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Project Scheduling

Project Scheduling. The Critical Path Method (CPM). Cost Analyses Using The Critical Path Method (CPM). The critical path method (CPM) is a deterministic approach to project planning. Completion time depends only on the amount of money allocated to activities.

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Project Scheduling

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  1. Project Scheduling The Critical Path Method (CPM)

  2. Cost Analyses Using The Critical Path Method (CPM) • The critical path method (CPM) is a deterministic approach to project planning. • Completion time depends only on the amount of money allocated to activities. • Reducing an activity’s completion time is called crashing.

  3. Normal and CrashTimes and Costs • There are two extreme values for the completion times and costs to consider for each activity. • Normal completion time (TN) when the “usual” or normal Cost (CN) is spent to complete the activity. • Crash completion time (TC), the theoretical minimum possible completion time when an amount (CC) is spent to complete the activity. • If any amount between CN and CC is spent, the activity completion time is reduced proportionately. • If more than CC is spent, the completion time will not be reduced below TC.

  4. Determining the Time and Cost of an Activity • The maximum time reduction for an activity is R = TN – TC. • This maximum time reduction is achieved by spending E = CC – CN extra dollars. • Any percentage of the maximum extra cost E spent to crash an activity, yields the same percentage reduction of the maximum time savings.

  5. Example • An activity under normal conditions cost CN = $2000 and takes TN = 20 days. • A maximum time reduction down to a TC = 12 day completion time can be achieved by spending CC = $4400. • Here R = 20-12 = 8 days and E = $4400 - $2000 = $2400. Marginal cost $2400/8 = $300 per day. What would it cost to complete the activity in 17days? Days reduced = 20 – 17 = 3. Extra cost will be 3($300) = $900 Activity will cost $2000 + $900 = $2900 How long would it take to complete the activity if $2600 were spent? Extra money spent = $2600 - $2000 = $600. Days reduced = = 600/300 = 2 Activity will take 20 - 2 = 18 days.

  6. CPM -- Meeting a Deadline at Minimum Cost • When a deadline to complete a project cannot be met using normal times, additional resources must be spent to crash activities to reduce the project completion time from that achieved using normal costs. • CPM can use linear programming to: • MIN Total Extra Cost Spent • So that: • The deadline is met • No activity is crashed more than its maximum crash amount • The activities are performed in accordance with the precedence relations

  7. Baja Burrito Restaurants – Meeting a Deadline at Minimum Cost • Baja Burrito (BB) is a chain of Mexican-style fast food restaurants. • It is planning to open a new restaurant in 19 weeks. • Management wants to • Study the feasibility of this plan, • Study suggestions in case the plan cannot be finished by the deadline.

  8. Baja Burrito Restaurants – When all the activities are crashed to the maximum, the restaurant will open 17 weeks at crash cost of $300,000. Without spending any extra money, the restaurant will open in 29 weeks at a normal cost of $200,000. *Determined by the PERT-CPM template

  9. Baja Burrito Restaurants –Network presentation E O I K B A F G J M N C H P L D

  10. Baja Burrito Restaurants –Marginal costs For Activity A R = TN – TC = 5 – 3 = 2 E = CC – CN = 36 – 25 = 11 Marginal Cost M = 11/2 =$5.50

  11. Baja Burrito Restaurants –Linear Programming • Linear Programming Approach • Variables Xj = start time for activity j. Yj = the amount of crash in activity j. • Objective Function Minimize the total additional funds spent on crashing activities. • Constraints • The project must be completed by the deadline date D. • No activity can be reduced more than its Max. time reduction. • Start time of an activity takes place not before the finish time of all its immediate predecessors.

  12. The Linear Programming Model • Xj = start time for activity j • Yj = the amount of crash in activity j Min 5.5YA+10YB+2.67YC+4YD+2.8YE+6YF+6.67YG+10YH+5.33YI+12YJ+4YK+5.33YL+1.5YN+4YO+5.33YP Minimize total crashing costs

  13. Deadline and Maximum Crash Time Constraints ST £ X ( FIN ) 19 Min 5.5YA+10YB+2.67YC+4YD+2.8YE+6YF+6.67YG+10YH+5.33YI+12YJ+4YK+5.33YL+1.5YN+4YO+5.33YP Meet the deadline Maximum time reductions YA ≤ 2.0 YB ≤ 0.5 YC ≤ 1.5 YD ≤ 1.0 YE ≤ 2.5 YF ≤ 0.5 YG ≤ 1.5 YH ≤ 0.5 YI ≤ 1.5 YJ ≤ 0.5 YK ≤ 1.0 YL ≤ 1.5 YM ≤ 1.5 YN ≤ 2.0 YO ≤ 1.5 YP ≤ 1.5

  14. Example of Precedence Constraints E 4-YE O F I N I S H M 3-YM Min 5.5YA+10YB+2.67YC+4YD+2.8YE+6YF+6.67YG+10YH+5.33YI+12YJ+4YK+5.33YL+1.5YN+4YO+5.33YP Analysis of Activity O O’s Start Time  E’s Start Time + E’s duration XO  XE + (4-YE) O’s Start Time  M’s Start Time + M’s duration XO  XM + (3-YM)

  15. Complete Set ofPrecedence Constraints F I N I S H Activity start time ≥ Finish time of immediate predecessors Min 5.5YA+10YB+2.67YC+4YD+2.8YE+6YF+6.67YG+10YH+5.33YI+12YJ+4YK+5.33YL+1.5YN+4YO+5.33YP XB³XA+(5 – YA) XC³XA+(5 – YA) XD³XA+(5 – YA) Xe³XA+(5 – YA) XF³XA+(5 – YA) XB³XB+(1 – YB) XF³XC+(3 – YC) XG³XF+(1 – YF) .. . X(FIN)³XN+(3 – YN) X(FIN)³XO+(4 – YO) X(FIN)³XP+(4 – YP) All xj’s and yj’s ≥ 0

  16. Click Solve INPUT Activity Names, Time/Cost Data, Project Deadline, and Immediate Predecessors CPM-DEADLINE TEMPLATE Select Solver

  17. Operating Within a Fixed Budget • CPM can also be applied to situations where there is a fixed budget. • The objective now is to minimize the project completion time given this budget. • Of course if the budget = sum of the normal costs, no crashing can be done and the minimum completion time of network with normal times is the minimum project completion time • But if the budget exceeds the total of the normal costs, decisions must be made as to which activities to crash.

  18. The New CPM Model CPM - DEADLINE Minimize 5.5YA + 10YB + 2.67YC + 4YD + 2.8YE + 6YF + 6.67YG + 10YH + 5.33YI + 12YJ + 4YK + 5.33Y L+ 1.5YN + 4YO + 5.33YP CPM - BUDGET X(FIN) 5.5YA + 10YB + 2.67YC + 4YD + 2.8YE + 6YF + 6.67YG + 10YH + 5.33YI + 12YJ + 4YK + 5.33Y L+ 1.5YN + 4YO + 5.33YP The only change is that the deadline constraint in the previous model is now the objective, and the objective in the previous model becomes the first constraint. s.t. X(FIN) £ 19 Minimize s.t. £25 The other constraints of the crashing model remain the same.

  19. Click Solve Add END node The predecessors for END are nodes without successors INPUT Activity Names, Time/Cost Data, Maximum Budget, and Immediate Predecessors CPM-BUDGET TEMPLATE Call Solver

  20. Review • CPM assumes the percent time reduction of an activity is proportional to the percent of the maximum added cost • Linear programming formulation for: • Min cost to meet a deadline • Min completion within a fixed budget • CPM-Deadline and CPM-Budget templates

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