3 A Linear Programming Approach Critical Path MethodA Linear Programming Approach
4 Determine the critical path and the critical activities. Example Problem12/4/2017The Build-Rite Construction Company has identified the following ten activities that take place in building a house.ActivityImmediate PredecessorsExpected Time (days)1Walls and Ceiling25Foundation-3Roof Timbers4Roof SheathingElectrical Wiring6Roof Shingles87Exterior SidingWindows9Paint6, 7, 1010Inside Wall Board8, 5Determine the critical path and the critical activities.The example description is modified from the exercises described in Moore and Weatherford, Decision Modelling, Pearson 2001.
5 Solution Critical activities: 1, 2, 3, 4, 6, 9 12/4/2017SolutionActivityEarly Start ScheduleLate Start ScheduleTotal SlacksESEFLSLF138210413512141862171516923Critical activities: 1, 2, 3, 4, 6, 9The project manager should adjust accordingly the budgets and resource allocations to avoid any delay on these activities.
7 A LP Model for CPM analysis 12/4/2017Objective Functionminimize the project duration.Precedence ConstraintsPredecessorSuccessorDuration of the Predecessor21354687910Constraintse.g. activity 6 precedes activity 9Decision Variablesstart times of the activities
8 AON network & LP Model Nodes Decision Variables Precedence Constraints 12/4/2017NodesDecision VariablesPrecedence ConstraintsArcsOptimal SolutionLongest Path92135874106Note that an alternative LP model can be derived from the AOA network.Critical activities can then be identified via sensitivity analysis.
9 Precedence Constraints 12/4/2017ParametersThe start time of the projectThe (expected) times of the activitiesPrecedence Relations of two activitiesActivityTime (days)1Walls and Ceiling52Foundation3Roof Timbers4Roof SheathingElectrical Wiring6Roof Shingles87Exterior SidingWindows9Paint10Inside Wall BoardProject Start TimePrecedence ConstraintsPredecesorSuccessor21345687910
10 12/4/2017A Linear Programming Approach for Critical Path Analysis A Spreadsheet ImplementationActivityTime (days)Start timeFinish time1Walls and Ceiling5-2Foundation3Roof Timbers4Roof SheathingElectrical Wiring6Roof Shingles87Exterior SidingWindows9Paint10Inside Wall BoardProject Start TimeObjective ValueStart timeFinish time3810131221152323 days
11 Early start schedule Late start schedule EF38101312211523Early start scheduleAny activity will be started at its earliest start time.Late start scheduleAny activity will be started at its latest start time.LSLF3810131418211623
12 Early Start/Late Start Schedule 12/4/2017Critical ActivitiesSince the total slacks can be determined using the early start schedule and the late start schedule, the critical activities can be identified as well.Early Start/Late Start ScheduleTotal SlacksActivityESEFLSLF138210413512141862171516923Critical activities: 1, 2, 3, 4, 6, 9
13 Demo To enable the solver in EXCEL 2010 You may find the solver in 12/4/2017DemoTo enable the solver in EXCEL 2010File Options Add-Ins Select “Solver Add-in” Go Select “Solver Add-in” OKYou may find the solver inData SolverObjective FunctionDecision VariablesConstraintsUse simplex method forthe LP models
14 A Linear Programming Approach CrashingA Linear Programming Approach
15 SEEM3530 Tutorial 2 Project Management 12/4/2017Example ProblemBuild-Rite’s engineers have calculated the cost of completing each activity. Their results are given below.ActivityNormalTimeCostCrash TimeCrash Cost1550372220301548613217456552940102234e.g. Cost for Activity 1How much would it cost to complete the project within 22 days? 21 days? 20 days?...
16 Solution: Time-Cost Trade-Off ProjectCostThe normal schedule obtained using CPMeach activity is performed at its lowest cost and at a normal duration.Project DurationThe crashing process has revealed a relationship between the cost and the schedule of the project, which allows us to prepare our budget by considering the possible trade-offs between cost and time.
18 A LP Model for Crashing with a fixed project due date 12/4/2017A LP Model for Crashing with a fixed project due dateMinimize the cost for crashingPrecedence ConstraintsMax. Clashed DaysNo. of days to crashProject due dateStart times of the activities.
19 12/4/2017A Linear Programming Approach for Crashing A Spreadsheet ImplementationCrashingActivityCrashing (days)Normal TimeNormal CostCrash TimeCrash CostMax. Crash DaysCost per Crash Day1?5503722112030101548613217456552940223412Project CostObj. ValueCrashing (days)21484288372.1
22 Immediate Predecessors ExampleThe working hours requirements of the activities are estimated. They are described below.ActivityImmediate PredecessorsExpected Time (days)Work hours1Walls and Ceiling2530 hrsFoundation-322 hrsRoof Timbers8 hrs4Roof Sheathing16 hrsElectrical Wiring6 hrs6Roof Shingles84 hrs7Exterior SidingWindows12 hrs9Paint6, 7, 1010Inside Wall Board8, 5The example question is modified from Project Management (Shtub, Bard, Globerson) Exercise 10.1
23 Resource levelingThe reallocation of slacks in activities to minimize fluctuations in resource requirement profile.The resource profile before leveling.large resource fluctuationOverallocation of resourceThe resource profile after leveling.Minimized resource fluctuationNo delay in the project
25 1. Create a Project. File New Blank Project File Options ScheduleSet the working hours per day. E.g. 8 hours.The durations of activities (or tasks) are fixed in our case.
26 2. Input the task information Task Gantt ChartInput the task informationSelect all the tasks and then press “Auto Schedule”
27 3. Set the Project Start Date Project Project Information Statistics
28 3. Identify the critical path Task Gantt Chart Network DiagramGantt chart Add New Column “total slack”Note that the project can be finished within 23 days.
29 4. Add a renewable resource Task Gantt Chart Resource SheetIn the first row, inputResource Name: ManpowerType: workMax. Units: 100%Examples of renewable resourceManpowerMaterialsMachines
30 5. Type in the resource usage Input the work hours of the activitiesSelect the column right click Assign Resources Select “Manpower” AssignTaskWork hour130 hrs222 hrs38 hrs416 hrs56 hrs64 hrs7812 hrs910
32 7. Resource LevelingResource Leveling Options tick “level only within available slack.Resource level allFrustration is minimized.No over-allocationThe project duration remains the same (total slacks are reduced )Smaller ffrustration