A Hybrid Genetic Algorithm-TOPSIS-Computer Simulation Approach For Optimum Operator Assignment In Cellular Manufacturing Systems

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A hybrid genetic algorithm-TOPSIS-computer
simulation approach for optimum operator
assignment in cellular manufacturing systems
Ali Azadeh , Hamrah Kor & Seyed-Morteza Hatefi
To cite this article: Ali Azadeh , Hamrah Kor & Seyed-Morteza Hatefi (2011) A hybrid genetic
algorithm-TOPSIS-computer simulation approach for optimum operator assignment in cellular
manufacturing systems, Journal of the Chinese Institute of Engineers, 34:1, 57-74, DOI:
10.1080/02533839.2011.552966
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Journal of the Chinese Institute of Engineers
Vol. 34, No. 1, January 2011, 57–74
A hybrid genetic algorithm-TOPSIS-computer simulation approach for optimum operator
assignment in cellular manufacturing systems
Ali Azadehab*, Hamrah Korab
and Seyed-Morteza Hatefiab
a
Department of Industrial Engineering, Center of Excellence for Intelligent Base Experimental Mechanics,
University of Tehran, PO Box 11365-4563, Tehran, Iran; b
Department of Engineering Optimization Research,
College of Engineering, University of Tehran, PO Box 11365-4563, Tehran, Iran
(Received 1 April 2009; final version received 23 August 2009)
This article presents a decision-making approach based on a hybrid genetic algorithm (GA) and a technique for
order performance by similarity to ideal solution (TOPSIS) simulation (HGTS) for determining the most efficient
number of operators and the efficient measurement of operator assignment in cellular manufacturing systems
(CMS). The objective is to determine the labor assignment in a CMS environment with the optimum
performance. We use HGTS for getting near optimum ranking of the alternative with best fit to the fitness
function. Also, this approach is performed by employing the number of operators, average lead time of demand,
average waiting time of demand, number of completed parts, operator utilization, and average machine
utilization as attributes. Also, the entropy method is used to determine the weight of attributes. Furthermore,
values of attributes are procured by means of computer simulation. The unique feature of this model is
demonstration of efficient ranks of alternatives by reducing the distance between neighborhood alternatives. The
superiority and advantages of the proposed HGTS are shown through qualitative and qualitative comparisons
with TOPSIS, data envelopment analysis (DEA), and principal component analysis (PCA).
Keywords: hybrid; genetic algorithm; TOPSIS; simulation; decision making; entropy; optimization; CMS
1. Introduction
Cellular manufacturing systems (CMS) are typically
designed as dual resource constraint (DRC) systems,
where the number of operators is fewer than the total
number of machines in the system. The productive
capacity of DRC systems is determined by the com-
bination of machine and labor resources. Jobs waiting
to be processed may be delayed because of the non-
availability of a machine or an operator or both. This
fact makes the assignment of operators to machines an
important factor for determining the performance of
CMS and therefore, the development of a multi-
functional workforce a critical element in the design
and operation of CMS.
The efficiency of a many-inputs, many-outputs
decision-making unit (DMU) may be defined as a
weighted sum of its outputs divided by a weighted sum
of its inputs. This so-called ‘‘engineering ratio’’ is the
most popular of a number of alternative measures of
efficiency e.g., Doyle and Green (1994). Hsu-Shih
(2008) exploits incremental analysis or marginal anal-
ysis to overcome the drawbacks of ratio scales
utilized in various multi-criteria or multi-attribute
decision-making (MCDM/MADM) techniques. Yang
et al. (2007) presented two MADM methods in solving
the proposed case study. Both methods use the analytic
hierarchy process (AHP) to determine attribute
weights a priori. The first method is a TOPSIS and
the second method is fuzzy-based. Sarkis and Talluri
(1998) proposed an innovative framework for evaluat-
ing flexible manufacturing systems (FMS) in the
presence of both cardinal and ordinal factors. Ertay
(2002) proposed a framework based on multi-criteria
decision-making for analyzing a firm’s investment
justification problem in a normal and high mold
production technology to cope with the competition
in the global market. Ertay and Ruan (2005) proposed
a decision model for an operator assignment problem
in CMS using a two-phase procedure. In the first
phase, an empirical investigation is conducted using an
exploratory case study with the purpose of finding the
factors that affect the development and deployment of
labor flexibility. In the second phase, based on the
findings from the empirical investigation, a set of
propositions is translated into a methodology frame-
work for examining labor assignments. Zhang et al.
*Corresponding author. Email: aazadeh@ut.ac.ir; ali@azadeh.com
ISSN 0253–3839 print/ISSN 2158–7299 online
ß 2011 The Chinese Institute of Engineers
DOI: 10.1080/02533839.2011.552966
http://www.informaworld.com

(2006), via analyzing the character of an iRS/OS
problem, used the concept of multistage decision
making to formulate an efficient multi-objective
model for minimizing the makespan, balancing the
workload, and minimizing the total transition times
simultaneously by decomposing the problem into two
main phases. Chauvet et al. (2000) developed two
operator assignment problems in which the task times
are dependent on both the assigned task and the
assigned operator. The number of operators is greater
than the number of tasks and one operator can be only
assigned to one task. They aimed to minimize the
maximum completion time of all tasks. This problem is
also known as a bottleneck assignment problem.
Data envelopment analysis (DEA) is a non-para-
metric linear programming-based technique for mea-
suring the relative efficiency of a set of similar units,
usually referred to as DMUs. Because of its successful
implementation and case studies, DEA has achieved
much attention and widespread use by business and
academic researchers. Evaluation of data warehouse
operations (Mannino et al. 2008), selection of FMS
(Liu 2008), assessment of bank branch performance
(Camanho and Dyson 2005), and analysis of the firm’s
financial statements (Edirisinghe and Zhang 2007) are
examples of using DEA in various areas.
Wittrock (1992) developed a parametric preflow
algorithm to solve the problem of assigning human
operators to operations in a manufacturing system.
Also, Süer and Tummaluri (2008) present a three-
phase approach to assigning operators to various
operations in a labor-intensive cellular environment.
First, finding alternative cell configurations; second,
loading cells and finding crew sizes; and third, assign-
ing operators to operations. Bidanda et al. (2005)
discussed human-related issues in a cellular environ-
ment and presented the results of a survey they have
performed. Askin and Huang (2001) and Fitzpatrick
and Askin (2005) discussed forming effective teams in
cellular systems. Cesani and Steudal (2005) studied
labor flexibility in CMS, particularly in cell implemen-
tations allowing intra-cell operator mobility. Slomp
and Molleman (2000), Slomp et al. (2005) and
Molleman and Slomp (1999) discussed training and
cross-training policies and their impact on shop floor
performance. Nembhard (2001a, b), Nembhard and
Mustafa (2000) and Scott et al. (2001) proposed a
heuristic approach to assign workers to tasks based on
individual learning rates and discussed the correlation
between learning and forgetting rates. Jeff et al. (2001)
presented a discrete event simulation model to under-
stand the dynamics of learning and forgetting to
predict variable manufacturing costs and capacity
accurately.
Ayag and Özdemir (2006) present an AHP that is
used for machine tool selection problems due to the
fact that it has been widely used in evaluating various
kinds of MCDM problems in both academic research
and in practice. Also, Önüt et al. (2008) describe a
fuzzy TOPSIS-based methodology for evaluation and
selection of vertical CNC machining centers for a
manufacturing company in Istanbul, Turkey. The
criteria weights are calculated using the fuzzy AHP.
In fact, they introduced two phased methodology
based on fuzzy AHP and fuzzy TOPSIS for selecting
the most suitable machine tools. Recently, Rao (2006)
presented a material selection model using graph
theory and a matrix approach. However, the method
does not have a provision for checking consistency in
the judgments of relative importance of the attributes.
Further, the method may be difficult to deal with if the
number of attributes is more than 20.
This research is divided into three phases. In the
first phase, the proposed approach is described. In the
second phase, we use a simulation model for evaluation
of the identified scenarios. Finally, the scenarios are
translated into HGTS to find the best scenario.
1.1. Technique for order performance by similarity to
ideal solution
The Technique for Order Performance by Similarity to
Ideal Solution (TOPSIS) method, which is based on
choosing the best alternative having the shortest
distance to the ideal solution and the farthest distance
from the negative-ideal solution, was first proposed in
1981 by Hwang and Yoon (1981). The ideal solution is
the solution that maximizes the benefit and also
minimizes the total cost. On the contrary, the
negative-ideal solution is the solution that minimizes
the benefit and also maximizes the total cost. The
following characteristics of the TOPSIS method make
it an appropriate approach which has good potential
for solving decision-making problems:
. An unlimited range of cell properties and
performance attributes can be included.
. In the context of operator assignment, the
effect of each attribute cannot be considered
alone and must always be seen as a trade-off
with respect to other attributes. Any change
in, for instance, amount of demand, lead time
or operator utilization indices can change the
decision priorities for other parameters. In
light of this, the TOPSIS model seems to be a
suitable method for multi-criteria operator
assignment problems as it allows explicit
trade-offs and interactions among attributes.
58 A. Azadeh et al.

More precisely, changes in one attribute can
be compensated for in a direct or opposite
manner by other attributes.
. The output can be a preferential ranking of
the alternatives (scenarios) with a numerical
value that provides a better understanding of
differences and similarities between alterna-
tives, whereas other MADM techniques (such
as the ELECTRE methods e.g., Roy 1991,
1996) only determine the rank of each
scenario.
. Pair-wise comparisons, required by methods
such as the Analytical Hierarchy Process
(Saaty 1990, 2000), are avoided. This is
particularly useful when dealing with a large
number of alternatives and criteria; the
methods are completely suitable for linking
with computer databases dealing with scenario
selection.
. It can include a set of weighting coefficients
for different attributes.
. It is relatively simple and fast, with a system-
atic procedure.
Hwang and Yoon (1981) introduced the TOPSIS
method based on the idea that the best alternative
should have the shortest distance from an ideal
solution. They assumed that if each attribute takes a
monotonically increasing or decreasing variation, then
it is easy to define an ideal solution. Such a solution is
composed of all the best attribute values achievable,
while the worst solution is composed of all the worst
attribute values achievable. The goal is then to propose
a solution which has the shortest distance from the
ideal solution in the Euclidean space (from a geomet-
rical point of view). However, it has been argued that
such a solution may need to have simultaneously the
farthest distance from a negative ideal solution (also
called nadir solution). Sometimes, the selected solution
(here candidate scenario) which has the minimum
Euclidean distance from the ideal solution may also
have a short distance from the negative ideal solution
as compared to other alternatives. The TOPSIS
method, by considering both the above distances,
tries to choose solutions that are simultaneously close
to the ideal solution and far from the nadir solution.
In a modified version of the ordinary TOPSIS method,
the ‘city block distance’, rather than the Euclidean
distance, is used so that any candidate scenario which
has the shortest distance to the ideal solution is
guaranteed to have the farthest distance from the
negative ideal solution.
The TOPSIS solution method consists of the
following steps:
(i) Normalize the decision matrix. The normali-
zation of the decision matrix is done using the
following transformation:
nij ¼
rij
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pm
i¼1 r2
ij
q ; i ¼ 1, 2, . . . , n, ð1Þ
where m is the number of scenarios (DMUs), n the
number of criteria, and rij an element of the decision
matrix.
(ii) Multiply the columns of the normalized deci-
sion matrix by the associated weights. The weighted
and normalized decision matrix is obtained as
Vij ¼ nijW0
j i ¼ 1, 2, . . . , m; j ¼ 1, 2, . . . , n, ð2Þ
where W0
j represents weight of the jth criteria and V the
weighted normalized decision matrix.
(iii) Determine the ideal and nadir ideal solutions.
The ideal and the nadir value sets are determined,
respectively, as follows:
fVþ
1 , Vþ
2 , . . . , Vþ
6 g
¼ fð½MaxÞ
¢
i½vij ¢ji 2 k¢,
½Min
¢
i½VðijÞj j 2 k0
ji ¼ 1, 2, . . . m
and j ¼1, 2, . . . , ng, ð3Þ
fVþ
1 , V
2 , . . . , V
6 g
¼ fð½MinÞ ¢ i½vij ¢ji 2 k¢,
½Max
¢
i½Vðij Þj j 2 k0
ji ¼ 1, 2, . . . m
and j ¼1, 2, . . . , ng, ð4Þ
where K is the index set of benefit criteria and K0
the
index set of cost criteria.
(iv) Measure distances from the ideal and nadir
solutions. The two Euclidean distances for each
alternative are, respectively, calculated as
Sþ
i ¼
X
n
j¼1
ðvij vþ
j Þ
( )0:5
;
i ¼ 1, 2, . . . , m; j ¼ 1, 2, . . . n, ð5Þ
S
i ¼
X
n
j¼1
ðvij v
j Þ
( )0:5
;
i ¼ 1, 2, . . . , m; j ¼ 1, 2, . . . n: ð6Þ
Remark: In the so-called ‘block TOPSIS’ method, the
two distances are obtained as
Sþ
i ¼
X
n
j¼1
jvij vþ
j j and S
i ¼
X
n
j¼1
jvij v
j j: ð7Þ
Journal of the Chinese Institute of Engineers 59

(v) Calculate the relative closeness to the ideal
solution. The relative closeness to the ideal solution
can be defined as
Cj ¼
S
i
Sþ
i þ S
i
, i ¼ 1, 2, . . . , 0 m, 0 ci 1: ð8Þ
The higher the closeness means the better the rank.
The methods for assessing the relative importance
of criteria must be well defined.
For solving MADM problems, it is generally
necessary to know the relative importance of each
criterion. It is usually given as a set of weights, which
are normalized, and which add up to one. The
importance coefficients in the MADM methods refer
to intrinsic ‘weight’. Some papers deserve mention
because they include information concerning the
methods that have been developed for assessing the
weights in an MADM problem, these are Refs. Olson
(2004), Simos and Gestion (1990), and Roy (1991). The
entropy method is the method used for assessing the
weight in a given problem because, with this method,
the decision matrix for a set of candidate scenarios
contains a certain amount of information. In other
words, the entropy method works based on a
predefined decision matrix. Since there is, in scenario
selection problems, direct access to the values of the
decision matrix, the entropy method is the appropriate
method. Entropy, in information theory, is a criterion
for the amount of uncertainty, represented by a
discreet probability distribution, in which there is
agreement that a broad distribution represents more
uncertainty than does a sharply packed one. The
entropy idea is particularly useful for investigating
contrasts between sets of data. The entropy method
consists of the following procedure:
(i) Normalizing the decision matrix
pij ¼
rij
Pm
i¼1 rij
i ¼ 1, 2, . . . m, j ¼ 1, 2, . . . , n: ð9Þ
(ii) Calculating the entropy with data for each
criterion, the entropy of the set of normalized out-
comes of the jth criterion is given by
Ej ¼ k
X
m
½ pij lnð pijÞ i ¼ 1, 2, . . . , m, j ¼ 1, 2, . . . , n:
ð10Þ
Meanwhile, the k ¼ 1
ln m is a constant to make sure that
the ej value is between 0 and 1.
Using the entropy method, it is possible to combine
the scenario designer’s priorities with those of sensi-
tivity analysis. Final weights defined are a combination
of two sets of weights. The first is the set of objective
weights that are derived directly from the nature of the
design problem using the entropy method, and with no
regard to the designer’s desires. The second is the set of
subjective weights that are defined by the scenario
designer’s preferences to modify the previous weights
and find the total weights. When the scenario designer
finds no reason to give preference to one criterion
over another, the principle of insufficient reason
suggests that each one should be equally preferred.
W0
j ¼
dj
Pn
j¼1 d0
j
, 8j, ð11Þ
where dj¼1 Ej is the degree of diversity of the
information involved in the outcomes of the jth
criterion.
The value j is: j ¼ 1, 2, . . . , m otherwise, if the
scenario designer wants to add the subjective weight
according to the experience, particular constraint of
design and so on, the weight factor is revised as
wj ¼
jw0
j
Pn
j¼1 jw0
j
, 8j: ð12Þ
In this article, the revised Simos method (Shanian
and Savadogo 2006) has been used to define the
subjective weights in a given problem by the following
algorithm:
(1) The non-normalized subjective weights (1) . . .
(r) . . . (
n) associated with each class of equally
placed criteria, arranged in the order of increasing
importance. The criterion or group of criteria identi-
fied as being least important is assigned the score of 1,
i.e., (1) ¼ 1.
(2) The normalized subjective weight: j is desig-
nated the normalized weight of criterion i such that:
X
n
i¼1
j ¼ 100:
It is concluded that the introduced combined
weighting scheme is important for decision-making
problems. It can take into account both the nature of
conflicts among criteria and the practicality of the
decisions. This opportunity reflects the advantage of
more controllable design selections. The entropy
approach can be used as a good tool in criteria
evaluation. This possibility makes the entropy method
very flexible and efficient for scenario design.
60 A. Azadeh et al.

1.2. Genetic algorithm
GA is a part of evolutionary computing, which is
rapidly growing in the area of artificial intelligence
(AI). Also, it was inspired by Darwin’s theory of
evolution. Simply said, problems are solved by an
evolutionary process resulting in a best (fittest) solu-
tion (survivor). In GA, the solution is repeatedly
evolved until the best solution is fixed. For using GA,
the solution must be represented as a genome (or
Chromosome). The GA then creates a population of
solutions and applies genetic operators such as muta-
tion and crossover to evolve the solutions in order to
find the best one(s).
The general outline of GA is summarized below:
Algorithm 1: Genetic algorithm
Step 1: Generate random population with n chromo-
somes by using symbolic representation scheme (suit-
able size of solutions for the problem).
Step 2: Evaluate the fitness function of each chro-
mosome x in the population by using the proposed
objective functions.
Step 3: Create a new population by iterating loop
highlighted in the following steps until the new
population is complete.
(i) Select two parent chromosomes from a pop-
ulation according to their fitness from Step 2.
Those chromosomes with the better fitness will
be chosen.
(ii) With a preset crossover probability, crossover
will operate on the selected parents to form
new offspring (children). If no crossover is
performed, offspring are observed to be the
exact copies of parents. Here, multi-point
crossover is used while partially matched
crossover is employed for Problem.
(iii) With a preset mutation probability, mutation
will operate on new offspring at each gene.
Chosen genes are swapped to perform
mutation.
(iv) Place new offspring in the new population.
Step 4: Deliver the best solution in the current
population. If the end condition is satisfied, stop.
Step 5: Go to Step 2.
According to the description of GAs in the above,
the proposed method should be specially designed in
accordance with the nature of the problem. Therefore,
aspects including chromosome representation, fitness
evaluation, parent selection, crossover (reproduction),
and mutation will be tailor-made for the problem
(Haupt and Haupt 1998).
The mentioned model has a few limitations.
Because the variables are constrained to integer
values, the model is difficult to solve for a large
number of scenarios and machines due to the compu-
tational complexity. Also, this model does not offer cell
designers the flexibility to change objective functions
and constraints. In this section, we use an approach
using GAs developed by Ebrahimipour et al. (2007), to
solve the simultaneous multi-criteria decision and
operator assignment problem.
1.2.1. Representation and initialization
The initialization operator is used to create the initial
population by filling it with randomly generated
individuals. Each individual is a representative of the
problem solution which is identified by its digit string.
The deletion operator deletes all members of the old
population who cannot contribute as influential par-
ents for the next generation.
1.2.2. Evaluation
This stage involves checking the individuals to see how
well they are able to satisfy the objectives in the
problem. The fitness operator quantifies the total
characters of each chromosome (individual) in the
population. The evaluating fitness operator assesses
the value of fitness function of each chromosome in
order to satisfy the objectives based on maximum or
minimum level.
1.2.3. Crossover and mutation
Crossover is aimed at exchanging bit strings between
two parent chromosomes. The crossover used in this
model is one cut-point method. For example, the
parent chromosomes are randomly selected and the cut
point is then randomly selected at position 2 as follows:
½6 3 5 4j3 2 5 6
½2 1 3 6j5 6 2 0:
Also, offspring are formed by exchanging the end parts
of their parents, as follows:
½6 3 5 4j5 6 2 0
½2 1 3 6j3 2 5 6:
In this article, we use multi-point crossover based
on a single point.

Mutation is performed as random perturbation.
Any gene in the chromosome may be randomly
selected to be mutated, at a preset rate. The mutation
operator for the cell design problem is designed to
perform random exchange. For a selected gene mk, it
will be replaced by a random integer with [1, upper
bound]. An example is given as follows:
½4 6 5 2 1 5:
In this chromosome, the fourth gene is selected for
mutation. The value of the gene is replaced by five.
After mutation: ½4 6 5 5 1 5.
1.2.4. Selection
In this study, Tournament selection is used to select
pairs for mating. Tournament selection is a recent
approach that closely mimics mating competition in
nature. The procedure is to pick a small subset of
chromosomes randomly (two or three) from the
mating pool, and the chromosome with the lowest
cost in this subset becomes a parent. The tournament
repeats for every parent needed. Threshold and tour-
nament selection make a nice pair, because the
population never needs to be sorted. Tournament
selection works best for larger population sizes because
sorting becomes time-consuming for large populations.
2. The hybrid GA-TOPSIS simulation
The hybrid GA-TOPSIS-simulation (HGTS) is an
extremely efficient approach for selection of the
optimum operator allocations in CMS. First, we
define the scenarios by considering all available con-
ditions. In this stage, we study the CMS environment.
Then, we define significant scenarios based on the
number of working shifts and operators. A simulation
model is developed and run to identify the efficiency of
each scenario. To avoid bias, each scenario is run 30
times. After this stage, we have a data set about all
scenarios which shows the value of each criterion.
Then, by HGTS, we solve the problem. The steps of
HGTS are presented as follows:
(1) By the entropy method, we define the weight
factor for the each criterion. This step can be ignored if
the weight factors are available.
(2) In the next step, the procured data from the
previous section is used by GA for initialization
process.
(3) After determining the weight factor, TOPSIS
method is used to solve the problem to define the best
scenarios.
(4) In the fourth step, GA uses the output of
TOPSIS as input. In fact, the initial population for GA
is the TOPSIS solution.
(5) The problem is finally solved by GA and
checked by TOPSIS. Moreover, we define the best
ranking of scenarios.
The details of how HGTS works in practice are
shown in the next section (Section 3). Figure 1 presents
the proposed HGTS approach for optimum operator
assignment.
3. Empirical illustration
Manned cells are a very flexible system that can adapt
to changes in the customer’s demand or other changes
quite easily and rapidly in the product design. The cells
described in this study are designed for flexibility, not
line balancing. The walking multi-functional operators
permit rapid rebalancing in a U-shaped manner. The
considered cell has eight stations and can be operated
by one or more operators, depending on the required
output for the cell. The times for the operations at the
stations do not have to be balanced. The balance is
achieved by having the operators walk from station-to-
station. The sum of operation times for each operator
is approximately equal. In other words, any division of
stations that achieves balance between the operators
is acceptable. Operators perform scenario movements
in cells. Once a production batch size arrives at a cell,
it is divided to transfer batch sizes. Transfer
batch size is the transfer quantity for intra-cell move-
ments of parts. The ability to rebalance the cell quickly
to obtain changes in the output of the cell can be dem-
onstrated by the developed simulation model.
The existing manned cell example for a case
model is presented in Figure 2 (Azadeh and
Anvari 2006).
Alternatives for reducing the number of operators
in the cell are as follows:
(1) Eight operators (one operator for each
machine).
(2) Seven operators (two operators handling
two machines and one operator for each of
the rest).
(3) Six operators (two operators handling
two machines and one operator for each of
the rest).
(4) Five operators (three operators handling two
machines and one operator for each of the rest).
(5) Four operators (each operator handling two
machines).
(6) Six operators (one operator for three machines,
others with one machine each).
62 A. Azadeh et al.

(7) Four operators (one operator handling five
machines, three operators with one machine
each).
(8) Three operators (two operators handling three
machines each and one operator handling two
machines).
(9) Five operators (one operator to four machines
and one operator for each of others).
(10) Three operators (one operator to four
machines and two operators handling two
machines each).
(11) Three operators (one operator to four
machines and one operator to three machines
and one operator to one machine).
(12) Two operators (one operator for four
machines).
In simulation experiments, when the machines are
assigned to the operators, the cycle time of the
bottleneck resource is chosen as close as possible to
the cycle time of the operator. The developed model
Define CMS scenarios
Generation of outputs data by
computer simulation
Solving operator assignment
problem by TOPSIS
Identification of weight factors
from Entropy method
Final assignment: By utilizing GA structure
in TOPSIS
Identifying CMS model
Define attributes weight factor by
Entropy method
TOPSIS solution as initial
population for GA Approach
Solve the CMS problem by GA
Figure 1. The overview of the integrated HGTS approach.
D: Decoupler station
Operator
Operator movement when not working
Operator movement with parts
Figure 2. The existing manned cell example for the case
model.

includes the some assumptions and constraints as
follows:
. The self-balancing nature of the labor assign-
ment accounts for differences in operator
efficiency.
. The machines have no downtime during the
simulated time.
. The time for the operators to move between
machines is assumed to be zero. The machines
are all close to each other.
. The sum of the multi-function operation times
for each operator is approximately equal.
. There is no buffer for the station work.
As mentioned, outputs collected from simulation
model are the average lead time of demand, the
average of waiting time of demand, average operator
and machine utilization, and the number of completed
parts per annum. The results of the simulation exper-
iment are used to compare the efficiency of the
alternatives. Each labor assignment scenario considers
three shifts with 1, 2, or 3 shifts per day. Moreover, a
flexible simulation model (Figure 3) is built by Visual
SLAM (Pritsker 1995), which incorporates all 36
scenarios for quick response and results.
Furthermore, ability to rebalance the cell quickly to
obtain changes in output of cell can be demonstrated
by the simulation model. So, in the developed model,
different demand levels and part types have been taken
into consideration.
System performance was monitored for different
workforce levels and shifts by means of simulation. In
simulation experiments, each of the demanded parts
had a special type and level. The types of parts that the
cell can produce and levels of demand within the cell
were determined as two and three by experiment. The
time processing of jobs for each of the stations works is
related to the part type. The objective of scenarios
consists of reducing the number of operators in the
cell, and observing how the operation is distributed
among the operators. After deletion of transient state,
the 36 scenarios were executed for 2000 h (250 working
days, each day composed of three shifts, each shift
consisting of 8 h of operation). Each scenario was also
replicated 30 times to insure that reasonable estimates
of means of all outputs could be obtained.
The Table 1 shows the output of the simulation
model.
3.1. Application of HGTS approach
A total of 36 scenarios were selected as the core of
our study. The values of the six indices for the
operator assignment are presented in Table 1. The
main structure of HGTS in this study is based on
the assumption that the best scenario is identified
with the indices in which each is the maximum value
of its possible values. Therefore, for the scenario 36,
a scenario with the best possible attribute, called our
Figure 3. Simulation model of operators’ allocation in CMS.
64 A. Azadeh et al.

goal in the problem, illustrates the maximum present
abilities in operator assignment. To achieve the
appropriate rank (array), every possible array, com-
prising 36 scenarios, is considered as a 64-bit chromo-
some. Then, in accordance with the sequence in the
chromosome, the total distance among the first sce-
nario which can be scenario from 1 to 36 and
our goal (scenario 36), the second scenario with the
first and the next with upper-ranked scenario are
calculated.
Respectively, the total distance mentioned above is
a variable, dependent on the scenarios’ positions in the
array. Consequently, in each chromosome, we can find
a new value for total distance. Undoubtedly, the best
sequence of the scenarios is an array which has the
minimum total distance with high internal cohesion
among its scenarios. In fact, our fitness function is a
multivariate combination in which its most prominent
components are total distance and variance. The above
genetic concepts are achieved through a set of well-
defined steps as follows:
Step 1: Normalize the index vectors. The six attrib-
utes must be normalized and have same order to be
used in HGTS. Indices X1, X2, and X3 have opposite
order than the rest of the indices.
Step 2: Standardize the indices X1 X6. The indices
are standardized and shown in Table 2. They are
standardized through predefined mean and standard
deviation for each index.
Table 1. Simulation results for the case model (decision matrix).
X1 X2 X3 X4 X5 X6
Scenarios
Number of
operators
Average lead
time of demand
Average waiting
time of demand
Number of
completed parts
Operator
utilization
Average machine
utilization
Sen1-1 8 303.41 136.35 675 20.19 43.59
Sen1-2 16 251.75 110.21 815 7.65 16.08
Sen1-3 24 213.07 118.69 740 10.80 22.68
Sen2-1 7 269.95 118.80 585 15.77 28.55
Sen2-2 14 232.12 88.63 655 10.20 18.54
Sen2-3 21 220.35 95.50 870 4.03 7.06
Sen3-1 6 267.90 101.12 595 28.90 43.13
Sen3-2 12 231.43 86.08 740 28.38 43.91
Sen3-3 18 213.70 97.71 690 10.00 15.86
Sen4-1 5 305.55 239.57 565 50.85 65.20
Sen4-2 10 245.62 105.61 885 16.71 21.98
Sen4-3 15 227.37 82.83 655 6.81 8.96
Sen5-1 4 281.89 115.80 515 38.55 42.56
Sen5-2 8 255.15 101.38 710 24.38 28.86
Sen5-3 12 230.73 85.77 635 7.99 8.61
Sen6-1 6 284.90 87.99 615 21.73 34.14
Sen6-2 12 247.50 116.91 740 19.98 31.90
Sen6-3 18 225.47 103.73 645 11.48 18.24
Sen7-1 4 333.03 129.15 545 35.89 42.05
Sen7-2 8 261.38 97.55 840 20.51 23.40
Sen7-3 12 242.44 102.74 800 13.99 15.99
Sen8-1 3 329.63 100.47 570 40.65 36.83
Sen8-2 6 258.76 107.42 665 23.60 20.99
Sen8-3 9 246.91 201.60 685 20.85 18.43
Sen9-1 3 389.24 152.93 445 60.25 59.75
Sen9-2 6 343.72 165.53 740 47.70 46.86
Sen9-3 9 305.07 152.91 670 25.35 25.00
Sen10-1 3 337.32 114.61 530 53.45 46.60
Sen10-2 6 299.19 238.46 850 32.20 28.54
Sen10-3 9 263.20 204.81 720 23.45 21.23
Sen11-1 3 363.18 196.82 535 60.20 55.70
Sen11-2 6 276.19 110.25 615 29.15 26.61
Sen11-3 9 261.57 107.71 680 23.25 21.95
Sen12-1 2 361.91 185.12 505 64.88 49.63
Sen12-2 4 329.36 248.24 875 74.33 54.90
Sen12-3 6 291.53 106.71 660 31.13 22.15

Step 3: Define the production module. This module
is defined to create and manipulate the 50-
individual population by filling it with randomly
generated individuals. Each individual is defined by a
64-bit string.
Step 4: Define recombination module which com-
prises four sections:
. Tournament selection operator chooses indi-
viduals with probability 80% from the popu-
lation for reproduction.
This is considered a popular type of selection
method in HGTS. The basic concept in a
tournament is that the best string in the
population will win both its tournaments,
while the worst will never win, and thus will
never be selected. However, in this study, the
other kinds of selection methods named sigma
scaling and rank selection are considered in
order to determine the best method.
. Uniform crossover operator which combines
bits from the selected parents with the prob-
ability of 85%.
. Mutation operator consists of making (usually
small) alterations to the values of one or more
genes in a chromosome.
. Regeneration operator which is used to create
100-individual generations.
Step 5: Define evaluation module:
The fitness function to determine the goodness of
each individual based on the objectives is defined by
Table 2. Standardized matrix for the six indices.
Scenarios
Number of
operators
Average lead
time of demand
Average waiting
time of demand
Number of
completed parts
Operator
utilization
Average machine
utilization
Sen1-1 0.184 0.550 0.113 0.010 0.444 0.825
Sen1-2 1.289 0.560 0.438 1.246 1.140 0.979
Sen1-3 2.762 1.390 0.259 0.584 0.965 0.546
Sen2-1 0.368 0.169 0.257 0.785 0.689 0.161
Sen2-2 0.921 0.981 0.892 0.167 0.998 0.818
Sen2-3 2.210 1.234 0.747 1.731 1.341 1.570
Sen3-1 0.552 0.213 0.629 0.696 0.039 0.794
Sen3-2 0.552 0.996 0.946 0.584 0.010 0.846
Sen3-3 1.657 1.377 0.701 0.142 1.009 0.994
Sen4-1 0.737 0.596 2.287 0.961 1.256 2.242
Sen4-2 0.184 0.691 0.535 1.864 0.637 0.592
Sen4-3 1.105 1.083 1.014 0.167 1.186 1.446
Sen5-1 0.921 0.087 0.320 1.403 0.574 0.757
Sen5-2 0.184 0.487 0.624 0.319 0.212 0.141
Sen5-3 0.552 1.011 0.952 0.343 1.121 1.469
Sen6-1 0.552 0.152 0.906 0.520 0.359 0.205
Sen6-2 0.552 0.651 0.297 0.584 0.456 0.058
Sen6-3 1.657 1.124 0.574 0.255 0.927 0.837
Sen7-1 0.921 1.186 0.039 1.138 0.426 0.724
Sen7-2 0.184 0.353 0.704 1.467 0.427 0.499
Sen7-3 0.552 0.760 0.595 1.113 0.788 0.985
Sen8-1 1.105 1.113 0.643 0.917 0.690 0.381
Sen8-2 0.552 0.409 0.496 0.078 0.255 0.657
Sen8-3 0.000 0.664 1.487 0.098 0.408 0.825
Sen9-1 1.105 2.393 0.462 2.021 1.777 1.884
Sen9-2 0.552 1.415 0.727 0.584 1.081 1.039
Sen9-3 0.000 0.585 0.462 0.034 0.158 0.394
Sen10-1 1.105 1.278 0.345 1.270 1.400 1.022
Sen10-2 0.552 0.459 2.263 1.555 0.222 0.162
Sen10-3 0.000 0.314 1.555 0.407 0.263 0.641
Sen11-1 1.105 1.833 1.386 1.226 1.775 1.619
Sen11-2 0.552 0.035 0.437 0.520 0.053 0.289
Sen11-3 0.000 0.349 0.490 0.054 0.275 0.594
Sen12-1 1.289 1.806 1.140 1.491 2.034 1.221
Sen12-2 0.921 1.107 2.469 1.775 2.558 1.566
Sen12-3 0.552 0.294 0.511 0.123 0.162 0.581
66 A. Azadeh et al.

the total distance and variance that can be shown by
dtotal ¼
X
6
j¼1
ðxG1j xmax j Þ2
X
36
i¼2
X
6
j¼1
ðxGij xGi1 jÞ2
#1
2
,
Sch ¼
1
36
X
36
j¼1
ðdi
dÞ2
#1
2
,
FðChromosomeÞ ¼ dtotal Sch,
ð13Þ
where i is the number of scenarios (DMUs) and j the
number of criteria.
In the above-mentioned formula, dTotal indicates
the total distance between adjacent scenarios.
The evaluation of operator assignment is the ability
of each chromosome to satisfy the objective function.
Therefore, our motivation to obtain the best array in
this problem is to minimize the fitness function
mentioned above. After producing 1000 generations,
we reach the best fitness function value, 295.362,
related to the chromosome which can be shown by the
sequence at Table 3.
3.2. Implementation of TOPSIS
For the first step of this methodology, the decision
matrix (Table 1), representing the performance values
of each alternative with respect to each criterion,
Table 3. Simulation results for the case model (decision matrix).
X1 X2 X3 X4 X5 X6
Scenarios
Number of
operators
Average lead
time of demand
Average waiting
time of demand
Number of
completed parts
Operator
utilization
Average machine
utilization
Sen1-1 8 303.41 136.35 675 20.19 43.59
Sen1-2 16 251.75 110.21 815 7.65 16.08
Sen1-3 24 213.07 118.69 740 10.80 22.68
Sen2-1 7 269.95 118.80 585 15.77 28.55
Sen2-2 14 232.12 88.63 655 10.20 18.54
Sen2-3 21 220.35 95.50 870 4.03 7.06
Sen3-1 6 267.90 101.12 595 28.90 43.13
Sen3-2 12 231.43 86.08 740 28.38 43.91
Sen3-3 18 213.70 97.71 690 10.00 15.86
Sen4-1 5 305.55 239.57 565 50.85 65.20
Sen4-2 10 245.62 105.61 885 16.71 21.98
Sen4-3 15 227.37 82.83 655 6.81 8.96
Sen5-1 4 281.89 115.80 515 38.55 42.56
Sen5-2 8 255.15 101.38 710 24.38 28.86
Sen5-3 12 230.73 85.77 635 7.99 8.61
Sen6-1 6 284.90 87.99 615 21.73 34.14
Sen6-2 12 247.50 116.91 740 19.98 31.90
Sen6-3 18 225.47 103.73 645 11.48 18.24
Sen7-1 4 333.03 129.15 545 35.89 42.05
Sen7-2 8 261.38 97.55 840 20.51 23.40
Sen7-3 12 242.44 102.74 800 13.99 15.99
Sen8-1 3 329.63 100.47 570 40.65 36.83
Sen8-2 6 258.76 107.42 665 23.60 20.99
Sen8-3 9 246.91 201.60 685 20.85 18.43
Sen9-1 3 389.24 152.93 445 60.25 59.75
Sen9-2 6 343.72 165.53 740 47.70 46.86
Sen9-3 9 305.07 152.91 670 25.35 25.00
Sen10-1 3 337.32 114.61 530 53.45 46.60
Sen10-2 6 299.19 238.46 850 32.20 28.54
Sen10-3 9 263.20 204.81 720 23.45 21.23
Sen11-1 3 363.18 196.82 535 60.20 55.70
Sen11-2 6 276.19 110.25 615 29.15 26.61
Sen11-3 9 261.57 107.71 680 23.25 21.95
Sen12-1 2 361.91 185.12 505 64.88 49.63
Sen12-2 4 329.36 248.24 875 74.33 54.90
Sen12-3 6 291.53 106.71 660 31.13 22.15

is computed by a simulation model. Next, these
performance values are normalized by Equation 2. In
Step 3, the normalized matrix is multiplied with the
criteria weights calculated by the entropy method
(Table 4). The step of defining the ideal solution
consists of taking the best values of alternatives and
using similar principles, obtaining the negative-ideal
solution by taking the worst values of alternatives.
Subsequently, the alternatives are ranked with respect
to their relative closeness to the ideal solution (Table 5).
4. Results and discussion
According to the above results, scenario 12-2 (one
operator for four machines, two shifts per day) is the
most efficient one. The second best scenario is the
scenario 12-1, which is similar to scenario 12-2 only
with one shift per day. The third best scenario is 9-1
with five operators (one operator for four machines
and one operator per machine for the rest with one
shift per day). Table 5 presents the rankings of the
proposed HGTS versus TOPSIS for the best 18
rankings.
4.1. Verification and validation
DEA and principal component analysis (PCA) are
used to verify and validate the results of the proposed
HGTS. DEA and PCA are among the most powerful
tools in multivariate analysis. However, we show that
the proposed algorithm has several advantages over
these methods. These are discussed in the following
sections. First, mathematical models of DEA and PCA
are discussed and their efficiency scores and ranking
are evaluated. Then, their ranking scores together with
TOPSIS are compared with the proposed HGTS.
4.1.1. Data envelopment analysis
The two basic DEA models are CCR based on
Charnes, Cooper and Rhodes (Charnes et al. 1978)
and BCC based on Banker, Charnes and Cooper
(Banker et al. 1984) with constant returns to scale and
variable returns to scale, respectively. DMUo is
Table 5. The rankings of HGTS versus TOPSIS.
TOPSIS solution HGTS solution
Scenario Rank Scenario Rank Scenario Rank Scenario Rank
35 1 8 19 31 1 8 19
34 2 27 20 35 2 27 20
25 3 20 21 25 3 30 21
31 4 4 22 34 4 24 22
28 5 33 23 28 5 33 23
10 6 30 24 10 6 20 24
26 7 24 25 26 7 11 25
22 8 11 26 19 8 4 26
13 9 17 27 13 9 17 27
19 10 21 28 22 10 21 28
7 11 15 29 7 11 15 29
29 12 5 30 23 12 5 30
36 13 12 31 36 13 12 31
32 14 2 32 32 14 2 32
16 15 18 33 16 15 18 33
23 16 9 34 29 16 9 34
14 17 3 35 14 17 3 35
1 18 6 36 1 18 6 36
Score 303.406 Score 295.362
Table 4. Entropy-weighted coefficients.
Attribute Index Dj Weight
Number of operator W1 0.046273 0.296356
Average lead time of
demand
W2 0.003721 0.023834
Average waiting time of
demand
W3 0.016075 0.102952
Number of completed
parts
W4 0.003838 0.024582
Operator utilization W5 0.05315 0.340401
Average machine
utilization
W6 0.033082 0.211876
68 A. Azadeh et al.

assigned the highest possible efficiency score o 1 that
constraints allow from the available data by choosing
the optimal weights for the output and inputs. If
DMUo receives the maximal value o ¼ 1, then it is
efficient, but if o 5 1, it is inefficient, since with its
optimal weights, another DMU receives the maximal
efficiency. Basically, the model divides the DMUs into
two groups, efficient (o ¼ 1) and inefficient (o 5 1),
by identifying the efficient in the data. The original
DEA model is not capable of ranking efficient units
and therefore it is modified to rank efficient units
(Andersen and Petersen 1993).
The original fractional CCR model (14) evaluates
the relative efficiencies of 36 scenarios ( j ¼ 1, . . . , 36),
each with 3 inputs (average lead-time of demands,
average waiting time of demands and the number of
the operators (in a working day) and 3 outputs
(average operator/machine utilization and numbers of
completed parts per year) denoted by x1j, x2j, x3j, y1j,
and y2j and y3j, respectively. This is done by maximiz-
ing the ratio of weighted sum of output to the weighted
sum of inputs:
max J0 ¼
P3
r ryr0
P3
i¼1 vixi0
s:t: ¼
P3
r ryrj
P3
i¼1 vixij
1 j ¼ 1, 2, . . . 36
r, vi 0 i ¼ 1, . . . 3, r 1, . . . 3:
ð14Þ
In model (14), the efficiency of DMUo is o and ur
and vi are the factor weights. However, for computa-
tional convenience, the fractional programming model
(14) is re-expressed in linear program (LP) form as
follows:
max J0 ¼
X
3
r¼1
ryr0,
s:t: ¼
X
3
r¼1
ryrj
X
3
i¼1
vjxij 0,
vixi0 ¼ 1,
r, vi i ¼ 1, . . . 3, r ¼ 1, . . . 3,
ð15Þ
where is a non-Archimedean infinitesimal introduced
to insure that all the factor weights have positive values
in the solution. The model (16) evaluates the relative
efficiencies of 36 scenarios ( j ¼ 1, . . . , 36), respectively,
by minimizing inputs (average lead-time of demands,
average waiting time of demands, and the number of
operators (in a working day) when inputs are constant.
The dual of LP model for input-oriented CCR is given
as follows:
e0 ¼ min ,
s:t: xi0
X
36
j¼1
jxrj, i ¼ 1, . . . 3,
yr0
X
36
j¼1
jyrj, r ¼ 1, . . . 3,
j 0:
ð16Þ
The output-oriented CCR model is given as
follows:
e0 ¼ max ,
s:t: xi0
X
36
j¼1
jxij, i ¼ 1, . . . 3,
yr0
X
36
j¼1
jyrj, r ¼ 1, . . . 3,
j 0:
ð17Þ
If
P
j ¼ 1 ( j ¼ 1, . . . , 36) is added to model (16),
the BCC model is obtained which is input oriented and
its return to scale is variable:
e0 ¼ min ,
s:t: xi0
X
36
j¼1
jxij, i ¼ 1, . . . 3,
yr0
X
36
j¼1
jyrj, r ¼ 1, . . . 3,
X
36
j¼1
j ¼ 1,
j 0:
ð18Þ
The output-oriented BCC model is shown in
Equation (18).
e0 ¼ max ,
s:t: xi0
X
36
j¼1
jxij, i ¼ 1, . . . 3,
yr0
X
36
j¼1
jyrj, r ¼ 1, . . . 3,
X
36
j¼1
j ¼ 1,
j 0:
ð19Þ
However, the LP model (16) does not allow for
ranking of efficient units as it assigns a common index
of one to all the efficient scenarios in the data set.
Therefore, the dual model (16) was modified by
Andersen and Petersen for DEA-based ranking

purposes, as follows (Andersen and Petersen 1993):
e0 ¼ min ,
s:t: xi0
X
36
j¼1, j 06¼1
jxij, i ¼ 1, . . . 3,
yr0
X
36
j¼1, j 06¼1
jyrj, r ¼ 1, . . . 3,
j 0:
ð20Þ
Model (20), which excludes DMUo, is under evalua-
tion from the input–output constraints so that the
efficient units are assigned an index of greater than one
and the index for inefficient units is identical to that of
model (3). Model (20) is used to determine the DEA
ranking in this article. An insufficient number of
scenarios for a DEA model would tend to rate all
DMUs 100% efficient, because of an inadequate
number of degrees of freedom. A proper scenario
number is required for identifying a true performance
frontier. A rule of thumb for maintaining an adequate
number of degrees of freedom when using DEA is to
obtain at least two DMUs for each input or output
measure. The results from solving the DEA would
generate those performance frontiers that then become
the final candidate designs.
4.1.2. Principle component analysis
The objective of PCA is to identify a new set of
variables such that each new variable, called a princi-
pal component, is a linear combination of origi-
nal variables. Second, the first new variable y1
accounts for the maximum variance in the sam-
ple data and so on. Third, the new variables
(principal components) are uncorrelated. PCA is per-
formed by identifying the eigenstructure of the covari-
ance or the singular value decomposition of the
original data.
Here, the former approach will be discussed. It is
assumed that there are 9 variables (indexes) and 36
DMUs, and djir ¼ yrj/xij (i ¼ 1, . . . ,3; r ¼ l, . . . ,3) repre-
sents the ratios of individual output (average operator/
machine utilization and numbers of completed parts
per year) to individual input (average lead-time of
demands, average waiting time of demands, and the
number of the operators (in a working day)) for each
DMUj ( j ¼ l, . . . , 36). Obviously, the bigger the djir, the
better the performance of DMUj in terms of the rth
output and the ith input. Now let djk ¼ djir, where
k ¼ 1, . . . , 9 and 9 ¼ 3 3. We need to find some
weights that combine those nine individual ratios of
di for DMUj. Consider the following 36 9
data matrix composed by djk: D ¼ (dl, . . . , d5)369
with each row representing nine individual ratios
of di for each DMU and each column representing
a specific output/input ratio. That is,
dk ¼ (dk1, . . . , dk36)T
. The PCA is employed here to
find new independent measures (principal components)
which are, respectively, different linear combinations
of dl, . . . , d9 so that the principal components can be
combined by their eigenvalues to obtain a weighted
measure of djk. The PCA process of D is carried out as
follows:
Step 1: Calculate the sample mean vector d
and
covariance matrix S.
Step 2: Calculate the sample correlation matrix R.
Step 3: Solve the following equation.
jr lpj ¼ 0 ð21Þ
We obtain the ordered p characteristic roots
(eigenvalues) 1 2 9 with
P
j ¼ 9
( j ¼ 1, . . . , 9) and the related p characteristic vectors
(eigenvectors) (lm1, lm2 . . . , lm9) (m ¼ 1, . . . , 9). Those
characteristic vectors compose the principal compo-
nents Yi. The components in eigenvectors are, respec-
tively, the coefficients in each corresponding Yi:
Ym
X
p
j¼1
lmj ^
xij for m ¼ 1, 2, . . . 9 and i ¼ 1, 2, . . . , 36:
ð22Þ
Step 4: Calculate the weights (wi) of the principal
components and PCA scores (zi) of each DMU
(i ¼ 1, . . . , 36). Furthermore, the z vector (z1, . . . , z9)
where zj shows the score of jth DMUs is given by
Zi ¼
X
9
j1
wiYj i ¼ 1, 2, . . . , 36: ð23Þ
The DEA and PCA methods are applied to the
data set of 36 DMUs. The DEA results show that 12
out of the 36 DMUs are relatively efficient. However,
exact ranking cannot be obtained for these DMUs. In
order to improve the discriminating power of DEA, the
Andersen and Petersen (1993) model was utilized.
Also, PCA rankings of 36 DMUs with respect to 9
indicators (output/input) were obtained. DEA effi-
ciency scores and PCA scores together with rankings of
DMUs are shown in Table 6.
70 A. Azadeh et al.

4.1.3. Correlation analysis
Table 7 shows a correlation between HGTS and other
methods, namely, DEA and principal component
analysis (PCA). Table 7 also reports the results of
nonparametric statistical tests of the relationship
between the stated techniques which result in the
rejection of H0 at 0.01 levels.There is a high correlation
between HGTS and TOPSIS. Also, correlation
between HGTS and PCA is very high (0.809),
which shows that HGTS results are reasonable.
Spearman’s rho correlations comparing these methods
imply that results of all methods except DEA are
not statistically different. Thus, there is a direct
relationship between HGTS, DEA, PCA, and TOPSIS
in terms of data sets generated by computer simulation
with respect to the 36 scenarios. Particularly, the
Spearman test statistic rs 4 0.75 indicates a strong
direct relationship. Also, we apply analysis of variance
(ANOVA) to show if the proposed approach pro-
duces greater efficiency scores than DEA, PCA,
and TOPSIS.
4.1.4. Analysis of variance
ANOVA was used to evaluate the effects of the
optimum operator assignment in CMS model.
DMU’s efficiencies were considered for HGTS model
in comparison with DEA, PCA, and TOPSIS methods.
First, it was tested whether efficiencies have the same
behavior in HGTS (1), DEA (2), PCA (3), and
TOPSIS (4) models. Furthermore, it is tested whether
the null hypothesis H0: 1 ¼ 2 ¼ 3 ¼ 4 was to be
accepted. It was concluded that the four treatments
differ at ¼ 0.05. Furthermore, the least significant
difference (LSD) method is used to compare the pairs
of treatment means 1, 2, 3 and 4. That is H0:
i ¼ j for all i 6¼ j . The results of LSD revealed that at
¼ 0.05, 1 4 2, 3, 4 and 2 4 3, 4 also 3 ¼ 4
and hence treatment 1 (HGTS) produces a significantly
greater efficiency than other treatments. The advan-
tages of the HGTS model with respect to efficiencies
are shown in Tables 8 and 9.
Table 6. Results of DEA and PCA for 36 labor-assignment
scenarios.
Scenario
DEA
score DEA rank Zpca
PCA
rank
Sen1-1 0.88 25 0.137262 14
Sen1-2 0.87 27 1.93488 33
Sen1-3 0.97 16 0.42223 18
Sen2-1 0.78 35 1.59225 30
Sen2-2 0.84 30 1.40223 28
Sen2-3 0.92 18 1.22056 26
Sen3-1 1.03 11 1.131111 10
Sen3-2 1.32 3 1.717474 7
Sen3-3 0.87 26 2.13083 35
Sen4-1 1.26 4 2.613937 4
Sen4-2 0.86 28 2.79368 36
Sen4-3 0.88 24 1.77313 32
Sen5-1 0.98 14 0.80638 12
Sen5-2 0.93 17 0.19444 17
Sen5-3 0.84 29 2.01194 34
Sen6-1 1.01 12 0.51764 13
Sen6-2 0.88 23 0.58377 21
Sen6-3 0.79 34 1.52258 29
Sen7-1 0.89 22 0.73232 23
Sen7-2 1.10 8 1.36003 9
Sen7-3 0.91 21 1.62194 31
Sen8-1 1.16 6 2.035429 6
Sen8-2 0.92 19 0.46171 19
Sen8-3 0.82 32 1.33913 27
Sen9-1 1.12 7 3.171607 2
Sen9-2 1.10 9 0.946035 11
Sen9-3 0.69 36 0.7505 24
Sen10-1 1.17 5 2.964201 3
Sen10-2 0.97 15 0.50993 20
Sen10-3 0.82 33 1.18616 25
Sen11-1 0.99 13 1.652479 8
Sen11-2 1.08 10 0.059721 15
Sen11-3 0.83 31 0.7177 22
Sen12-1 1.62 1 3.735966 1
Sen12-2 1.42 2 2.161883 5
Sen12-3 0.91 20 0.10925 16
Table 8. ANOVA results of HGTS versus other methods.
Source DF
Sum of
squares
Mean
square F p-value
Block 3 243.972 81.324 112.08 0.000
Treatment 35 80.65 2.3044 3.18 0.000
Error 105 76.18 0.7256
Total 143 400.8147
Table 7. Non-parametric Spearman (rs) correlations
analysis.
HGTS DEA PCA
TOPSIS 0.987 0.574 0.833
HGTS 0.542 0.809
DEA 0.728

4.1.5. Qualitative comparison
We have proved that the proposed approach of this
study provides good correlation with other robust
multivariate methods. We have also shown that HGTS
provides higher efficiency scores than previous meth-
ods through ANOVA and LSD. A comparative study
between HGTS and other methods is presented in
Table 10. All methods can solve problems with
multiple inputs and outputs, but HGTS can locate
the best DMU with both specified and unspecified
priori weights. Clearly, the hybrid model is capable of
solving decision-making models with great flexibility
and it consequently demonstrates efficient ranking of
alternatives.
5. Conclusion
This article presented a decision-making approach
based on a HGTS for determining the most efficient
number of operators and the efficient measurement of
operator assignment in CMS. The objective was to
determine the labor assignment in a CMS environment
with the optimum performance. We used HGTS for
obtaining near optimum ranking of the alternative in
accordance with fitness function. Also, this approach
was performed by employing the number of operators,
average lead time of demand, average waiting time of
demand, number of completed parts, operator utiliza-
tion, and average machine utilization as attributes.
Entropy method was used to determine the weight of
attributes. Furthermore, values of the attributes were
procured by means of computer simulation. The
unique feature of this model is demonstration of
efficient ranks of alternatives by reducing the distance
between neighborhood alternatives. The superiority
and advantages of the proposed HGTS were shown
through a comparative study composed by TOPSIS,
DEA, and PCA.
HGTS used was introduced as a powerful method
for ranking the scenarios in the operator assignment
problem based on the attributes discussed in this
article. Also, the TOPSIS approaches verified our
findings. But TOPSIS is not able to present efficient
ranking of scenarios for complicated problems.
Furthermore, the HGTS approach is capable of
ranking the alternatives by near optimum fitness
function and it determines the best solution with
minimum distance.
Table 9. Multiple comparisons between HGTS and other methods.
(I) VAR00001 (J) VAR00001
Mean difference (IJ)
Lower bound
Standard error
Upper bound
Significance
Lower bound
95% confidence interval
Upper bound Lower bound
LSD HGTS DEA 2.40335(*) 0.24957 0.000 1.9099 2.8968
PCA 3.38752(*) 0.24957 0.000 2.8941 3.8809
TOPSIS 2.89899(*) 0.24957 0.000 2.4056 3.3924
DEA HGTS 2.40335(*) 0.24957 0.000 2.8968 1.9099
PCA 0.98417(*) 0.24957 0.000 0.4907 1.4776
TOPSIS 0.49564(*) 0.24957 0.049 0.0022 0.9891
PCA HGTS 3.38752(*) 0.24957 0.000 3.8809 2.8941
DEA 0.98417(*) 0.24957 0.000 1.4776 0.4907
TOPSIS 0.48853 0.24957 0.052 0.9820 0.0049
TOPSIS HGTS 2.89899(*) 0.24957 0.000 3.3924 2.4056
DEA 0.49564(*) 0.24957 0.049 0.9891 0.0022
PCA 0.48853 0.24957 0.052 0.0049 0.9820
Note: *The mean difference is significant at the 0.05 level.
Table 10. Distinct features of HGTS versus other methods.
Multiple
inputs
Multiple
outputs
No need for
prior assignment
of weights
Ranking
capability
Specification of
weights for
each indicator
Flexibility on
the basis of
assigning weights
TOPSIS
p p
–
p p
–
HGTS
p p p p p p
PCA
p p p p
– –
DEA
p p p p
– –
72 A. Azadeh et al.

The decision matrix is introduced for selecting the
appropriate scenario for the operator assignment in
CMS. The weighted coefficients are obtained for every
attribute by making use of the entropy method. The
decision matrix and weighted coefficients are taken as
the inputs for ordinary TOPSIS. These models list
candidate scenarios from the best to the worst, taking
into account all scenario selection criteria including
attributes. Methods that determine both the score and
the rank of each candidate scenario may be preferred
over methods that provide only the ranks of scenarios.
The score option can provide better insight for
designers and it takes into account both the differences
and similarities of the candidate scenarios. HGTS can
be considered an efficient tool to enhance the accuracy
of the final decision in designing a CMS system, as we
have shown in this article.
Nomenclature
Ci the relative closeness of ith candidate
material to the ideal solutions
Ej the entropy value for jth attribute
J the set of decision attributes
k constant of the entropy equation
K set of benefit criteria
K0
set of cost criteria
m the number of scenarios
n the number of criteria
nij an element of the normalized decision
matrix
pij an element of the decision matrix in the
normalized mode for entropy method
rij an element of the decision matrix
rþ
j the best value of jth attribute
r
j the worst value of jth attribute
Sþ
i distance of design to the ideal solution for
the ith candidate material
S
i distance of design from the negative ideal
solution for the ith candidate
s the number of outputs
V weighted normalized decision matrix
Vij an element of the weighted normalized
decision matrix
Vþ
j ideal solution for jth attribute
V
j negative ideal solution for jth attribute
vi the weight given to input i
w0
j the weight coefficient of jth attribute
wj balanced weight coefficient of jth attribute
X the vector of optimization variables
Xj jth attribute in the decision matrix
Xij the amount of input i produced by unit j
Yrj the amount of output r produced by unit j
r The weight given to output r
j the priority of jth attribute comparing with
others
jo the efficiency score of DMU0
a non-Archimedean infinitesimal
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