Journal of Mathematical Modelling and Application
2010, Vol. 2, No.2, 60-86
Penalty approaches for Assignment Problem with single side constraint via
Genetic Algorithms
Jayanta Majumdar
Department of Mathematics, Durgapur Government College, Durgapur-713214, India,
[email protected]
Asok Kr. Bhunia
Department of Mathematics, The University of Burdwan, Burdwan-713104, India,
[email protected]
Abstract
The goal of this article is to investigate the applicability of Genetic Algorithms (GAs) to solve
Assignment Problem with Single Side Constraint (APSSC) due to either time restriction or budgetary
restriction, etc. For this purpose, two different models of APSSC are formulated – one for
deterministic cost/time parameters and another for imprecise cost/time parameters. To handle the side
constraint in solving each of these models, two new optimization problems are formulated using two
different penalty function techniques. Then the reduced problems are solved by using elitist genetic
algorithm (EGA). This algorithm employs some new features on initialization, pair-wise careful
comparison among feasible and infeasible solutions using tournament selection in conjunction with
two heuristic operators one for making feasible solution from the infeasible one and the other for
improving feasible solution. To illustrate the models, a set of test problems generated randomly are
solved and the computational statistics of each model regarding objective function values, generations,
computational times and number of objective function evaluations are compared.
Keywords: Assignment problem with side constraint; Penalty approach; Genetic algorithm
1. Introduction
The basic structure of classical Assignment Problem (AP) is the minimization of an objective
function involving cost/time subject to the one-to-one correspondence between a set of tasks and a set
of agents. This problem is perhaps one of the first fundamental and well-studied problems in the
optimization literature due to its many applications such as facility location, personnel scheduling, task
assignment, job shop loading and manpower scheduling. The classical AP is also significant because
of its use as a sub-problem to more complex 0-1 optimization problems.
Besides the classical APs, many real-life problems may, however, contain an extra side
constraint (in conjunction with the usual assignment constraints and the 0-1 integrality conditions),
such as budgetary limitations, or time restrictions, etc., that affect the assignment decisions. Due to
this fact, AP with side constraints has been studied extensively in the literature. Gupta and Sharma
(1981), Aggarwal (1985), Mazzola and Neebe (1986), Aboudi and Jornsten (1990), Aboudi and
Nemhauser (1991) and Leishout and Volgenant (2007) proposed different branch and bound (B&B)
algorithms. On the other hand, Rosenwein (1991) developed a method for obtaining improved lower
bounds for B&B algorithm that employs Lagrangian decomposition (LD) instead of sub-gradient
optimization. Kennington and Mohammadi (1994) proposed a heuristic algorithm based on bisection
search after solving a series of AP instances. Punnen and Aneja (1995) developed a tabu search
heuristic using strategic oscillation, multiple start and randomized short-term memory as a means of
diversification of search paths. Among the above mentioned works, Leishout and Volgenant (2007),
Rosenwein (1991), and Kennington and Mohammadi (1994) formulated and solved APSSC whereas
Mazzola and Neebe (1986) and Aboudi and Jornsten (1990) formulated and solved AP with more than
one side constraint.
To the best of our knowledge, among all the aforesaid works, fixed (deterministic) real
numbers have been used as the cost/time parameters. However, in real-life situations, these numbers
might not be fixed rather be imprecise as cost/time for performing a task/job by a facility
61
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
(machine/person) may vary due to different reasons. To represent such imprecise numbers, stochastic,
fuzzy and fuzzy-stochastic approaches may be used. In stochastic approach, the
coefficients/parameters are viewed as random variables with known probability distributions. On the
other hand, in fuzzy approach, the parameters, constraints and goals are viewed as fuzzy sets/fuzzy
numbers. It is also assumed that their membership functions are known. Again, in fuzzy-stochastic
approach, some parameters are viewed as fuzzy sets and others, as random variables. However, it is
not always easy for a decision maker to specify the appropriate membership function for fuzzy
approach, exact probability distribution of a parameter for stochastic approach and both for fuzzystochastic approach. For these reasons, we have represented such imprecise numbers using a more
general approach by means of interval valued numbers. Thus, generally, real-life problems involve
interval valued cost/time parameters. In order to solve this type of interval APSSC, order relations
between two interval valued numbers are essential. In the existing literature, very few researchers have
proposed the definitions of interval order relations in different ways. Among them, one may refer to
the works of Moore (1979), Ishibuchi & Tanaka (1990), Chanas and Kuchta (1996), Kundu (1997),
Sengupta and Pal (2000) and Majumdar and Bhunia (2007). However, their definitions are not
complete in all respects. Recently, Mahato and Bhunia (2006) developed complete definitions of
interval order relations for both minimization and maximization problems with respect to optimistic
and pessimistic decision makers’ preference.
To solve constrained optimization problems using GAs or classical optimization techniques,
penalty function techniques are most popular among the researchers during last few decades.
However, the most difficult aspect of the penalty function approach is to find appropriate penalty
parameters required to guide the search towards the constrained optimum (Deb (2000)). This often
requires users to experiment with different values of penalty parameters (Deb (2000)). Moreover,
penalty parameters are required to make the constraint violation values of the same order as the
objective function value.
Genetic Algorithms (GAs) initiated by John Holland in early 1970s, are robust search and
optimization technique based on the principles derived from the dynamics of natural evolution and
genetics. Unlike traditional heuristics and some metaheuristics like tabu search, GAs work with a
population of feasible solutions (known as chromosomes) iteratively by applying three basic genetic
operators, viz. selection/ reproduction, crossover and mutation in each iteration (called generation)
until a termination criterion is satisfied.
In this paper, two models of APSSC, viz. fixed and interval type have been formulated and
solved using elitist GA (EGA). Here, the interval type model has been formulated using the definitions
due to Mahato and Bhunia (2006) of interval order relations for minimization problems. In order to
handle the side constraint for infeasible solutions, two different penalty techniques, viz. Big-M
Penalty (BMP) technique and Parameter Free Penalty (PFP) technique have been used so that the
given APSSC has been converted to a reduced minimization problem. After that, to solve the reduced
minimization problems, EGA has been proposed. Our proposed EGA incorporates some new features
on initialization, tournament selection and two heuristic operators, viz. make_feasible (that converts
infeasible solutions to feasible ones) and improve_feasible (that improves the fitness value of feasible
solutions). Finally, extensive computational results of our different EGA versions (based on different
crossovers) using both the penalty techniques on all over 28 (25 for the first model and 3 for the
second model) randomly generated test problems as well as their comparative analyses have been
reported.
2. Interval arithmetic and order relations between intervals
Generally, an interval valued number is defined by its lower and upper limits as
A = [ aL , aR ] = { x : aL ≤ x ≤ aR , x ∈ R}
where aL and aR are the lower and upper limits respectively and R, the set of all real numbers. The
interval A is also defined by its centre and radius as
A = ac , aw = { x : ac − aw ≤ x ≤ ac + aw , x ∈ R}
Jayanta Majumdar and Asok Kr. Bhunia
62
where ac = (aL + aR)/2 and aw = (aR −aL)/2 are respectively the centre and width of interval A. Now, we
shall discuss some interval arithmetic.
Definition 1: Let * ∈ {+, −,., ÷} be a binary operation on the set of real numbers. If A and B are two
closed intervals, then
A * B = {a * b : a ∈ A, b ∈ B}
defines a binary operation on the set of closed intervals. In the case of division, it is assumed that
0∈ B.
The operations on intervals may be explicitly calculated (for two interval numbers
A = [ aL , aR ] = ac , aw and B = [bL , bR ] = bc , bw ) from Definition 1 as:
A + B = [ aL , aR ] + [bL , bR ] = [ aL + bL , aR + bR ] ,
(1)
A + B = ac , aw + bc , bw = ac + bc , aw + bw ,
(2)
kA = k [ aL , aR ] = [ kaL , kaR ] for k ≥ 0 ,
(3)
= [ kaR , kaL ] for k < 0,
kA = k ac , aw = kac , k aw
(4)
(5)
where k is a real number.
In our paper, we shall use only equations (1) and (3).
Next, we shall discuss the order relations for finding the decision makers’ preference between
interval cost(s)/time(s) of minimization problems.
Let the uncertain cost(s)/time(s) from two alternatives be represented by two closed intervals
A = [ aL , aR ] = ac , aw and B = [bL , bR ] = bc , bw respectively. It is also assumed that the cost/time
of each alternative lies in the corresponding interval. These two intervals A and B may be of the
following three types:
Type–I: Both the intervals are disjoint.
Type–II: Intervals are partially overlapping.
Type–III: One interval is contained in the other.
Optimistic decision making
For optimistic decision making, the decision maker expects the lowest time/cost ignoring the
uncertainty.
According to Mahato and Bhunia (2006) the order relations of interval numbers for
minimization problems in case of optimistic decision making are as follows:
Definition 2: Let us define the order relation ≤O min between A = [aL, aR] and B = [bL , bR ] as
A ≤O min B ⇔ aL ≤ bL
A <O min B ⇔ A ≤O min B ∧ A ≠ B
Pessimistic decision making
For pessimistic decision making, the decision maker expects the minimum cost/time for
minimization problems according to the principle “Less uncertainty is better than more uncertainty”.
According to Mahato and Bhunia (2006) the order relations of interval numbers for
minimization problems in case of pessimistic decision making are as follows:
Definition 3: Let us define the order relation ≤ p min between A = [ aL , aR ] = ac , aw
B = [bL , bR ] = bc , bw as
and
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
63
A < p min B ⇔ ac < bc for Type-I and Type-II intervals
A < p min B ⇔ ( ac ≤ bc ) ∧ ( aw < bw ) for Type-III intervals.
However, for Type – III intervals with ( ac < bc ) ∧ ( aw > bw ) , the pessimistic decision cannot be taken.
Here, the optimistic decision is to be considered.
3. Problem formulation
An APSSC can be considered as a 0–1 integer programming problem. Here two different
models of APSSC have been formulated which are as follows:
3.1. Model – I (Fixed (deterministic) type)
Let Cij be the fixed cost/time required when task i ( i = 1, 2,..., n ) is assigned to j-th
(j=1,2,…,n) facility (machine/agent), rij , the associated resource usage of this assignment and b is the
available capacity of the resource. Then the model can be stated as follows:
n
n
∑∑ C x
Minimize Z =
ij ij
(6)
= 1,
j = 1, 2,..., n
(7)
= 1,
i = 1, 2,..., n
(8)
i =1 j =1
n
subject to
∑x
i =1
ij
n
∑x
j =1
n
and
ij
n
∑∑ r x
i =1 j =1
where
ij ij
≤b
(9)
xij ∈ {0,1} , i, j = 1, 2,..., n
(10)
3.2. Model – II (Interval type)
Here only Cij s are assumed as intervals of the form ⎡CLij , CRij ⎤ . Then the model of this type
⎣
⎦
can be stated as follows:
Minimize Z =
n
n
∑∑ ⎡⎣C
i =1 j =1
Lij
, CRij ⎤⎦ xij
(11)
subject to the same constraints and restrictions as in Model – I.
4. GA based Penalty techniques
The objective of penalty techniques is to compute the fitness value for the infeasible solution
(due to violation of the side constraint, viz. the constraint (9)) so that the constrained minimization
problem is converted to a reduced minimization problem. However, the fitness value of a feasible
solution is nothing but its objective function value. Here we have used two different types of penalty
techniques as follows:
4.1. Big-M Penalty (BMP) technique
Here, a large positive value (say, M, in case of interval form it is taken as [M, M]) is blindly
assigned to the fitness value for the infeasible solution and so
( fitness_value )infeasible = M ( in case of optimization problem with fixed coefficients )
Jayanta Majumdar and Asok Kr. Bhunia
64
= [ M, M ] ( in case of optimization problem with interval coefficients )
4.2. Parameter Free Penalty (PFP) technique
Unlike BMP technique, here, the fitness value of the worst feasible solution in the population
is added with the amount of constraint violation and the same is assigned to the fitness value for the
infeasible solution and so
⎛ n n
⎞
fitness_value
=
fitne
ss_value
+
(
)infeasible (
)worst_feasible ⎜ ∑∑ rij xij − b ⎟
⎝ i =1 j =1
⎠
Thus, after conversion using the above two penalty techniques, the corresponding reduced
problems of the two models can be stated as follows:
Model – I
Minimize Ẑ = Z + θ
where in case of BMP technique,
θ = 0,
= −Z + M ,
if solution is feasible
if solution is infeasible
and in case of PFP technique,
θ = 0,
if solution is feasible
⎛ n n
⎞
= − Z + f worst _ feasible + ⎜ ∑∑ rij xij − b ⎟ , if solution is infeasible
⎝ i =1 j =1
⎠
(Here f stands for fitness_value)
Model – II
Minimize Ẑ = Z + θ
where in case of BMP technique,
θ = [ 0, 0] ,
if solution is feasible
= − Z + [ M , M ] , if solution is infeasible
and in case of PFP technique,
θ = [0,0] ,
if solution is feasible
⎡
⎛n n
⎞
⎛n n
⎞⎤
= −Z + ⎢ fworst _ feasible + ⎜ ∑∑rij xij − b ⎟ , fworst _ feasible + ⎜ ∑∑rij xij − b ⎟⎥ , if solution is infeasible
⎝ i=1 j=1
⎠
⎝ i=1 j=1
⎠⎦⎥
⎣⎢
subject to the same constraints (excluding (9)) and restrictions (for both models) mentioned earlier.
5. Elitist Genetic Algorithm (EGA)
The structure of our developed elitist GA (EGA) for solving APSSC involving n2 integer
variables xij (whose values are either 0 or 1) has been shown below:
procedure EGA;
begin
generate initial population P0 (a set of chromosomes);
evaluate fitness of population P0 ;
obtain best found chromosome from P0 ;
initialize generation counter : t ← 0 ;
while termination criterion not satisfied do
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
65
increase generation counter : t ← t + 1 ;
select above average chromosomes from Pt −1 and form Pt with multiple copies of better
chromosomes using tournament selection;
create offspring from randomly selected parent chromosomes of Pt by crossover and replace
corresponding chromosomes;
eventually mutate randomly selected chromosome of Pt and replace the corresponding
chromosome;
evaluate fitness of new population Pt ;
obtain best found chromosome from Pt ;
compare best found chromosomes of Pt and Pt −1 and replace the worst result of Pt by the
best found result of Pt −1 if it is better than that of Pt ; [elitist operation]
end while
print best found result;
end
We shall now discuss the different processes/operators of EGA in details.
5.1. Chromosome representation and initialization
For proper and successful functioning of GA, the designing of an appropriate chromosome of
solutions to the problem is an important task. There are different ways for representation of
chromosomes. In the proposed EGA, matrix representation of chromosomes (Majumdar & Bhunia
(2007)) has been used.
After appropriate selection of chromosome representation, the next step is to initialize the
chromosomes which will take part in the artificial genetics. Here, to create the initial population of
GA, a combination of two heuristics has been used successively for the two halves of the whole
population. The first heuristic is the same as has been described in Majumdar and Bhunia (2007),
whereas for the second heuristic, the pseudocode has been shown below:
procedure second_heuristic;
begin
compute average of the accumulated resources:
⎛
⎞
A = ⎜ ∑ ∑ rij ⎟ n 2 ;
⎝ i∈N j∈N ⎠
for i ← 1 to n do
rowi ← 0;
for j ← 1 to n do
S j ← 0;
end for (j)
end for (i)
randomly select a j ∈ N ;
for count ← 1 to n do
{
⎡⎣ N = {1, 2,......, n}⎤⎦
}
i* ← i ∈ N rowi = 0 and rij ≤ A
Sj ← i ;
*
[ S j indicates the facility to which j-th job is assigned]
rowi* ← 1 ;
break;
search for an yet unselected i ∈ N rowi = 0;
Jayanta Majumdar and Asok Kr. Bhunia
66
if such an i exists then
S j ← i;
rowi ← 1;
break;
end if
j ← j + 1;
if j > n − 1 then
j ← j − n;
end if
end for (count)
end
5.2. Tournament selection
The selection process is one of the most important factors in GA. This process is dependent on
the very well known evolutionary principle “Survival of the fittest”. The primary objective of this
operator is to emphasize on the average solutions from the population for the next generation. Here
tournament selection has been used as our task is to solve the constrained optimization problem. In this
selection, a group of chromosomes/individuals from the population is chosen randomly and the best
individual in this group is selected as parents for the next generation. This process is repeated until the
population size number of chromosomes is selected. In this study, tournament selection has been used
to select the better chromosome/individual from randomly selected two chromosomes/individuals.
When comparing two chromosomes/individuals, one can have the following three possible situations:
1. Both are feasible. In this case, the chromosome/individual with a better fitness value is
selected.
2. One is feasible and the other is infeasible. In this case, the feasible chromosome/individual
is selected.
3. Both are infeasible. In this case, any one chromosome/individual is selected.
It may be noted that the selection of better chromosome/individual for problems of Model-II
(interval type) has been done using the definitions (Definition 2 and 3) of order relations between two
interval numbers as the fitness value of each chromosome/individual is interval valued for those
problems.
5.3. Crossover and Mutation
After the selection process, the resulting chromosomes take part in the crossover operation to
produce possibly better offspring to improve the current population. This operation operates on two or
more parent chromosomes (solutions) at a time and produces offspring by combining the features of
the parent chromosomes (solutions). In this operation, expected [ pc ⋅ psize ] ( [ ] denotes the integral
value) number of chromosomes will take part. In our EGA, three existing crossovers, viz. modified
form of whole arithmetical crossover (MWAX), matrix binary crossover (MBX) and row exchange
crossover (REX) have been implemented. Moreover, for MWAX and MBX, as most of the offspring
chromosomes (solutions) will be infeasible in terms of violation of the constraints (7) or (8), a repair
algorithm (as suggested by Majumdar and Bhunia (2007, 2006) has been embedded after the said
crossovers.
After successful completion of crossover operation, the next genetic operation is the mutation.
The main objective of the mutation operator is to introduce the genetic diversity of the population.
This operator is used to enhance the fine tuning capabilities of the system. It is implemented to a single
chromosome only with low probability. Here, row exchange mutation (REM) has been used where
two randomly selected or adjacent rows for a randomly chosen chromosome have been interchanged.
Due to this change, it is ensured that the feasibility of the chromosome will be maintained.
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
67
5.4. Additional heuristic operators
In conjunction with the usual GA operators discussed earlier, two heuristic operators have been
implemented in our algorithm of which the first one is make_feasible that will make feasible solution
from the infeasible (in terms of violation of (9)) one and the second is improve_feasible that will
upgrade the fitness value of a feasible solution. The poseudocodes of these operators have been
successively shown below:
procedure make_feasible;
begin
select an infeasible chromosome from the population;
compute average of the accumulated resource requirement:
n
n
A = R n , R = ∑∑ rij ;
i =1 j =1
S j =i
[ S j indicates the facility to which j-th job is assigned]
for i ← 1 to n do
search for j ′ ∈ N S j ′ = i and rij ′ > A ;
i ′ ← i;
break;
for j ← 1 to n do
{
}
j * ← j ∈ N j ≠ j ′ and Ci′j = minimum and ri′j < A
end for (j)
{
}
i * ← i ∈ N S j* = i ;
S j* ← i′;
if ri* j* > ri′j* then
(
)
(
)
R ← R − ri* j* − ri′j* ;
else
R ← R + ri′j* − ri* j* ;
end if
S j′ ← i* ;
if ri′j ′ > ri* j ′ then
(
)
(
)
R ← R − ri′j′ − ri* j′ ;
else
R ← R + ri* j′ − ri′j′ ;
end if
if R ≤ b break;
end for (i)
end
procedure improve_feasible;
begin
select an infeasible chromosome from the population;
Jayanta Majumdar and Asok Kr. Bhunia
68
compute accumulated resources requirement:
n
n
R = ∑∑ rij ;
[ S j indicates the facility to which j-th job is assigned]
i =1 j =1
S j =i
for count ← 1 to n/2 do
randomly select a j ∈ N ;
N ← N − { j} ;
{
}
i′ ← i ∈ N S j = i ;
for i ← 1 to n do
{
}
i* ← i ∈ N i ≠ i′ and Cij = minimum ;
end for (i)
*
*
search for an unselected j ∈ N S j∗ = i ;
if such an j * exists then
if Cij < Ci′j and R − ri′j − ri* j* + ri* j + ri′j* ≤ b then
S j ← i* ;
S j* ← i′;
N ← N − { j*} ;
R ← R − ri′j − ri* j* + ri* j + ri′j* ;
end if
end if
end for (count)
end
6. Computational results and discussion
To illustrate the proposed two models of APSSC, we have solved 28 problems (25 for the
Model –I and 3 for Model –II ) generated randomly with the help of the following three different
versions of our developed EGA based on different crossover operators:
EGA-1: GA with MWAX
EGA-2: GA with MBX
EGA-3: GA with REX
To conduct the experiments, we have coded the different versions of EGA in C/C++ and
tested on a Pentium IV (3.0 GHz processor, 1GB RAM) PC under LINUX environment.
For
problems of Model –II, results of computation have been done separately for the case of optimistic and
pessimistic decision making. Two classes of test problem have been considered for Model – I: random
problems and negatively correlated problems. On the other hand, only random problem instances have
been considered for Model – II.
For random problems of Model – I, both Cij and rij values have been considered from a
uniform distribution of random integers in the interval [50, 150] and for those of Model– II, the CLij
and CRij values have been chosen in the following way:
CLij = a uniformly distributed random integer in [50, 150]
CRij = CLij + a uniformly distributed random integer in [1, 3]
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
69
On the other hand, for negatively correlated problems of Model – I, Cij values have been
chosen in the same way as those for random problems while rij values have been calculated as
follows:
{
rij = minimum 50 + ( Cij )
max
}
− Cij + a uniformly distributed random integer in [ 0, 10] , 150
⎡⎛
The right-hand side of the resource constraint, viz. b has been set as b = ⎢⎜
n
n
∑∑ r
⎣⎢⎝ i =1
j =1
ij
⎞ ⎤
⎟ n ⎥ , where
⎠ ⎥⎦
[ a ] is the greatest integer not exceeding the value a. For each value of n = 10, 15, 20, 30 and 40, five
different problems have been generated and solved for Model – I. But for Model – II, one problem for
each value of n = 10, 15 and 20 have been solved. The results reported in Tables I – VIII have been
obtained from 10 independent runs (trials) per problem and also for each EGA version with different
sets of random numbers.
In the experiments, the following GA parameter values have been considered:
population size ( psize ) = 10n;
maximum number of generations ( mgen ) = 50 (for Model – I) and
= 100 (for Model – II);
probability of crossover ( pc ) = 0.8;
probability of mutation ( pm ) = 0.2.
The following observations obtained by BMP technique and PFP technique for each of the 25
random and negatively correlated problems of Model – I using different EGA versions have been
displayed in Tables I – VI:
the best found objective value ( Objbest );
average objective value ( Objavg );
the generation at which the best solution is first seen ( Gen best );
time for the best solution ( Tbest );
average running time of GAs ( Tavg ) and
number of objective function evaluation for the best solution ( FE best ).
Again, the computational results of the best found objective value ( Objbest ), the minimum,
maximum and average number of generation and CPU time where the best solution is found ( Gen best
and Tbest ) and the average number of objective function evaluation for the best solution ( FE best (Avg))
using BMP technique and PFP technique respectively for different EGAs with respect to optimistic
decision making and pessimistic decision making separately have been presented in Table-VII and
Table-VIII.
From the computational results for random problems of Model – I shown in Tables I, II and
III, it is observed that
(i)
Objbest values in PFP technique are less or equal to those in BMP technique for 72%,
76% and 76% of the cases respectively using EGA-1, EGA-2 and EGA-3.
(ii) Objavg values in PFP technique are greater than those in BMP technique for 64%,
52% and 40% of the cases respectively using EGAs.
(iii) For EGA-1 and EGA-3, the Tbest values in PFP technique are less or equal to those in
BMP technique for 72% and 80% of the cases respectively. On the other hand, for
EGA-2, the result is somewhat surprising, viz. 100%.
Jayanta Majumdar and Asok Kr. Bhunia
Table – I Computational results for random problems of Model – I using EGA-1
n (i ) ∗
BMP Technique
PFP Technique
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
10(1)
611
615.9
9
0.64
0.64
1000
611
617.3
6
0.26
0.268
700
10(2)
647
647
1
0.64
0.64
200
647
647
1
0.26
0.26
200
10(3)
616
616.4
1
0.25
0.445
200
616
616.2
1
0.25
0.256
200
10(4)
591
591
1
0.66
0.752
200
591
591.9
3
0.27
0.288
400
10(5)
631
631
1
0.13
0.298
200
631
631
1
0.25
0.262
200
15(1)
886
886
6
0.77
0.995
1050
886
887.5
3
0.82
0.836
600
15(2)
928
937.3
4
0.74
0.819
750
929
938.6
8
0.79
0.805
1350
15(3)
864
864.4
1
0.71
0.796
300
864
864
1
0.71
0.722
300
15(4)
853
853
1
0.70
0.708
300
853
853
1
0.70
0.708
300
15(5)
874
874.3
5
0.77
1.071
900
874
879.3
5
0.73
0.762
900
20(1)
1128
1131.4
11
1.79
2.617
2400
1128
1133.2
10
1.71
1.813
2200
20(2)
1153
1161.3
7
2.25
2.476
1600
1150
1162.4
4
2.03
2.155
1000
20(3)
1116
1124.9
14
2.75
2.945
3000
1105
1125.1
6
1.89
1.957
1400
20(4)
1180
1199.9
19
4.47
4.965
4000
1177
1206.9
12
2.70
2.824
2600
20(5)
1173
1173.7
16
1.76
1.871
3400
1173
1173.7
6
1.68
1.738
1400
70
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
71
Table – I Computational results for random problems of Model – I using EGA-1 (Contd.)
n (i ) ∗
BMP Technique
PFP Technique
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
30(1)
1618
1627.9
17
5.13
8.365
5400
1626
1632.2
16
4.32
7.157
5100
30(2)
1677
1697.1
14
2.70
6.012
4500
1688
1705.1
17
4.49
4.802
5400
30(3)
1635
1647.7
14
2.20
5.078
4500
1639
1647.9
14
3.36
3.901
4500
30(4)
1670
1699
13
2.25
6.09
4200
1676
1700
15
5.36
5.459
4800
30(5)
1643
1658.8
18
4.46
7.598
5100
1642
1663.9
16
5.42
5.562
5100
40(1)
2227
2262.3
29
0.31
0.325
12000
2223
2250.7
24
0.18
0.20
10000
40(2)
2185
2238.8
29
0.31
0.399
12000
2182
2228
29
0.20
0.208
12000
40(3)
2316
2317.2
29
0.31
0.377
12000
2253
2301.4
29
0.19
0.192
12000
40(4)
2212
2296.7
29
0.31
0.374
12000
2243
2273.1
29
0.20
0.207
12000
40(5)
2281
2319.6
29
0.31
0.387
12000
2299
2322.3
29
0.20
0.207
12000
Jayanta Majumdar and Asok Kr. Bhunia
Table – II Computational results for random problems of Model – I using EGA-2
n (i ) ∗
BMP Technique
PFP Technique
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
10(1)
611
614.5
3
0.31
0.338
400
611
613.8
1
0.19
0.198
200
10(2)
647
647
1
0.29
0.294
200
647
647
1
0.02
0.029
200
10(3)
616
616.8
1
0.30
0.303
200
616
616.8
1
0.02
0.022
200
10(4)
591
595
14
0.32
0.351
1500
591
595.1
3
0.21
0.217
400
10(5)
631
631
3
0.29
0.308
400
631
631
2
0.18
0.193
300
15(1)
886
886
14
0.95
1.029
2250
886
888.8
9
0.12
0.127
1500
15(2)
920
936.1
17
1.75
1.842
2700
929
938.6
12
1.50
1.52
1950
15(3)
864
864
2
0.85
0.871
450
864
865.2
2
0.07
0.079
450
15(4)
853
853
1
0.84
0.855
300
853
853
1
0.07
0.078
300
15(5)
874
874.5
13
0.87
1.026
2100
870
875.4
5
0.52
0.526
900
20(1)
1118
1128.9
19
2.60
2.915
4000
1128
1135.7
10
0.19
0.213
2200
20(2)
1160
1168.8
15
2.49
2.526
3200
1160
1168.8
12
0.25
0.276
2600
20(3)
1118
1123.9
10
1.01
2.411
2200
1126
1126
5
0.17
0.207
1200
20(4)
1185
1190.3
20
4.82
5.056
4200
1182
1195.2
15
1.80
1.832
3200
20(5)
1173
1173.5
13
2.05
2.125
2800
1173
1177.6
7
1.09
1.105
1600
72
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
73
Table – II Computational results for random problems of Model – I using EGA-2 (Contd.)
n (i ) ∗
BMP Technique
PFP Technique
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
30(1)
1623
1638.9
19
4.93
5.249
6000
1623
1634.5
2
0.68
0.971
900
30(2)
1671
1721.4
13
2.83
3.486
4200
1656
1716.7
3
0.99
1.002
1200
30(3)
1639
1655.3
19
4.64
6.015
6000
1639
1656.1
2
0.80
0.852
900
30(4)
1705
1719.9
14
2.81
2.882
4500
1689
1703.5
8
1.12
1.214
2700
30(5)
1673
1690.7
14
10.06
10.807
4500
1636
1666.1
9
5.75
5.796
3000
40(1)
2224
2262.5
29
0.37
0.383
12000
2238
2274.8
23
0.25
0.270
9600
40(2)
2252
2288.7
20
0.31
0.333
8400
2169
2206.9
12
0.28
0.292
5200
40(3)
2312
2343.1
15
0.27
0.361
6400
2266
2311.9
28
0.20
0.224
11600
40(4)
2258
2320
26
0.35
0.37
10800
2203
2251.4
26
0.29
0.350
10800
40(5)
2289
2356.7
23
0.33
0.349
9600
2235
2299.2
30
0.20
0.208
12400
Jayanta Majumdar and Asok Kr. Bhunia
Table – III Computational results for random problems of Model – I using EGA-3
n (i ) ∗
BMP Technique
PFP Technique
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
10(1)
611
611.7
4
0.27
0.285
500
611
616.6
1
0.20
0.201
200
10(2)
647
647
1
0.26
0.263
200
647
647
1
0.20
0.200
200
10(3)
616
616.2
1
0.28
0.291
200
616
616.3
1
0.20
0.200
200
10(4)
591
596.3
7
0.29
0.317
800
591
591.8
3
0.22
0.238
400
10(5)
631
634.4
9
0.29
0.343
1000
631
631
1
0.20
0.207
200
15(1)
897
915.1
19
1.05
1.07
3000
886
886
9
0.51
0.577
1500
15(2)
942
942.5
7
0.75
0.81
1200
929
939
5
0.52
0.559
900
15(3)
864
871.2
7
0.76
0.763
1200
864
864.4
2
0.71
0.722
450
15(4)
853
853
1
0.69
0.704
300
853
853
1
0.70
0.709
300
15(5)
867
873.3
4
0.90
0.932
750
870
874.9
4
0.80
0.815
750
20(1)
1133
1139
12
1.88
1.892
2600
1128
1143.2
9
1.78
1.788
2000
20(2)
1197
1232.6
5
0.12
1.924
1200
1154
1162.7
3
0.11
2
800
20(3)
1105
1117.3
15
1.81
2.093
3200
1120
1125.1
13
1.78
1.874
2800
20(4)
1226
1251
13
1.78
1.902
2800
1185
1204.6
13
1.95
2.269
2800
20(5)
1174
1177.4
5
1.75
1.996
1200
1173
1173.8
4
1.67
1.726
1000
74
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
75
Table – III Computational results for random problems of Model – I using EGA-3 (Contd.)
n (i )
BMP Technique
PFP Technique
∗
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
Objbest
Objavg
30(1)
1663
1697.1
13
7.30
7.363
4200
1647
1687.8
30(2)
1696
1780.1
13
6.13
10.182
4200
1742
30(3)
1662
1688.7
11
9.59
9.529
3600
30(4)
1724
1767.2
10
8.96
9.243
30(5)
1658
1679.7
16
12.04
40(1)
2307
2387.5
14
40(2)
2308
2397.8
40(3)
2413
40(4)
40(5)
Gen best
Tbest
Tavg
FE best
10
3.40
9.394
3300
1787.5
13
3.16
9.676
4200
1641
1689.8
11
8.36
8.808
3600
3300
1745
1786.9
6
5.05
7.911
2100
12.574
5100
1666
1685.9
11
9.16
9.586
3600
0.22
0.253
6000
2323
2380.1
27
0.29
0.296
11200
6
0.18
0.209
2800
2290
2360.7
12
0.32
0.378
5200
2512.9
29
0.32
0.332
12000
2407
2477.9
15
0.18
0.279
6400
2430
2468.6
23
0.27
0.286
9600
2427
2518
29
0.49
0.529
12000
2384
2499.1
18
0.25
0.273
7600
2354
2427.3
29
0.36
0.496
12000
Jayanta Majumdar and Asok Kr. Bhunia
76
(iv) The Tavg values for all instances in PFP technique are less or equal to those in BMP
technique using EGA-1 excepting for a single instance (viz. 20(3)) and are less than
using EGA-2. However, for EGA-3, the result is quite different. Here Tavg in PFP
(v)
technique are larger than those in BMP technique for 9 instances.
For all the EGAs using BMP as well as PFP techniques, the Gen best values in PFP
technique are less or equal to those in BMP technique for all instances of n = 10, 15,
20 and 30 while for n = 40, this is not always true. Moreover, for all problems of n =
10, 15, 20 and 30, Gen best ≤ 20 and when n = 40, Gen best ≤ 30.
(vi) The FE best values in PFP technique are less or equal to those in BMP technique for
84%, 92% and 84% of the cases respectively for EGA-1, EGA-2 and EGA-3.
Likewise, from the results of negatively correlated problems of Model – I shown in Tables IV,
V and VI, it is observed that
(i)
Objbest values in PFP technique are less or equal to those in BMP technique for 60%
and 76% of the cases respectively using EGA-1 and EGA-2, while for EGA-3, the
result is somewhat different. Here Objbest values in PFP technique are larger than
those in BMP technique for 18 cases.
(ii) As for the random problem instances, here also the Objbest values in PFP technique
are larger than those in BMP technique for 64%, 44% and 92% of the cases
respectively using EGA-1, EGA-2 and EGA-3.
(iii) For EGA-1 and EGA-2, the Tbest values in PFP technique are less or equal to those in
BMP technique for 64% and 84% of the instances respectively. On the other hand, the
results for EGA-3 are the following:
for n = 10, 15, 20 and 30
Tbest
<< Tbest
PFP
BMP
(iv)
Tavg values in PFP technique are less or equal to those in BMP technique for 68% and
(v)
92% of the cases respectively for EGA-1 and EGA-2. For EGA-3, the former are
results are far lesser than the later for 88% of the cases.
Regarding Gen best values same comparison are there for EGA-2 and EGA-3 as in the
random problem instances. On the contrary, for
EGA-1, Gen best
PFP
≥ Gen best
BMP
in all the cases except for a single case, viz. the problem 20(3).
(vi) For EGA-2 and EGA-3, the FE best values in PFP technique are all less or equal to
those in BMP technique for 100% and 92% of the cases respectively. But, for EGA-1,
the former are greater or equal to the later in all the cases.
Again, from the results for random problems of Model – II shown in Tables VII and VIII it
has been seen that
(i) The Objbest values (for n = 10 and 20) in BMP as well as PFP techniques as obtained
from different EGAs remain the same, viz. [675, 691] and [1126, 1167] respectively
and for n = 15, the value, viz. [937, 962] in PFP technique using EGA-3 is the better,
in case of optimistic decision making. On the other hand, in pessimistic case, although
the Objbest values in both BMP and PFP techniques are the same, viz. [675, 691] (for
n = 10) using different EGAs, for n = 15 and n = 20, the values, viz. [930, 959] (using
EGA-1 and EGA-3) and [1126, 1170] (using EGA-3) of PFP technique are the better.
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
77
Table – IV Computational results for negatively correlated problems of Model – I using EGA-1
n (i ) ∗
BMP Technique
PFP Technique
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
10(1)
932
932.4
3
0.20
0.212
400
932
932.6
6
0.20
0.212
700
10(2)
708
710.2
7
0.20
0.221
800
708
709.4
11
0.20
0.213
1200
10(3)
929
929.7
2
0.20
0.211
300
929
930.2
2
0.20
0.199
300
10(4)
958
962.6
1
0.22
0.211
200
960
963.2
6
0.20
0.204
700
10(5)
971
973.8
2
0.20
0.206
300
973
976.5
4
0.20
0.205
500
15(1)
930
935.6
5
0.54
0.549
900
937
937
6
0.54
0.547
1050
15(2)
1039
1040.5
7
0.54
0.552
1200
1032
1039.3
12
0.56
0.559
1950
15(3)
883
885.7
4
0.57
0.574
750
885
887.2
9
0.56
0.568
1500
15(4)
1463
1468.2
4
0.54
0.628
750
1463
1469
6
0.56
0.585
1050
15(5)
1302
1305.7
7
0.62
0.58
1200
1303
1306.9
10
0.58
0.586
1650
20(1)
1364
1371.3
13
1.58
1.507
2800
1368
1371.9
16
1.68
1.414
3400
20(2)
1518
1521.6
17
1.78
1.527
3600
1519
1523
19
1.75
1.482
4000
20(3)
1616
1620.9
19
1.69
1.666
4000
1615
1621
24
1.74
1.566
5000
20(4)
1252
1254.8
13
1.54
1.541
2800
1252
1268.2
15
1.54
1.575
3200
20(5)
1526
1531.6
16
1.67
1.563
3400
1525
1529.5
18
2.24
1.9
3800
Jayanta Majumdar and Asok Kr. Bhunia
Table – IV Computational results for negatively correlated problems of Model – I using EGA-1 (Contd.)
n (i ) ∗
BMP Technique
PFP Technique
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
30(1)
2060
2064.8
14
5.39
5.045
4500
2057
2063.6
17
7.02
5.77
5400
30(2)
2292
2295.3
19
6.09
5.786
6000
2293
2296.9
20
6.18
5.247
6300
30(3)
1885
1926.9
17
6.80
5.28
5400
1877
1913.3
20
6.17
4.804
6300
30(4)
2001
2004.2
14
4.19
5.286
4500
1998
2003.3
18
7.12
6.223
5700
30(5)
1748
1792.6
12
4.49
5.024
3900
1745
1789
18
5.39
5.5
5700
40(1)
2914
2974.9
29
0.24
0.22
12000
2913
2974.5
29
0.23
0.215
12000
40(2)
2562
2656.2
29
0.23
0.225
12000
2559
2735.8
29
0.22
0.207
12000
40(3)
2431
2528.1
29
0.23
0.218
12000
2556
2609.2
29
0.23
0.219
12000
40(4)
2324
2423.5
29
0.23
0.221
12000
2331
2443
29
0.23
0.231
12000
40(5)
2377
2533
29
0.23
0.225
12000
2465
2480.6
29
0.23
0.23
12000
78
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
79
Table – V Computational results for negatively correlated problems of Model – I using EGA-2
n (i ) ∗
BMP Technique
PFP Technique
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
10(1)
932
932
3
0.23
0.241
400
932
932.6
3
0.24
0.241
400
10(2)
708
709.8
12
0.26
0.249
1300
708
708.7
11
0.24
0.247
1200
10(3)
929
929.9
8
0.24
0.245
900
929
930.2
6
0.24
0.243
700
10(4)
958
961.7
14
0.30
0.258
1500
957
963.2
7
0.25
0.24
800
10(5)
971
972.6
6
0.27
0.253
700
971
972
2
0.24
0.252
300
15(1)
926
934.2
18
0.82
0.678
2850
930
936.3
7
0.66
0.636
1200
15(2)
1032
1036.9
8
0.65
0.68
1350
1032
1037.6
6
0.66
0.681
1050
15(3)
883
883.8
14
0.68
0.743
2250
883
886.8
10
0.69
0.719
1650
15(4)
1463
1466.6
17
0.74
0.891
2700
1463
1467.3
16
0.73
0.8
2550
15(5)
1302
1305.6
8
0.68
0.909
1350
1304
1305.7
7
0.74
0.76
1350
20(1)
1366
1371.3
17
2.34
1.987
3600
1018
1253.8
14
2.16
1.931
3000
20(2)
1513
1518.3
18
2.47
2.225
3800
1022
1417.9
14
1.88
1.985
3000
20(3)
1615
1623.2
17
2.94
2.356
3600
1614
1622.1
13
2.08
2
2800
20(4)
1252
1255.8
16
2.14
2.351
3400
1252
1253.9
12
1.91
2.018
2600
20(5)
1529
1534.1
18
2.04
2.104
3800
1043
1486.7
15
2.01
1.952
3200
Jayanta Majumdar and Asok Kr. Bhunia
Table – V Computational results for negatively correlated problems of Model – I using EGA-2 (Contd.)
n (i ) ∗
BMP Technique
PFP Technique
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
30(1)
2062
2068.1
9
6.93
7.118
2700
1786
2041.5
7
5.44
6.273
2400
30(2)
2301
2308.2
9
6.87
7.147
2700
2304
2306.8
7
4.92
5.405
2400
30(3)
1885
2013.5
12
9.67
9.757
3900
1521
1917.2
6
7.92
8.685
2100
30(4)
2005
2011
19
12.10
12.223
6000
2006
2011.1
15
6.77
6.908
4800
30(5)
1570
1775.4
12
7.67
7.944
3900
1789
1798.3
8
6.70
6.779
2700
40(1)
2907
2934.8
29
0.29
0.294
1200
1878
2696.5
20
0.24
0.242
840
40(2)
2544
2642.4
29
0.29
0.292
1200
2476
2636.5
29
0.28
0.287
1200
40(3)
2445
2581
29
0.30
0.307
1200
1801
2438.4
29
0.29
0.297
1200
40(4)
2333
2431.5
29
0.29
0.293
1200
2337
2420.6
27
0.29
0.283
1200
40(5)
2483
2587.5
29
0.29
0.304
1200
1843
2414.5
19
0.23
0.276
800
80
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
81
Table – VI Computational results for negatively correlated problems of Model – I using EGA-3
n (i ) ∗
BMP Technique
PFP Technique
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
10(1)
932
932.9
20
0.22
0.229
2100
932
932.9
10
0.04
0.043
1100
10(2)
709
717.2
6
0.21
0.231
700
719
719.9
2
0.03
0.037
300
10(3)
929
929.8
3
0.20
0.211
400
929
932.6
3
0.03
0.033
400
10(4)
958
961.3
16
0.22
0.232
1700
964
966.3
2
0.03
0.032
300
10(5)
971
973.5
13
0.24
0.241
1400
973
976.7
12
0.20
0.225
1300
15(1)
930
942.8
13
0.61
0.662
6000
946
960.4
8
0.08
0.095
1350
15(2)
1041
1047.1
12
0.59
0.628
1950
1047
1103.8
10
0.19
0.198
1650
15(3)
888
895.2
20
0.79
0.883
3150
896
903.7
7
0.09
0.194
1200
15(4)
1470
1474.6
18
0.54
0.596
2850
1474
1482
14
0.10
0.182
2250
15(5)
1308
1311.4
18
0.62
0.652
2850
1308
1314.2
5
0.08
0.177
900
20(1)
1379
1386
20
1.42
1.518
4200
1388
1399.8
18
0.26
0.269
3800
20(2)
1528
1529.6
20
1.45
1.686
4200
1523
1533.1
14
0.24
0.257
3000
20(3)
1622
1624.5
20
1.52
1.676
4200
1625
1630
13
0.24
0.255
2800
20(4)
1258
1316.2
20
1.67
1.682
4200
1401
1415.5
6
0.19
0.197
1400
20(5)
1535
1542.2
18
1.62
1.657
3800
1543
1556.5
8
0.22
0.25
1800
Jayanta Majumdar and Asok Kr. Bhunia
Table – VI Computational results for negatively correlated problems of Model – I using EGA-3 (Contd.)
n (i ) ∗
∗
BMP Technique
PFP Technique
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
Objbest
Objavg
Gen best
Tbest
Tavg
FE best
30(1)
2079
2086.5
15
4.76
4.808
4800
2086
2170.1
8
0.60
0.649
2700
30(2)
2305
2312.5
14
4.64
4.829
4500
2318
2324.6
7
0.58
0.654
2400
30(3)
2043
2050.4
11
5.52
6.151
3600
2054
2159.7
7
0.57
0.679
2400
30(4)
2015
2025.7
13
3.97
5.673
4200
2024
2036.4
4
0.52
0.723
1500
30(5)
1786
1845.1
17
6.65
6.679
5400
1921
1943.4
5
0.54
0.703
1800
40(1)
3075
3142.2
29
0.23
0.292
12000
3065
3181.7
29
0.22
0.296
12000
40(2)
2697
2863.6
28
0.22
0.292
11600
2714
2880.1
29
0.22
0.185
12000
40(3)
2818
2925.8
29
0.22
0.298
12000
2754
2902.1
27
0.21
0.299
11200
40(4)
2627
2791.4
29
0.23
0.283
12000
2774
2824.8
26
0.21
0.292
10800
40(5)
2900
2922.2
16
0.18
0.191
6800
2772
2991.7
21
0.20
0.205
8800
problem size n (with different problem numbers i )
82
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
83
Table – VII Computational results for random problems of Model – II using BMP Technique
Objbest
n•
GA
Opt
10
15
20
Genbest
Tbest
Opt
Pes
Pes
Opt
FEbest (Avg)
Pes
Min
Max
Avg
Min
Max
Avg
Min
Max
Avg
Min
Max
Avg
Opt
Pes
GA-1
[675, 691]
[675, 691]
1
9
5
2
2
2
0.01
0.15
0.08
0.05
0.06
0.057
600
300
GA-2
[675, 691]
[675, 691]
2
4
2.6
4
5
4.2
0.03
0.07
0.05
0.10
0.14
0.112
300
520
GA-3
[675, 691]
[675, 691]
3
4
3.4
2
8
4.4
0.05
0.07
0.06
0.05
0.21
0.113
440
540
GA-1
[946, 975]
[945,975]
9
48
16.2
10
77
40.14
0.37
1.92
0.66
0.70
11.63
4.64
258.3
6171.4
GA-2
[946, 975]
[945,975]
3
31
8.3
5
68
31.16
0.13
1.30
0.35
0.35
10.61
4.29
1395
4825
GA-3
[946, 975]
[947,974]
11
18
14.5
27
78
62.5
0.11
0.18
0.14
1.53
4.23
2.21
2325
9525
GA-1
[1126, 1167]
[1142, 1182]
46
85
67.75
2
9
4.2
0.57
1.11
1.85
0.35
1.54
0.72
13750
1040
GA-2
[1126, 1167]
[1142, 1182]
20
74
41.25
3
13
7.6
0.29
1.10
0.50
0.53
2.26
1.33
8450
1720
GA-3
[1126, 1167]
[1131, 1170]
20
55
38.33
19
88
54.16
2.40
5.35
3.72
10.61
10.65
10.62
7866.7
11300
Jayanta Majumdar and Asok Kr. Bhunia
84
Table – VIII Computational results for random problems of Model – II using PFP Technique
Objbest
n•
GA
Opt
10
15
20
Genbest
Tbest
Opt
Pes
Pes
FEbest (Avg)
Opt
Pes
Min
Max
Avg
Min
Max
Avg
Min
Max
Avg
Min
Max
Avg
Opt
Pes
GA-1
[675, 691]
[675, 691]
3
3
3
4
4
4
0.00
0.01
0.007
0.01
0.02
0.014
400
500
GA-2
[675, 691]
[675, 691]
4
4
4
4
5
4.1
0.06
0.07
0.068
0.01
0.02
0.018
500
600
GA-3
[675, 691]
[675, 691]
3
3
3
3
4
3.6
0.00
0.02
0.008
0.00
0.02
0.012
400
460
GA-1
[946, 975]
[930,959]
22
70
36.88
8
76
41.7
0.13
0.10
1.02
0.19
13.94
2.231
5681.25
6405
GA-2
[946, 975]
[945,975]
14
64
27.11
11
12
11.5
0.10
0.40
15.03
0.10
0.11
0.103
4383.3
1875
GA-3
[937, 962]
[930,959]
15
18
16.5
31
37
34.25
4.63
6.38
5.505
0.56
2.00
1.175
2625
5287.5
GA-1
[1126, 1167]
[1142, 1182]
33
95
57
7
11
9.4
0.78
2.21
1.332
0.14
0.21
0.182
11600
2080
GA-2
[1126, 1167]
[1142, 1182]
20
69
44.5
5
7
6.2
0.02
1.05
0.535
0.11
0.16
0.133
9100
1440
GA-3
[1126, 1167]
[1126, 1170]
18
54
30.6
6
95
31.83
0.35
2.57
1.163
3.74
3.75
3.742
6333.3
6566.7
• problem size
Penalty approaches for Assignment Problem with single side constraint via Genetic Algorithms
85
Average Gen best and Tbest values (for optimistic decision) in PFP technique are less
than those in BMP technique for the EGAs except for EGA-2, when n = 10 and 20.
For n = 15, the results are contradictory. Here, all the former results are greater than
those of the later. On the other hand, for pessimistic decision, all the Tbest values for
PFP technique are far lesser than those for BMP technique in maximum cases and the
Gen best values in PFP technique are less than those in BMP technique for EGA-2 and
EGA-3.
(iii) For optimistic decision making, the average FE best values in PFP technique are lesser,
greater and lesser respectively than those in BMP technique for EGA-1, EGA-2 and
EGA-3 respectively when n = 10, 20 and are only greater for all EGAs when n = 15.
Whereas, for pessimistic case, the average FE best values in PFP technique are lesser
than those in BMP technique using all the EGAs for n = 15 and n = 20.
(ii)
7. Conclusions
For the first time, we have developed an elitist genetic algorithm (EGA) to solve the
Assignment Problem with Single Side Constraint (APSSC) for both deterministic and interval
objectives. Here, to formulate the problems with interval objectives, existing interval order relations
for minimization problems are used with respect to optimistic as well as pessimistic decision makers’
point of view. To evaluate the fitness of infeasible solutions in terms of the violation of the side
constraint, two different penalty techniques, viz. Big-M Penalty (BMP) technique and Parameter
Free Penalty (PFP) technique are considered in our developed algorithm. The main idea behind these
methods is that if a solution is infeasible, one will never bother to compute its fitness value as because
it does not make any sense due to the fact that the infeasible solution simply cannot be implemented
practically. Moreover, pair-wise careful comparisons among feasible and infeasible solutions are made
using tournament selection so as to provide a search towards the feasible optimum. Regarding the
comparison of BMP and PFP, no definite conclusion can be made depending upon all the observed
parameters of our experiments. However, analyzing all the points of observations presented in Section
6, PFP technique is proved better than BMP technique in majority of the cases for the developed two
models of APSSC.
From the analysis of our experiments, it is observed that our EGA with new features on GA
initialization, tournament selection and two heuristic operators, viz. make_feasible and
improve_feasible performs well in terms of the best solution, number of generations required, small
time and reasonable number of function evaluation.
We believe that our EGA with two different penalty techniques will be effective to solve other
constrained optimization problems including those with several constraints, e.g. Generalized
Assignment Problems (GAPs), etc. For future research, one may develop even more efficient GA to
solve APSSC with interval resources and/or flexible resource capacities.
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