Project SchedulingProject Scheduling

The Critical Path Method (CPM)

Cost Analyses Using Cost Analyses Using The Critical Path Method (CPM)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.crashing.

• 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.

Normal and CrashNormal and CrashTimes and CostsTimes and Costs

Determining the Time and Determining the Time and Cost of an ActivityCost of an Activity

• The maximum time reduction for an activity is R = TR = TN N – T– TCC.

• This maximum time reduction is achieved by spending E = CE = CCC – C – CNN 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.

ExampleExample• 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.

How longwould it taketo complete theactivity if $2600 were spent?

Marginal cost $2400/8 =

$300 per day.

Extra money spent = $2600 - $2000 = $600.

Days reduced = = 600/300 = 2

Activity will take20 - 2 = 1818 days.

What wouldit cost tocomplete theactivity in 17days?

Days reduced = 20 – 17 = 3.

Extra cost will be3($300) = $900

Activity will cost$2000 + $900 = $2900$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 -- CPM -- Meeting a Deadline at Meeting a Deadline at MinimumMinimum Cost 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 – Baja Burrito Restaurants – Meeting a Deadline at Minimum Meeting a Deadline at Minimum

CostCost

Baja Burrito Restaurants –Baja Burrito Restaurants –

Without spending any extra money, the restaurant will open in29 weeks at a normal cost of $200,000.

When all the activities are crashed tothe maximum, the restaurant will open 17 weeks at crash cost of $300,000.

*Determined by the PERT-CPM template

Baja Burrito Restaurants –Baja Burrito Restaurants –Network presentationNetwork presentation

A

D

C

B

E

F G

I

H

L

O

J NM

K

P

Baja Burrito Restaurants –Baja Burrito Restaurants –Marginal costsMarginal costs

For Activity AR = TN – TC = 5 – 3 = 2E = CC – CN = 36 – 25 = 11Marginal Cost M = 11/2 =$5.50

• Linear Programming Approach– Variables

Xj = start time for activity j.Yj = the amount of crash in activity j.

– Objective FunctionMinimize 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 –Baja Burrito Restaurants –Linear Linear ProgrammingProgramming

The Linear Programming ModelThe Linear Programming ModelXj = start time for activity jYj = the amount of crash in activity jMinimize total crashing costsMin 5.5YA+10YB+2.67YC+4YD+2.8YE+6YF+6.67YG+10YH+

5.33YI+12YJ+4YK+5.33YL+1.5YN+4YO+5.33YP

Min 5.5YA+10YB+2.67YC+4YD+2.8YE+6YF+6.67YG+10YH+5.33YI+12YJ+4YK+5.33YL+1.5YN+4YO+5.33YP

Maximum timereductions

YA ≤ 2.0YB ≤ 0.5YC ≤ 1.5YD ≤ 1.0YE ≤ 2.5YF ≤ 0.5YG ≤ 1.5YH ≤ 0.5

19FINX )(ST

Meet thedeadline

Deadline and Maximum Crash Deadline and Maximum Crash Time ConstraintsTime Constraints

YI ≤ 1.5YJ ≤ 0.5YK ≤ 1.0YL ≤ 1.5YM ≤ 1.5YN ≤ 2.0YO ≤ 1.5YP ≤ 1.5

FINISH

Min 5.5YA+10YB+2.67YC+4YD+2.8YE+6YF+6.67YG+10YH+5.33YI+12YJ+4YK+5.33YL+1.5YN+4YO+5.33YP

Example of Precedence ConstraintsExample of Precedence Constraints

Analysis of Activity O

E4-YE

M3-YM

O

O’s Start Time E’s Start Time + E’s duration

O’s Start Time M’s Start Time + M’s durationXO XM + (3-YM)

XO XE + (4-YE)

Min 5.5YA+10YB+2.67YC+4YD+2.8YE+6YF+6.67YG+10YH+5.33YI+12YJ+4YK+5.33YL+1.5YN+4YO+5.33YP

Complete Set ofComplete Set ofPrecedence ConstraintsPrecedence Constraints

XBXA+(5 – YA)XCXA+(5 – YA)XDXA+(5 – YA)XeXA+(5 – YA)XFXA+(5 – YA)XBXB+(1 – YB)XFXC+(3 – YC)XGXF+(1 – YF)

.

.

.

X(FIN)XN+(3 – YN)X(FIN)XO+(4 – YO)X(FIN)XP+(4 – YP)

FINISH

Activity start time ≥Finish time of

immediate predecessors

All xj’s and yj’s ≥ 0

CPM-DEADLINE TEMPLATE

Select Solver

ClickSolve

INPUTActivity Names, Time/Cost Data, Project Deadline, and Immediate Predecessors

Operating Within a Fixed BudgetOperating 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.

2525

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

The other constraints of the crashing model remain the same.

The New CPM ModelThe 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.

Minimizes.t.

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

CPM - DEADLINE

CPM - BUDGET

CPM-BUDGET TEMPLATE

INPUTActivity Names, Time/Cost Data, Maximum Budget, and Immediate Predecessors

Add END nodeThe predecessors for ENDare nodes without successors

Call Solver

ClickSolve

ReviewReview

• 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