*2.2.3 Working principle*

All MCDM methods share some similar working principles upto certain extent, which are as follows:

1.Selection of Criteria:


2. Selection of Alternatives:


3. Selection of method to provide weightage to the criteria:


#### *2.2.4 Flowchart for decision making*

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

**Figure 1.** *Flowchart for decision making.*

#### *2.3.1 Algorithms for cross entropy method*

**Step 1**: Feature weight *βij* is calculated for *i th* alternative and *j th* criterion as

$$\beta\_{\vec{\imath}\vec{\jmath}} = \frac{a\_{\vec{\imath}\vec{\jmath}}}{\sum\_{i=1}^{m} a\_{\vec{\imath}\vec{\jmath}}^{2}}, (\mathbf{1} \le \mathbf{i} \le \mathbf{m}, \mathbf{1} \le \mathbf{j} \le \mathbf{n})$$

**Step 2**: The output entropy ε<sup>j</sup> of the jth factor

$$\begin{aligned} \varepsilon\_{\circ} &= -\kappa \sum\_{i=1}^{m} \left( \beta\_{ij} \ln \beta\_{ij} \right), (1 \le \mathbf{j} \le \mathbf{n}) \\\\ &\kappa = \frac{1}{\ln m} \end{aligned}$$

**Step 3**: Calculation of variation coefficient of jth factor *ξ <sup>j</sup>*

$$\xi\_j = |1 - e\_j|$$

**Step 4**: Calculation of weight of the entropy *w <sup>j</sup>*

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

*Comparison of Cross-Entropy Based MCDM Approach for Selection of Material in Sugar… DOI: http://dx.doi.org/10.5772/intechopen.98242*

#### **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:

$$\begin{array}{ccccccccc} & \mathbf{C\_1} & \mathbf{C\_2} & \mathbf{C\_3} & & \mathbf{C\_n} \\ & A\_1 & a\_{12} & a\_{13} & \cdots & a\_{1n} \\ A\_2 & a\_{21} & a\_{22} & a\_{23} & \cdots & a\_{2n} \\ M = A\_3 & a\_{31} & a\_{32} & a\_{33} & \cdots & a\_{3n} \\ & \vdots & \vdots & \vdots & \ddots & \vdots \\ A\_m & \begin{bmatrix} \vdots & \vdots & \vdots & \ddots & \vdots \\ a\_{m1} & a\_{m2} & a\_{m3} & \cdots & a\_{mn} \end{bmatrix} \end{array}$$

$$W = \begin{bmatrix} w\_1 & w\_2 & w\_3 & \cdots & w\_n \end{bmatrix}$$

Here, A1 , A2 , A3 … … Am are the alternatives which are available and supposed to be ranked by decision maker C1 , C2 , C3 … … Cn are the criteria which will govern ranking of the alternatives. *aij* shows the performance of alternative *Ai* on the basis of *Cj* and *w <sup>j</sup>* 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 *nij* � �:

$$n\_{\vec{\eta}} = \frac{a\_{\vec{\eta}}}{\sum\_{i=1}^{m} a\_{\vec{\eta}}}$$

**Step 2**: Calculation of weighted normalize decision matrix *Wij* � �:

$$\mathcal{W}\_{\vec{\eta}} = \mathfrak{n}\_{\vec{\eta}} \ast w\_{\vec{\jmath}}$$

Where *w <sup>j</sup>* is the weightage of criterion *Cj*.

**Step 3**: Calculation of *S*<sup>þ</sup> and *S*�:

*S*<sup>þ</sup> and *S*� are the summation of weighted normalized value that are evaluated for benefit criteria as well as non-benefit criteria.

$$\mathcal{S}\_i^+ = \sum\_{j=1}^n \mathcal{W}\_{ij} . (\mathbf{i} = \mathbf{1}, 2, 3 \dots \mathbf{m}),$$

Where *Wij* is the weighted normalize elements for all the benefit criteria

$$\mathfrak{S}\_i^- = \sum\_{j=1}^n W\_{ij}. (\mathbf{i} = \mathbf{1}, \mathbf{2}, \mathbf{3} \dots \mathbf{m})$$

Where *Wij* is the weighted normalize elements for all the non-benefit criteria

*Comparison of Cross-Entropy Based MCDM Approach for Selection of Material in Sugar… DOI: http://dx.doi.org/10.5772/intechopen.98242*

**Step 4**: Evaluating relative weightage of each alternative Qi:

$$Q\_i = \mathbb{S}\_i^+ + \frac{\sum\_{i=1}^m \mathbb{S}\_i^-}{\mathbb{S}\_i^- \quad \sum\_{i=1}^m \frac{1}{\mathbb{S}\_i^-}}$$

**Step 5**: Determining the priority order (Pri):

$$Pr\_i = \frac{Q\_i}{\max \quad Q\_i}$$

Maximum value of *Pri* is given maximum priority and ranked 1, second largest value of *Pri* 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 *nij* � �:

$$n\_{\vec{\eta}} = \frac{a\_{\vec{\eta}}}{\sqrt{\sum\_{i=1}^{m} a\_{\vec{\eta}}^2}}$$

**Step 2**: Calculation of weighted normalize decision matrix *Wij* � �:

$$\mathcal{W}\_{\vec{\eta}} = \mathcal{w}\_{\vec{f}} \times \mathfrak{n}\_{\vec{\eta}}.$$

**Step 3**: Evaluating of Priorities (Qi):

$$Q\_i = \sum\_{j=1}^{n} W\_{ij}$$

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

**Step 4**: Ranking of alternatives:

Maximum value of the variable Qi is provided the maximum priority and ranked 1, second largest of Qi 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* <sup>∗</sup> *<sup>j</sup> and f* <sup>Λ</sup> *j*

$$f\_{\stackrel{\cdot}{\cdot j}}^{\*} = \text{Min}(a\_{\stackrel{\cdot}{\cdot j}}), (\text{ } j = \text{1, 2, 3, } \dots n)$$

$$f\_{\stackrel{\circ}{j}}^{\Lambda} = \mathsf{Max}(a\_{\stackrel{\circ}{j}}), (\stackrel{\circ}{j} = \mathbf{1}, \mathbf{2}, \mathbf{3}, \dots \mathbf{n}),$$

Where *aij* stands for elements of decision matrix **Step 2:** Calculation of relative matrix *Rij* � �

$$R\_{\vec{\eta}} = \frac{f\_j^\*-a\_{\vec{\eta}}}{f\_j^\*-f\_j^{\Lambda}}$$

**Step 3:** Calculation of weighted normalized decision matrix *Wij* � �

$$\mathcal{W}\_{\vec{\eta}} = \mathcal{R}\_{\vec{\eta}} \times \mathcal{w}\_{\vec{\jmath}}$$

**Step 4:** Calculation of *γ <sup>j</sup>* by the concept of Manhattan distance.

$$\mathcal{Y}\_{\vec{j}} = \sum\_{j=1}^{n} W\_{\vec{j}}, (i = 1, 2, 3, \dots m)$$

**Step 5 :** Calculation of *δ <sup>j</sup>* by the concept of Chebyshev distance.

$$\delta\_{\vec{j}} = \text{Max}\left(W\_{\vec{\eta}}\right), (i = 1, 2, 3 \dots m).$$

**Step 6:** Calculation of priority values *ρ <sup>j</sup>* .

$$\rho\_j = \frac{V\left(\mathbf{y}\_j - \mathbf{y}^\*\right)}{\left(\mathbf{y}^\Lambda - \mathbf{y}^\*\right)} + (\mathbf{1} - V) \frac{\left(\delta\_j - \delta\_j^\*\right)}{\left(\delta\_j^\Lambda - \delta\_j^\*\right)}$$

$$\begin{aligned} \text{Where, } \boldsymbol{\chi}^{\*} &= \min \left( \boldsymbol{\chi}\_{j} \right) \\ \boldsymbol{\chi}^{\Lambda} &= \max \left( \boldsymbol{\chi}\_{j} \right) \\ \delta\_{j}^{\*} &= \min \left( \delta\_{j} \right) \\ \delta\_{j}^{\Lambda} &= \max \left( \delta\_{j} \right) \\ V &= \frac{n+1}{2n}, \text{ n is the no. of criterion} \end{aligned}$$

**Step 7:** Ranking of various possible alternatives

According to values of *ρ <sup>j</sup>* , 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 (nij)

*Comparison of Cross-Entropy Based MCDM Approach for Selection of Material in Sugar… DOI: http://dx.doi.org/10.5772/intechopen.98242*

$$n\_{\vec{\eta}} = \frac{a\_{\vec{\eta}}}{\sqrt{\sum\_{i=1}^{m} a\_{\vec{\eta}}^2}}$$

**Step 2**: Calculation of weighted normalize decision matrix (Wij)

$$\mathcal{W}\_{\vec{\eta}} = \mathcal{w}\_{\vec{\jmath}} \times \mathfrak{n}\_{\vec{\eta}}$$

Where *w <sup>j</sup>* is the weight of the criterion *C<sup>j</sup>* . **Step 3:** Calculation of Positive Ideal Solution (Pis) and Negative Ideal Solution (Nis)

$$P\_{ii} = \begin{cases} \text{Max} \left( W\_{\vec{\eta}} \right), (\; j = 1, 2, 3, \dots \; n) \text{for benefit criteria} \\ \text{Min} \left( W\_{\vec{\eta}} \right), (\; j = 1, 2, 3, \dots \; n) \text{for non-benefit criteria} \end{cases}$$

$$N\_{ii} = \begin{cases} \text{Max} \left( W\_{\vec{\eta}} \right), (\; j = 1, 2, 3, \dots \; n) \text{for non-benefit criteria} \\ \text{Min} \left( W\_{\vec{\eta}} \right), (\; j = 1, 2, 3, \dots \; n) \text{for benefit criteria} \end{cases}$$

**Step 4:** Calculation of separation measures *S*<sup>þ</sup> *m* � � for Pis and *S*� *m* � � for Nis

$$\mathcal{S}\_{m}^{+} = \sqrt{\sum\_{j=1}^{n} \left( W\_{ij} - P\_{is} \right)^{2}}, \left( 1 \le i \le m, \, 1 \le j \le n \right)$$

$$\mathcal{S}\_{m}^{-} = \sqrt{\sum\_{j=1}^{n} \left( W\_{ij} - N\_{is} \right)^{2}}, \left( 1 \le i \le m, \, 1 \le j \le n \right)$$

Separation measures are measured using Euclidean distance method. **Step 5:** Calculation of relative closeness to the ideal solution ð Þ *RCis*

$$RC\_{\rm is} = \frac{\mathcal{S}\_{\rm m}^{-}}{\mathcal{S}\_{\rm m}^{+} + \mathcal{S}\_{\rm m}^{-}}$$

**Step 6:** Arrangment of the RCis 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 (nij)

$$n\_{\vec{\eta}} = \frac{a\_{\vec{\eta}}}{\sqrt{\sum\_{i=1}^{m} a\_{\vec{\eta}}^2}}$$

Where aij is the performance of value of alternative Ai on the basis of criterion Cj. **Step 2:** Calculation of Positive Ideal Solution (Pis) and Negative Ideal Solution (Nis)

$$P\_{\vec{n}} = \begin{cases} \text{Max}(n\_{\vec{\eta}}), (\; j = 1, 2, 3, \dots \; n) for \,\text{benefit} \,\text{critical} \\\ \text{Min}(n\_{\vec{\eta}}), (\; j = 1, 2, 3, \dots \; n) for \,\text{non} \,-\,\text{benefit} \end{cases}$$

$$N\_{\vec{n}} = \begin{cases} \text{Max}(n\_{\vec{\eta}}), (\; j = 1, 2, 3, \dots \; n) for \,\text{non} \,-\,\text{benefit} \,\text{critical} \\\ \text{Min}(n\_{\vec{\eta}}), (\; j = 1, 2, 3, \dots \; n) for \,\text{benefit} \,\text{critical} \end{cases}$$

**Step 3:** Calculation of separation measures *S*<sup>þ</sup> *m* � � for Pis and *S*� *m* � � for Nis

$$\begin{aligned} \mathcal{S}\_m^+ &= \sqrt{\sum\_{j=1}^n w\_j \left( n\_{\vec{\imath}\vec{\jmath}} - P\_{\vec{\imath}\vec{\imath}} \right)^2}, \left( \mathbf{1} \le \mathbf{i} \le m \right) \\\\ \mathcal{S}\_m^- &= \sqrt{\sum\_{j=1}^n w\_j \left( n\_{\vec{\imath}\vec{\jmath}} - N\_{\vec{\imath}\vec{\imath}} \right)^2}, \left( \mathbf{1} \le \mathbf{i} \le m \right) \end{aligned}$$

Separation measures are measured using Euclidean distance method. **Step 4:** Calculation of relative closeness to the ideal solution (RCis)

$$RC\_{\rm is} = \frac{\mathcal{S}\_{\rm m}^{-}}{\mathcal{S}\_{\rm m}^{+} + \mathcal{S}\_{\rm m}^{-}}$$

**Step 5:** Arrangment of the RCis 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**.

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


*Comparison of Cross-Entropy Based MCDM Approach for Selection of Material in Sugar… DOI: http://dx.doi.org/10.5772/intechopen.98242*


#### **Table 1.**

*List of alternatives and the selection criteria.*


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

### **4.3 Weighting of criteria**

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

### **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**).


#### **Table 2.**

*Decision matrix.*


**Table 3.**

*Table of weights of the criteria.*


**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**).

#### **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**).

#### **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**).


**Table 5.**

*Table for ranking of alternatives by MOORA method.*


#### **Table 6.**

*Table for ranking of alternatives by VIKOR method.*


*Comparison of Cross-Entropy Based MCDM Approach for Selection of Material in Sugar… DOI: http://dx.doi.org/10.5772/intechopen.98242*

**Table 7.**

*Table for ranking of alternatives by TOPSIS method.*


**Table 8.**

*Table for ranking of alternatives by modified 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**).

#### **5. Comparative analysis**

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

### **5.1 Comparison by rs**

Spearman's rank correlation coefficient (rs) is the method of comparing the ranks of alternatives obtained from different test. Using rs, the similarity between two sets of rankings can be measured. The value of rs 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 rs varies from -0.9 to 1.0. The value of rs 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.
