Comparison of Cross-Entropy Based MCDM Approach for Selection of Material in Sugar Industry

Abstract

One of the major problems faced in engineering is the selection of the material which is most suitable for the product. Material selection from a large number with diverse mechanical, physical and chemical properties and choosing the best material which is the most satisfying for making the job is a very complex process. Material selection is important as it determines durability, reliability and cost of the product. Selection of suitable material which gives maximum performance with minimum cost is often observed to be a multi-criterion decision-making (MCDM) problem with different objectives. This chapter presents an integrated approach to select the suitable material to be used as base of induction cookware which can give maximum performance with minimum cost. In the integrated approach the weights of the criteria are computed using the cross-entropy method and ranking of the alternatives is done using the different MCDM methods. The methods are further illustrated with an example and the result obtained from different cross-entropy MCDM methods are compared for finding the most suitable method for serving the purpose.

1. Introduction

In this present techno-economic scenario material selection poses one of the major challenges in industries. Selection of the improper material adversely affects the reputation of an organization and also reduces the profitability. Selection of the proper material is a step in the process of product design where it aims in increasing the performance with minimum cost. Material selection is also important from the perspective of sustainability of industries [1, 2].

In the recent years, traditional materials were replaced by more advance materials due to their mechanical and chemical properties. The materials were used for manufacturing from complex geometry to long lasting products. With the large number of readily available materials the process of material selection is done with the help of multi-criteria decision-making (MCDM) process.

Moreover, MCDM is the study concerned with optimal decision making and the modelling of deterministic systems. Its focus in the interdisciplinary field of application, clutching a wide range of quantitative techniques. Whereas, Industrial Engineering is about the enhancement, refinement, and installation of integrated systems of personnel, material, and equipment. Industrial engineering is also about processing of the information. MCDM integrated industrial engineering provide a rational approach to engineering and managerial problem solving through deliberate application of scientific methods. In the MCDM model, the set of available materials are called alternatives among which the best material is selected on the basis of the certain properties called criteria.

In practical scenario MCDM addresses the performance of different alternatives on the basis of information and resource limitations of a company or industry or organization, working towards the establishing of beneficial policies. The function of the decision maker is to guide the engineers, managers and administration by processing the information available in the industries.

1.1 Entropy based Multi-Criteria Decision Making (MCDM)

In today’s hi-tech engineering world, MCDM have evolved as one of the most important tools of decision making in a complex situation. MCDM also help in taking decisions in a situation where there is little or no chance of any altercation. In the present socio-economic world where a decision of selecting the best is effected by a large number of criteria, MCDM plays a very important role in such aspects. Decision making is done based on various criteria which might be important equally or not. From the last statement it can be said that every criterion are weighted which help the decision maker in taking the decision. One of the major problems faced in MCDM problem is the assigning of weights to the criterion. Some of the different techniques of assigning weights to the criteria are 5Ws and H method, fuzzy method, cross-entropy method etc. Cook [3] and Vesna Čančer [4] used 5Ws and H technique; Kumar and Gag , Amiri et al., Kemal Vatansever and Yiğit Kazançoğlu , Keshavarz Ghorabaee et al., used fuzzy for determining the weights of different criteria. ZOU Zhi-hong et al. [5], Wei Liu and Jin Cui [6], Chia-Chang Hung and Liang-Hsuan Chen [7], Farhad Hosseinzadeh Lotfi and Reza Fallahnejad [8], Yuguo Qi et al. [9], Peiyue et al. [10], Kshitij Dashore et al. [11], Deepa Joshi and Sanjay Kumar [12], Anhai Li et al. [13], Harish Garg et al. [14], Zhang-peng Tian et al. [15], Elham Ebrahimi et al. [16], Harish Garg [17], Javier Martínez-Gómez et al. [18] used cross-entropy method for determining the weightage of the criteria for solving MCDM problem.

1.2 Application of MCDM in material selection

With increasing choice of materials and large number of manufacturing process available to the designers, the selection of an optimal material have become more complex and more challenging than before [19]. In order to address the issue of material selection researchers like Ashby proposed MCDM as one of the best tools. Ashby et al. [20] have identified three material selection strategies which are (a) free searching based on quantitative analysis, (b) checklist/questionnaire based on expertise capture, and (c) inductive reasoning and analog procedure. A large number of literature exist for selecting the suitable material for a product. Based on the Ashby work a lot of researches is carried out in this respect. Out of which some are reviewed. Milani et al. [21] studied the ways in which different criteria transformation techniques effects the result in TOPSIS method for selecting gear material. In the year 2006, R.V. Rao and J.P. Davim [22] developed a combined AHP-TOPSIS model for selecting material to be used in non-heat-treatable cover material. Shanian A and Savadogo O. [23, 24], successfully applied TOPSIS in selecting material for a particular product design with maximum performance and minimum cost. Again in the same year Shanian A and Savadogo O. implemented the ELECTRE-I method for selecting material to be used in Bipolar Plates for Polymer Electrolyte Fuel Cells Applications. In this chapter, ELECTRE-I gave the same result with or without negative criterion. In 2008, Sharif Ullah and Harib [25] proposed an intelligent method for selecting material where informations regarding the design configurations, working conditions and design-relevant information were not known. According to Karande and Chakraborty [26], material selection is also an important factor for a product to strive in the competition in market because improper material selection may result in failure to fulfil customer and manufacturer requirements. In 2013a, 2013b, Shankar Chakraborty and Prasenjit Chatterjee [27, 28], observed that in case of material selection the ranking performance of VIKOR is far better than the TOPSIS and PROMETHEE methods. In the chapter the authors also concluded that the best and the worst is solely dependent on the weights of the criteria. They further added that time for selecting the most suited material could be reduced by identifying the criterion with maximum weight. In their second chapter COPRAS and ARAS methods were termed as the most appropriate method for gear material selection as the result obtained from both the method are quite similar and also both the techniques are fool proof techniques.

1.3 Motivation

Decision making theory plays a vital role where decisions have to be taken in cases the performance parameter differs from each other by a very small margin. There are different decision making methods which takes in account different mathematical concepts such as the best alternative is one which in a best way can compromise the conflicting scenario (VIKOR), the alternative which is farthest from the non-benefit criteria is the best alternative (MOORA), the alternative having the highest relative weightage for the benefit criteria is the best alternative (COPRAS) and so on. Hence when a decision is taken with different MCDM methods, the result may differ for each method. Alternative which is best by one method may not be the best by some other method. In problems like material selection each wrong decision is associated with some penalty. For such cases the weights of the criteria are so calculated that it reduces the penalty for not choosing the actual best alternative instead of choosing the predicted best alternative. Moreover, selecting materials in sugar industry is well known MCDM problem. But, the literatures fail to answer which method to apply for material selection and the risk involved for choosing a method. The literatures also do not explain the accuracy degree to which two or more selection methods would agree to the same decision. The main aim of this chapter is find a material suitable for manufacturing equipment in the sugar industry by different MCDM methods and also to find the degree to which all the methods would agree to the decision.

1.4 Novelties

Lot of researchers have worked in the field of MCDM and developed novel approaches for precisely selecting alternatives. Some of the novelties developed in this chapter are as follows:

1. Application of CE integrated MCDM approaches are used for selecting material in sugar industry.

2. Comparison between the results obtained from CE based MCDM techniques.

3. Based on the comparison a predictive model is developed that can forecast the result by different CE integrated MCDM approaches.

1.5 Structure of the chapter

The chapter is organized into 7 sections. The first section is introduction which describes the importance of the material selection and the comprehensive literature related to the topic. Section 2 describes the preliminary concept of the methodologies used for the study. Section 3 summarises the steps of the MCDM methods and section 4 describes the case study that is considered for the present study. Section 5 is the comparative study of the result obtained from various MCDM methods. Section 6 discusses the summary of the findings and section 7 concludes the chapter.

2. Preliminaries

2.1 Multi-criteria decision making (MCDM)

Definitions 1: Decision can be defined as the rationale conclusions after evaluating the circumstances.

Definition 2: In perspective of cognitive science, selection of a particular action or a certain belief that is a subset of the alternatives is knows as Decision Making.

The process of selection of an action that belong to the subset of the alternatives is done by considering all criteria’s which can either alter the judgement of same or dissimilar extent is called as Multi-Criteria Decision Making. In MCDM technique, the existing possibilities are the subset of alternatives or else, these are the choices out of which the finest possibility is to be selected by analysing various factors which could influence the outcome. All the alternatives are ranked in ascending order beginning from the finest to the worst and on the basis of factors which can influence the conclusions, these factors are known as criteria. Criteria can be defined as the factors which have direct influence on decision. In MCDM technique, there are number of criteria and which each of them effects the judgement and conclusion either to the similar or dissimilar extent.

2.2 Steps involved in decision making

2.2.1 GOFER method for taking decision

During 1980s, psychologist Leon Munn and some of his colleagues developed a technique called GOFER specifically for decision making [29]. The acronym for GOFER stands for a 5 decision making steps.

G: Goal - Selection of material

O: Options - Set of readily available materials

F: Facts - Performance with respect to the criteria

E: Effect - How the material selection are effected?

R: Review - Reviewing distinct facts and data associated to the options.

2.2.2 DECIDE method for taking decision

In the year 2008, another decision making technique was developed by Kristina Guo known as DECIDE [30]. This method consist of six parts namely:

D: Define the problem i.e. selection of material.

E: All the criteria and factors are established on basis of which the decision is made i.e. enlisting the criteria.

C: Enlisting the various available materials.

I: Discard the materials having high cost to benefit ratio.

D: A plan is developed for selecting the finest alternative.

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

See Figure 1.

Figure 1.

Flowchart for decision making.

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 minimization. Y. R. Rubinstien was the first to suggest this approach in 1999 by extending his previous work done in 1997.

2.3.1 Algorithms for cross entropy method

Step 1: Feature weight ${\beta }_{\mathit{ij}}$ is calculated for $i^{\mathit{th}}$ alternative and $j^{\mathit{th}}$ criterion as

$${\beta }_{\mathit{ij}}=\frac{a_{\mathit{ij}}}{{\sum }_{i=1}^m{a}_{\mathit{ij}}^2},\left(1\le \mathrm{i}\le \mathrm{m},1\le \mathrm{j}\le \mathrm{n}\right)$$

Step 2: The output entropy ε_j of the j^th factor

$${\epsilon }_j=-\kappa \sum _{i=1}^m\left({\beta }_{\mathit{ij}}{\mathit{ln\beta }}_{\mathit{ij}}\right),\left(1\le \mathrm{j}\le \mathrm{n}\right)$$

$$\kappa =\frac{1}{lnm}$$

Step 3: Calculation of variation coefficient of j^th factor ${\xi }_j$

$${\xi }_j=\left|1-{\epsilon }_j\right|$$

Step 4: Calculation of weight of the entropy $w_j$

$$w_j=\frac{{\xi }_j}{{\sum }_{j=1}^n{\xi }_j}$$

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 estimating 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-hong et 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] developed 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 Qi et 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 Dashore et 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 Li et al. [13] in their chapter applied entropy based MCDM methodsfor optimal selection of cutting tool material. Harish Garg et 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 Tian et al. [15], developed a CE based decision making model to deal with interval valued neutrosophic sets. In the year 2016, Elham Ebrahimi et al. [16] compared the result obtained from fuzzy COPRAS and CE-COPRAS to evaluate the customer-company relationship. Javier Martínez-Gómez et 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 and deliver good accuracy while evaluating the weights of the feature attribute. MCDM problem can thus be expressed as a matrix:

$$\begin{array}{ccccc}{\mathrm{C}}_1& {\mathrm{C}}_2& {\mathrm{C}}_3& & {\mathrm{C}}_n\end{array}$$

$$M=\begin{array}{c}{A}_1\\ A_2\\ A_3\\ ⋮\\ A_m\end{array}\left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \cdots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \cdots & a_{2n}\\ a_{31}& a_{32}& a_{33}& \cdots & a_{3n}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& a_{m3}& \cdots & a_{\mathit{mn}}\end{array}\right]$$

$$W=\left[\begin{array}{ccc}{w}_1& w_2& \begin{array}{ccc}{w}_3& \cdots & w_n\end{array}\end{array}\right]$$

Here, A₁ , A₂ , A₃ … … A_m are the alternatives which are available and supposed to be ranked by decision maker C₁ , C₂ , C₃ … … C_n are the criteria which will govern ranking of the alternatives. $a_{\mathit{ij}}$ shows the performance of alternative $A_i$ on the basis of $C_j$ and $w_j$ 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 performance 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 matrix $\left(n_{\mathit{ij}}\right)$:

$$n_{\mathit{ij}}=\frac{a_{\mathit{ij}}}{{\sum }_{i=1}^m{a}_{\mathit{ij}}}$$

Step 2: Calculation of weighted normalize decision matrix $\left(W_{\mathit{ij}}\right)$:

$$W_{\mathit{ij}}=n_{\mathit{ij}}\ast w_j$$

Where $w_j$ is the weightage of criterion $C_j$.

Step 3: Calculation of $S^{+}$ and $S^{-}$:

$S^{+}$ and $S^{-}$ are the summation of weighted normalized value that are evaluated for benefit criteria as well as non-benefit criteria.

$$S_i^{+}=\sum _{j=1}^n{W}_{\mathit{ij}}.\left(\mathrm{i}=1,2,3\dots \mathrm{m}\right)$$

Where $W_{\mathit{ij}}$ is the weighted normalize elements for all the benefit criteria

$$S_i^{-}=\sum _{j=1}^n{W}_{\mathit{ij}}.\left(\mathrm{i}=1,2,3\dots \mathrm{m}\right)$$

Where $W_{\mathit{ij}}$ is the weighted normalize elements for all the non-benefit criteria

Step 4: Evaluating relative weightage of each alternative Q_i:

$$Q_i=S_i^{+}+\frac{{\sum }_{i=1}^m{S}_i^{-}}{\begin{array}{cc}{S}_i^{-}& {\sum }_{i=1}^m\frac{1}{S_i^{-}}\end{array}}$$

Step 5: Determining the priority order (Pr_i):

$${\mathit{Pr}}_i=\frac{Q_i}{\begin{array}{cc}\mathit{max}& Q_i\end{array}}$$

Maximum value of ${\mathit{Pr}}_i$ is given maximum priority and ranked 1, second largest value of ${\mathit{Pr}}_i$ 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 matrix $\left(n_{\mathit{ij}}\right)$:

$$n_{\mathit{ij}}=\frac{a_{\mathit{ij}}}{\sqrt{{\sum }_{i=1}^m{a}_{\mathit{ij}}^2}}$$

Step 2: Calculation of weighted normalize decision matrix $\left(W_{\mathit{ij}}\right)$:

$$W_{\mathit{ij}}=w_j\times n_{\mathit{ij}}$$

Step 3: Evaluating of Priorities (Q_i):

$$Q_i=\sum _{j=1}^n{W}_{\mathit{ij}}$$

Priorities is the difference between the sum of benefit criteria and non-benefit criteria.

Step 4: Ranking of alternatives:

Maximum value of the variable Q_i is provided the maximum priority and ranked 1, second largest of Q_i is provided the second priority and ranked 2 and so on.

3.3 The VIKOR method

VIKOR method, developed for evaluating decision making problems with conflicting as well as non-commensurable criteria by Serafim Opricovic. This method assumes accepts compromise with conflicting resolution. VIKOR methods ranks various alternatives and evaluates the solution called as compromise which is the closest value to the ideal.

Step 1: Calculation of $f_j^{\ast }\mathit{\text{and}}{f}_j^{\mathrm{\Lambda }}$

$$f_j^{\ast }=\mathit{Min}\left(a_{\mathit{ij}}\right),\left(j=1,2,3,\dots n\right)$$

$$f_j^{\mathrm{\Lambda }}=\mathit{Max}\left(a_{\mathit{ij}}\right),\left(j=1,2,3,\dots n\right)$$

Where $a_{\mathit{ij}}$ stands for elements of decision matrix

Step 2: Calculation of relative matrix $\left(R_{\mathit{ij}}\right)$

$$R_{\mathit{ij}}=\frac{f_j^{\ast }-a_{\mathit{ij}}}{f_j^{\ast }-f_j^{\mathrm{\Lambda }}}$$

Step 3: Calculation of weighted normalized decision matrix $\left(W_{\mathit{ij}}\right)$

$$W_{\mathit{ij}}=R_{\mathit{ij}}\times w_j$$

Step 4: Calculation of ${\gamma }_j$ by the concept of Manhattan distance.

$${\gamma }_j=\sum _{j=1}^n{W}_{\mathit{ij}},\left(i=1,2,3,\dots m\right)$$

Step 5 : Calculation of ${\delta }_j$ by the concept of Chebyshev distance.

$${\delta }_j=\mathit{Max}\left(W_{\mathit{ij}}\right),\left(i=1,2,3\dots m\right)$$

Step 6: Calculation of priority values ${\rho }_j$.

$${\rho }_j=\frac{V\left({\gamma }_j-{\gamma }^{\ast }\right)}{\left({\gamma }^{\mathrm{\Lambda }}-{\gamma }^{\ast }\right)}+\left(1-V\right)\frac{\left({\delta }_j-{\delta }_j^{\ast }\right)}{\left({\delta }_j^{\mathrm{\Lambda }}-{\delta }_j^{\ast }\right)}$$

Where, ${\gamma }^{\ast }=\mathit{min}\left({\gamma }_j\right)$

    ${\gamma }^{\mathrm{\Lambda }}=\mathit{max}\left({\gamma }_j\right)$

    ${\delta }_j^{\ast }=min\left({\delta }_j\right)$

    ${\delta }_j^{\mathrm{\Lambda }}=max\left({\delta }_j\right)$

    $V=\frac{n+1}{2n}$, n is the no. of criterion

Step 7: Ranking of various possible alternatives

According to values of ${\rho }_j$, the alternative values are ranked from ascending order. Here, the smallest is the best alternative and the largest is considered as the worst alternatives.

3.4 The TOPSIS method

Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) was developed in 1981 by Hwang and Yoon. The primary objective of TOPSIS was to determine the finest alternatives by minimizing the positive-ideal solution’s distance and maximizing the negative-ideal solution’s distance [31]. All the various alternative solutions shall be ranked on the basis of their closeness to ideal solution i.e., the closest alternative to the ideal is considered as the best solution whereas, the least close alternative to the ideal is considered as the worst solution. Steps governing the ranking of the alternatives by TOPSIS method are:

Step 1: Calculation of normalized decision matrix (n_ij)

$$n_{\mathit{ij}}=\frac{a_{\mathit{ij}}}{\sqrt{{\sum }_{i=1}^m{a}_{\mathit{ij}}^2}}$$

Step 2: Calculation of weighted normalize decision matrix (W_ij)

$$W_{\mathit{ij}}=w_j\times n_{\mathit{ij}}$$

Where $w_j$ is the weight of the criterion $C^j$_.

Step 3: Calculation of Positive Ideal Solution (P_is) and Negative Ideal Solution (N_is)

$$P_{\mathit{\text{is}}}=\left\{\begin{array}{l}\mathit{Max}\left(W_{\mathit{ij}}\right),\left(j=1,2,3,\dots .n\right)\mathit{\text{for benefit criteria}}\\ \mathit{Min}\left(W_{\mathit{ij}}\right),\left(j=1,2,3,\dots .n\right)\mathit{\text{for non}}-\mathit{\text{benefit criteria}}\end{array}\right.$$

$$N_{\mathit{\text{is}}}=\left\{\begin{array}{l}\mathit{Max}\left(W_{\mathit{ij}}\right),\left(j=1,2,3,\dots .n\right)\mathit{\text{for non}}-\mathit{\text{benefit criteria}}\\ \mathit{Min}\left(W_{\mathit{ij}}\right),\left(j=1,2,3,\dots .n\right)\mathit{\text{for benefit criteria}}\end{array}\right.$$

Step 4: Calculation of separation measures $\left(S_m^{+}\right)$ for P_is and $\left(S_m^{-}\right)$ for N_is

$$S_m^{+}=\sqrt{{\sum }_{j=1}^n{\left(W_{\mathit{ij}}-P_{\mathit{\text{is}}}\right)}^2},\left(1\le i\le m,1\le j\le n\right)$$

$$S_m^{-}=\sqrt{{\sum }_{j=1}^n{\left(W_{\mathit{ij}}-N_{\mathit{\text{is}}}\right)}^2},\left(1\le i\le m,1\le j\le n\right)$$

Separation measures are measured using Euclidean distance method.

Step 5: Calculation of relative closeness to the ideal solution $\left({\mathit{RC}}_{\mathit{\text{is}}}\right)$

$${\mathit{RC}}_{\mathit{\text{is}}}=\frac{S_m^{-}}{S_m^{+}+S_m^{-}}$$

Step 6: Arrangment of the RC_is values in descending order and ranking from the largest value to the smallest value.

3.5 The modified TOPSIS method

The Modified TOPSIS method is a revised version of the TOPSIS model. In the Modified TOPSIS model the Pis and Nis do not depends on the weighted decision matrix. Steps for ranking alternatives by Modified TOPSIS method is as follows:

Step 1: Calculation of normalized decision matrix (n_ij)

$$n_{\mathit{ij}}=\frac{a_{\mathit{ij}}}{\sqrt{{\sum }_{i=1}^m{a}_{\mathit{ij}}^2}}$$

Where a_ij is the performance of value of alternative A_i on the basis of criterion C_j.

Step 2: Calculation of Positive Ideal Solution (P_is) and Negative Ideal Solution (N_is)

$$P_{\mathit{\text{is}}}=\left\{\begin{array}{l}\mathit{Max}\left(n_{\mathit{ij}}\right),\left(j=1,2,3,\dots .n\right)\mathit{\text{for benefit criteria}}\\ \mathit{Min}\left(n_{\mathit{ij}}\right),\left(j=1,2,3,\dots .n\right)\mathit{\text{for non}}-\mathit{\text{benefit criteria}}\end{array}\right.$$

$$N_{\mathit{\text{is}}}=\left\{\begin{array}{l}\mathit{Max}\left(n_{\mathit{ij}}\right),\left(j=1,2,3,\dots .n\right)\mathit{\text{for non}}-\mathit{\text{benefit criteria}}\\ \mathit{Min}\left(n_{\mathit{ij}}\right),\left(j=1,2,3,\dots .n\right)\mathit{\text{for benefit criteria}}\end{array}\right.$$

Step 3: Calculation of separation measures $\left(S_m^{+}\right)$ for P_is and $\left(S_m^{-}\right)$ for N_is

$$S_m^{+}=\sqrt{{\sum }_{j=1}^n{w}_j{\left(n_{\mathit{ij}}-P_{\mathit{\text{is}}}\right)}^2},\left(1\le i\le m\right)$$

$$S_m^{-}=\sqrt{{\sum }_{j=1}^n{w}_j{\left(n_{\mathit{ij}}-N_{\mathit{\text{is}}}\right)}^2},\left(1\le i\le m\right)$$

Separation measures are measured using Euclidean distance method.

Step 4: Calculation of relative closeness to the ideal solution (RC_is)

$${\mathit{RC}}_{\mathit{\text{is}}}=\frac{S_m^{-}}{S_m^{+}+S_m^{-}}$$

Step 5: Arrangment of the RC_is values in descending order and ranking from the largest value to the smallest value.

4. Application

4.1 Material selection in sugar industry

A few researchers and product designer have studied the failure rate in the sugar industrial equipment. From their study they have found that in India, the failure due to corrosion of the equipments’ cost a sum of about US$250 million [32]. A comparison of better corrosion resistance material was done in [33]. [34] has suggested the use anti-corrosive medium such as sulphanilamide, sulphapyridine and sulphathiazole for better performance. The corrosion effect of the sugar-cane juice on the carbon steel roll was studied in [35] along with the effect of austenitic stainless steel on the welded carbon steel roll. In [36], the authors studied the characteristics and corrosion behaviour of high Chromium White Iron. [37] studied the abrasion corrosion test for Iron-Chromium-Carbon shielded metal arc wielding for its used in the sugar industry whereas [38] studied the wear mechanism ploughed by silica in sugar cane roller shell.

4.2 The problem

The problem that mostly faced by the designers is to choose the suitable material for manufacturing equipment in the sugar industry. The different materials and the selection criteria are taken from [39] which are given in the form of Table 1.

Alternatives	Criteria
J4 (A^1)	Yield strength (C^1)	Ultimate tensile strength (C^2)
JSLAUS (A^2)	Percentage of elongation (C^3)	Hardness (C^4)
204Cu (A^3)	Cost (C^5)	Corrosion resistance (C^6)
409M (A^4)	Wear resistance (C^7)
304 (A^5)

Table 1.

List of alternatives and the selection criteria.

The different criteria used for selecting the alternatives are as follows:

1. Yield strength (C¹): Yield strength is the most important criteria for material selection. It is a positive criterion. Yield strength does not allow the equipment to deform plastically.

2. Ultimate tensile strength (C²): It is a measure of material’s toughness. It is a positive criterion.

3. % of Elongation (C³): It is the measure to withstand the operating load. It is a positive criterion.

4. Hardness (C⁴): It is the measure to resist plastic deformation due to the applied force. It is a positive criterion.

5. Cost (C⁵): It is the monetary value for purchase of the material. More the cost of material, lesser the chance to buy the material. Hence, it is a negative criterion.

6. Corrosion resistance (C⁶): It is the measure of the material ability to reduce the binding energy in metals. It is a positive criterion.

7. Wear resistance (C⁷): It is the measure of the material ability to resist the wearing. It is a positive criterion

The value of the properties is listed in Table 2 and it acts as the decision matrix for selection of materials.

Alternatives	Criteria
	C^1	C^2	C^3	C^4	C^5	C^6	C^7
A^1	382	728	48	98	112	0.16	2.75
A^2	420	790	58	97	210	0.31	2.63
A^3	415	795	55	96	120	0.05	2.50
A^4	270	455	32	78	184	0.40	4.00
A^5	256	610	60	86	89	0.01	2.59

Table 2.

Decision matrix.

4.3 Weighting of criteria

Criteria are weighted by Cross Entropy method (Table 3).

Criteria	C^1	C^2	C^3	C^4	C^5	C^6	C^7
Weight	0.09872	0.09669	0.09926	0.08166	0.1261	0.4034	0.09418

Table 3.

Table of weights of the criteria.

4.4 Ranking by COPRAS method

According to COPRAS method alternative 4 i.e. 409M carbon alloy is the best alternative for manufacturing of equipment in sugar industry (Table 4).

Material	$S_i^{+}$	$S_i^{-}$	$Q_i$	${\mathit{Pr}}_i$	Rank
A^1	0.16619	0.0197534	0.19027	70.72	3
A^2	0.23815	0.0370377	0.25098	93.28	2
A^3	0.12303	0.0211644	0.14549	54.08	4
A^4	0.25441	0.0324521	0.26906	100.00	1
A^5	0.09213	0.0156969	0.12242	45.50	5

Table 4.

Table for ranking of alternatives by COPRAS method.

4.5 Ranking by MOORA method

According to the MOORA method alternative 4 i.e. 409M carbon alloy is the best alternative for manufacturing of equipment in sugar industry (Table 5).

Material	$Q_i$	Pri	Rank
A^1	0.292	70.94	3
A^2	0.384	93.25	2
A^3	0.216	52.41	4
A^4	0.412	100.00	1
A^5	0.167	40.65	5

Table 5.

Table for ranking of alternatives by MOORA method.

4.6 Ranking by VIKOR method

According to the VIKOR method alternative 5 i.e. 304 carbon alloy is the best alternative for manufacturing of equipment in sugar industry (Table 6).

Material	${\gamma }_j$	${\delta }_j$	${\rho }_j$	Rank
A^1	0.487	0.155	0.360	3
A^2	0.808	0.310	0.870	5
A^3	0.421	0.097	0.218	2
A^4	0.605	0.403	0.815	4
A^5	0.182	0.099	0.004	1

Table 6.

Table for ranking of alternatives by VIKOR method.

4.7 Ranking by TOPSIS method

According to TOPSIS method alternative 4 i.e. 409M carbon alloy is the best alternative for manufacturing of equipment in sugar industry (Table 7).

Material	$S_m^{+}$	$S_m^{-}$	${{RC}}_{{\text{is}}}$	Rank
A^1	0.183	0.123	0.401	3
A^2	0.084	0.230	0.732	2
A^3	0.266	0.058	0.179	4
A^4	0.052	0.296	0.850	1
A^5	0.297	0.052	0.150	5

Table 7.

Table for ranking of alternatives by TOPSIS method.

4.8 Ranking by modified TOPSIS method

According to Modified TOPSIS method alternative 4 i.e. 409M carbon alloy is the best alternative for manufacturing of equipment in sugar industry (Table 8).

Material	$S_m^{+}$	$S_m^{-}$	${{RC}}_{{\text{is}}}$	Rank
A^1	0.183	0.123	0.599	3
A^2	0.084	0.230	0.268	2
A^3	0.266	0.058	0.821	4
A^4	0.052	0.296	0.150	1
A^5	0.297	0.052	0.850	5

Table 8.

Table for ranking of alternatives by modified TOPSIS method.

5. Comparative analysis

The 5 different MCDM techniques are compared using the Spearman’s rank correlation coefficient (r_s).

5.1 Comparison by r_s

Spearman’s rank correlation coefficient (r_s) is the method of comparing the ranks of alternatives obtained from different test. Using r_s, the similarity between two sets of rankings can be measured. The value of r_s usually lies between -1 and 1, where the value of 1 denotes a perfect match between two rank orderings. Table 10 shows the Spearman’s rank correlation coefficient values when the rankings of the material alternatives as obtained employing all the considered MCDM methods are compared between themselves and also with respect to the rank ordering.

From the table it is observed that the value of r_s varies from -0.9 to 1.0. The value of r_{s for} a MCDM method when compared with itself is always 1. Hence from the Table 10 we can conclude the rank obtained from different MCDM models may or may not be in agreement with each other.

5.2 Comparison by R²

R² is a statistic that will give some information about the goodness of fit of a model. In regression analysis, the coefficient of determination is a statistical measure of how well the regression line approximates the real data points. R² lies in between 0 to 1. If the value R² is 1 then it interprets that the predicted value is exactly equal to the actual value. When the result obtained from different MCDM methods are compared with each other the value of R² is shown in Table 11.

When the results obtained from different MCDM techniques are compared using regression analysis the maximum value of R² obtained is 1 among different combinations of COPRAS, MOORA, TOPSIS and Modified TOPSIS and the least value is 0.81 between VIKOR and COPRAS, MOORA and Modified TOPSIS. From the Table 11 it can be concluded that there is some penalty for choosing a wrong MCDM method for ranking of alternatives. Figures 2–11 are the comparative figure of different MCDM techniques.

Figure 2.

Comparison of result obtained from COPRAS and MOORA.

Figure 3.

Comparison of result obtained from COPRAS and VIKOR.

Figure 4.

Comparison of result obtained from COPRAS and TOPSIS.

Figure 5.

Comparison of result obtained from COPRAS and modified TOPSIS.

Figure 6.

Comparison of result obtained from MOORA and VIKOR.

Figure 7.

Comparison of result obtained from MOORA and TOPSIS.

Figure 8.

Comparison of result obtained from MOORA and modified TOPSIS.

Figure 9.

Comparison of result obtained from VIKOR and TOPSIS.

Figure 10.

Comparison of result obtained from VIKOR and modified TOPSIS.

Figure 11.

Comparison of result obtained from TOPSIS and modified TOPSIS.

6. Discussion

As the weights of criteria is the most important parameter for the decision maker to choose the best alternative from a bunch of alternatives. So, in order to compute the best alternative, different integrated cross entropy based MCDM are implemented. The result obtained from different MCDM techniques is then compared with the existing result [39]. The result thus obtained from VIKOR method matches with the existing literature whereas the result obtained from other methods fail to match. From the Table 9 and Figure 12, it was observed that the 4 out of the 5 Multi-Criteria Decision Making methods except VIKOR gives exactly the same result. Hence, it is validated and it can be conclude that 80% of the time alternative 4 is the best alternative for the given problem. But, result obtained from VIKOR matches with the [39]. Therefore, a need of comparative analysis arose. The different cross entropy based MCDM methods are compared using Spearman’s Rank Correlation Coefficient. From the comparative analysis, the value of r_s is tabulated in the Table 10. The regression coefficient value R² is tabulated in Table 11. From both the table it was found that the result obtained from the VIKOR method strongly disagrees with the result that obtained from the COPRAS, MOORA, TOPSIS and modified TOPSIS. Whereas the result obtained from COPRAS, MOORA, TOPSIS and modified TOPSIS are a perfect match. The reason behind this is that ranking of alternatives totally based on the values of the criteria of the alternatives. If the values of the criteria are changed then there is huge probability of the change in rank of the alternatives.

Material	COPRAS	MOORA	VIKOR	TOPSIS	Modified TOPSIS
A^1	3	3	3	3	3
A^2	2	2	5	2	2
A^3	4	4	2	4	4
A^4	1	1	4	1	1
A^5	5	5	1	5	5

Table 9.

Ranking of alternatives by different MCDM methods.

Figure 12.

Ranking by different MCDM methods.

	COPRAS	MOORA	VIKOR	TOPSIS	Modified TOPSIS
COPRAS	1	1	-0.9	1	1
MOORA		1	-0.9	1	1
VIKOR			1	-0.9	-0.9
TOPSIS				1	1
Modified TOPSIS					1

Table 10.

Spearman’s rank correlation coefficient.

	COPRAS	MOORA	VIKOR	TOPSIS	Modified TOPSIS
COPRAS	1	1	0.81	1	1
MOORA		1	0.81	1	1
VIKOR			1	0.81	0.81
TOPSIS				1	1
Modified TOPSIS					1

Table 11.

Table of value of R².

6.1 Benefits of the present study in industrial information

Benefits of implementing the present study in the industry

• Decisions regarding operational research could be made with confidence.

• The present study can provide opportunities for improvement.

• Archive critical historical data for analysis and reference

• When everyone has unfettered access to the exact and precise predictive model then decision could be taken easily and in short duration which shall act in the favour of the industry.

7. Conclusion

In the present study, an approach to select the best material from a bunch of materials under different criteria is implemented. Although many researches have been conducted in this field, yet no literature exist which can pin point the method to be used for a certain problem. In the present study with the aim of choosing the best alternative, the criteria are weighted using the cross entropy method. Based on their weights the alternatives are ranked and it was observed that the ranking of alternatives is the same for 4 methods out of the 5. From the present study it can be concluded that the alternative 4 i.e 409M carbon alloy is the best alternative.

Acknowledgments

The authors like to extend their heartiest gratitude to the reviewers for their valuable advises and would also like to thank Production Engineering Department of N.I.T Agartala for helping in all possible ways to carry out the work.

List of abbreviations

MCDM	Multi-criteria decision-making
MAGDM	Multi-Attribute Group Decision Making
AHP	Analytical hierarchy process
CE	Cross entropy
COPRAS	Complex Proportional Assessment
MOORA	Multi Objective Optimization on the Basis of Ratio Analysis
VIKOR	VlseKriterijumska Optimizacija I Kompromisno Resenje
TOPSIS	Technique for Order Preference by Similarity to Ideal Solution