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. crashing.Reducing an activity’s completion time is called crashing.

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

Determining the Time and Cost of an Activity R = T N – T CThe maximum time reduction for an activity is R = T N – T C. E = C C – C NThis maximum time reduction is achieved by spending E = C C – C N 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.

Example An activity under normal conditions cost C N = $2000 and takes T N = 20 days. A maximum time reduction down to a T C = 12 day completion time can be achieved by spending C C = $4400. Here R = = 8 days and E = $ $2000 = $2400. How long would it take to complete the activity if $2600 were spent? Marginal cost $2400/8 = $300 per day. Extra money spent = $ $2000 = $600. Days reduced = = 600/300 = 2 Activity will take = 18 days. 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 $2900 $ $900 = $2900

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 CPM -- 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. Baja Burrito Restaurants – Meeting a Deadline at Minimum Cost

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

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

Baja Burrito Restaurants – Marginal costs For Activity A R = T N – T C = 5 – 3 = 2 E = C C – C N = 36 – 25 = 11 Marginal Cost M = 11/2 =$5.50

Linear Programming Approach –Variables X j = start time for activity j. Y j = 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. Baja Burrito Restaurants – Linear Programming

The Linear Programming Model X j = start time for activity j Y j = the amount of crash in activity j Minimize total crashing costs Min 5.5Y A +10Y B +2.67Y C +4Y D +2.8Y E +6Y F +6.67Y G +10Y H Y I +12Y J +4Y K +5.33Y L +1.5Y N +4Y O +5.33Y P

Maximum time reductions Y A ≤ 2.0 Y B ≤ 0.5 Y C ≤ 1.5 Y D ≤ 1.0 Y E ≤ 2.5 Y F ≤ 0.5 Y G ≤ 1.5 Y H ≤ FINX  )( ST Meet the deadline Deadline and Maximum Crash Time Constraints Y I ≤ 1.5 Y J ≤ 0.5 Y K ≤ 1.0 Y L ≤ 1.5 Y M ≤ 1.5 Y N ≤ 2.0 Y O ≤ 1.5 Y P ≤ 1.5

FINISHFINISH Min 5.5Y A +10Y B +2.67Y C +4Y D +2.8Y E +6Y F +6.67Y G +10Y H Y I +12Y J +4Y K +5.33Y L +1.5Y N +4Y O +5.33Y P Example of Precedence Constraints Analysis of Activity O E 4-Y E M 3-Y M O O’s Start Time  E’s Start Time + E’s duration O’s Start Time  M’s Start Time + M’s duration XOXO  X M + (3-Y M ) XOXO  X E + (4-Y E )

Min 5.5Y A +10Y B +2.67Y C +4Y D +2.8Y E +6Y F +6.67Y G +10Y H Y I +12Y J +4Y K +5.33Y L +1.5Y N +4Y O +5.33Y P Complete Set of Precedence Constraints X B  X A +(5 – Y A ) X C  X A +(5 – Y A ) X D  X A +(5 – Y A ) X e  X A +(5 – Y A ) X F  X A +(5 – Y A ) X B  X B +(1 – Y B ) X F  X C +(3 – Y C ) X G  X F +(1 – Y F ).. X(FIN)  X N +(3 – Y N ) X(FIN)  X O +(4 – Y O ) X(FIN)  X P +(4 – Y P ) FINISHFINISH Activity start time ≥ Finish time of immediate predecessors All x j ’s and y j ’s ≥ 0

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

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.

 25 Minimize 5.5Y A + 10Y B Y C + 4Y D + 2.8Y E + 6Y F Y G + 10Y H Y I + 12Y J + 4Y K Y L + 1.5Y N + 4Y O Y P The other constraints of the crashing model remain the same. The New CPM Model s.t. X(FIN)  19 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. Minimize s.t. X(FIN) 5.5Y A + 10Y B Y C + 4Y D + 2.8Y E + 6Y F Y G + 10Y H Y I + 12Y J + 4Y K Y L + 1.5Y N + 4Y O Y P CPM - DEADLINE CPM - BUDGET

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

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