Project Scheduling

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Project Scheduling

• The Critical Path Method (CPM)

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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 activitys completion time is called
  crashing.

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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.

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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.

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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.
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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

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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.

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Baja Burrito Restaurants
Determined by the PERT-CPM template
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Baja Burrito Restaurants Network presentation
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Baja Burrito Restaurants Marginal costs
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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.

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The Linear Programming Model
Xj start time for activity j Yj the amount
of crash in activity j
Min 5.5YA10YB2.67YC4YD2.8YE6YF6.67YG10YH5
.33YI12YJ4YK5.33YL1.5YN4YO5.33YP
Minimize total crashing costs
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Deadline and Maximum Crash Time Constraints
Min 5.5YA10YB2.67YC4YD2.8YE6YF6.67YG10YH5
.33YI12YJ4YK5.33YL1.5YN4YO5.33YP
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
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Example of Precedence Constraints
Min 5.5YA10YB2.67YC4YD2.8YE6YF6.67YG10YH5
.33YI12YJ4YK5.33YL1.5YN4YO5.33YP
Analysis of Activity O
Os Start Time ? Es Start Time Es duration
XO
? XE (4-YE)
Os Start Time ? Ms Start Time Ms duration
XO
? XM (3-YM)
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Complete Set ofPrecedence Constraints
Min 5.5YA10YB2.67YC4YD2.8YE6YF6.67YG10YH5
.33YI12YJ4YK5.33YL1.5YN4YO5.33YP
Activity start time Finish time of immediate
predecessors
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 xjs and yjs 0
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CPM-DEADLINE TEMPLATE
INPUT Activity Names, Time/Cost Data, Project
Deadline, and Immediate Predecessors
Select Solver
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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.

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The New CPM Model
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.
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The other constraints of the crashing model
remain the same.
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INPUT Activity Names, Time/Cost Data, Maximum
Budget, and Immediate Predecessors
CPM-BUDGET TEMPLATE
Call Solver
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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