TOP 1%
E: Select the best material
2.2.3 Working principle
All MCDM methods share some similar working principles upto certain extent, which are as follows:
1.Selection of Criteria:
• All noted criteria must be in correlation to the alternatives.
• The criteria are to be well-prepared along with the decision.
• The criteria must have some relevance either equally or alike.
• The criteria must not be dependent on each other in any sort of way.
2.Selection of Alternatives:
• The alternatives which have been selected must be real in nature.
• The alternatives which have been selected must be available.
3. Selection of method to provide weightage to the criteria:
• Outranking Method –An outranking relation is to be built using a series of pairwise assessments of the alternatives.
• Compensatory Method–Here, strengths and capabilities are embraced over the weakness.
2.2.4 Flowchart for decision making SeeFigure 1.
2.3 Cross entropy (CE) method
In problems related to MCDM technique, the hardest job is to accurately assign weights to the various criteria with respect the ranked alternatives. Therefore, Cross Entropy Methods is often used to assign weights to the criteria. The cross entropy methods is nothing else but a generic form of a well-known Monte Carlo simulation that is used in complex estimation and optimization problems for error minimiza- tion. Y. R. Rubinstien was the first to suggest this approach in 1999 by extending his previous work done in 1997.
Comparison of Cross-Entropy Based MCDM Approach for Selection of Material in Sugar… DOI: http://dx.doi.org/10.5772/intechopen.98242
2.3.1 Algorithms for cross entropy method
Step 1: Feature weightβij is calculated forith alternative and jth criterion as
βij ¼ aij
Pm
i¼1a2ij, 1ð ≤i≤m, 1≤j≤nÞ Step 2: The output entropyεjof the jthfactor
εj ¼ κXm
i¼1
βijlnβij
, 1ð ≤j≤nÞ
κ¼ 1 lnm
Step 3: Calculation of variation coefficient of jthfactorξj ξj ¼1 εj
Step 4: Calculation of weight of the entropywj
wj ¼ ξj Pn
j¼1ξj
Figure 1.
Flowchart for decision making.
6
Advances in Decision Making
2.4 Application of CE based MCDM techniques in engineering problem Cross Entropy is an important method for determining the weights of the criteria. The penalty for selecting a non-best alternative over the best is less when criteria are weighted using the Cross Entropy method. In the year 1997, Y. R.
Rubinstein first developed an adaptive variance minimization algorithm for esti- mating probabilities of rare events for stochastic networks which was later in the year 1999 was modified for solving combinatorial optimization problems. Then later the Cross Entropy method was used along with the MCDM problems for minimizing the penalty for not choosing the best alternative.
A lot of researches have been conducted where Cross Entropy method is used along with the MCDM method for decision making. Some of the literatures are reviewed and presented. In the year 2006, ZOU Zhi-honget al. [5] applied CE method to determine the weightage of different criteria for evaluating water quality in a fuzzy environment. Wei Liu and Jin Cui [6], applied CE method along with MCDM model for evaluation of sustainable development of China’s sport. Farhad Hosseinzadeh Lotfi and Reza Fallahnejad [8], proposed a method where entropy method can be used for for weighting different criteria of non-deterministic data such as interval valued data. Chia-Chang Hung and Liang-Hsuan Chen [7] devel- oped a fuzzy TOPSIS decision model where weights of the criteria are calculated with the entropy method and the alternative are represented by intuitionistic fuzzy sets. In the year 2010, Yuguo Qiet al.[9] proposed a model where evaluation of power network structure is done by entropy based MCDM method under fuzzy environment. This method is a combination of both subjectivity and objectivity, and provides good platform for quantitative as well as qualitative analysis. Kshitij Dashoreet al. [11] compared the results obtained from different MCDM techniques where the weights of the criteria are evaluated using CE method. The authors concluded that the same best alternative is obtained from TOPSIS, SAW and WPM methods.
2.5 Recent work of CE based MCDM
Some of the recent CE based MCDM works that have been reviewed are also presented in this chapter. In the year 2015, Anhai Liet al. [13] in their chapter applied entropy based MCDM methodsfor optimal selection of cutting tool material.
Harish Garget al. proposed a CE based Multi-Attribute Group Decision Making (MAGDM). The model thus proposed gives a useful way for dealing fuzzy MAGDM within attribute weights efficiently and effectively. Zheng-peng Tianet al. [15], developed a CE based decision making model to deal with interval valued
neutrosophic sets. In the year 2016, Elham Ebrahimiet al. [16] compared the result obtained from fuzzy COPRAS and CE-COPRAS to evaluate the customer-company relationship. Javier Martínez-Gómezet al. [18] developed a MCDM model which includes compromised weighting method composes of Analytical Hierachy Process and Entropy method. The authors successfully applied CE- based MCDM method for material selection.
3. Different CE-MCDM techniques
From a set of alternatives, best quantitative solution is evaluated using ranking solution and is provided by MCDM process. In this research work, cross entropy method is applied due to the reason that is highly reliable for measuring information Comparison of Cross-Entropy Based MCDM Approach for Selection of Material in Sugar… DOI: http://dx.doi.org/10.5772/intechopen.98242
and deliver good accuracy while evaluating the weights of the feature attribute.
MCDM problem can thus be expressed as a matrix:
C1 C2 C3 Cn
M¼ A1
A2
A3
⋮ Am
a11 a12 a13 ⋯ a1n
a21 a22 a23 ⋯ a2n
a31 a32 a33 ⋯ a3n
⋮ ⋮ ⋮ ⋱ ⋮
am1 am2 am3 ⋯ amn
2 6 6 6 6 6 6 6 6 6 6 4
3 7 7 7 7 7 7 7 7 7 7 5
W ¼½w1 w2 w3 ⋯ wn
Here, A1, A2, A3 … … Amare the alternatives which are available and supposed to be ranked by decision maker C1, C2, C3 … … Cnare the criteria which will govern ranking of the alternatives.aijshows the performance of alternativeAion the basis ofCj andwj is the weight of the criterion.
3.1 The complex proportional assessment (COPRAS) method
In 1994, Zavadskas and Kaklauskas presented the COPRAS method which is a reference ranking method for ranking different alternatives [28]. Alternative’s per- formance is primarily considered in COPRAS method with respect to various criteria. Therefore, the method aims to select the finest decision considering the ideal-best as well as the ideal-worst solutions. Steps used to rank those alternatives by using COPRAS method are as follows:
Step 1: Calculation of normalized decision matrixnij : nij ¼ aij
Pm i¼1aij
Step 2: Calculation of weighted normalize decision matrixWij : Wij ¼nij∗wj
Wherewj is the weightage of criterionCj. Step 3: Calculation ofSþandS :
SþandS are the summation of weighted normalized value that are evaluated for benefit criteria as well as non-benefit criteria.
Sþi ¼Xn
j¼1
Wij:ði¼1, 2, 3…mÞ
WhereWijis the weighted normalize elements for all the benefit criteria Si ¼Xn
j¼1
Wij:ði¼1, 2, 3…mÞ
WhereWijis the weighted normalize elements for all the non-benefit criteria
8
Advances in Decision Making
Step 4: Evaluating relative weightage of each alternative Qi: Qi¼Sþi þ
Pm i¼1Si Si Pm
i¼1 1
Si
Step 5: Determining the priority order (Pri):
Pri ¼ Qi max Qi
Maximum value ofPriis given maximum priority and ranked 1, second largest value ofPri is given second priority and ranked 2 and so on.
3.2 The MOORA method
MOORA (Multi Objective Optimization on the Basis of Ratio Analysis) was developed by Brauers in 2004 for solving different complex and conflicting decision matrix. Performance measures of alternatives with respect to different criteria are represented by the decision matrix of MOORA. Steps governing the ranking of different alternatives by MOORA methods are:
Step1: Calculation of normalized decision matrixnij : nij ¼ aij
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm
i¼1a2ij q
Step 2: Calculation of weighted normalize decision matrixWij : Wij ¼wjnij
Step 3: Evaluating of Priorities (Qi):
Qi¼Xn
j¼1
Wij
Priorities is the difference between the sum of benefit criteria and non-benefit criteria.