Modality of teaching learning based optimization algorithm to reduce the consistency ratio of the pairwise comparison matrix in analytical hierarchy processing
Abstract
This paper presents an approach to improve the consistency of pairwise comparison matrix in analytical hierarchy process (AHP) using teaching learning based optimization (TLBO) algorithm. The purpose of this proposed approach to minimize the consistency ratio (CR). Consistency test for the comparison matrix in AHP have been studied rigorously since AHP was introduced in 1970s. However, existing approaches are either too complicated or difficult. Most of them could not preserve the original judgments provided by an expert. To improve the consistency ratio (CR), this research work proposes a simple, effective and efficient method which will minimize the CR to almost zero while preserving the judgment values in pairwise comparison matrix. The correctness of the proposed method is proved by applying it to two real world case studies reported in literature, namely new product design selection and material selection (work tool combination). The experimentation shows that the proposed approach is efficient and accurate to satisfy the consistency requirements of AHP.
Keywords
Analytical hierarchy process (AHP) Pairwise comparison matrix Teaching learning based optimization (TLBO) Consistency ratio1 Introduction
In past three decades number of methods have been proposed and developed which uses the pairwise comparison matrix for solving the multiple criteria decision making (MCDM) between finite alternatives (Keeney and Raiffa 1976). To resolve the qualitative and quantitative factor for the decision makers in the MCDM, Saaty in 1970s has proposed an analytical hierarchy process (AHP) (Saaty 2001, 2003, 2005, 2006). An AHP has been successfully applied to the wide variety of real world applications (Li and Ma 2007; Cao et al. 2008; Dong et al. 2008; Iida 2009, Peng et al. 2012; Lin et al. 2011; Peng et al. 2011a, b; Rao 2007, 2013; Borkar et al. 2016). In AHP the pairwise comparison matrix consist of judgments expressed on a numerical scale of 1–9 by decision maker based on their expertise and experiences. Consistency is one of the most important issues in AHP, meanwhile the consistency ratio is hard to obtain when a large number of criteria is evaluated. In some cases this comparison matrix would be inconsistent due to the limitations of expertise and experiences. Some existing approaches are difficult and complicated to either revise the comparison matrix or to preserve the original comparison matrix. What matters in AHP is how to construct a pairwise judgment matrix with consistency ratio small enough. Saaty in (Saaty 1980) proposes a consistency index \({\text{CI}}~={\text{ }}(\lambda \max ~  1)/\left( {{\text{n}}  1} \right)\) and a consistency ratio CR = CI/RI. In Saaty’s opinion, the consistency ratio (CR) of less than 0.1 is acceptable. But it is difficult to construct such a judgment matrix with satisfactory CR because of the complexity of the decision problem and the limited ability of human thinking. There are two approaches by which inconsistent matrices can be made consistent: (1) the decision maker will go for reassessment process to get the new comparison matrix which is consistent. This process is quite time consuming as the reassessment is to be repeated until the matrix is consistent. (2) Modify the value of comparison matrix by a method until the consistency ratio is satisfied. This approach has taken attention of many researchers to modify the inconsistent pairwise comparison matrix (Cao et al. 2008; Lin et al. 2011; Costa 2011).
The population based heuristic algorithms have two important groups: evolutionary algorithms (EA) and swarm intelligence (SI) based algorithms. Some of the recognized evolutionary algorithms are: Genetic Algorithm (GA), Evolution Strategy (ES), Evolution Programming (EP), Differential Evolution (DE), Bacteria Foraging Optimization (BFO), Artificial Immune Algorithm (AIA), etc. Some of the well known swarm intelligence based algorithms are: Particle Swarm Optimization (PSO), Shuffled Frog Leaping (SFL), Ant Colony Optimization (ACO), Artificial Bee Colony (ABC), Fire Fly (FF) algorithm, etc. Few of the above optimization technique have been modeled to reduce the consistency ratio of AHP and researchers modified the inconsistent matrix using above intelligent techniques to prepare the consistency matrix. Genetic algorithm (GA) has been used by (Lin et al. 2008) and (Costa 2011) to obtain the consistent matrices. A research using particle swarm optimization (PSO) and taguchi method is presented in (Yang et al. 2012) to solve the inconsistent comparison matrix. Tahuchi method is used to reduce the number of experiments required for tuning the control parameters of PSO. This approach has improved the previous research using genetic algorithm (Lin et al. 2008). Besides considering CR must be less than 0.1, (Lin et al. 2008) and (Yang et al. 2012) also determine two aspects, difference index (Di) aspect which represent the distance matrix and consistent ratio aspect. These two aspects are combined in the overall index (OI). They proposed method repairing the inconsistent matrix while minimizing the OI (Girsang et al. 2014aa, b) has implemented ant colony optimization (ACO) approach to solve inconsistency problem. ACO is used to enhance the minimal deviation matrix and to enhance the minimal consistent ratio.
The objective of this research work is to propose the simple, efficient and accurate approach to reduce the consistency ratio of the pairwise comparison matrix. In order to reduce the consistency ratio, we have applied the teaching learning based optimization (TLBO) to tune the judgment values of the comparison matrix.
This research work includes:

A simple, efficient and accurate approach to reduce the consistency ratio.

Modality of TLBO to tune the elements of pairwise comparison matrix, while preserving the judgments made by an expert.

Novel approach for identifying variables (judgment elements) and deciding lower and upper bound for it.

Several case real world cases studies were evaluated to present the robustness of the proposed method.
The contribution of this research work includes:

A hybrid model which will tune the parameters of multiattribute decision making methods using multiobjective decision making approach.

Decision making using algorithmic parameter less optimization technique while preserving the expert’s judgment.
The remaining parts of this paper are organized as follows. Why teaching learning based optimization is incorporated in this research work and details of it are presented in Sect. 2. The proposed method for tuning the judgments of the pairwise comparison matrix and reducing the consistency ratio is presented in Sect. 3. Two real world case studies: new product design selection and work tool selection (material selection) are presented in Sect. 4. Finally conclusion is provided.
2 Teaching learning based optimization
Some of the best and well known metaheuristics techniques developed over the last three decades to solve the many engineering optimization problems are: Genetic Algorithm (Goldberg 1989), Artificial Immune Algorithm (AIA) (Farmer et al. 1986), Ant Colony Optimization (ACO) (and Stutzle 2004), Particle Swarm Optimization (PSO) (Kennedy and Eberhart 1995), Differential Evolution (DE) (Efren et al. 2010), Harmony Search (HS) (Geem et al. 2001), Bacteria Foraging Optimization (BFO) (Passino 2002), Artificial Bee Colony (ABC) (Karaboga 2005) etc. The above algorithms require common controlling parameters like population size, number of generations, elite size, etc. Besides the common control parameters, these algorithms require their own algorithmspecific control parameters. Various studies has been carried out to either enhance the existing optimization algorithm (Liu and Tang 1999; Chakraborty et al. 2011; Shi et al. 2007) or to hybridize the existing algorithms (Karen et al. 2006; Yildiz 2009).
The main limitations of the above mentioned algorithms are that different control parameters are required for functional working of these algorithms. The proper selection of these parameters is the crucial step and it is important. In case of GA, controlling parameters are population size, crossover rate, mutation rate etc. Similarly, In PSO, it uses inertia weight, cognitive and social parameters. Similarly, ABC requires number of employed, scout and onlookers bees etc. HS requires harmony memory, pitch adjustment rate, etc. Continuous research has been carried to develop an optimization algorithm which does not require any algorithm specific parameters (Rao 2016). Teaching Learning Based Optimization (TLBO) is the optimization method which works on the philosophy of teaching and learning. In TLBO, teacher is considered to be highly learned person and he/she shares his/her knowledge with the learners.
2.1 TLBO algorithm
A teaching learning process inspired optimization algorithm, TLBO is recently proposed by (Rao et al. 2012a; Rao 2013b, 2015; Rao and Patel 2012b, 2013b, c). TLBO mimics the teaching–learning ability of the teacher and learners in classroom. In TLBO, population is group of learners, design parameters are the different subjects offered to learners and fitness value of the optimization problem is the learners result. The best solution in the entire population is termed to be teacher. The working of TLBO algorithm is divided into two phases, ‘Teacher phase’ and ‘Learner phase’. The working of both phases is explained below.
2.1.1 Teacher phase
2.1.2 Learner phase
Accept \({X_{new}}\) if it gives better function value.
3 Proposed method
In this research work, a novel framework is proposed to get the consistent pairwise comparison matrix, which includes problem definition formulation, conversion of qualitative values to the quantitative values, the normalization using the beneficiary attributes and nonbeneficiary attributes. A novel approach is presented in this research work to select the variables for TLBO along with the proposed mechanism to decide the lower and upper bound of the variables. Modality of TLBO is employed in this research work to get the optimal judgment values of the comparison matrix and to get the minimum consistency ratio (CR).
3.1 Step 1: the objective is to assess the given alternatives based on considered attributes
The decision table, given in Table 1, shows alternatives, A_{i} (for i = 1, 2, …, n), attributes, T_{j} (for j = 1, 2, …, m), weights of attributes, w_{j} (for j = 1, 2, …, m) and the measures of performance of alternatives, C_{ij} (for i = 1, 2, …, n; j = 1, 2, …, m). Given multi attribute decision making method and the decision table information, the task of the decision maker is to find the best alternative and/or to rank the entire set of alternatives. To consider all possible attributes in decision problem, the elements in the decision table must be normalized to the same units.
Decision table in MADM methods
Alternatives  Attributes (weights)  

T_{1} (w_{1})  T_{2} (w_{2})  T_{3} (w_{3})  –  –  T_{m} (w_{m})  
A_{1}  C_{11}  C_{12}  C_{13}  –  –  C_{14} 
A_{2}  C_{21}  C_{22}  C_{23}  –  –  C_{24} 
A_{3}  C_{31}  C_{32}  C_{33}  –  –  C_{34} 
–  –  –  –  –  –  – 
–  –  –  –  –  –  – 
A_{n}  C_{n1}  C_{n2}  C_{n3}  –  –  C_{nm} 
3.2 Step 2: compute the normalized decision matrix:
The attributes can be considered as beneficial or nonbeneficial. Normalized values are calculated by (C_{ij})_{K}/(C_{ij})_{L}, where (C_{ij})_{K} is the measure of the attribute for the Kth alternative, and (C_{ij})_{L} is the measure of the attribute for the Lth alternative that has the highest measure of the attribute out of all alternatives considered. This ratio is valid for beneficial attributes only. A beneficial attribute (e.g., efficiency) means its higher measures are more desirable for the given decisionmaking problem. By contrast, nonbeneficial attribute (e.g., cost) is that for which the lower measures are desirable, and the normalized values are calculated by (C_{ij})_{L}/(C_{ij})_{K} (Rao 2007). In reality, measure of performance (C_{ij}) can be crisp, fuzzy and/or linguistic. The decision makers can appropriately make use of any of the eight scales suggested (Chen and Hwang 1992). For example, 5 and 11point scale and the corresponding crisp scores of the fuzzy numbers are given in Table 2 (Rao 2007, Chap. 4).
Conversion of linguistic terms into fuzzy scores (5 and 11point)
5point scale  11point scale  

Linguistic term  Assigned crisp score  Linguistic term  Assigned crisp score 
Low  0.115  Exceptionally low  0.0455 
Below average  0.295  Extremely low  0.1364 
Average  0.495  Very low  0.2273 
Above average  0.695  Low  0.3182 
High  0.895  Below medium  0.4091 
Medium  0.5000  
Above medium  0.5909  
High  0.6818  
Very high  0.7727  
Extremely high  0.8636  
Exceptionally high  0.9545 
3.3 Step 3: construct a pairwise comparison matrix using relative importance scale:
Saaty in (Saaty 1980) has provided a fundamental scale of AHP for entering judgments, wherein attributes are compared with itself, so main diagonal values will always be 1. The numbers 3, 5, 7, and 9 are corresponds to judgments ‘moderately important’, ‘strongly important’, ‘very strongly important’ and ‘absolutely important’ respectively. The numbers 2, 4, 6, and 8 are used when there is compromise between the above mentioned values. Assuming T attributes, the pairwise comparison of attribute i with attribute j results in T_{M × M} where t_{ij} denotes the comparative importance of i attribute with j. In the matrix T_{M × M}, t_{ij} = 1 when i = j and b_{ji } = 1/b_{ij}.
\(\begin{array}{*{20}{l}} {{{\text{T}}_1}} \\ {{{\text{T}}_2}} \\ {{{\text{T}}_3}} \\  \\  \\ {{{\text{T}}_{\text{M}}}} \end{array}\mathop {\left[ {\begin{array}{*{20}{l}} 1&{{{\text{t}}_{12}}}&{{{\text{t}}_{13}}}&  &  &{{{\text{t}}_{1{\text{M}}}}} \\ {{{\text{t}}_{21}}}&1&{{{\text{t}}_{23}}}&  &  &{{{\text{t}}_{{\text{2M}}}}} \\ {{{\text{t}}_{31}}}&{}&1&  &  &{{{\text{t}}_{{\text{3M}}}}} \\  &  &  &  &  &  \\  &  &  &  &  &  \\ {{{\text{t}}_{{\text{M}}1}}}&{{{\text{t}}_{{\text{M}}2}}}&{{{\text{t}}_{{\text{M}}3}}}&  &  &1 \end{array}} \right]}\limits^{\begin{array}{*{20}{l}} {{{\text{T}}_1}}&{{{\text{T}}_2}}&{{{\text{T}}_3}}&  &  &{{{\text{T}}_{\text{M}}}} \end{array}}\)
3.4 Step 4: identification of variables for TLBO
In this research work, a new way of identification of required variable for teaching learning based optimization algorithm is proposed. This identification process preserves the relative importance of attribute i with all other attributes. The algorithm for identification of variables is as follows:
[X] be the vector which hold all the identified variables using above algorithm.
3.5 Step 5: deciding lower and upper bounds for variables
Very crucial aspect of TLBO is identification of lower and upper bound for the identified variables. Here variables values are the relative importance of attribute i to attribute j. The pairwise comparison matrix is supposed to be prepared by domain expert having deep knowledge about the problem definition and this pairwise comparison matrix leads to the weights of the considered attributes. But in most of the cases even an expert makes slightly wrong judgment may result in 1–10% of error in decision making. Here in this research work we are assuming that even an expert can make 10% of judgment error. In general lower bound will be the judgment value −1 and upper bound will be the judgment value +1. The algorithm for deciding lower and upper bound for variable is as follows:
3.6 Step 6: minimize the consistency ratio (CR) using TLBO
Let the T_{M × M} be the pairwise comparison matrix prepared in step 3 and [X] be the array of identified variables and [L] and [U] be the array of lower and upper bound respectively for the identified variables. Teaching learning based optimization algorithm is applied in this research work minimize the consistency ratio by optimally tuning the identified variables values with specified bound. This approach will preserve an expert judgment and also it reduces the CR significantly which result in best decision making. Goal is to obtain the optimal pairwise comparison matrix OT_{M × M} from T_{M × M} by using TLBO. Algorithm to minimize CR using TLBO is as follows:
1.  Initialize the population (i.e. learners’), variables of the optimization problem (here variables are identified in step 4) and lower and upper bound for the identified variables 
2.  Select the best learner of each subject as a teacher for that subject and calculate mean result of learners in each subject 
3.  Evaluate the difference between current mean result and best mean result according to Eq. (1) by utilizing the teaching factor (TF) [Eq. (2)]. 
4.  Update the learner’s knowledge with the help of teacher’s knowledge according to Eq. (3) 
5.  Update the learner’s knowledge by utilizing the knowledge of some other learner according to Eqs. (4) and (5) 
6.  Repeat the procedure from 2 to 5 until the termination criterion is met (CR <0.1) 
The mentioned algorithm has to be applied on single objective function: Minimize CR()
a.  Find the normalized weight (w_{j}) of each attribute by calculating geometric mean and normalizing geometric mean of rows in comparison matrix T_{M × M} 
b.  Calculate the matrices A3 = A1 * A2 and A4 = A3/A2 
c.  Determine the Eigen value λmax which is average of A4 
d.  Calculate the consistency index CI = (λmax − M)/(M − 1) where M is the number of attributes 
e.  Calculate the consistency ration CR = CI/RI where RI is the random index 
The above algorithm results in optimal pairwise comparison matrix OT_{M × M}.
3.7 Step 7: evaluate the consistency ratio for the optimized relative importance matrix
3.8 Step 8: evaluate alternative to obtain the overall performance score for the alternatives
4 Case studies
To evaluate the robustness of the proposed research work, we have applied the proposed method to two real world case studies and the objective data of the alternatives are taken from literature. Case study 1 is about selection of new product design taken from (Besharati et al. 2006), case study 2 is about material selection (work tool combination) taken from (Enache et al. 1995).
4.1 Case study 1: evaluation of product designs
In case of product design, the selection of final design for the production is the crucial stage. Selection of best design results in overall success of the product in the market. A casebased reasoning model is presented in (Haque et al. 2000) for engineers and managers. The probability of success of the product design is introduced in (Suh 2001). For the innovative product design, creativitybased design model is presented in (Hsiao and Chou 2004). A detailed framework for how several factors affect the making of new product design in presented in (Ozer 2005), they have also provided the guidelines for reducing the negative impacts. An ideascreening method for new product design in presented in (Lo et al. 2006, Kulak and Kahraman 2005) with a group of decision makers having imprecise, uncertain and inconsistent preferences. This model provides authors with consistent information. A sensitivity analysis for new product design in presented in (Maddulapalli et al. 2007) with implicit value function.
To demonstrate the proposed model, the case study presented by (Besharati et al. 2006) is considered in this research work. Both performance related and market related attribute are considered. They have considered a problem of design and the selection of power electronic device based on three attributes namely: manufacturing cost, junction temperature and thermal cycles to failure. Ten alternatives were considered.
Step 1 The objective is to assess the alternative, i.e., ten alternative product designs were considered based on considered attributes: junction temperature (JT), cycles to failure (CF) and manufacturing cost (MC) (refer the objective data presented in Table 3).
objective data of product design
Design no.  Junction temperature (°C)  Cycles to failure  Manufacturing cost ($) 

1  126  22,000  85 
2  105  38,000  99 
3  138  14,000  65 
4  140  13,000  60 
5  147  10,600  52 
6  116  27,000  88 
7  112  32,000  92 
8  132  17,000  75 
9  122  23,500  85 
10  135  15,000  62 
Step 2 Normalization of the objective data.
The quantitative values of product design selection problem are normalized. In this example, CF is a beneficiary attribute whereas JT and MC are the nonbeneficiary attributes. The values for these attributes are normalized and are presented in Table 4.
Normalized data of the product design selection problem
Design no.  Junction temperature (°C)  Cycles to failure  Manufacturing cost ($) 

1  0.8333  0.5789  0.6118 
2  1  1  0.5223 
3  0.7609  0.3684  0.8 
4  0.75  0.3421  0.8667 
5  0.7143  0.2789  1 
6  0.9052  0.7105  0.5909 
7  0.9375  0.8421  0.5652 
8  0.7955  0.4474  0.6933 
9  0.8607  0.6184  0.6118 
10  0.7778  0.3947  0.8397 
Step 3 Determine the relative importance of various attributes under consideration with respect to objective.
Let the decision maker/domain expert select the following exercise depending upon the requirements. The assigned values for this example are taken from (Rao 2007, 2013).
\(\begin{array}{*{20}{l}} {{\text{JT}}} \\ {{\text{CF}}} \\ {{\text{MC}}} \end{array}\mathop {\left {\begin{array}{*{20}{l}} 1&{1/3}&{1/5} \\ 3&1&{1/3} \\ 5&3&1 \end{array}} \right}\limits^{\begin{array}{*{20}{l}} {{\text{JT}}}&{{\text{CF}}}&{{\text{MC}}} \end{array}}\)
CF is considered moderately important than JT, so judgment value 1/3 is assigned to JT over CF. MC is considered as strongly important than JT, so judgment value 1/5 is assigned to JT over MC. MC is considered as strongly important than CF, so judgment value 1/3 is assigned to CF over MC
From the original pairwise comparison matrix, it is noted that the distinct values for JT over CF and MC is 1/3 and 1/5, so it will be treated as variables. Only single value is noted for CF over MC i.e. 1/3, so it will be considered as third variable. Here we have taken only distinct values of say attribute i over all other attributes, this will preserve the expert judgment and also reduces the number of variable count required for TLBO.
Step 5 Deciding lower and upper bounds for variables:
The lower and upper bounds for the identified variables (step 4) are presented in Table 5. Detailed algorithm for deciding these bounds is provided in step 5 of Sect. 3.
lower and upper bound for \(\left[ {{X_i}} \right]\)
[X_{i}]  1/3  1/5  1/3 

Lower bound  1/2  1/4  1/2 
Upper bound  1/4  1/6  1/4 
Step 6 Obtain the optimal relative importance matrix using TLBO: \(\begin{array}{*{20}{l}} {{\text{JT}}} \\ {{\text{CF}}} \\ {{\text{MC}}} \end{array}\mathop {\left {\begin{array}{*{20}{l}} {1.0000}&{0.4472}&{0.2243} \\ {2.2361}&{1.0000}&{0.4981} \\ {4.4581}&{2.0076}&{1.0000} \end{array}} \right}\limits^{\begin{array}{*{20}{l}} {{\text{JT}}}&{{\text{CF}}}&{{\text{MC}}} \end{array}}\)
TLBO algorithm is applied several times to check for any further improvement in the consistency ratio and it is observed that the global optimum solution for the problem under consideration is obtained in fifth generation (refer Fig. 1).
Step 7 Evaluate the consistency ratio for the optimized relative importance matrix
JT  CF  MC  Weights  A3  A4  λmax  CI  CR  

JT  1.0000  0.4472  0.2243  0.1299  0.3896  2.9997  3  −1.0432e05  −2.0062805 
CF  2.2361  1.0000  0.4981  0.2898  0.8693  3  
MC  4.4581  2.0076  1.0000  0.5803  1.7411  3.0002 
The geometric mean of the matrix which is obtained in step 6 results in the normalized weights of each attribute are JT = 0.1299, CF = 0.2898, and MC = 0.5803. The value of λmax is 3 and CR = −2.0062805, which is very much less (almost zero) than the allowed CR value of 0.1. The obtained CR value using TLBO is much less as compared to CR value of 0.0370 if it is computed from original pairwise comparison matrix. Thus, there is good consistency obtained by using TLBO for the expert’s pairwise comparison matrix.
Step 8 Evaluate alternative/obtain the overall performance score for the alternatives.
The values of product designs are calculated. The scores of the each alternative, A_{i} by considering the original pairwise matrix and by optimal pairwise comparison matrix is presented in Table 6. From both the scores it is observed that, product design 5 will be the first choice, but now product design 2 will be the second choice which was previously at fourth position. Product design number 3, 4, 7, 8, 9 are also having now new ranking. As the ranking obtained using TLBO is with almost zero percent error, so it will be treated as final ranking for alternatives
Alternatives scores and rank for product design selection
Alternatives score with  

Product design no.  Original pairwise comparison matrix (step 3)  Optimal pairwise comparison matrix obtained using TLBO (step 6)  
Score  Rank  Score  Rank  
1  0.6265  10  0.6310  10 
2  0.6976  4  0.7245  2 
3  0.6844  5  0.6699  6 
4  0.7190  2  0.6995  4 
5  0.7838  1  0.7540  1 
6  0.6547  7  0.6664  7 
7  0.6757  6  0.6938  5 
8  0.6405  8  0.6353  9 
9  0.6396  9  0.6460  8 
10  0.7177  3  0.7022  3 
From the Fig. 1, it is noted that the global optimal solution is obtained in fifth generation where CR is 5.15e006 which is almost zero. The number of generations used to run the TLBO algorithm is 10. Algorithm is allowed to run ten times in order to check the best result and population size of 50 were considered
4.2 Case study 2: machinability evaluation
A process of removal of material using cutting tools is machining and the manufacturing industries strive for minimum production cost and maximum rate of production. A manufacturing process generally consists of several phases like process design, planning, machining and quality control wherein machinability is related to process planning and machining operations. While product design machinability of materials need to be taken into consideration where best material is to be chosen from finite set of materials based on design and functionality satisfaction. Material selection is an important task in the process of product design, for overall reducing the production costs. Here the machinability refers to, selection of the best material from available set of materials which also satisfy the required product design and functionality. This selection of material depends upon manufacturer’s interest and on other aspects. The machining process is affected by number of variables (input and output). The most common input variables are machine tool, cutting tool, cutting conditions, work material properties, cutting fluid properties and output variables are cutting tool life, cutting force, power consumption, metal removal rate, noise, vibrations, cutting temperature, etc (Rao 2007). From the literature it is noted that the criteria in general for machinability evaluation of different work materials include tool wear rate, cutting force, tool life, power consumption, etc (Arunachalam and Mannan 2000; Ong and Chew 2000; Dravid and Utpat 2001; Kim et al. 2002; Boubekri et al. 2003; Rao 2005, Salak et al. 2006; Morehead et al. 2007).
Enache et al. (1995) carried out several experiments on titanium alloys using various cutting tools, and presented a mathematical model for machinability. Here six work tools are under consideration and evaluation will be done with three attributes namely tool wear rate (TWR), specific energy consumed (SEC) and surface roughness (SR). TWR, SEC and SR are considered to be nonbeneficiary attributes, means lower values are desired.
Step 1 The objective is to assess the alternative, i.e., work tools 1–6 based on considered attributes: TWR, SEC and SR (Refer the objective data presented in Table 7).
Objective data of material selection problem
Work tool combination  Tool wear rate (m/min)  Specific energy consumed (N)  Surface roughness (μm) 

1  0.0610  219.7400  5.8000 
2  0.0930  3523.7200  6.3000 
3  0.0640  2693.2100  6.8000 
4  0.0280  761.4600  5.8000 
5  0.0340  1593.4800  5.8000 
6  0.0130  2849.1500  6.2000 
Step 2 Normalization of the objective data:
The quantitative values of work material selection problem are normalized. In this example, TWR, SEC and SR are nonbeneficial attributes. The values for these attributes are normalized and are presented in Table 8.
Normalized data of attributes of work material selection problem
Work tool  TWR  SEC  SR 

1  0.2131  1  1 
2  0.1398  0.0624  0.9206 
3  0.2031  0.0816  0.8529 
4  0.4643  0.2886  1 
5  0.3824  0.1379  1 
6  1  0.0771  0.9355 
Step 3 Determine the relative importance of various attributes under consideration with respect to objective:
Let the decision maker/domain expert select the following exercise depending upon the requirements. The assigned values for this example are for demonstration purposes [taken from (Rao 2007)] only and it is to be decided by a domain expert.
\(\begin{array}{*{20}{l}} {{\text{TWR}}} \\ {{\text{SEC}}} \\ {{\text{SR}}} \end{array}\mathop {\left {\begin{array}{*{20}{l}} 1& 5& 7 \\ {1/5}& 1& 3 \\ {1/7}& {1/3}& 1 \end{array}} \right}\limits^{\begin{array}{*{20}{l}} {{\text{TWR}}}& {{\text{SEC}}}& {{\text{SR}}} \end{array}}\)
Step 4 Identification of variable for TLBO:
\(\left[ {{X_i}} \right]=[5,~7,~3]\)
Step 5 Deciding lower and upper bounds for variables (Table 9).
lower and upper bounds for variables
[X_{i}]  5  7  3 

Lower bound  4  4  2 
Upper bound  6  8  4 
Step 6 Obtain the optimal relative importance matrix using TLBO:
\(\begin{array}{*{20}{l}} {{\text{TWR}}} \\ {{\text{SEC}}} \\ {{\text{SR}}} \end{array}\mathop {\left {\begin{array}{*{20}{l}} {1.0000}&{4.0711}&{7.9768} \\ {0.2456}&{1.0000}&{2.0065} \\ {0.1254}&{0.4984}&{1.0000} \end{array}} \right}\limits^{\begin{array}{*{20}{l}} {{\text{TWR}}}&{{\text{SEC}}}&{{\text{SR}}} \end{array}}\)
Step 7 Evaluate the consistency ratio for the optimized relative importance matrix.
The normalized weights of each attribute are TWR = 0.7289, SEC = 0.1805, and SR = 0.0907. The value of λmax is 3.0001 and CR = 6.0501e005, which is much much less (almost zero) than the allowed CR value of 0.1. Thus, there is good consistency obtained by using TLBO for the experts pairwise comparison matrix.
Step 8 Evaluate alternative/obtain the overall performance score for the alternatives.
The values of work tools are calculated (refer Eq. 8). Score comparison for original pairwise matrix and optimal pairwise matrix using TLBO is presented in Table 10. From the Table 10, it is observed that, work tool 6 will the first choice and also other rankings are also same.
Alternatives scores and rank for material selection problem
Alternatives score with  

Work tool combination  Original pairwise comparison matrix (step 3)  Optimal pairwise comparison matrix obtained using TLBO (step 6)  
Score  Rank  Score  Rank  
1  0.4251  3  0.4265  3 
2  0.1884  6  0.1966  6 
3  0.2328  5  0.2401  5 
4  0.4746  2  0.4811  2 
5  0.3863  4  0.3942  4 
6  0.8209  1  0.8276  1 
From the Fig. 2, it is noted that the global optimal solution is obtained in fifth generation where CR is 6.05e005 which is almost zero. The number of generations used to run the TLBO algorithm is 10. Algorithm is allowed to run ten times in order to check the best result and population size of 50 were considered.
The proposed approach can be applied on several decision making problems and some of the Industrial Applications are as follows (Rao 2007, 2011; Borkar et al. 2016): Evaluation of product design, Material selection for given engineering applications, Machinability Evaluation of Work Materials Cutting Fluid Selection for a Given Machining Application, Evaluation and Selection of Modern Machining Methods, Evaluation of Flexible Manufacturing Systems, Machine Selection in a Flexible Manufacturing Cell, Failure Cause Analysis of Machine Tools, Robot Selection for a Given Industrial Application, Integrated Project Evaluation and Selection, Selection of Rapid Prototyping Process in Rapid Product Development, Optimal route selection problem etc.
5 Conclusion
This paper presents a simple and effective method, to improve the inconsistent comparison matrix to obtain the almost zero consistent ratio (CR). The proposed approach is built based on teaching learning based optimization (TLBO) algorithm, which was proved to be algorithmic parameters less and successful for solving several optimization problems. To improve the consistency ratio (CR) while preserving the judgment values in pairwise comparison matrix, modality of TLBO is proposed and applied. The variable identification algorithm is proposed in this research work. Selection of bounds for variables is also proposed. The correctness of the proposed method is proved by applying it to two real world case studies reported in literature, namely new product design selection and material selection (work tool combination). In case of product design selection problem, alternative 5 was ranked first by using conventional and proposed approaches but for rank 2, alternative 2 was selected instead of alternative 4. The proposed approach can be applied to variety of alternative selection problems as enlisted above. The result shows the TLBO algorithm is the potential method to solve the inconsistent pairwise comparison matrix in AHP. The other intelligent algorithm like particle swarm optimization, ant colony optimization, genetic optimization or any of the advance optimization technique can be used to solve these selection problems.
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