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Geoderma

journal homepage: www.elsevier.com/locate/geoderma Application of SAW, TOPSIS and fuzzy TOPSIS models in cultivation priority planning for maize, rapeseed and soybean crops

Javad Seyedmohammadia,, Fereydoon Sarmadianb, Ali Asghar Jafarzadeha, Mohammad Ali Ghorbanic,d, Farzin Shahbazia

a

Soil Science Department, Faculty of Agriculture, University of Tabriz, Tabriz, Iran b

Soil Science Department, College of Agriculture and Natural Resources, Faculty of Agricultural Engineering and Technology, University of Tehran, Karaj, Iran c

Water Engineering Department, Faculty of Agriculture, University of Tabriz, Tabriz, Iran d

Engineering Faculty, Near East University, 99138 Nicosia, North Cyprus, Mersin 10, Turkey ARTICLE INFO

Handling Editor: A.B. McBratney

Keywords:

Cultivation priority

Fuzzy AHP

Fuzzy TOPSIS

SAW

TOPSIS

ABSTRACT

Cultivation priority planning is a very important and vital step in suitable and sustainable revenue of agricultural land. The growth of urban areas and industrial intensification has contributed to a reduction in valuable agri- cultural lands and to various environmental impacts including climate change. This reduction in agricultural land severely impacts food production and food security. In order to effectively address this issue, spatial analytical and optimization methods based on evaluating multiple criteria decision are needed to evaluate the capability and suitability of available lands for current and future food production. The objective of this study is to implement the GIS and multi-criteria decision analysis (MCDA) techniques as an improved method of multi- criteria decision making for evaluating areas suitable for cultivation priority planning of maize, rape and soy- bean crops. For this purpose, 12,000 ha land which is located in Ardabil province, west-north of Iran was in- vestigated by excavation of 167 soil profiles and 313 augers. After soil sampling and analysis, soils were clas- sified in Aridisols. 24 soil series and 66 land units were identified and separated in study area. The several criteria had limitation for maize, rape and soybean cultivation in studying lands which the most limiting eva- luation criteria including soil depth, slope, climate, pH, electrical conductivity, exchangeable sodium percen- tage, calcium carbonate and gypsum were selected for usage in prioritization models by principal component analysis and multi-dimensional scaling methods. Selected criteria were very important in growth of maize, rape and soybean. Simple additive weighting (SAW), Technique for Order of Preference by Similarity to Ideal Solution

(TOPSIS) and Fuzzy TOPSIS methods were used for cultivation priority planning of maize, rape and soybean crops in land units. Analytical Hierarchy Process (AHP) and Fuzzy AHP approaches were used to determine weight values of the criteria. Multivariate variance analysis proves significant difference among three methods at 0.05 probability level. With attention to allocated scores by prioritization models, crops cultivation priority was determined as maize, rape and soybean in land units, respectively and maize crop was preferred to other plants. The statistical analysis results with regard to mean comparison derived from least significant difference (LSD) test showed that Fuzzy TOPSIS method set cultivation priority planning of maize, rape and soybean crops for land units more accurately than the others, due to fuzzy TOPSIS method used appropriate values of criteria weights, twin comparing nature of alternative (crop) from positive and negative ideal, data standardization, mathematical equations and matrixes as well as fuzzy logic relations and principles for calculation of process performing. This study emphasizes the successful application of MCDA in dealing with complicated issues in the context of cultivation priority planning management. It is anticipated that, the integration of this developed framework in the planning policies of cultivation priority in developing countries as an effective tool for in- tegrated regional land use planning can help in conducting better control over soil, land and environment losses. 1. Introduction

Land consolidation is considered as the most effective land management planning approach for solving land fragmentation, a problem that prevents rational agricultural development and rural sustainable development more generally (Demetriou, 2016). The http://dx.doi.org/10.1016/j.geoderma.2017.09.012

Received 4 November 2016; Received in revised form 27 June 2017; Accepted 10 September 2017

Corresponding author.

E-mail address: adg91o@r.postjobfree.com (J. Seyedmohammadi). Geoderma 310 (2018) 178–190

0016-7061/ © 2017 Elsevier B.V. All rights reserved. MARK

decline of worthy agricultural land, as a result of constant urban and industrial growth, directly affects the ability to produce food at a large scale. So, agricultural production must be moved to other available land or land currently used for other purposes in order to meet global food demand (Lambin and Meyfroidt, 2011). This problem is more ag- gravated by the effects of global climate change and regional sensitivity increases (Montgomery et al., 2016). Moreover, increases in the fre- quency of risks such as drought, flooding, soil degradation, and regional shifts in crop production expense can seriously change economic mar- kets, trade, and socioeconomic development (Schmidhuber and Tubiello, 2007). Consequently, urbanization and climate can severely impact on global food security (Montgomery et al., 2016). Analysis of agricultural planning includes the consideration of a number of factors, including natural system constraints, compatibility with existing land uses, existing land use policies, and the availability of community facilities. The suitability techniques analyze the interaction between location, development actions, and environmental elements to classify the units of observation according to their suitability for a par- ticular use (Malczewski, 2004). In reality, not all the conflicting objec- tives due to economic development, community or conservation interests are always taken into consideration, which could cause to political and manipulative, decisions. Modern planning theories such as commu- nicative planning and actor-network theory focus on the fact that effec- tive planning decisions should essentially consider all participants with a variety of discourse types and values (Mosadeghi et al., 2015). This en- courages approaches for integrating very heterogeneous data, making them available to the various stakeholders to allow them to make more informed and less subjective decisions (Greene et al., 2010). In the 1960s, the first multi-criteria decision making (MCDM) techniques were developed to facilitate difficulties in conforming dif- ferent ideas and managing large amounts of complicated information in the decision-making process (Zopounidis and Pardalos, 2010). These capabilities have encouraged planners to combine MCDM with other planning tools such as geographical information system (GIS). Multi- criteria decision making involves a multi-stage process of i) defining objectives, ii) choosing the criteria to measure the objectives, iii) spe- cifying alternatives, (iv) assigning weights to the criteria, and (v) ap- plying the appropriate mathematical algorithm for ranking alternatives. MCDM allows accommodating the need for unbiased integration of modern planning objectives for independent identification and ranking of the most suitable planning solutions (Mosadeghi et al., 2009). These spatial MCDM techniques are able to improve the transpar- ency and analytic difficulty of the land use decisions (Hajkowicz and Collins, 2006). Practical applications of such spatial MCDM techniques have become more widespread in land suitability studies (Chang et al., 2008; Chen et al., 2010; Greene et al., 2010; Arciniegas et al., 2011; Kordi and Brandt, 2012; Elaalem, 2012; Akinci et al., 2013). Recent studies which show application of MCDM techniques in identifying the extent of future land-use zones are rare at local scale (Mosadeghi et al., 2013). The majority of previous MCDM applications mainly focus on using MCDM to rank the priority of predefined management options or planning scenarios (Xevi and Khan, 2005; Hajkowicz and McDonald, 2006; Ananda and Herath, 2008; Hajkowicz, 2008). Spatial MCDM, however, can be used not only to rank the priority of options and performing scenario analysis, but also to provide insight into the spatial extent of the alternatives (Arciniegas et al., 2011). This capability can help local land use planners to identify land use zones for future agri- culture and urban development. It can be particularly useful in situa- tions where planning instruments do not provide prescriptive guideline for local planning decisions.

Advanced MCDM methods including SAW, AHP, TOPSIS, Fuzzy set theory and Random set theory provide more sophisticated algorithms to process uncertain or inaccurate data (Zhang and Achari, 2010; Mosadeghi et al., 2015; Nguyen et al., 2015; Prakash and Barua, 2015; Wang, 2015; He et al., 2016; Kaliszewski and Podkopaev, 2016; Montgomery et al., 2016; Onat et al., 2016). The Fuzzy set theory techniques are considered the most common techniques for dealing with imprecise and uncertain problems (Zarghami et al., 2008; Zhang and Achari, 2010; Mosadeghi et al., 2013, 2015; Montgomery et al., 2016). Most of the empirical studies have applied Fuzzy techniques without a comparative analysis to study whether using more sophisti- cated techniques like Fuzzy AHP will correctly make a significant dif- ference comparing conventional AHP. On the other hand, the few stu- dies that have done comparative analysis in land suitability applications

(Ertugrul and Karakasoglu, 2008; Elaalem, 2012; Kordi and Brandt, 2012; Elaalem, 2013) have mainly focused on arithmetic aspects such as differences in criteria weights, option rankings, or the effects of in- troducing uncertainty into their models. This need for comparative analyses carries an even greater imperative in the context of applying spatial MCDM methods to real-world cultivation priority planning de- cisions, where transparency and simplicity of the decision-making model is a key element during consultation with the stakeholders

(Mosadeghi et al., 2015).

Multiple criteria decision analysis studies use a multitude of criteria and weights derived from expert knowledge in a spatial context and using geospatial datasets (Yu et al., 2011). Multiple criteria decision analysis outputs can be used for planning purposes and to facilitate decision-making processes and tools (Stauder, 2014; Malczewski and Rinner, 2015). Multi-criteria analysis has also been used for the de- velopment of spatial decision support systems to assist decision makers in addressing complex spatial problems and to analyze the trade-offs between alternatives for a given problem (Montgomery et al., 2016). Maize, rape and soybean are important, strategic and principal as well as the major crops that are found in agricultural production sys- tems. Cereals represent the major source of dietary protein for humans

(Rótolo et al., 2015), and production figures for 2012 show that maize represented 34% of total global cereal production (FAO/STAT, 2014). Cereals such as maize are a key source of genetic material for food production, and integrate the net benefits of natural and human systems interaction through managed agro-ecosystems that we call agriculture

(Rótolo et al., 2015). Soybean is grown world-wide as an important staple and commercial crop. The reserved area for planting soybean around the world is 99,501,101 ha (FAO/STAT, 2009). Soybean ac- counted for 56% of production of the main world oilseed crops in 2011 with a total production of 251.5 million tons (ASA, 2012). Rape seed contains both high oil and protein content. Rape is one of the main winter grain crops in Iran and world (Kamkar et al., 2014). The vast majority of maize, rape and soybean oil is used in the food industry, about one third in spreads and cooking oil and about two thirds in the commercial food service sector. These crops meal, the main by product of crushed seeds, is used as a high protein feed for intensive livestock, mainly in the pig, poultry and dairy industries.

Maize grows in the temperature range of 14–40 C with optimum range 18–32 C on many types of soils. It grows in the pH range 5.2–8.5 and optimum 5.8–7.8. Any yield reduction doesn't occur at an electrical conductivity (EC) < 1 dSm−1 (IIASA and FAO, 2012). Optimum values of available phosphorus and potassium for maize growth are 14 mg kg−1 and 220 mg kg−1,respectively(Gheibi et al., 2014). Soils are suitable for maizeproductionthathaveexchangeablesodiumpercentage(ESP)<6%, gypsum<4% and calcium carbonate<15% in useful soil depth (IIASA and FAO, 2012). The mean temperature range for a proper growth of oilseed rape is 8–30 C, the optimum range being 12–22 C. Rape can be produced on a wide variety of soils with optimum values of pH, EC, ESP, gypsum, calcium carbonate, available phosphorus and potassium, 5.6–7, < 2 dSm−1,< 8%,< 2%,< 12%,> 18mgkg−1 and>195 mg kg−1, respectively (IIASA and FAO, 2012; Noorgholipour et al., 2014). The temperature range for the growth of soybean is 15–40 C. The growth is optimal at temperature between 20 and 30 C. Soybean can be grown on soils with vast variety that have optimal values of pH 5.5–7.5, EC<5.5 dSm−1, ESP<8%, gypsum<0.2%, calcium carbonate<15% (IIASA and FAO, 2012), available phosphorus > 13 mg kg−1 and potassium> 160 mg kg−1 in useful soil depth (Warncke et al., 2004). J. Seyedmohammadi et al. Geoderma 310 (2018) 178–190 179

Accordingly, the present research uses a case study to compare the outcomes of Simple Additive Weighting (SAW), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) and Fuzzy TOPSIS with regard to weights derived from Analytical Hierarchy Process

(AHP) and Fuzzy AHP in cultivation priority planning for maize (Zea mays), oilseed rape (Brasica napus L.) and soybean (Glycine max.) crops in the northwest of Iran, Ardabil province, Dasht-e-Moghan region. Studies on use of the SAW, TOPSIS and fuzzy TOPSIS models for cul- tivation priority planning are very scarce. Application of these MCDA methods is a novelty in land evaluation planning due to their special algorithm. In this way, we also used GIS technique, a comprehensive data set on crops ecological requirements, climatic, topographic, soil data and statistical analysis.

2. Material and methods

2.1. Description of the study area

The study area (about 12,000 ha) is located in the Dasht-e-Moghan region (Ardabil province, northwest, Iran) (Fig. 1), between 39 21 to 39 28 N latitude and 47 34 to 47 48 E longitude. This area has three major physiographic units: upper terraces and plateaus, alluvial fans and piedmont plain. The soil temperature and moisture regimes are

“Thermic” and “Aridic border on xeric”, respectively with regard to Newhall program (USDA, 2012). According to the US Soil Taxonomy System the dominant soils of the study area are Aridisols (Soil Survey Staff, 2014b). This area is characterized with a little rain in winter and a relatively hot arid in summer. The amount of annual rainfall is low and mostly falls in spring and autumn. The maximum rainfall is recorded in October, reaching about 34.3 mm and the mean annual precipitation is 268.5 mm in Pars-Abad station. Temperature is high during the summer months and relatively low in winter with annual mean 15.2 C. The hottest temperature is recorded in July, reaching about 27.2 C and the coldest month is January, reaching about 3.7 C. Potential evaporation is low (20.2 mm/month) in December and January when the tem- perature is comparatively low. Potential evaporation value is high

(185.5 mm/month) in July when the temperature is comparatively high. The altitude is 183 to 310 m. The area was formed in the latter part of the Miocene and the beginning of the Pliocene and Eocene periods, the surface of the area is essentially occupied by formations from the Quaternary (Ghorbani, 2013).

2.2. Processing of digital image and elevation contour layer Processing of digital images for Landsat 8 Thematic Mapper (TM) satellite image (path 167, row 33) with a spatial resolution of 30 m acquired during 2015 was performed using Erdas Imagine software. The original scan line corrector image has been replaced with estimated values based on histogram-matched scenes to improve the utility of the scan line corrector data. The TM image was stretched using linear 5%, smoothly filtered, and their histograms were matched. The used image was geometrically corrected using a rectification method. Elevation contour lines and points were used to generate raster Digital Elevation Model (DEM) with resolution of 5 5 m using ArcGIS 10.3.1 software. The physiographic units were distinguished using the processed sa- tellite image, DEM and Google earth image of study area and classified into groups. DEM map was smoothly filtered then converted to slope map.

2.3. Fieldwork, laboratory analyses and preparation of soil and land units' maps

Field studies were carried out to identify the physiographic units and the reality of soils interpretation. For this purpose, soil sampling points e.g. soil profiles were dug with between distances 1000 m. Soil changes between two profiles were controlled by an auger, if soil changes were considerably auger convert to profile. Finally, a total of 480 observation points including 167 soil profiles and 313 augers were excavated to check the accuracy of mapping units. A detailed mor- phological description of the studied soil profiles was elaborated on the basis outlined by Soil Survey Staff (2012). Soil samples have been collected from all of the horizons soil profiles and analyzed for physical and chemical characteristics using the standard analytical methods as described in Soil Survey Staff (2014a).

Soil samples were air dried, well crushed, sieved and particles < 2 mm were used for chemical analyses. Soil color in both wet and dry samples, particle size distribution, soil bulk density, exchangeable so- dium percentage (ESP), electrical conductivity (EC), soil reaction (pH), organic carbon content (OC), calcium carbonate, gypsum, available potassium, available phosphorus and cation exchange capacity (CEC) were determined by Munssel color charts, hydrometric method, clod method, ammonium acetate using flame photometer, saturation soil paste, soil water suspension 1:2, the modified Walkley and Black method, volumetrically using calcimetery method, acetone method, Fig. 1. Location of the study area and soil profiles. J. Seyedmohammadi et al. Geoderma 310 (2018) 178–190 180

extracted in the 1.0 N ammonium acetate solution (pH = 7) using flame photometer, extracted in 0.5 N NaHCO3 solution (pH 8.5) using spectrophotometer and sodium oxalate at a pH of 8.2, respectively. Finally, delineation of soil taxonomic units was determined using soil data information derived from soil profiles sampling analyzes and information of processed satellite and Google earth images. Prepared slope polygon map was overlaid with a soil taxonomic units' map which produced land units (mapping units) map, in this way the soil taxo- nomic unit delineation was incidentally protected. 2.4. Selection of the diagnostic criteria

For assessing of land units, effective criteria for crops (maize, oil- seed rape and soybean) growth should be selected. The number of these criteria should be eight according to land suitability evaluation prin- ciples (Sys et al., 1991). For this purpose, Multi-dimensional Scaling

(MDS) and Principal Component Analysis (PCA) were used for criteria selection. MDS method advises for separation similar criteria by Eu- clidean distance (Wang, 2012). In this study, MDS method was per- formed on limiting criteria for crops growth and eight criteria were selected with regard to Euclidean distance. Also, PCA method was used for selection of important criteria on crop yield (Da Sheng et al., 2011). In PCA analysis principal components that have Eigen value > 1, coefficient's maximum values of criteria for these principal components were distinguished as effective criteria.

2.5. Determination of criteria weights

2.5.1. AHP

The weights of criterion were calculated in four steps in pair-wise comparison matrix i.e. (1) formation of judgments, (2) calculation of assigned ranks, (3) preparation of normalized pair-wise comparison matrix and finally, (4) calculation of weights. Judgments of ranks were formed based on expert opinion and compared in pair-wise comparison matrix. The pair-wise comparison analysis helps for decision makers to assign different levels of importance of criteria involved in crops prioritization. Priority determination analysis studies have assigned criteria weights according to their relative importance and land char- acteristics (Elaalem, 2012). The cell values of pair-wise comparison matrix were divided by sum of the column to obtain the cell values in normalized pair-wise comparison matrix and averaged in row to cal- culate the weights of criterion (Akinci et al., 2013). These, calculated weights were scaled from 0 to 1 in ascending order to maintain hier- archy according to their importance in land suitability for wheat cul- tivation. The accuracy of the calculated weights pair-wise comparison matrix depends on the consistency of judgments of ranking the cri- terion. Consistency ratio measures logical inconsistency of the judg- ments and facilitate identification of possible errors (Cengiz and Akbulak, 2009). Saaty (2008) suggests acceptable consistency ratio values up to 0.1. Therefore, it is suggested that pair-wise comparison matrix should be revised according to improved judgment, if the con- sistency ratio excides the upper limit 0.1.

2.5.2. Fuzzy AHP

The membership function for a triangular fuzzy number (TFN) Ã shown in Fig. 2 on space μ [0, 1] with basis denoted as (l, m, u) can be defined as follows in Eq. (1):

=

−

−

−

−

μA

xl

lxm

mxu

otherwise

0,

,

,

0,

x

xl

ml

xu

mu (1)

The parameters (l, m, u) are real numbers which indicate the lowest possible value, the most favorable value, and the highest possible value

(l < m < u), respectively that describe a fuzzy case. The operational rules (algebraic operations) for two TFNs à = (a1,a2,a3) and B = (b1, b2,b3) are:

à +B = (a 123,a,a ) + (b 123,b,b ) =(a 1 + b 12,a + b 23,a + b 3 ) à = B (a 123,a,a ) (b 123,b,b ) = (a 1 b 12,a b 23,a b 3 )

=

à − 1 (1 a, 1 a, 1 a )

321

where + denotes extended summation of two TFNs, and denotes the extended multiplication.

The traditional AHP (Saaty, 1980) is extensively used in MCDA. In comparison with other methods, it is more easily understood in context of mathematical calculations. Despite its wide and successful applica- tions, it has always been criticized for its inability in managing un- certainty resulting from relating whole numbers to decision makers understanding. It is thus ineffective in application to ambiguous deci- sion problems which are prevalent in real world (Javanbarg et al., 2012). Therefore, the integration of fuzzy set theory with traditional AHP, which employs the principles of decomposition; pair-wise com- parisons; priority vector aggregation and synthesis (Taylan et al., 2014; Prakash and Barua, 2015) is vital in dealing with sources of uncertainty. It has a potential to map the perceptions, incomplete information and approximations to produce decisions by employing the membership functions. The use of fuzzy extension of AHP, Fuzzy AHP, is proper in solving the hierarchical rating of fuzzy decision problems. It allows the decision maker to concentrate on specific sub-criteria to make pair-wise comparison among the criteria that have the same root depending on their position on the hierarchy structure which yields the relative tra- deoff in the form of comparison matrices (Del Vasto-Terrientes et al., 2015). The extent analysis of Fuzzy AHP can be used to derive the priority weights from fuzzy comparison matrices. The following steps explain the procedure:

Step 1. Translating the linguistic terms used by decision makers to express the comparative judgments among the main criteria with respect to the overall goal, and evaluation criteria with respect to their main criterion into TFNs. The arrangement of the comparison matrices will be as below (Nadaban et al., 2016; Zyoud et al., 2016):

==

Aa

lmu

lmu

lmu

lmu

lmu lmu (1,1,1) (1,1,1) (1,1,1)

͠ ij n n

nnn

nnn

nnnnnn

12 12 12

21 21 21

111

222

111222 (2)

The range of used values in Fuzzy AHP utilizes the scale that is shown in Table 1 (Tesfamariam and Sadiq, 2006).

Step 2. For the aggregation of the preferences of t (decision makers) in order to build final pair-wise comparison matrix, the following method is used (Zyoud et al., 2016):

Fig. 2. Illustration of fuzzy triangular number (TFN). J. Seyedmohammadi et al. Geoderma 310 (2018) 178–190 181

=

== =

=

W LwMwUw

Lw min Lw Mw Mw Uw max Uw ij ij ij ij

ij t ijt ij T t

T

ijt ij t ijt

1

1 (3)

Wij= the triangular fuzzy weight of the ith criterion in comparison with the jth criterion.

Step 3. Determining the weights for the involved criteria. The synthetic pair-wise comparison matrix is computed using geometric mean method. Geometric mean ri is defined as:

= …

…

…

= rLwLw LwMwMw Mw

Uw Uw Uw

rLwMwUw i ij ij ijn ij ij ijn

ij ij ijn

igm gm gm

12 12

12

nn

n

11

1

(4)

gm: geometric mean.

Step 4. The weight for each criterion is determined. This is done by normalizing the matrix.

=

+…

+

+…

+

+…

+

−

−

−

−

wrrr r

w

Lw Uw Uw Mw

Mw Mw

Uw Lw Lw cccc cn

c

gmc gmc gmcn gmc

gmc gmcn

gmc gmc gmcn

1112

1

1

11

1

1

1

1

11

1

(5)

wc1: fuzzy weight (triangular fuzzy number) of first criterion, c1 = first criterion in comparison matrix.

2.6. Determination methods of cultivation priority 2.6.1. Simple additive weighting (SAW)

SAW is one of the most often used techniques for resolving spatial decision analysis problems. The decision maker directly assigns weights of relative importance to each attribute. A total score is then obtained for each alternative (crop) by multiplying the importance weight as- signed for each attribute by the scaled value given to the alternative for that attribute and summing the products over all attributes. The nor- malized value for positive criteria is calculated as fallow: n == …= …

g

g

ij imjn 1,, 1,,

ij

max (6)

And for negative criteria:

n == …= …

g

g

ij imjn 1,, 1,,

min

ij (7)

Final score = ( wn g ij ) w gij = 1

ij (8)

where gij: criterion value, gmax: maximum value for each positive cri- terion, gmin: minimum value for each negative criterion, nij: normalized value.

2.6.2. TOPSIS

The TOPSIS (technique for order performance by similarity to ideal solution) was first developed by Hwang and Yoon (1981). According to this technique, the best alternative would be the one that is nearest to the positive ideal solution and farthest from the negative ideal solution

(Ertugrul and Karakasoglu, 2007). The positive ideal solution is a so- lution that maximizes the benefit criteria and minimizes the cost cri- teria, whereas the negative ideal solution maximizes the cost criteria and minimizes the benefit criteria (Wang and Elhag, 2006). In short, the positive ideal solution is composed of all best values attainable from the criteria, whereas the negative ideal solution consists of all worst values attainable from the criteria (Wang, 2008). There have been lots of studies in the literature using TOPSIS for the solution of MCDM pro- blems (Dagdeviren et al., 2009). The computational steps of the TOPSIS method are presented in the following steps (Lozano et al., 2016; Onat et al., 2016):

Step 1. Establishing a performance decision matrix:

=

A

aa

aa

aa

a

a

a

ij

mm

n

n

mn

11 12

21 22

12

1

2

(9)

Step 2. Calculating the normalized decision matrix. The normalized value rij is calculated as follows and normalized decision matrix as Eq. 11:

=

=… =…

=

r

a

a

ij ; ( im 1, 2,, ); ( jn 1, 2,, )

ij

i

m

1 ij

2

(10)

=

R

rr

rr

rr

r

r

r

ij

mm

n

n

mn

11 12

21 22

12

1

2

(11)

Step 3. Calculating the weighted normalized decision matrix. The weighted normalized value vij is calculated as follows:

= =

V RW

vv

vv

vv

v

v

v

ij ij n n

mm

n

n

mn

11 12

21 22

12

1

2

(12)

where wj is the weight of the jth criterion or attribute and j = W = 1 n

1 j

.

Step 4. Determining the positive ideal (Ai

+) and negative ideal (Ai

−)

Table 1

Fuzzy numbers used for making pair-wise comparisons. Relative importance aFuzzy scale bDefinition Explanation 1 (1, 1, 1) Equal importance Intermediate values between two adjacent judgments 3 (3 −Δ

c,3,3 +Δ) Weak importance Experience and judgment slightly favor one activity over another 5 (5 −Δ,5,5 + Δ) Essential or strong importance Experience and judgment strongly favor one activity over another 7 (7 −Δ,7,7 + Δ) Demonstrated importance One activity is strongly favored and demonstrated in practice 9 (8, 9, 9) Demonstrated importance One activity is strongly favored and demonstrated in practice 2, 4, 6, 8 (x −Δ,x,x + Δ) Intermediate values between two adjacent judgments When compromise is needed 1

x

(1/(x + Δ), x, 1/(x −Δ))

1

9

(1/9, 1/9, 1/8)

a The intensity of importance definition is in accordance with the description proposed by Saaty (1980). b Minimum, most likely, and maximum values.

c

Δ is a fuzzification factor.

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