints Cell $AB$8 $AB$9 $AB$10 $AB$11 $AB$12 $AB$13 $AB$14 $AB$15 $AB$16 $AB$17 $AB$18 $AB$19 $AB$20 $AB$21 $AB$22 $AB$23 $AB$24 $AB$25 $AB$26 $AB$27 $AB$28 $AB$29 $AB$30 $AB$31 $AB$32 $AB$33 $AB$34 $AB$35 $AB$36 $AB$37 $AB$38 Name 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 Crash limit A Crash limit B Crash limit C Crash limit D Crash limit E Crash limit F Crash limit G Crash limit H Crash limit I Crash limit J Crash limit K Crash limit L Crash limit M Project finish Final Shadow Value Price 4.0 818.2 8.0 818.2 9.0 818.2 16.5 0.0 16.5 0.0 6.0 818.2 11.0 750.0 11.0 0.0 11.0 0.0 11.0 68.2 4.0 750.0 8.0 750.0 10.0 0.0 6.0 750.0 16.0 68.2 16.0 0.0 5.0 818.2 0.0 -818.2 4.0 -68.2 4.5 -484.8 4.0 -68.2 5.5 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 41.0 -818.2 Constraint R.H. Side 4 8 9 9 9 6 11 11 11 11 4 8 4 6 16 14 5 0 4 4.5 4 5.5 2 2 4 3 8 7 2.5 1.5 41 Allowable Increase 0 0 0 7.5 7.5 0 4 2 6 0 4 4 6 4 0 2 0 4 5.5 5.5 5.5 1E+30 0 1E+30 1E+30 1E+30 1E+30 1E+30 1E+30 1E+30 5.5 Allowable Decrease 4 5.5 5.5 1E+30 1E+30 5.5 0 7.5 20 2 0 0 1E+30 0 2 1E+30 5.5 0 0 0 0 0 2 2 4 3 8 7 2.5 1.5 0
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SMS ID: 110387
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3.
3.1
Analysis of Project Solution: Sensitivity Analysis: Project Objective
Sensitivity analysis has been generated from model solution that also provides the upper limit and lower limit of the variable coefficient of the objective function within solution would remain optimum. As per above project taken, the final value of variable Cb in the objective function is 4.0. The current coefficient of the variable is 750, allowable increase is 0 and allowable decrease is Infinity. It indicates that current solution would remain optimum if crash cost per unit of time for activity B varies from 750 to Infinity. Reduced cost of the non-basic variables whose value is zero in the optimum solution provides the information about how much objective coefficients of these variables should be reduced to have a positive value of those variables in the optimum solution. As per the above project, reduced cost of a current non-basic variable Ce is 68. It proves that the current coefficient of this variable, which is now 818 must reduce 68 (that means the coefficient would be 68 or below) to get a basic (positive) value of this variable in the optimum solution.
3.2
Sensitivity Analysis: Right Hand Side constraints
Sensitivity of Right hand side of the constraints provides information regarding the status of the constraints; which of these constraints are binding (fully utilized) or non-binding. Binding constraints having a value in the shadow price column, other than zero, means how much contribution these binding constraints will provide individually in the objective function, if the value of the right hand side of these constraints is increased by 1 unit. In the above project, for the precedence constraint A?B, if the duration of the activity A is increased by one unit (1 day) from 4 to 5 days, the overall cost of the project will increase by $750. The value is valid for an increase of up to 1.5 days and decrease by 0.5 days. In contrast, for the Crash constant for activity C (Cc), an increase in duration of this crash time by one unit (1 day) from 4.5 to 5.5 days will cause overall cost of the project to be reduced by $333. The value is valid for an increase of up to 0.5 days and decrease by 1.5 days. Therefore, the conclusion is that liner programming model has helped Mr Iyer (RTS Division of Reliance communications Ltd.) to reach the optimization of project costing, where as sensitivity analysis has helped to provide some flexibility to complete the project assigned.
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PGCBM-21
SMS ID: 110387
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doc_291675499.doc
Project Assignment- XLRI
OPERATIONAL RESEARCH (OR)
INDIVIDUAL ASSIGNMENT- Take a problem from your own organization and apply the concepts taught in the class
NAME: RAJESH KUMAR VERMA SMS ID: 110387 SID NO: RB12044 BATCH: PGCBM-21 FACULTY: DR. SUPRIYA KUMAR DE STUDY CENTER: DAKC, NAVI MUMBAI
RAJESH KUMAR VERMA
PGCBM-21
SMS ID: 110387
PAGE:
1
Table of Contents
1. PROJECT PROBLEM: ............................................................................................................................................................3 2. PROJECT SOLUTION:............................................................................................................................................................4 2.1 MANUAL APPROACH ...........................................................................................................................................................5 2.2 LINEAR PROGRAMMING APPROACH.........................................................................................................................................5 2.2.1 Decision Variables:.................................................................................................................................................6 2.2.2 Objective Function:.................................................................................................................................................6 2.2.3 Constraints:.............................................................................................................................................................6 2.3 EXCEL SOLUTION................................................................................................................................................................9 2.3.1 Excel Solution by Solver- Answer Report: .........................................................................................................10 3. ANALYSIS OF PROJECT SOLUTION:..............................................................................................................................14 3.1 SENSITIVITY ANALYSIS: PROJECT OBJECTIVE........................................................................................................................14 3.2 SENSITIVITY ANALYSIS: RIGHT HAND SIDE CONSTRAINTS.......................................................................................................14
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PROJECT PROBLEM: A Reliance technical Software (RTS) Development Project team has got a assignment to develop software for Wireless business of Reliance. This Project is very important for the RTS division because it is very crucial software which will save 10% of running cost in wireless business. RTS division has been given whole end to end responsibility for this software package from Requirement Analysis till final Deployment and installation. Wireless business has communicated recently to the project manager of RTS 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. RTS division management has decided to assign this project to Mr Iyer as the Project Manager for this Project and given target to plan to present within a week time. Mr. Iyer has come up with the following plan.
Tas k ID A B C D E F Name Effor ts 4 16 9 6 33 4 No of Resour ces utilized 1 2 1 1 3 1 Durati on 4 8 9 6 11 4 Time Savin gs 0 4 4.5 4 5.5 2 A B C D E Predeces sors Cost $1,500.0 0 $3,000.0 0 $1,500.0 0 $1,500.0 0 $4,500.0 0 $1,500.0 0 $1,500.0 0 $3,000.0 0 $3,000.0 0 $3,000.0 0 $3,000.0 0 $1,500.0 0 $1,500.0 0 $30,000. 00 Critic al Yes Yes Yes Yes Yes Yes
1.
Scope of work Software Requirements Design-High Level Design-Low Level Development/Codi ng Unit Testing Specification and Plan Integration Testing Specification and Plan Unit analysis Testing Integration Testing Training Material User Documentation Deployment Post Deployment Review
G H I J K L M
4 16 12 32 42 5
1 2 2 2 2 1
4 8 6 16 14 5 3 64
2 4 3 8 7 2.5 1.5
E F H,G C,E C,E I,J,K L Total Cost
No Yes Yes No No Yes Yes
3 1 Total Project Duration
Table-1: PROJECT PLAN- START WITH Durations and Effort of work are here is in Man Days. RAJESH KUMAR VERMA PGCBM-21 SMS ID: 110387 PAGE: 3
Plan Activities/Tasks A-B-C-D-E-F-H-I-L-M are on Critical Path and the Project duration comes to 64 man days. Total project Cost comes to $30,000. On the finalization of project plan with wireless business, it has been decided that the project is very crucial and duration of 64 days for the project is very high and not acceptable. The wireless business requested that to reduce the project duration by at least 20 days with no extra or minimal extra cost impact. RTS management (Mr Iyer) agreed during the meeting that he would try to reduce the duration and will come up with the new plan within 3 days time.
2.
PROJECT SOLUTION:
Mr Iyer has planned and reduced the project time by a technique called Crashing, which takes care to reduce the duration on a minimal extra cost. Crashing activity started, Mr Iyer 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 Man Days.
No of Resour ces utilized 1 2 1 1 3 1 Pr ed ec es so rs Cra sh Cos t 1,5 00 6,0 00 3,0 00 4,5 00 9,0 00 3,0 00 3,0 00 6,0 00 6,0 00 Extr a Cos t 3,0 00 1,5 00 3,0 00 4,5 00 1,5 00 1,5 00 3,0 00 3,0 00 Cras h Cost/ Day $0 $750 $333 $750 $818 $750
Ta sk ID A B C D E F
Name
Effort s
Durati on
Crash. Durati on 4 4 4.5 2 5.5 2
Time Savin gs 0 4 4.5 4 5.5 2
Cost
Crit ical
Scope of work Software Requirements Design-High Level Design-Low Level Development/Co ding Unit Testing Specification and Plan Integration Testing Specification and Plan Unit analysis Testing Integration Testing
4 16 9 6 33 4
4 8 9 6 11 4
A B C D E
1,50 0 3,00 0 1,50 0 1,50 0 4,50 0 1,50 0 1,50 0 3,00 0 3,00 0
Yes Yes Yes Yes Yes Yes
G H I
4 16 12
1 2 2
4 8 6
2 4 3
2 4 3
E F H, G
$750 $750 $1,00 0
No Yes Yes
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J K L M
Training Material User Documentation Deployment Post Deployment Review
32 42 5 3
2 2 1 1
16 14 5 3 64
8 7 2.5 1.5
8 7 2.5 1.5 Total Cost
C, E C, E I,J, K L
Total Project Duration
3,00 0 3,00 0 1,50 0 1,50 0 30,00 0
6,0 00 9,0 00 3,0 00 3,0 00
3,0 00 6,0 00 1,5 00 1,5 00
$375 $857 $600 $1,00 0
No No Yes Yes
Table-2: Plan with Crash Time and Crash Cost against each activity. Attempted for two approaches to the problem:
2.1
Manual Approach
As per the demand and requirement of wireless group Mr Iyer has decided to crashing project time, 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 for various activities. The cost-time can be computed using the following formula: Crash Cost – Normal Cost Slope = -------------------------------------Normal Time –Crash Time Then next step is to identify that activity on the critical path with the smallest crash cost per time period. This activity will be crashed to the point at which RTS division and wireless management’s desired deadline. 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 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.
2.2
Linear programming approach
Linear programming is a decision making tool under certain situation. The basic assumption of this approach is that to know some relevant data with certainty. Basic data requirements are as follows: 1) To know the project activity time and the precedence sequence. 2) To know what extent an activity can be crashed. 3) Crash cost per unit of time for all activities. The above requirements are explained and calculated in Table-2.
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Now variable of the problem are:
2.2.1 Decision Variables:
To decide variable for each activity “i”, the following two variables can be defined. Ti = Time at each activity “i”, starts. Ci = Number of periods in days by which an activity “i”, is crashed.
2.2.2 Objective Function:
Objective function here is to minimize the total cost of crashing the project down to 44 (64-20 = 44) days. Crash Cost per activity is computed using following formula: Crash Cost – Normal Cost -------------------------------Normal Time –Crash Time Objective function can be expressed as follows. MIN (Z) = $0Ca+ $750Cb +$333Cc +$750Cd +$818Ce +1000Ci+375Cj+ $857Ck + $600Cl +$1000Cm +$750Cf +$750Cg +$750Ch
2.2.3 Constraints:
These constraints can be classified in to three categories. 1) Precedence Constraints: The activities of a project are interrelated and the starting of some activities is dependent upon the completion of some other activities; so we need to establish work sequence of the activities through constraints. For this problem the precedence constraints are defined as below:
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Table-3: Precedence Relationship of the Activities
Task ID A B C D E F G H I J K L M Name Scope of work Software Requirements Design-High Level Design-Low Level Development/Coding Unit Testing Specification and Plan Integration Testing Specification and Plan Unit analysis Testing Integration Testing Training Material User Documentation Deployment Post Deployment Review Predeces sors
A B C D E E F H,G C,E C,E I,J,K L
Fig-1: FLOW DIAGRAM
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Table-4: Precedence Constraints
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 Tc >= Tb + 8-Cb Td>= Tc+9-Cc Tk >= Tc+9-Cc Tj >= Tc+9-Cc Te >=Td+6-Cd Tf >= Te+11-Ce Tk >= Te+11-Ce Tg>= Te+11-Ce Tj >= Te+11-Ce Th >= Tf + 4- Cf Ti >= Th + 8-Ch Ti >= Tg + 4-Cg Tl >= Ti + 6-Ci Tl >= Tj + 16-Cj Tl >= Tk + 14-Ck Tm >= Tl + 5-Cl i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. i.e. Tl – Tk + Ck>=14 Tm - Tl + Cl >=5 Tb-Ta+ Ca >=4 Tc-Tb+ Cb >=8 Td - Tc + Cc >=9 Tk - Tc + Cc>=9 Tj – Tc + Cc>=9 Te – Td + Cd>=6 Tf - Te + Ce >=11 Tk – Te + Ce >=11 Tg - Te + Ce >=11 Tj – Te + Ce >=11 Th - Tf + Cf >= 4 Ti – Th + Ch >=8 Ti - Tg + Cg >=4 Tl – Ti + Ci >=6 Tl – Tj + Cj >=16
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). Mathematically IT CAN BE EXPRESSED AS: Tb >= Ta + 4-Ca OR IT CAN BE Tb-Ta+ Ca >=4 In the similar fashion, precedence relationship is shown in all other activities are explained in Table-4 2) Crash Time Constraints: Now we can reduce the time to complete an activity by increasing the resources or by improving the productivity. But here it is not possible to reduce the time required to complete an activity after a certain threshold limit. Such intention will result in superfluous resources employment which will be an inefficient approach and because of this allowable time to crash an activity has a limit. As required now 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:
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Table-4: Crash Time Constraints
Ca <= 0 Cb <= 4 Cc <= 4.5 Cd <= 4 Ce < = 5.5 Cf <= 2 Cg <= 2 Ch <= 4 Ci <= 3 Cj <= 8 Ck <= 7 Cl <= 2.5 Cm <= 1.5
3) Project Completion Constraints: The project completion constraint will recognize completion of last activities) must take place before the project deadline date. In this project, activity M is the last activity in the project and starts at Tm. The normal duration for activity M is 3 days, and Cm denotes the number of days by which its duration can be crashed. Therefore the actual duration of activity M is (3-Cm), and its completion time is (Tm+3Cm). This can be explained as: Tm+3-Cm <= 44 i.e. Tm-Cm <= 41 (The normal project duration is 64. As per Wireless business, it should be reduced by 20 days, hence the last activity M completion time should be <= 64-20 = 44 days.] 4) Non-Negativity Constraints: All Ti and Ci >= 0
2.3
Excel Solution
Tale-5: Excel Solution
TA TB TC TD TE TF TG TH Ti Tj Tk Tl Tm CA CB CC CD CE CF CG CH Ci Cj Ck Cl Cm
Start Start Start Start Start Start Start Start Start Start Start Start Start Crash C rash Crash C rash Crash C rash Crash C rash Crash Crash Crash Crash Crash A B C D E F G H I J K L M A B C D E F G H I J K L M 0.0 4.0 8.0 1 2.5 1 4.5 20.0 20.0 22.0 30.0 20.0 20.0 36.0 41 .0 0.0 $0 4.0 $750 4.5 $333 4.0 $750 5.5 $81 8 2.0 $750 0.0 $750 0.0 0.0 0.0 0.0 0.0 0.0
$750 $1 ,000 $375
$857 $600 $1 ,000 1 3500
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In the above table-5 the activities B, C, D, F and K are required to be crashed by 4.0, 4.5, 4.0, 0.5 and 7.0 days respectively to complete the project within stipulated 44 days. The total cost for crashing will be $13500 2.3.1 Excel Solution by Solver- Answer Report:
M ro o E el 1 .0 A s er R o ic s ft xc 1 nw ep rt W rk h o s eet: [exc s eet.xls o er E tries el h ]S lv n R o C ep rt reated 0 -0 -2 1 1 :3 :5 : 2 6 02 6 0 2
T rg t C ll (M ) a e e in C ell $ B6 A $ C sh co ra st
N e am
O in V e rig al alu 150 3 0 .0
F al V e in alu 150 3 0 .0
A ju b C lls d sta le e C ell N e am $ $ B5 S lu n v lu S rt A o tio a e ta $ $ C5 S lu n v lu S rt B o tio a e ta $ $ D5 S lu n v lu S rt C o tio a e ta $ $ E5 S lu n v lu S rt D o tio a e ta $$ F5 S lu n v lu S rt E o tio a e ta $ $ G5 S lu n v lu S rt F o tio a e ta $ $ H5 S lu n v lu S rt G o tio a e ta $ 5 I$ S lu n v lu S rt H o tio a e ta $$ J5 S lu n v lu S rt I o tio a e ta $ $ K5 S lu n v lu S rt J o tio a e ta $$ L5 S lu n v lu S rt K o tio a e ta $ $ M5 S lu n v lu S rt L o tio a e ta $ $ N5 S lu n v lu S rt M o tio a e ta $ $ O5 S lu n v lu C sh A o tio a e ra $ $ P5 S lu n v lu C sh B o tio a e ra $ $ Q5 S lu n v lu C sh C o tio a e ra $ $ R5 S lu n v lu C sh D o tio a e ra $ $ S5 S lu n v lu C sh E o tio a e ra $$ T5 S lu n v lu C sh F o tio a e ra $ $ U5 S lu n v lu C sh G o tio a e ra $ $ V5 S lu n v lu C sh H o tio a e ra $ $ W5 S lu n v lu C sh I o tio a e ra $$ X5 S lu n v lu C sh J o tio a e ra $ $ Y5 S lu n v lu C sh K o tio a e ra $$ Z5 S lu n v lu C sh L o tio a e ra $ A5 A $ S lu n v lu C sh M o tio a e ra
O in V e rig al alu 0 .0 4 .0 8 .0 1 .5 2 1 .5 4 2 .0 0 2 .0 6 2 .0 2 3 .0 0 2 .0 0 2 .0 0 3 .0 6 4 .0 1 0 .0 4 .0 4 .5 4 .0 5 .5 2 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0
F al V e in alu 0 .0 4 .0 8 .0 1 .5 2 1 .5 4 2 .0 0 2 .0 6 2 .0 2 3 .0 0 2 .0 0 2 .0 2 3 .0 6 4 .0 1 0 .0 4 .0 4 .5 4 .0 5 .5 2 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0 0 .0
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Constraints Cell $AB$8 $AB$9 $AB$10 $AB$11 $AB$12 $AB$13 $AB$14 $AB$15 $AB$16 $AB$17 $AB$18 $AB$19 $AB$20 $AB$21 $AB$22 $AB$23 $AB$24 $AB$25 $AB$26 $AB$27 $AB$28 $AB$29 $AB$30 $AB$31 $AB$32 $AB$33 $AB$34 $AB$35 $AB$36 $AB$37 $AB$38
Name 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 Crash limit A Crash limit B Crash limit C Crash limit D Crash limit E Crash limit F Crash limit G Crash limit H Crash limit I Crash limit J Crash limit K Crash limit L Crash limit M Project finish
Cell Value 4.0 8.0 9.0 18.5 16.5 6.0 11.0 13.0 17.0 11.0 4.0 8.0 4.0 6.0 16.0 14.0 5.0 0.0 4.0 4.5 4.0 5.5 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 41.0
Formula $AB$8>=$AD$8 $AB$9>=$AD$9 $AB$10>=$AD$10 $AB$11>=$AD$11 $AB$12>=$AD$12 $AB$13>=$AD$13 $AB$14>=$AD$14 $AB$15>=$AD$15 $AB$16>=$AD$16 $AB$17>=$AD$17 $AB$18>=$AD$18 $AB$19>=$AD$19 $AB$20>=$AD$20 $AB$21>=$AD$21 $AB$22>=$AD$22 $AB$23>=$AD$23 $AB$24>=$AD$24 $AB$25<=$AD$25 $AB$26<=$AD$26 $AB$27<=$AD$27 $AB$28<=$AD$28 $AB$29<=$AD$29 $AB$30<=$AD$30 $AB$31<=$AD$31 $AB$32<=$AD$32 $AB$33<=$AD$33 $AB$34<=$AD$34 $AB$35<=$AD$35 $AB$36<=$AD$36 $AB$37<=$AD$37 $AB$38<=$AD$38
Status Binding Binding Binding Not Binding Not Binding Binding Binding Not Binding Not Binding Binding Binding Binding Binding Binding Binding Binding Binding Binding Binding Binding Binding Binding Binding Not Binding Not Binding Not Binding Not Binding Not Binding Not Binding Not Binding Binding
Slack 0.0 0.0 0.0 9.5 7.5 0.0 0.0 2.0 6.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 0 0 0 0 0 2 4 3 8 7 2.5 1.5 0
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2.3.2 Excel Solution by Solver- Sensitivity Report
Microsoft Excel 11.0 Sensitivity Report Worksheet: [excel sheet.xls]Solver Entries Report Created: 02-06-2012 16:31:48
Adjustable Cells Cell $B$5 $C$5 $D$5 $E$5 $F$5 $G$5 $H$5 $I$5 $J$5 $K$5 $L$5 $M$5 $N$5 $O$5 $P$5 $Q$5 $R$5 $S$5 $T$5 $U$5 $V$5 $W$5 $X$5 $Y$5 $Z$5 $AA$5 Name Solution value Start A Solution value Start B Solution value Start C Solution value Start D Solution value Start E Solution value Start F Solution value Start G Solution value Start H Solution value Start I Solution value Start J Solution value Start K Solution value Start L Solution value Start M Solution value Crash A Solution value Crash B Solution value Crash C Solution value Crash D Solution value Crash E Solution value Crash F Solution value Crash G Solution value Crash H Solution value Crash I Solution value Crash J Solution value Crash K Solution value Crash L Solution value Crash M Final Reduced Value Cost 0.0 818.2 4.0 0.0 8.0 0.0 12.5 0.0 14.5 0.0 20.0 0.0 20.0 0.0 22.0 0.0 30.0 0.0 20.0 0.0 20.0 0.0 36.0 0.0 41.0 0.0 0.0 0.0 4.0 0.0 4.5 0.0 4.0 0.0 5.5 0.0 2.0 0.0 0.0 750.0 0.0 0.0 0.0 250.0 0.0 306.8 0.0 39.0 0.0 600.0 0.0 181.8 Objective Coefficient 0 0 0 0 0 0 0 0 0 0 0 0 0 0 750 333.3333333 750 818.1818182 750 750 750 1000 375 857.1428571 599.9999999 1000 Allowable Increase 1E+30 1E+30 1E+30 1E+30 1E+30 68.18181818 68.18181818 68.18181818 750 68.18181818 68.18181818 818.1818182 818.1818182 818.1818182 68.18181818 484.8484849 68.18181818 38.96103891 0 1E+30 68.18181818 1E+30 1E+30 1E+30 1E+30 1E+30 Allowable Decrease 818.1818182 818.1818182 68.18181818 68.18181818 68.18181818 38.96103891 0 0 38.96103891 38.96103891 0 38.96103891 181.8181815 1E+30 1E+30 1E+30 1E+30 68.18181818 1E+30 750 0 250.0000001 306.8181818 38.96103891 599.9999999 181.8181816
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Constraints Cell $AB$8 $AB$9 $AB$10 $AB$11 $AB$12 $AB$13 $AB$14 $AB$15 $AB$16 $AB$17 $AB$18 $AB$19 $AB$20 $AB$21 $AB$22 $AB$23 $AB$24 $AB$25 $AB$26 $AB$27 $AB$28 $AB$29 $AB$30 $AB$31 $AB$32 $AB$33 $AB$34 $AB$35 $AB$36 $AB$37 $AB$38 Name 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 Crash limit A Crash limit B Crash limit C Crash limit D Crash limit E Crash limit F Crash limit G Crash limit H Crash limit I Crash limit J Crash limit K Crash limit L Crash limit M Project finish Final Shadow Value Price 4.0 818.2 8.0 818.2 9.0 818.2 16.5 0.0 16.5 0.0 6.0 818.2 11.0 750.0 11.0 0.0 11.0 0.0 11.0 68.2 4.0 750.0 8.0 750.0 10.0 0.0 6.0 750.0 16.0 68.2 16.0 0.0 5.0 818.2 0.0 -818.2 4.0 -68.2 4.5 -484.8 4.0 -68.2 5.5 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 41.0 -818.2 Constraint R.H. Side 4 8 9 9 9 6 11 11 11 11 4 8 4 6 16 14 5 0 4 4.5 4 5.5 2 2 4 3 8 7 2.5 1.5 41 Allowable Increase 0 0 0 7.5 7.5 0 4 2 6 0 4 4 6 4 0 2 0 4 5.5 5.5 5.5 1E+30 0 1E+30 1E+30 1E+30 1E+30 1E+30 1E+30 1E+30 5.5 Allowable Decrease 4 5.5 5.5 1E+30 1E+30 5.5 0 7.5 20 2 0 0 1E+30 0 2 1E+30 5.5 0 0 0 0 0 2 2 4 3 8 7 2.5 1.5 0
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3.
3.1
Analysis of Project Solution: Sensitivity Analysis: Project Objective
Sensitivity analysis has been generated from model solution that also provides the upper limit and lower limit of the variable coefficient of the objective function within solution would remain optimum. As per above project taken, the final value of variable Cb in the objective function is 4.0. The current coefficient of the variable is 750, allowable increase is 0 and allowable decrease is Infinity. It indicates that current solution would remain optimum if crash cost per unit of time for activity B varies from 750 to Infinity. Reduced cost of the non-basic variables whose value is zero in the optimum solution provides the information about how much objective coefficients of these variables should be reduced to have a positive value of those variables in the optimum solution. As per the above project, reduced cost of a current non-basic variable Ce is 68. It proves that the current coefficient of this variable, which is now 818 must reduce 68 (that means the coefficient would be 68 or below) to get a basic (positive) value of this variable in the optimum solution.
3.2
Sensitivity Analysis: Right Hand Side constraints
Sensitivity of Right hand side of the constraints provides information regarding the status of the constraints; which of these constraints are binding (fully utilized) or non-binding. Binding constraints having a value in the shadow price column, other than zero, means how much contribution these binding constraints will provide individually in the objective function, if the value of the right hand side of these constraints is increased by 1 unit. In the above project, for the precedence constraint A?B, if the duration of the activity A is increased by one unit (1 day) from 4 to 5 days, the overall cost of the project will increase by $750. The value is valid for an increase of up to 1.5 days and decrease by 0.5 days. In contrast, for the Crash constant for activity C (Cc), an increase in duration of this crash time by one unit (1 day) from 4.5 to 5.5 days will cause overall cost of the project to be reduced by $333. The value is valid for an increase of up to 0.5 days and decrease by 1.5 days. Therefore, the conclusion is that liner programming model has helped Mr Iyer (RTS Division of Reliance communications Ltd.) to reach the optimization of project costing, where as sensitivity analysis has helped to provide some flexibility to complete the project assigned.
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