The Critical Path Model

Published Date: 02 Nov 2017

1. Introduction

Project scheduling problem is essential research and

application area within engineering project management.

Project scheduling problem is to determine the

schedule of allotting activities so as to balance the total

cost and the completion time. The project is completed

when activities are finished within time windows (release

time and due date) can be associated with each activity

of the project.In real-world many activities are uncertain

due to the vagueness of duration and they are dependent

on each other[1][2].

Researchers studied the project scheduling problem with

uncertain activity duration times,modelled them in probability

theory.Some projects, in which the activities may

processed many times before and historical data of the

activity duration times are sufficiently available using

that data for deciding the duration of activity,but this

research uncertainty of the activity duration times can

be described by using probability distributions via statistical

techniques. However, for the projects, in which

the activities may seldom or never be performed before

and therefore are lack of statistical data, the duration

times can only be described by stochastic variables. It is

notable that all papers studying project scheduling problem

with stochastic activity duration times just resolved

problems in optimizing the deadline under resource or

cost limits but not considered time-dependences between

activities.Some papers considered only dependencies not

the uncertainty of activities duration.

This paper is to study the project scheduling problem

with activity duration times that are random and timedependent.

We proposed a way of forming the critical

path and construct a method that is suitable of

finding the critical path in problem of the random and

time-dependent activity duration time project scheduling

problem.Then present three stochastic models to address

various requirements arising in the problem.We focusing

on new models by invoking techniques of stochastic simulation

and genetic optimization.Also study the impact

of the genetic operators and other parameters of genetic

algorithms(GA’s)being applied to the random and time

dependent project scheduling models.

2. Related Work

Pioneeringwork by Freeman[3], Sobel, Golenko-Ginzburg

and Gonik[4] , Kotiah and Wallace, Loostma, MacCrimmon

and Ryavec, Parks and Ramsing, Elmaghraby[5],

and Valls [6] have studied project-scheduling problem

with random activity duration times but these studies

not considered a time-dependence component. There

have been a suite of studies addressing project scheduling

with time-dependent activity duration times[7]. Examples

L. E. Drezet and J. C. Billaut include projectscheduling

problems where resources are employees and

activity requirements are time dependent, and E. Poder

1

considered time-dependent activities requirements for

project-scheduling problems[8]. These activity duration

times should also be stochastic due to the sheer complexity

and the iterative,concurrent and evolutionary nature

characteristics of development of software itself.

Stochastic-scheduling models for software projects without

any time dependence have also been introduced by

F. Padberg[9][10] . However, in most of the cases, the

activity duration times of a software project scheduling

are commonly both random and time dependent.

Decision makers sometimes are interested in the cost,

which satisfies the total cost of fund constraints with the

given confidence level _ and fulfils time limits in practice

with a given confidence level _. Chance-constrained

programming (CCP) by A.Charnes and W.Cooper,is a

sound alternative used to build the model[11]. The latest

version of CCP was provided by Liu[13], and the application

of CCP to random project-scheduling problem

(RPSP) was discussed by Ke and co-workers[14]. They

also practice, decision makers may wish to maximize the

chance when some management targets are given. Such a

problem can be solved by dependentchance programming

(DCP), a method for optimizing the chance function of

an uncertain environment event.

3. Programmer’s design

We consider a random and time-dependent activities

project scheduling problem which has the objective

of minimizing total cost under completion time limits.

Precedence relations between each of the activity

enforce that an activity may not be started before

all its predecessors are finished. Random and timedependent

activities schedule detection consists of three

main phases, 1) Finding critical path 2) Choose stochastic

simulation model 3) Apply GA for optimization

3.1. Mathematical Model

Objective is to minimize the cost under some completion

time limit.We assume that each activity can be processed

only if all the foregoing activities are finished;each

activity should be processed without interruption;there

are no dummy activities;and all activity duration times

are assumed to be stochastic.The structure of the project

is described by activity-on-edge (AOE) network, where

nodes stand for the milestones, while the arcs represent

the activities of the project.

Generally, a project can be described by a directed

acyclic graph. Let G = (V,A) be a directed acyclic

graph representing a project, where V = {1, 2, ..., n + 1}

is the set of nodes,A is the set of arcs, and (i, j)_A is the

arc of the graph G from node i to node j.

In order to model the project scheduling problem, we

first introduce the following indices and parameters:

m: the index of the several times, where m_{1, 2, ..., hij}

and hij is the several times that depart from node i to

j.

Bm

ij : the events that the activity starting with m along

the arc(i, j).

Pr

_

Bm

i(i+1)

_

= PN(i−1)i

k=1 Pr

_

Bk

i(i−1)

_

.P r

_

Bm

i(i+1) | (Bk

i(i−1)

_

_ij : random duration time of activity represented by

(i, j) in A and _ = {_m

i |_m

i = μmi

+ Tm

i(i+1)};

Let D be the path set of the project, k_D. Suppose

that Tj(_, k) is the arrival time of node j by activity

(i, j), (i, j)_A. Then the completion time of the total

project can be determined in the form

T(_) = max{

Ph(n−1)n

m=1 Pr(Bm

(n−1)n)._m

n−1 − μ0}.

To build the models of the random time project scheduling,

we assume that T0 is the deadline of the project,

the total cost of fund of the project is denoted by

C(_, k), and k is the critical path of the project.Critical

path time is equivalent to the completion time of entire

2

Figure 1: Overall processing flow.

project.

3.2. Data independence and Data Flow architecture

We are taking activities as input and its durations with

different probability as well as multiple dependency conditions.

That is activity with random and/or time dependent

in nature. Using activity on edge structure to represent

all activity and milestones in the network form.

3.2.1. Critical path model

From the network we first finding out the critical

path by using critical path model(CPM),it finding the

longest path from source to destination node.It is the

direct generalization of critical path analysis to the

uncertain domains. In this approach we constructs the

membership function for the stochastic set of critical

path lengths and the random activity criticality by

solving a series of mathematical programming problems.

Total critical path time is considered as completion time

of project.In critical path model suppose (i, j)_A is the

activity,from node i to j there are several paths at a

time we are taking only one path say it as m. At each

Figure 2: Critical Path

event we are calculating the probability of taken path

with respect to it’s previous event,adding this value in

further event calculus.

Pr(Bm

i(i+1)|Bk

(i−1)i) =

R tm

i(i+1)

tm−1

i(i+1)

_m(μmi

+ Tm

i(i+1))dt

where _m(μmi

+Tm

i(i+1)) is the distribution function from

node i to (i + 1) by starting at m,

μmi

is the depart time of node i and its value range from

tmi

to tm (i+1),

Tm

i(i+1) is the actual activity duration time over the

arc(i, (i + 1)).

Probability is expressed as

Pr(Bm12) = 1, tm−1

12 < μ0 < tm12,m"{1, 2, ..., h12}

Pr(Bm12) = 0, otherwise,m"{1, 2, ..., h12}

......

Pr(Bm

i(i+1)) = Ph(i−1)i

m=1 Pr(Bm

(i−1)i).P r{tm−1

i(i+1) _ _m

i−1 _

tmi

(i+1)},m"{1, 2, ..., hi(i+1)}, 1 _ i _ n − 1

.......

Pr(Bm

(n−1)n) = Ph(n−2)(n−1)

m=1 Pr(Bm

(n−2)(n−1)).P r{tm−1

(n−1)n) _

_m

n−2 _ tm (n−1)n},m"{1, 2, ..., h(n−1)n}.

Consider L(P1n) is arrival time of last node n,and

T(P1n) is duration time of it.Then

L(P1n) = Ph(n−1)n

m=1 Pr(Bm

(n−1)n)._m

n−1

T(P1n) = L(P1n) − μ0

3.2.2. Stochastic simulation

Then manager dynamically choosing the one of the

stochastic simulation model according to different types

of optimization requirement.Stochastic simulation models

provides another decision-making criterion for those

decision-makers concern the risk of some unfavourable

event occurring.In real-world project manager may want

3

to control the project cost within the finance budget and

meanwhile complete the project before the time limit.

A)Expected cost model ECM optimize expected objectives

with with some expected constraints.Since in

stochastic project scheduling the project cost can not be

minimized directly,it is natural to minimize the expected

cost under the expected completion time constraint.

We can build an expected cost model as

min E[C(_, k)]

E[max(Ph(n−1)n

m=1 Pr(Bm

(n−1)n)._m

n−1 − μ0)] _ T0

B)Probability cost model It solve problems with the request

that constraints should hold with at least some

given confidence levels.Manager may just want to obtain

the optimization goals with stochastic constraints holding

at least some predetermined confidence level _.

min{ ¯ C|Pr{C(_, k) _ ¯ C} _ _}

Pr{max(Ph(n−1)n

m=1 Pr(Bm

(n−1)n)._m

n−1 − μ0) _ T0} _ _

Where _,_ be the predetermined confidence level and ¯ C

is the minimum cost also called as probability cost of

project.

C)Maximum probability model When some management

targets are given for that decision makers may wish

to maximize the chance.Such condition problem can be

solved by dependent chance programming(DCP).Hence

decision-maker to maximize the credibility that the total

cost does not exceed some given budget under the

constraint that the credibility of completing the project

before the given date should be larger than or equal to a

predetermined confidence level _.

maxPr{C(_, k) _ C0}

Pr{max(Ph(n−1)n

m=1 Pr(Bm

(n−1)n)._m

n−1 − μ0) _ T0} _ _

3.2.3. GA for Optimization

Once the stochastic functions have been established,

we can use GA for an optimization.In

these models,we consider the random time project

scheduling which has objective of minimize total

cost under some completion time limits.Firstly

we initialise the chromosome using path of G

with source v0 to destination node vh+1,represent

as (v0, v1, v2, ...., vh, vh+1),it is easily show that

(v0, v1)"A, (v1, v2)"A, ....., (vh−1, vh)"A, (vh, vh+1)"A.

Different paths choosing different nodes and

arcs,according to that length of the chromosome is

changing but not exceeds the maximum limit of length

which is the total number of nodes in the actual network.

We use k = kij |(i, j)"A,where kij = 1 that means

arc(i, j) is in the path and if kij = 0 means arc(i, j) not

in the path.

1)Initialize the chromosome: Initializing feasible chromosome

as follows:

Step 1: Set y = 0.

Step 2: Randomly select an index m,which satisfies

(vy,m)"A.

Step 3: y y + 1 and vy = m

step 4: Repeat second and third step until vy = vh+1.

Step 5: Obtain a chromosome v0, v1, v2, ...., vh, v(h+1).

Then apply different genetic algorithm operations line

crossover,mutation and selection for increasing diversity

of the population.

2)Crossover:Proposed scheme using the same point(SP)

and different point(DP) crossover operations.

Apply first SP crossover operation when two

chromosomes chosen with at least one common

gene(node)except source and destination nodes.

Suppose that P1 = (v0, v

0

1, v

0

2, ..., vh, vh+1) and

P2 = (v0, v

0

1, v

0

2, ..., v

0

h, vh+1) are two chromosomes

having feasible paths from nodes v0 to vh+1.

If there are common nodes in chromosomes,then we

randomly choose one node say vi = v

0

i and exchange the

further path of these nodes.If no common node then do

nothing in SP operation.

When no common gene in paths then we can use the

4

DP crossover operation. After each crossover operation

checking feasibility, if any one of two newly generated

chromosomes is not feasible then reject the crossover

operation.Operations shown in Fig.3

3)Mutation:In proposed scheme we implemented two

different mutation operations,one-point(OP) mutation

and two-point(TP) mutation,respectively.

OP operation is as follows:

Step 1: Randomly choose integer from set {1, 2, ..., h}

and it denoted by i.

Step 2: By using process of chromosome initialization

generate path from node vi to node vh+1.

Step 3: Obtain new chromosome

(v0, v1, v2, ...., vi, v

0

i+1, ..., v

0

h, vh+1).

TP operation is as follows:

Step 1: Randomly choose integer from set {1, 2, ..., h}

while smaller number denoted by i and larger integer is

denoted by j.

Step 2: By using process of chromosome initialization

generate path from node vi to node vj .

Step 3: Obtain new chromosome

(v0, v1, v2, ...., vi, v

0

i+1, ..., v

0

j−1, vj , ..., vh, vh+1).

4)Selection:We implemented roulette wheel and tournament

selection mechanism

one-point and two-point mutation,and roulette wheel

and tournament selection schemes.

Four parameters for this algorithm are giving:the population

size of one generation Pop − size,the probability

of crossoverPc,the probability of mutationPm,and simulation

cycles Sim − cycles.

3.3. Turing Machine

Representing the different states of project and there flow

as follows:

State q0-graph read

State q1-node dependence

State q2-probability calculation for activity

State q3-condition calculation for activity

Figure 7: Turing machine.

Figure 8: ECM-Comparative Analysis.

State q4-Aggregate of probability and condition add with

previous value on path

State q5-ECM stochastic simulation

State q6-PCM stochastic simulation

State q7-MCM stochastic simulation

State q8-optimization

out put is best schedule represent in the form of critical

path.

4. Results and Discussion

We given the times,dependencies and several times for

each pair of nodes.Suppose we are requested to finish

project within due date of 120 days.

We can build expected cost model as:

minE[C(_, k)]

Subjected to E[T(_, k)] _ 120

To solve this model,stochastic simulation and GA are embedded,

after this run result are shown in table We can

observed that all costs differ little from each other.To

compare the difference between these costs,we introduce

a parameter called relative error which referred as "Error"

in the table. The relative error find by using formula

(actual value - optimal value)/optimal value * 100,where

optimal value is the minimal one of all costs in table.In

5

Figure 3: SP crossover(vi = v0

i ).

Figure 4: DP crossover(vi = v0

j ).

Figure 5: OP mutation.

Figure 6: TP mutation.

6

table relative error is only 0.660 percent even though

adopting different parameters in the ECM.

5. Conclusion

We have used the activity with random as well as timedependences.

We attempted to find the critical path

in random and time-dependent project scheduling problem.

Formulated the objective of minimizing the total cost

with some completion time limits.Also constructed three

types of stochastic programming models to satisfy different

requirements of decision makers,as well as studied

the effect of genetic operators and parameters on schedule.

In future study resource constraint can be included.

Further studies can also add the different optimization