Network Scheduling with Limited Resources
Introduction  <ul><li>Early start schedule (EST ) or Late start  schedule (LST)generated by two sets of data  </li></ul><u...
Schedule  Graph  , Unlimited  Resources Labor available Project completion time  18 10 12 6 6 1
Let assume following constraint 10 labor available on any one day  Schedule  Graph , Resource  Limit  Of  10  Men  Cont.. ...
Complexity Of Project Scheduling With Limited Resource <ul><li>Scheduling project  with  limited resource is combinatorial...
Heuristic Programs  <ul><li>Heuristic  – standard procedure  to solve problem  but not rule </li></ul><ul><li>Resource Lev...
Heuristic method for Resource Leveling of Project Schedule <ul><li>Utilize resources  at  more constant rate </li></ul><ul...
Early start schedule  graph Manpower loading chart  Maximum slack 7 days so postpone activity 9  Peak requirement 24 Numbe...
Schedule  Graph  Manpower loading  chart . Peak requirement reduce to 15  Cont..
Schedule  Graph  Manpower  Loading Chart  Activity 8 is delay by full slack of 2 days Peak requirement occur at days 1, 2 ...
Schedule  Graph  Manpower  Loading Chart. Best result obtain by delaying activity 3 by 2 days  Smoothen  histogram Cont..
Smoothing   Peak Resource Requirement
Heuristic Method for Resource Allocation in Project Scheduling <ul><li>Allocate resources serially in time </li></ul><ul><...
Day 1  - only one job 7 is available to start in this period and there  are  sufficient  men  to  schedule it. Schedule jo...
Day 2-  Job 1,3,4,5 can be started on day 2 , but there are not enough men to schedule them Then second heuristic is appli...
Day 3-  continue job 3(slack =0); 7 men remain continue job 4(slack =0); 3men remain continue job 1(slack =3); 2men remain...
Day 4 Continue job 4 (slack =0); 6 men remain  Continue job 1 (slack =3); 5 men remain  Job 6  has no slack and hence  is ...
Cont.. Day Manpower/ resource requirement
Day 5 Continue job 6 (slack=0); 4 men remain  Schedule Job 2 (slack =0); 2 men remain Schedule Job 1 ( slack = 0); 1 men r...
Cont.. Day Manpower
Day 7  Continue Job 1 (slack =1); 9 Men remain  Schedule Job 5 ( slack =0); 4 Men remain  Postpone job 9 ( slack =1) Day 8...
Day 9–  Schedule job 9 ( slack =0); 1 man remain .  Postpone job 8 (slack =2) Day 10 –  Schedule job 10 ( slack =0);  no m...
Day 12— Continue job 10 (slack =1); no men remain  Postpone Job 8( slack =0) Day 13— Schedule  job 8 ( slack =0); 2 men re...
Other Heuristic Approaches <ul><li>First schedule the critical activity </li></ul><ul><li>If it fails to do so then schedu...
<ul><li>If activity can not  schedule at minimum crew size then its Early start date is delayed  and schedule it on next d...
<ul><li>Add on Unused Resource     </li></ul><ul><li>list  activities  which required resources and  which have crew size...
Conceptual Problem of Critical Path Analysis When Resource Are Limited
Early start schedule Late Start schedule  Cont.. Day Resource Day Resource 1 2 3 4 5 18 10 10 8 7 1 2 3 4 5 4 4 5 18 22
If resources are limited up to 10 men then optimal early start schedule  and Late start schedule is as follows Early start...
Late Start schedule  Day Resource 1 2 3 4 5 6 10 10 8 10 7 8
<ul><li>O bservations </li></ul><ul><li>Slack concept is a measure of flexibility in a project schedule </li></ul><ul><li>...
Conclusion <ul><li>Ordinary PERT and CPM </li></ul><ul><li>No explicit  consideration for resources </li></ul><ul><li>2.  ...
<ul><li>3. Resource leveling  program   </li></ul><ul><li>Using  resource requirement  profile of  EST schedule  , the pr...
<ul><li>4. Resource allocation Program     </li></ul><ul><li>Fixed amount of resource  allocated to available  jobs accor...
Integer Programming Formulation – Project scheduling with limited Resource <ul><li>S- Resource  s= 1,2,3,……………,m </li></ul...
<ul><li>Objective function  </li></ul><ul><li>Minimize  </li></ul>where k is some number such  that
<ul><li>Variables : </li></ul><ul><li>= activity of job j on day d ;  </li></ul><ul><li>Constrained = 1 ( if job j is acti...
2 .  Job will be performed : 3. Capacity of shop will not be exceeded  4.No job will be started before its predecessors  a...
5 .  No jobs will be split  :
 
 
 
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Network scheduling with limited_resources- revised1

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Network scheduling with limited_resources- revised1

  1. 1. Network Scheduling with Limited Resources
  2. 2. Introduction <ul><li>Early start schedule (EST ) or Late start schedule (LST)generated by two sets of data </li></ul><ul><li>Activity precedence relation </li></ul><ul><li>Activity Duration (based on PERT or CPM ) </li></ul><ul><li>Assumption - Resources available in unlimited supply </li></ul><ul><li>In practical situations  fixed manpower , certain number of machine , and limited budget </li></ul>
  3. 3. Schedule Graph , Unlimited Resources Labor available Project completion time 18 10 12 6 6 1
  4. 4. Let assume following constraint 10 labor available on any one day Schedule Graph , Resource Limit Of 10 Men Cont.. 10 10 10 10 6 6 1
  5. 5. Complexity Of Project Scheduling With Limited Resource <ul><li>Scheduling project with limited resource is combinatorial problem (to find the best possible discrete solution) </li></ul><ul><li>Large combination of activity start time , each combination representing a different schedule </li></ul><ul><li>Integer programming can be applied to solve the problem </li></ul>
  6. 6. Heuristic Programs <ul><li>Heuristic – standard procedure to solve problem but not rule </li></ul><ul><li>Resource Leveling Program – </li></ul><ul><li>It reduces peak resource requirement </li></ul><ul><li>smooth out period assignment within a constraint on project duration </li></ul><ul><li>Resource Allocation Program – </li></ul><ul><li>It allocates resources to project activities in an attempt to find the shortest project schedule consistent with fixed resource limit </li></ul><ul><li>Project duration may change </li></ul>
  7. 7. Heuristic method for Resource Leveling of Project Schedule <ul><li>Utilize resources at more constant rate </li></ul><ul><li>Scheduler may use activity slack as means of smoothing peak resource requirement </li></ul><ul><li>Draw EST and LST schedule </li></ul><ul><li>Concentrate on Activity having maximum slack </li></ul>
  8. 8. Early start schedule graph Manpower loading chart Maximum slack 7 days so postpone activity 9 Peak requirement 24 Number over activity indicates the activity number as well as resource.
  9. 9. Schedule Graph Manpower loading chart . Peak requirement reduce to 15 Cont..
  10. 10. Schedule Graph Manpower Loading Chart Activity 8 is delay by full slack of 2 days Peak requirement occur at days 1, 2 Cont..
  11. 11. Schedule Graph Manpower Loading Chart. Best result obtain by delaying activity 3 by 2 days Smoothen histogram Cont..
  12. 12. Smoothing Peak Resource Requirement
  13. 13. Heuristic Method for Resource Allocation in Project Scheduling <ul><li>Allocate resources serially in time </li></ul><ul><li>Start on the first day and schedule all jobs possible , repeat for second day, and so on </li></ul><ul><li>When Several jobs compete for the same resource , give preference to the job having least slack </li></ul><ul><li>Reschedule non-critical jobs ,if possible , to free the resource for scheduling critical or non-slack jobs </li></ul>
  14. 14. Day 1 - only one job 7 is available to start in this period and there are sufficient men to schedule it. Schedule job 7 , 3 men remain available Only 10 men are available for the project on any day. Resource Day
  15. 15. Day 2- Job 1,3,4,5 can be started on day 2 , but there are not enough men to schedule them Then second heuristic is applied for scheduling the jobs with least slack first Schedule job 3( slack=0); 7 men remain Schedule job 4( slack=0); 3 men remain Schedule job 1( slack=3); 2 men remain Job 5 must be delayed , as there are just [2=10-(4+3+1)] men unassigned Postponed the job 5 ( slack =4) Cont..
  16. 16. Day 3- continue job 3(slack =0); 7 men remain continue job 4(slack =0); 3men remain continue job 1(slack =3); 2men remain Postpone job 5 (slack =3) Cont..
  17. 17. Day 4 Continue job 4 (slack =0); 6 men remain Continue job 1 (slack =3); 5 men remain Job 6 has no slack and hence is critical , but only 5 men are still unassigned Then 3 rd heuristic is applied – job 1 could be postponed without delaying the project Reschedule job 1 ; 6 men remain Schedule job 6 (slack =0); 0 men remain Postpone Job 5 (slack =2) Cont..
  18. 18. Cont.. Day Manpower/ resource requirement
  19. 19. Day 5 Continue job 6 (slack=0); 4 men remain Schedule Job 2 (slack =0); 2 men remain Schedule Job 1 ( slack = 0); 1 men remain Postpone job 5 (slack =1) Day 6 Continue Job 6 (slack =0); 4 Men remain . Continue Job 2 (slack =0); 2 Men remain . Continue Job 1 (slack =0); 1 Men remain . Postpone job 5 (slack =0) Now job 5 is critical but no active job can be postponed without delaying project Cont..
  20. 20. Cont.. Day Manpower
  21. 21. Day 7 Continue Job 1 (slack =1); 9 Men remain Schedule Job 5 ( slack =0); 4 Men remain Postpone job 9 ( slack =1) Day 8 Continue Job 5 (slack =0); 5 men remain Postpone job 9 (slack =0); No noncritical job can be rescheduled Postpone Job 8 (Slack =2) Day Manpower
  22. 22. Day 9– Schedule job 9 ( slack =0); 1 man remain . Postpone job 8 (slack =2) Day 10 – Schedule job 10 ( slack =0); no man remain. Postpone Job 8 (slack =1) Day 11 – Continue job 10 (slack =0); no men remain . Postpone Job 8 (slack =0)
  23. 23. Day 12— Continue job 10 (slack =1); no men remain Postpone Job 8( slack =0) Day 13— Schedule job 8 ( slack =0); 2 men remain Day 14 – Continue job 8 ( slack = 0); 2 men remain Cont.. Manpower Day Final schedule limited resource 1 2 3 4 5 6 7 8 9 10 11 12 13 14 7 7 7 10 9 9 6 6 9 10 10 10 8 8
  24. 24. Other Heuristic Approaches <ul><li>First schedule the critical activity </li></ul><ul><li>If it fails to do so then schedule it in next period </li></ul><ul><li>Rule for crew size selection  critical activity schedule first with maximum crew size ( i.e. crashing ) if it fails then </li></ul><ul><li>Schedule it at minimum crew size </li></ul>
  25. 25. <ul><li>If activity can not schedule at minimum crew size then its Early start date is delayed and schedule it on next day </li></ul><ul><li>For non-critical jobs also delayed if sufficient resources not available for scheduling </li></ul><ul><li>Borrow form active job- resources borrowed from the activity only when the stretching of the job will not delay the entire project </li></ul>Cont..
  26. 26. <ul><li>Add on Unused Resource  </li></ul><ul><li>list activities which required resources and which have crew size less than maximum </li></ul><ul><li>Job arrange in ascending order of their slack </li></ul><ul><li>Proceeding down the list and allocate resources </li></ul>Cont..
  27. 27. Conceptual Problem of Critical Path Analysis When Resource Are Limited
  28. 28. Early start schedule Late Start schedule Cont.. Day Resource Day Resource 1 2 3 4 5 18 10 10 8 7 1 2 3 4 5 4 4 5 18 22
  29. 29. If resources are limited up to 10 men then optimal early start schedule and Late start schedule is as follows Early start schedule Day Resource 1 2 3 4 5 6 10 10 10 9 7 7
  30. 30. Late Start schedule Day Resource 1 2 3 4 5 6 10 10 8 10 7 8
  31. 31. <ul><li>O bservations </li></ul><ul><li>Slack concept is a measure of flexibility in a project schedule </li></ul><ul><li>Slack depends on precedence ordering and resource availability </li></ul><ul><li>Resource limitation reduce the amount of slack in a schedule </li></ul><ul><li>Slack is conditional upon the scheduling rules for creating early start and late start schedules </li></ul>
  32. 32. Conclusion <ul><li>Ordinary PERT and CPM </li></ul><ul><li>No explicit consideration for resources </li></ul><ul><li>2. Scheduling program </li></ul><ul><li>For each resource , the program calculates a total resource requirement profile by summing period by period , requirements for that resource of all activities as they occur in an early start schedule </li></ul>
  33. 33. <ul><li>3. Resource leveling program  </li></ul><ul><li>Using resource requirement profile of EST schedule , the program attempts to reduce peak requirements by shifting slack jobs to nonpeak periods </li></ul><ul><li>Resource limits are not specified , but peak requirements are leveled as much as possible without delaying the specified due date. </li></ul>Cont..
  34. 34. <ul><li>4. Resource allocation Program  </li></ul><ul><li>Fixed amount of resource allocated to available jobs according to certain scheduling heuristic that determine which job will be postponed if total requirements for a given period exceed resources available </li></ul><ul><li>The completion date of the project may be pushed ahead to keep within specified resource limits </li></ul><ul><li>No heuristic program can guarantee an optimum schedule </li></ul>Cont..
  35. 35. Integer Programming Formulation – Project scheduling with limited Resource <ul><li>S- Resource s= 1,2,3,……………,m </li></ul><ul><li>d -day (time period) d=1,2,……………….,z </li></ul><ul><li>j -Job (activity) j= 1,2,………………,n </li></ul><ul><li>P- immediate predecessor of j ; p ε P j </li></ul><ul><li> = [ all immediate predecessors of j] </li></ul>
  36. 36. <ul><li>Objective function </li></ul><ul><li>Minimize </li></ul>where k is some number such that
  37. 37. <ul><li>Variables : </li></ul><ul><li>= activity of job j on day d ; </li></ul><ul><li>Constrained = 1 ( if job j is active ) or </li></ul><ul><li> 0 (if job j is inactive) </li></ul><ul><li>Constraints </li></ul><ul><li>0 ≤ X jd ≤ 1 (x is constrained to equal either 0 or 1); all jd </li></ul>
  38. 38. 2 . Job will be performed : 3. Capacity of shop will not be exceeded 4.No job will be started before its predecessors are completed d = 1,……,z AND j= 1,….,n except beginning job
  39. 39. 5 . No jobs will be split :
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