It is about Resource Constrained Project Scheduling Problem through Matlab.

1. PROJECT MANAGEMENT
THROUGH
GENETIC ALGORITHM

2.  PROJECT GUIDE:-
ARINDAM SINHA RAY
 SUBMITTED BY:-
AVAY MINNI
DEBADITYA SARKAR
RASHMI SAHA
RAJESH KUMAR MAHTO

3. CONTENTS
 INTRODUCTION
 GENETIC ALGORITHM
 RESOURCE CONSTRAINTS
 WHY GENETIC ALGORITHM?
 CASE STUDY
 ALGORITHM
 OUTPUT
 CONCLUSION & FUTURE WORK
 REFERENCE

4. INTRODUCTION
 A genetic algorithm (or GA) is a search technique
used in computing to find true or approximate
solutions to optimization and search problems.
 Genetic algorithms are categorized as global
search heuristics.
 Genetic algorithms are a particular class of
evolutionary algorithms that use techniques
inspired by evolutionary biology such as
inheritance, mutation, selection, and crossover
(also called recombination).

5. The basic genetic algorithm is as follows:
• [start] Genetic random population of n chromosomes (suitable
solutions for the problem)
• [Fitness] Evaluate the fitness f(x) of each chromosome x in the
population
• New population] Create a new population by repeating following steps
until the New population is complete
- [selection] select two parent chromosomes from a population
according to
their fitness ( the better fitness, the bigger chance to get selected).
- [crossover] With a crossover probability, cross over the parents to
form new offspring ( children). If no crossover was
performed, offspring is the exact copy of parents.
- [Mutation] With a mutation probability, mutate new offspring at each
locus (position in chromosome)
- [Accepting] Place new offspring in the new population.
• [Replace] Use new generated population for a further sum of the
algorithm.
• [Test] If the end condition is satisfied, stop, and return the best solution
in current population.
• [Loop] Go to step2 for fitness evaluation.

6. RESOURCE CONSTRAINTS
 A resource constraint is a limit on what can be done because of
limitations on what is available to do it.
 Resource abundance = shorter project duration and vice versa.
 First identify peaks of resource requirements.
 In practical, resources are finite . So , impractical for peak
resource needs.
 Ideally there should be even demand of resource for entire
project.
 This process of refining the plan to effectively manage and
schedule resources is called resource modeling .
-Resource Definition - Resource Allocation
-Resource Aggregation -Resource Leveling

7. WHY TO USE GENETIC ALGORITHM
 Liability
 Easy to discover global optimum.
 GA uses fitness function for evolution rather than
derivatives.
 Almost all conventional optimization technique
search from a single point but GAs always operate
on a whole population of points(strings).
 Easily modified for different problems.
 They perform very well for large scale optimization
problems.
 Can be employed for a wide variety of optimization
problems.

8. THE RESOURCE-CONSTRAINED PROJECT
SCHEDULING PROBLEM:-
 The RCPSP is one of the most complex scheduling problems. It
is considered as a generalization of many standard scheduling
problems.
 A RCPSP considers resources of limited availability and activities
of known durations and resources request. The problem consists
of finding a schedule of minimal duration by assigning a start time
to each activity such that precedence relation and the resource
availabilities are respected.
 The RCPSP can be defined as a combinational optimization
problem aims at finding a feasible solution by the help of a tuple
(V,p,E,R,B,b).
---- Activities consisting the project are identified by set
V={A0,….,An+1}. The set of non-dummy activities is identified by
Ai={A1,…...An}.
Durations are represented by vector p.
Precedence relation are given by E, such that Ai precedes
activity Aj.

9. ---Availabilities of resources are represented by a vector B.
such that Bk denotes the availability of Rk.
R is called unary or disjunctive resource.
Demands of activities for resources are abstracted by b.
 A schedule is a point S such that Si represents the strat time
of activity of Ai.
 A solution S is feasible if it is compatible with the precedence
constraints and resource constraints. Expressed as below
---- Sj-Si >= pi , for all (Ai,Aj) belongs to E (I)
Sum of bik <= Bk , for all Rk belongs to R and t >= 0. (II)
 The SCPSP is the problem of finding a non-preemptive
schedule S to precedence constraints (I) and resources
constraints (II).
Problem:
A RCPSP instance is given with n=8 real activities and |R|=1
resources with availabilities B = 5.

10. Ai A0 A1 A2 A3 A4 A5 A6 A7 A8 A9
pi 0 6 1 1 2 3 5 3 2 0
bi 0 2 1 3 2 1 2 3 1 0
Si 0 1 1 1 7 2 2 8 5 0
A1 A4 A7
A0 A2 A5 A9
A3 A6 A8

11. Mathematical Deduction:-
Min Z = 12A1+A2+3A3+4A4+3A5+10A6+9A7+3A8
Subject to,
3A3+A2 <= 5
A5+2A6+2A1 <= 5
2A6+2A1 <= 5
A8+2A4 <= 5
3A7 <= 5

12. FUNCTIONS USED FROM MATLAB LIBRARY

13. FIVE AUXILIARY FUNCTIONS USED:-
function e=vr(m,i)
%The i-th co-ordinate vector e in the Euclidean Space
 Create a matrix e of (m*1) where all the elements are Zero.
 Fill the i-th position of matrix e, with 1.
function d=delcols(d)
%here delete the duplicate column of the matrix d.
 Delete the repeated rows from the matrix d and return the
sorted one.
 Store the no. of columns of matrix d in n.

14.  Create an empty Matrix j.
 for k=1 to n
 Assign column wise elements of Matrix d into c.
 for l=k+1 to n
 if(Maximum value from column subtraction for(lth-cth)column
<= 100*eps)
 copy the lth column into j matrix.
 end if
 end for
 end for
 if j is not empty
 Sort each column of j matrix in ascending order.
 Delete all the columns except 1st one of matrix d.
 end if

15. function [row,mi]=MRT(a,b)
%The Minimum Ratio Test(MRT) performed on vector a & b.
%Output Parameters are:-
%row=index of the pivot row.
%mi=value of the smallest ratio.
 Store the length of the vector a into m.
 Create a Matrix C with elements 1 to m.
 Combine all the column into one column of Matrix a.
 Combine all the column into one column of Matrix b.
 If b>0,then assign the value of c into l.
 Array right division of lth column of a & b.
 Now, find the minimum from the Quotient matrix.
 Store the min. value in mi & the index no. in row matrix.
 Store the value of row-th index of l matrix into row.

16. function col=MRTD(a,b)
%The Maximum Ratio Test performed on vectors a and b.
%This function is called form within the function dsimplex.
%Output Parameter-:-
%col-index of the pivot column.
 Store the length of vector a into m.
 Create a Matrix C with elements 1 to m.
 Combine all columns into one column of matrix a.
 Combine all columns into one column of matrix b.
 If b<0,then assign the value of c into l.
 Array right division of l-th column of a & b.
 Now find the Max from quotient matrix.
 Store the max value in mi and index no. in column.
 Store the value of col-th index of l-th matrix into col.

17. function[m,j]=Br(d)
%Implementation of the Bland's rule applied to the array d.
%This function is called from the following functions:-
%Simplex2p,dsimplex,addconstr,simplex and cpa.
%Output parameters:-
%m=first negative no. in the array D.
%j=index of entry m.
 Find any less than zero elements in d and shows their index in
vector matrix.
 if ind is not empty
 Store the index no. of first negative no. in j.
 Store the first negative no. of matrix d in m.
 else
 create an empty matrix m.
 create an empty matrix j.
 end else-if

18. function vert=feassol(A,B)
%Basic feasible solution vert to the System of
constraints
%Ax=b, x>=0
%They are stored in column of matrix vert.
 Store the value of row and column of matrix A into
m & n respectively.
 Stop the warning by Matlab.
 Combine all the columns of b in one column.
 Declare an empty matrix vert.
 if (n>=m)//here column>=row
 Create a matrix t with all possible combination from
the range 1 to n taking m columns at a time.

19.  nv=n!/(n-m)!m!
 for i=1 to nv
 a)Create zero matrix of row n and column 1.
 b)(i)Choose the number from t-th column and
i-th row of matrix A.
 (ii)Divide the chosen element b & store the
quotient into x.
 if all(n>=0 &(n !=inf & n !=-inf))
 a)Assign n to the i-th row of t.
 b)copy the y matrix into vert matrix.
 end if
 end for
 else
 show error message.
 end else if

20.  if vert is not empty
 Delete the duplicate columns of vert matrix by
calling delcols function
 else
 create an empty matrix vert.
 end else if

21. Function vert = extrpts(A,rel,b)
%extreme points vert of the polyhedral set
%x=(x:Ax<=b or Ax>=b, x>=0)
%Inequality signs are =stored in the string rel e.g.
rel=‘<<>’ stands for <=,<= & >= respectively.
o Store the value of row & column of matrix A into m& n
respectively.
o Store the value of n in nlv
o For i=1 to m
o if rel =‘>’
o assign -1 to the i-th position of m*I mtrix by
calling vr function.
o copy the above matrix into A.
o Else
o assign 1 to the ith position of m*I matrix by calling
vr function.
o End else if

22.  if b(i)<0
 assign negative to all the ith row elements of A.
 assign negative to i-th position of b.
 end if
 end for
 Stop the warning by Matlab.
 Combine all the columns of b in one column.
 Declare an empty matrix vert.
 if (n>=m)//here column>=row
 Create a matrix t with all possible c combination
from the range 1 to n taking m columns at a time.

23.  nv=n!/(n-m)!m!
 for i=1 to nv
 a)Create zero matrix of row n and column 1.
 b)(i)Choose the number from t-th column and
i-th row of matrix A.
 (ii)Divide the chosen element b & store the
quotient into x.
 if all(n>=0 &(n !=inf & n !=-inf))
 a)Assign n to the i-th row of t.
 b)copy the y matrix into vert matrix.
 end if
 end for
 else
 show error message
 end else if

24.  Delete the duplicate columns of vert matrix by
calling delcols function.

25. OUTPUT

26. OUTPUT

27. OUTPUT

28. OUTPUT

29. OUTPUT

30. CONCLUSION & FUTURE WORK
 We have taken a resource constraint problem and
deduce it mathematically and formulate into linear
programming problem(LPP).
 We have found the basic feasible solutions and
extreme point from the LPP .
 Now we will assign fitness function to the feasible
solution and by using
SELECTION, REPRODUCTION, EVALUATION
and REPLACEMENT we have to create offspring.
 And from the offspring population by using best fit
,we will reach the optimal solution.

31. REFERENCES
 Introduction to Genetic Algorithm by
S.N.Shivanandam & S.N.Deepa.