Research Article Grey Weighted Sum Model for Evaluating ...downloads.hindawi.com/journals/mpe/2016/3824350.pdf · WSM method has been improved and applied in many areas over the years

Research ArticleGrey Weighted Sum Model for Evaluating BusinessEnvironment in West Africa

Moses Olabhele Esangbedo and Ada Che

School of Management Northwestern Polytechnical University Xirsquoan 710072 China

Correspondence should be addressed to Ada Che achenwpueducn

Received 9 January 2016 Revised 17 April 2016 Accepted 4 May 2016

Academic Editor Rafael Morales

Copyright copy 2016 M O Esangbedo and A CheThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

As West Africa investments grow the decision in which country to begin investment is of great importance to investors Thecomplexity of the criteria involved draws us to use a Multicriteria Decision-Making (MCDM) approach to address this problemIn this paper we use grey numbers in representing ranges of data and propose Grey Weighted SumModel (GWSM) for evaluatingand ranking of alternatives Sensitivity analysis is carried out considering wide ranges of uncertainties to verify the changes thatcan affect the results The Gambia is ranked the best country in West Africa The GWSM is highly recommended for long-terminvestors because GWSM considers the uncertainty of a business environment over a period of years Finally GWSM can be usedin conjunction with various weighting techniques putting the preferences of the investors into consideration

1 Introduction

West Africa is becoming part of the growing economy and itis a land of opportunitiesThe Economic Community ofWestAfrican States (ECOWAS) includes 15 member countries richinmineral resources andwithmillions of consumers As busi-nesses grow and want to expand their services to more coun-tries the country to begin investment is very important to theinvestors The ability to rank a country in relation to othersafter evaluating their business environment gives an investora quick insight in making a decision on where to invest itsresources for a good turnover A poor decision can easilylead to business failure and a huge loss to the investor Thusthe techniques and approaches for evaluating the businessenvironment of various countries are indispensable Not onlydoes the investor benefit from businesses but also it plays asignificant role in the growth of poorer countries [1]

For businesses to be successful there is a need to balanceshort-term and long-term goals in an uncertain businessenvironment like West AfricaThere are some levels of doubtfor an entrepreneur who has not invested in West AfricaSelecting the country to invest in West Africa becomes com-plicated across these countries considering several criteriaand the level of unequal uncertainties One of the symptomsof a poor decision is ignoring uncertainties [2] When doing

business inWest Africa uncertainties should be analysed andanticipated Every good Decision Maker (DM) knows thatthere are uncertainties investing in West Africa Howeverwhen these uncertainties are considered in decision-makingit increases the DMrsquos confidence on the decision outcome [3]

The World Bank (WB) launched the Doing BusinessProject (DBP) in 2002 making available annual objectivemeasures of business regulations and their enforcementacross 189 economies to the public The DBP delivers theranking of various countriesrsquo business environment annuallyThe data provided by DBPWB encourages countries towork towards better regulations and reform benchmarksAlso the data provides resources for academics journalistsresearchers and those interested in the business environmentof various countries [4] However the ranking values pro-vided by the DBP are annual values that do not reflect thebusiness environment performances over a period of yearsSince most businesses are not expected to survive for just ayear of operations it can be erroneous for investors to make adecision from a single year of objectivemeasures and rankingof the business environments especially for the long-terminvestment

The evaluation of business environment is addressedusing Multicriteria Decision-Making (MCDM) approachwhich can handle multiple conflicting criteria and closely

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3824350 14 pageshttpdxdoiorg10115520163824350

2 Mathematical Problems in Engineering

related alternatives Future data cannot be known from onlycurrent data but it is most likely that future data may bewithin the ranges of already known past and current datawhich are represented as interval grey numbers Althoughthe uncertainties of the future are absolute grey numbersare used to model the uncertainties of the future caused byincomplete and insufficient information inmaking an invest-ment decision This paper proposes an original approachcalled Grey Weighted Sum Model (GWSM) in evaluatingthe business environment in West Africa by accounting foruncertainties over a period of yearsThe contributions in thispaper are twofold Firstly we address a major problem notcovered by theDBP that is ranking countries over a period ofyears considering the degree of uncertainty in the country aswell as investors preferences represented as criteria weightsSecondly we extend the traditional Weighted Sum Model(WSM) by using grey numbers to represent performances ofevaluation criteria that may vary from time to time

This work is presented as follows Section 2 is LiteratureReview and Section 3 is GreyWeighted SumModel (GWSM)Section 4 presents the results and some discussions Thesensitivity analysis using GWSM is given in Section 5 Finallya conclusion is drawn in Section 6

2 Literature Review

There are several methods for solving and evaluating selec-tion problems in MCDM [5ndash7] Some of these methodsinclude the Analytical Hierarchy Process (AHP) developedbyThomas Saaty for evaluating alternatives by comparing allthe criteria of the alternatives [8] The Analytical NetworkProcess (ANP) was later introduced by Thomas Saaty as amethod to evaluate alternatives with interdependent decisioncriteria and alternatives [9] Zavadskas et al developed theCOmplex PRoportional ASsessment (COPRAS) for eval-uating alternatives which measure the complex efficiencyof an alternative as a proportion of the minimum andmaximum criteria values [10 11] Data Envelopment Analysis(DEA) is a method for evaluating the effective efficiencyof decision-making unit Shakouri et al [12] used the DEAin conjunction with Simple Additive Weighting (SAW) toassess power supply technologies Elimination and ChoiceExpressing Reality (ELECTRE) was developed as early as inthe 1960s [13] and has had more than four major iterativeimprovements to address the problem of not being ableto rank alternatives Several improvements that have beenintroduced to ELECTRE include ELECTRE I-IV ElECTRE-IS and ELECTREndashTRI [14] Technique for Order Preferencesby Similarity to an Ideal Solution (TOPSIS) was introducedby Hwang and Yoon [15] in selecting the best alternative as acompromise solution TOPSIS method selects the alternativewith the shortest distance to the ideal solution and farthestdistance to the negative ideal solution these distances areusually measured as Euclidean distances Evidential Reason-ing Approach (ERA) [16] evaluates alternatives with qualita-tive and quantitative criteria based on Dempster-Shafer (D-S) theory of evidence [17] with some level of ignorance GreyRelational Analysis (GRA) [18] is based on the Grey SystemTheory (GST) and uses the relative relational coefficient of

evaluation alternatives to the ideal alternative in determiningthe best alternative Multiattribute Utility Theory (MAUT)[19] is a classical method for evaluating alternatives by aggre-gating the satisfaction obtainable from every criterion mea-sured as utils Preference Ranking Organization Method forEnrichment of Evaluation (PROMETHEE) [20] was devel-oped by Jean-Pierre Brans which is an outranking methodResearchers have widely accepted PROMETHEE and theyhave developed it to a fully ranking technique Superiorityand Inferiority Ranking (SIR) [21] is a combination of theWSM and PROMETHEE where WSM is used for aggrega-tion of procedure and PROMETHEE for ranking proceduresSimple Multiattribute Rating Technique (SMART) [22] is anMCDM method in which the DM assigns points to everycriterion for all alternatives as well as the relative criteria forestimating the weights for evaluation SMART Extended toRanking (SMARTER) [23] is an improvement of SMART byusing rankings of the criteria for weight estimation

The relative importance of evaluation criteria for anMCDM problem is expressed using weights by the DM(s)With clear goal and alternatives the method of assigningweights to the evaluation criteria must be determined bymaking sure that it can produce themost accurate ratio of thecriteria based on the relative levels of importance Subjectiveor objective weighting methods [24] for MCDM problemscan be used in determining the weights For instance groupdecision-making process can be used in estimating theweights as well as the introduction of linguistic value becauseof the way humans think [25] Li et al [26] applied greyarithmetic mean weighting method in aggregating the groupweights of DMs in software selection Wang [27] appliedthis same technique to the selection of personnel using greylinguistic values for measuring the preferences of the groupDMs Similarly Bai and Sarkis Kose et al Kuang et aland Mehrjerdi [28ndash31] all used grey arithmetic mean incomputing the weights of criteria in a group decision-makingenvironment Kang et al [32] applied grey geometric mean inaggregating the weights of the DMs to determine the suitablemeeting location based on the service offered and the prefer-ences of the DMs Grey correlation operators can be used tocompute the relationships among all theDMspreferences andassign weights to the criteria based on the dependences in therelationship of the preferences [33] Other grey correlationoperators are Interval Grey Linguistic Variables OrderedWeighted Aggregation (IGLOWA) Interval Grey Linguis-tic Correlated Ordered Arithmetic Aggregation (IGLCOA)and Interval Grey Linguistic Correlated Ordered GeometricAggregation (IGLCOGA) [34] Jin et al [35] estimated theweights of the criteria using grey harmonic operators whichis based on the number of occurrences of theweights assignedby the group of DMs Esangbedo and Che [36] proposedthe Grey Rank Order Centroid (GROC) weights for groupdecision-making and applied it to evaluate African businessenvironment in a specific year rather than over a period ofseveral years addressed in this study based on the GRAmethod

The main techniques closely related to this paper areWSM and GST WSM can be described as a weighted linearcombination or scoring method [37 38] The WSM is also

Mathematical Problems in Engineering 3

called SAW and for consistency in this paper we use theterm WSM for both WSM and SAW [39 40] WSM is arepresentation of the DMrsquos preferences in a linear additivefunction [41] Triantaphyllou and Mann [42] examined theeffectiveness of decision-making methods and pointedWSMas one of the simplest methods for solving MCDM problemsThe best alternative is the one with the maximum scoresatisfying (1) after transforming all measurement criteria toa single dimension

119860lowast

WSM = max1⩽119894⩽119898

119899

sum

119895=1

119886119894119895119908119895 (1)

where119860lowastWSM is the score of the best alternative 119886119894119895represents

the value of the 119895th criterion of the 119894th alternative and 119908119895

corresponds to the weight of the 119895th criterion where 1 le 119894 le119898 and 1 le 119895 le 119899

From (1) the intuitive nature and ease of dealing withMCDM problems can be seen and this simplicity makesWSM one of the most important methods for both exploringpossible solutions for an MCDM problem and providingsolutions in comparison to other methods [5] A lot ofMCDM methods including the AHP and PROMETHEEuse weighted sum aggregation technique in selecting the bestalternative [38] Structuring of anMCDMproblem can easilybe done using WSM by making sure that the problem hasa clear objective criteria and alternatives The standardizedscores using the WSM have an equal relative order of magni-tude because the linear transformation is proportional to theraw data The simplicity of implementation provided by theWSM is a form of transparency of themodel to the DMs [43]WSM method has been improved and applied in many areasover the years Goh et al [44] introduced revised weightedmethod to WSM to reduce the effect of enormous dissimilardecision preferences on the decision results by eliminatingthe minimum and maximum rates on the decision criteriaduring the evaluation process Triantaphyllou and Lin [45]introduced fuzzy numbers andWSM combination approachIn simplifying the complexity and reducing the round-offerrors introduced by using fuzzy numbers inWSMModarresand Sadi-Nezhad [46] used preferences ratios in comparingfuzzy numbers Chou et al [40] proposed a fuzzy simpleadditive weighting system in determining the best locationfor facilities in a groupdecision-making environment Zavad-skas et al [47] combined weighted product model and WSMto form an evaluation method called Weighted AggregatedSum Product Assessment (WASPAS) Then Zavadskas et alfurther extended WASPAS to interval-valued intuitionisticfuzzy numbers as well as grey numbers because of the vague-ness in human judgement and preferences [48 49] Stanujkicand Zavadskas [50] introduced compensation coefficientvalues to WSM helping DMs select among the best rankingalternatives and the alternatives that best meet their prefer-ences Chen [51] applied WSM in group decision-making byusing interval type-2 fuzzy sets for linguistic and incompletepreference measurement Wang [52] addressed the drawbackof fuzzy multiplication in WSM by using relative preferencesto represent the weights of the criteria Zamri and Abdullah[53] combined linear programming and WSM in the context

of an interval fuzzy set for ranking alternatives Xu et al[54] proposed a discrete model for conflict-elimination fordetermining the weights of group experts in decision-makingand applied it to WSM in choosing the best alternative

GST is a mathematical concept of grey set developed byProfessor Deng [55] GST is a method capable of solvinguncertain problems with incomplete information and dis-crete data GST deals with information between a black anda white part which represent unknown and known infor-mation respectively GST has different sections that includeGRA grey decision grey programming grey prediction andgrey control Grey numbers are used in representing systemswith incomplete information GST has been applied in busi-ness management like project management and planning aswell as the stock market and portfolio selection [18 56 57]When considering MCDM with the advancement to greynumbers Xu and Sasaki [58] extended the technique of orderpreference by similarity to ideal and anti-ideal alternativesby measuring the closeness coefficient to the ideal alternativeusing grey numbers Li et al [59] used arithmetic mean forgroupweights of Grey PossibilityDegree (GPD) that providesthe position relationship between two grey numbers for rank-ing Turskis andZavadskas [60] applied grey number inAddi-tive Ratio Assessment (ARAS) for ranking ARAS is a ratioof the optimal value of the alternatives to the optimal valueof the ideal alternative resulting to utility degrees Liu et al[61] proposed a method for ranking interval numbers basedon a normal distribution and used it to optimize the miningmethods Mousavi et al [62] addressed the problem ofuncertainty in multicriteria optimization and compromisedsolution by using grey numbers in VIKOR method forranking material handling equipment VIKOR is an acronymfor the Slovenian phrase ldquoVlseKriterijumska Optimizacija IKompromisnoResenje in Serbianrdquowhichmeansmulticriteriaoptimization and compromise solution In contrast to TOP-SIS VIKOR calculates the relative alternative distance to theideal solution Oztaysi [63] combined grey number withAHPfor weighting the criteria and used TOPSIS ranking methodfor the selection of Content Management System (CMS)

Zavadskas et al [64] proposed SimpleAdditiveWeightingwith Grey (SAW-G) number and applied it in contractorselection for constructionworks Also SAW-Gwas applied inevaluating the performance of rural ICT centers in Iran [65]Nonetheless the SAW-G technique does not completely showthe degree of uncertainty in ranking because the boundarydistances between the lower and upper bounds of the greynumbers are not considered in the evaluation For examplesuppose the evaluated investment outcome measured inunits for the first and the second investment alternativesrepresented by grey numbers are otimes119868

1= [119894 7119894] and otimes119868

2=

[3119894 5119894] respectively That is the investment outcomes of thefirst alternative is between 119894 and 7119894 and the second alternativeis between 3119894 and 5119894 Using the SAW-G technique the whitevalue of these two investments is 4119894 Although these invest-ments have the same white value the degrees of uncertaintiesare not equal From the grey numbers the first investment hasa greater degree of uncertainty than the second investment Inother words we are more guaranteed about the outcome ofthe second investmentThe GWSM addresses this limitation


3 Grey Weighted Sum Model

GWSM is an extension of the WSM based on grey numbersFor this model we use an interval type of grey numbers tomodel the uncertainties A grey numberwith lower andupperbounds is called an interval grey number and it is representedas otimes119866 = [119892 119892] where 119892 and 119892 denote its lower and upperbounds respectively Some basic operations of two grey num-bers otimes119866 = [119892 119892] and otimes119867 = [ℎ ℎ] where ℎ is the lower boundand ℎ is the upper bound are as follows [66 67]

otimes119866 + otimes119867 = [119892 + ℎ 119892 + ℎ]

otimes119866 minus otimes119867 = [119892 minus ℎ 119892 minus ℎ]

otimes119866 times otimes119867 = [119892 119892] times [ℎ ℎ] = [119892ℎ 119892ℎ]

(2)

If 119892 is a white value that is a real number or crisp values

119892 times otimes119867 = [119892ℎ 119892ℎ] (3)

The traditional WSM makes use of real numbers forthe evaluation of alternatives Interval number is a formof uncertain values and GST deals with uncertain valuesthat are represented as interval grey numbers [68 69] Themain goal of considering uncertainty in decision-makingis to provide the DMs with a holistic view of the decisionproblem to deliberate and reason confidently [70] In dealingwith uncertainty in decision-making introducing GST toWSM is introducing reasonable slacks to the performanceof the alternative that will be used in the evaluation fordecision-making Business environment is dynamic forexample government policies natural environment taxesinterest and exchange rates do change GST is designed toanalyse systems with incomplete information and systemswhose ranges of performances are known These rangesare represented as grey numbers In the traditional WSMthe values of the criteria and weights are fixed values butfor some MCDM problems the performances of criteriamay vary within a range of values WSM cannot be usedto evaluate alternatives with varying performances of thedecision criteria The inability for the WSM to evaluatealternatives with uncertain criteria measurement representedas interval numbers is the limitation of the traditional WSMthat the GWSM addresses The primary concept of GWSMis using weighted grey numbers of the evaluation criteriafor the assessment of alternatives and putting the degree ofuncertainty into consideration using the boundary distanceof the criteria Its main procedure is explained as followsFirstly we construct a grey decision matrix and calculatea normalized grey decision matrix Then we aggregatethe weighted normalized grey decision matrix to obtain agrey value for all alternatives Next the boundary distancesof the alternatives are calculated and used in estimatingthe white values Lastly these white values are ranked todetermine the best alternative The GWSM steps are givenbelow

Step 1 (construct the grey decision matrix) The decisionmatrix119883 is represented by

119883 =(

otimes11990911

otimes11990912

sdot sdot sdot otimes1199091119899

otimes11990921

otimes11990922


d

otimes1199091198981

otimes1199091198982


) (4)

where otimes119883119894119895= [119909119894119895 119909119894119895] which represents the grey number of

the 119895th criterion of the 119894th alternative Also every alternativecan be written in a vector form

119883119894= (otimes119909

1198941 otimes1199091198942 otimes119909

119894119899) (5)

Step 2 (normalize the grey decision matrix) The normaliza-tion step is to make the criteria measurement in the samedirection Cost and benefits preferences are in two directionsof measurement For the benefits preferences that is thehigher the value the better the value they are normalized asfollows

otimes119909lowast

119894119895= [

119909119894119895minusmin

1le119894le119898119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

119909119894119895minusmin

1le119894le119898119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

]

(6)

For the cost preferences that is the smaller the value thebetter the value they are normalized as follows

otimes119909lowast

119894119895= [

max1le119894le119898

119909119894119895minus 119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

max1le119894le119898

119909119894119895minus 119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

]

(7)

Then a normalized decision matrix is constructed

119883lowast

=(

otimes119909lowast

11otimes119909lowast

12sdot sdot sdot otimes119909

lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

) (8)

In a vector form we define

119883lowast

119894= (otimes119909

lowast

1198941 otimes119909lowast

1198942 otimes119909

lowast

119894119899) (9)

Step 3 (determine the weights of the criteria) The weights119882used for the criteria are those assigned by experts or obtainedfrom theDMs using any of the techniques in Section 2 Eithergrey weights or white (crisp) weights that is weights in realnumbers can be used in this step Ideally grey weights shouldbe used when there is some level of uncertainties about theweights

For grey weights

119882 = (otimes1198821 otimes1198822 otimes119882

119899)119879

(10)


where otimes119882119895= [119908119895 119908119895] and 119908

119895and 119908

119895are the weightrsquos lower

and upper bounds of the 119895th criterionThe weights are scaledsuch that the summation of the upper bounds of the scaledweights is a unit value The weights are scaled using

otimes1198821015840

119895=

otimes119882119895

sum119899

119895=1119908119895

(11)

where otimes1198821015840119895= [1199081015840

119895 1199081015840119895] and sum119899

119895=11199081015840119895= 1

Therefore the scaled grey weight is

1198821015840

= (otimes1198821015840

1 otimes1198821015840

2 otimes119882

1015840

119899)119879

(12)

For white (crisp) weights

119882 = (1199081 1199082 119908

119899)119879

(13)

where 119908119895is the weight of the 119895th criteria Then after scaling

we have

1198821015840

= (1199081015840

1 1199081015840

2 119908

1015840

119899)119879

(14)

where 1199081015840119895= 119908119895sum119899

119895=1119908119895and sum119899

119895=11199081015840

119895= 1

Step 4 (aggregate the weighted decision matrix) This stepis the sum of the weighted normalized criteria for all thealternatives

119884 = 119883lowast

times1198821015840

(15)

For grey weights

119884 =(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

)(

(

otimes1198821015840

1

otimes1198821015840

2

otimes1198821015840

119899

)

)

(16)


119884 =(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

)(

(

1199081015840

1

1199081015840

2

1199081015840

119899

)

)

(17)

Therefore

119884 =(

otimes1199101

otimes1199102

otimes119910119898

) (18)

where otimes119910119894= [119910119894

119910119894] is a grey number which represents the

weighted sum of all the criteria for the 119894th alternative

Step 5 (obtain the white values of the alternatives) This stepis key in transforming the grey number to a white numberWhitenization is conducted using the following equation

119881119894= 119910119894

(1 minus 120582) + 119910119894120582 (19)

where the whitenization coefficient 120582 isin [0 1]

Step 6 (determine the boundary distance of the alternatives)In this step we calculate the degree of uncertainty 119878

119894 for the

119894th alternative 119878119894is defined as follows

119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

(20)

when 119901 = 1 the boundary distance is the Manhattandistance when 119901 = 2 the boundary distance is the Euclidiandistance and when 119901 = 3 the boundary distance is theMinkowski distance with 3rd degree

As 119878119894tends to zero theoretically the stability of the out-

come for the future investment tends to 100 In other wordsthere is no uncertainty

Step 7 (rank the alternatives) In ranking the alternatives theuncertainty of thewhite value obtained in Step 5 is consideredin calculating the rank scores Knowing the degree of uncer-tainty we can estimate the degree of certainty since the sumofboth degrees is a value of one The rank score is the productof the white value and the degree of certainty as follows

119911119894= 119881119894(1 minus 119878

119894) (21)

From the above formula as uncertainty decreases and thewhite value increases then the rank score increases resultingin a better rank value From the rank scores relative percent-age rank scores of all alternatives to the best score are alsocalculated The alternative with 100 is the best alternativePercentage rank scores are computed as follows

119885119894= [

119911119894

max (119911119894)] times 100 (22)

We note that the above procedure is similar to the oneproposed by Zavadskas et al [64 65] However we take intoaccount the degree of uncertainty and grey weights in ourprocedure It is not uncommon to see hybrid approaches insolving MCDM problems for example Fuzzy-AHP Fuzzy-TOPSIS Grey-AHP andGrey-TOPSIS [71ndash74]The approachwe propose above can be combined with different weightingapproaches to form several hybrid methods for instanceusing fuzzy weight method to form a fuzzy-GWSM and pair-wise comparison weighting method to form an AHP-GWSMmethod

4 Results and Discussion

In this study we use GWSM method for evaluating businessenvironment of West African countries Weights help torecognize the relative importance of the decision criteria andthey have a direct influence on the ranking of alternatives


Measuring the weights of criteria is part of the problem inthe evaluation of alternatives the DBP used equal weights inevaluating these countriesThe preferences of some investorsare used to estimate the weights of the evaluation criteriausing the GWSM method We use the region of West Africafor our sample and the data for these countries are obtainedfrom the DBP database which are then transformed to greynumbers The transformation is done by using the lowestand highest measurement values of these countries for everycriterion between the year 2008 and the year 2015 as the lowerand upper bounds of the grey number The grey numbersof these criteria and the weights are aggregated to give theweighted grey number for each country Ranking operationsare done using GWSM method and sensitivity analysis iscarried out to obtain ranges of ranking results

41 Evaluating Business Environment in West Africa For thiswork we focus on West African countries According toEncyclopaeligdia Britannica [75] the western African coun-tries include Benin Burkina Faso Cameroon Cape VerdeChad Cote drsquoIvoire Equatorial Guinea the Gambia GhanaGuinea Guinea-Bissau Liberia Mali Mauritania NigerNigeria Senegal Sierra Leone andTogo In total 19 countriesare evaluated These countries are indexed from 1 to 19 inalphabetical order Benin has an index number of 1 BurkinaFaso has an index number of 2 andTogohas an index numberof 19

The decomposition of a complex problem into a hierarchyis a very common method for solving problems similarto the ldquodivide and conquerrdquo approach Table 1 shows ahierarchy of the evaluation criteria given by the DBP [3676] The hierarch the goal is ldquoBusiness Environment ofCountriesrdquo The first-level hierarchy is the first-level criteriaand the second-level hierarchy is the second-level criteriaThe second-level criteria are indexed from 1 to 37 as givenin Table 1 The DBP gives a comprehensive explanation ofthese criteria at httpwwwdoingbusinessorgmethodologyThese countries are evaluated based on the criteria presentedby the DBP

Using the GWSM we present an evaluation of thebusiness environment in West Africa Following the steps inthe previous section the evaluation is presented below

Step 1 (construct the grey decision matrix) The decisioncriteria are shown in Table 1 There are in total 37 second-level criteria indexed from 1 to 37 The values of the second-level criteria obtained from the DPBWB database from 2008to 2015 are transformed into grey numbers using

otimes119909119894119895= [119909119894119895 119909119894119895] (23)

where

119909119894119895= min (V2008

119894119895 V2009119894119895

V2015119894119895

)

119909119894119895= max (V2008

119894119895 V2009119894119895

V2015119894119895

)

(24)

where V119894119895is the value for the second-level criteria 119895 for country

119894 given by the DBP for the year 2008 to year 2015 1 le

119894 le 19 and 1 le 119895 le 37 To simplify the notation wedefine 119883

119894= (otimes119909

1198941 otimes1199091198942 otimes119909

11989437) For example 119883

1=

([7 10] [12 34] [558 1981] [63 3542] [9 13] [111 317][24 38] [0 5] [0 172] [0 146549] [4 4] [120 120][117 119] [3 6] [1 1] [78 109] [0 0] [6 7] [1 1][3 5] [55 55] [270 270] [0 159] [0 264] [0 21] [7 8][25 34] [987 1101] [7 9] [25 38] [1222 1569] [750 825][647 647] [41 42] [4 4] [145 215] [167 226]) Otherdata are omitted here and the complete data are providedby the DBP at httpwwwdoingbusinessorgcustom-queryThe first element of the vector119883

1is otimes11990911

and it correspondsto the number of Procedures in Starting a Business in theRepublic of Benin with a lower bound of 7 and an upperbound of 10 All the elements of the vector 119883

119894have similar

corresponding lower and upper bounds to the second-levelcriteria of every country

Step 2 (normalize the grey decision matrix) A normalizedgrey decision matrix is constructed using (8)

For example 119883lowast1= ([05333 07333] [08841 09705]

[08418 09638] [0651 09938] [0871 1] [05351 0926][09846 09923] [04444 1] [0706 1] [00416 1] [1 1][0641 0641] [04537 04634] [01429 05714] [0166701667] [03391 04739] [0 0] [0 01667] [1 1] [0428607143] [03158 03158] [08767 08767] [0492 1] [023031] [09182 1] [06 08] [06667 0803] [09317 0951][05714 08571] [07273 0875] [08937 09353] [0618506706] [06473 06473] [04118 04706] [08 08] [0241703583] [04441 06011]) Other data are omitted here too

Step 3 (determine the weights of the criteria) Grey linguisticvalues are used to estimate the weights of 7DMs based onthe group aggregating technique used by some authors [26ndash31] The DMs are some Chinese investors that want to investin Africa A questionnaire in Chinese was designed to obtainthe weights of the criteria Grey linguistic values are used tomeasure the DMsrsquo preferences in weights as shown in Table 2The raw data can be found in [36]

Due to its simplicity of implementation the grey arith-metic mean method is used to aggregate the weights (119882)The weights are scaled such that the summation of the upperbounds of the scaled grey weights is a unit value After scalingusing (11) we have

1198821015840

= ([00144 00256] [00151 00265]

[00123 00229] [00123 00229] [00123 00229]

[00129 00237] [00123 00229] [0011 0021]

[00131 0024] [00104 00203] [00158 00274]

[00188 00313] [0015 00264] [00201 00331]

[0018 00303] [00144 00256] [00144 00256]

[00219 00354] [00238 00377] [00219 00354]

[00144 00256] [00115 00218] [00165 00284]

[00172 00294] [00144 00256] [00123 00229]


Table 1 Hierarchy of the evaluation criteria

Goal First-level criteria Second-level criteria Index (119896)

Business Environment of Countries

Starting a Business

Procedures 1Time 2Cost 3

Paid-in Minimum Capital 4

Dealing with Construction PermitsProcedures 5

Time 6Cost 7

Getting ElectricityProcedures 8

Time 9Cost 10

Registering PropertyProcedures 11

Time 12Cost 13

Getting Credit

Strength of Legal Rights Index 14Depth of Credit Information Index 15

Credit Registry Coverage 16Credit Bureau Coverage 17

Protecting Minority InvestorsExtent of Disclosure Index 18

Extent of Director Liability Index 19Ease of Shareholder Suits Index 20

Paying Taxes

Payments 21Time 22

Profit Tax 23Labour Tax and Contributions 24

Other Taxes 25

Trading Across Borders

Documents to Export 26Time to Export 27Cost to Export 28

Documents to Import 29Time to Import 30Cost to Import 31

Enforcing ContractsTime 32Cost 33

Procedures 34

Resolving InsolvencyTime 35Cost 36

Recovery Rate 37

Table 2 Grey linguistic values and their grey numbers

Linguistic values Grey weightsUnimportant [0 02]Somewhat important [02 04]Moderately important [04 06]Important [06 08]Extremely important [08 1]

[00178 00302] [00185 00311] [00123 00229]

[00171 00293] [00171 00293] [0018 00303]

[00171 00293] [0018 00303] [00112 00212]

[00137 00246] [00155 00272])119879

(25)


Step 4 (aggregate the weighted grey decision matrix) Theaggregated weights are calculated using (15)

119884 = (otimes1199101 otimes1199102 otimes1199103 otimes1199104 otimes1199105 otimes1199106 otimes1199107 otimes1199108

otimes 1199109 otimes11991010 otimes11991011 otimes11991012 otimes11991013 otimes11991014 otimes11991015 otimes11991016

otimes 11991017 otimes11991018 otimes11991019)119879

119884 = ([03222 0738] [02841 06751]

[03692 07732] [03045 06857] [02471 06091]

[03204 07043] [03298 06677] [03757 07671]

[03688 07927] [03307 06977] [02645 06564]

[03211 06929] [06929 03142] [03249 06658]

[031 06878] [02643 07138] [0295 07349]

[0263 07164] [03232 07098])119879

(26)

Step 5 (obtain the white values of the alternatives) From (19)where the center whitenization coefficient of 120582 = 05 is takenwe have 119881

1= 05301 119881

2= 04796 119881

3= 05712 119881

19=

05165 The results for Steps 5ndash7 are shown in Table 3

Step 6 (determine the boundary distance of the alternatives)UsingManhattan distance for measuring the distance of (20)we have 119878

1= 04157 119878

2= 03910 119878

3= 04040 119878

19=

03866

Step 7 (rank the alternatives) The rank scores 119911119894are calcu-

lated using (21) 1199111=03097 119911

2=02921 119911

3= 03405 119911

19=

03168 The percentage rank scores 119885119894are calculated using

(22)1198851=8906119885

2=8400119885

3=9791 119885

19=9111

Ranking these countries from the 1st position is as fol-lows the GambiagtCapeVerdegtGhanagt Equatorial Guineagt Mauritania gt Guinea gt Liberia gt Togo gt Cote drsquoIvoire gtNiger gt Benin gt Cameroon gtMali gt Burkina Faso gt Senegalgt Guinea-Bissau gt Chad gtNigeria gt Sierra Leone AlthoughEquatorial Guinea is ranked the 4th position it has the lowestdegree of uncertainty for investment outcome with a value of03379 Sierra Leone the least ranked country has the highestdegree of uncertainty with a value of 04533 as reflected by theboundary distance

42 Comparison between DBP and GWSM Results TheDBPuses equal weights for the evaluation of business environ-ment Equal weights may be regarded as an acceptablerepresentation of the criteria weights when the weights areunknown Nevertheless there is little or no disadvantagein measuring the weights of investors who intend to investin West Africa The investors can be regarded as the DMsTable 4 shows the cumulative performances and rankings ofWest African countries based on the evaluation method ofthe DBP from the year 2008 to 2015 [77] The DBP rankingsfor the year 2008 to 2015 are compared with the GWSM fromthe year 2008 to 2015 Figure 1 shows a clustered bar graphwith the results of these two methods where the rankings ofthe cumulative DBP GWSM using grey weights and GWSM

Table 3 GWSM rankings for the year 2008 to the year 2015

Index (119894) Countries 119881 119878 119911 119885 () Rankings1 Benin 05301 04157 03097 8906 11

2 BurkinaFaso 04796 03910 02921 84 14

3 Cameroon 04951 03812 03063 8809 124 Cape Verde 05712 04040 03405 9791 25 Chad 04281 03620 02731 7854 17

6 CotedrsquoIvoire 05124 03839 03157 9077 9

7 EquatorialGuinea 04987 03379 03302 9495 4

8 The Gambia 05714 03914 03477 100 19 Ghana 05808 04239 03346 9622 310 Guinea 05142 03669 03255 9361 6

11 Guinea-Bissau 04604 03920 028 8051 16

12 Liberia 0507 03718 03185 9159 713 Mali 05245 04206 03039 8738 1314 Mauritania 04953 03409 03265 9388 515 Niger 04989 03778 03104 8926 1016 Nigeria 0489 04495 02692 7741 1817 Senegal 0515 04400 02884 8293 1518 Sierra Leone 04897 04533 02677 7698 1919 Togo 05165 03866 03168 9111 8

using equal weights are represented as white solid black andcheckerboard fillings respectively

Firstly we compare the GWSM using grey weights withDBP method The ranking for each country changes usingboth methods except Cape Verde The Gambia is ranked the1st position using the GWSM and Ghana is the 1st positionusing the cumulative result of the DBP However Ghana isranked the 3rd position The ranking of Cape Verde the 2ndposition is unchanged using both methods The Gambia isranked the 3rd position using the cumulative DBP methodAlthough Sierra Leone is ranked the 4th position by the DBPmethod the GWSM ranks Sierra Leone the 19th positionbecause of the highest degree of uncertainty with a value of04533measured as the aggregated boundary distances SierraLeone has some very poor performance in some criteriafor instance the cost of starting a business the number ofprocedures in getting electricity and the cost of enforcingcontracts have grey values of [38 11807] [0 9] and [401495] respectively In comparison to the Gambia these greyvalues are [131 279] [0 5] and [46 46] Similarly Nigeria isranked the 18th position using the GWSMwhich has a degreeof uncertainty of 04533 A corresponding moderate degreeof uncertainty in Togorsquos business environment (03866) hasimproved its ranking from the 16th position using the cumu-lative DBP method to the 8th position using the GWSM

Secondly we make a comparison of both methods usingequal weights We acknowledge the importance of weights in


Table 4 DBP rankings for the year 2008 to the year 2015

Countriesyears 2008 2009 2010 2011 2012 2013 2014 2015 Total RankingsBenin 3918 3764 3867 4021 4275 4437 4661 511 34053 14Burkina Faso 3249 3731 4101 4228 4558 4587 4789 4836 34079 13Cameroon 3971 3924 4333 4538 4775 4879 5035 4985 3644 7Cape Verde 5073 5104 5332 5582 5858 5885 5803 5794 4443 2Chad 2832 2867 301 2972 3239 3256 3557 3725 25458 19Cote drsquoIvoire 4013 4027 4136 415 4418 4483 489 5226 35343 8Equatorial Guinea 4226 4225 4578 4509 4755 4789 4774 4901 36758 5The Gambia 4844 4862 4847 4877 499 516 5176 5481 40236 3Ghana 5943 607 6147 6361 6428 6778 6441 6524 50692 1Guinea 3575 3589 3723 3597 3859 4148 4654 4742 31887 17Guinea-Bissau 326 3237 3346 3362 4098 4212 4355 4321 3019 18Liberia 3754 4123 422 4228 4647 4853 4656 4661 35142 10Mali 3673 3832 443 4746 4903 5024 5206 5259 37073 4Mauritania 3933 4098 4142 4252 4491 4502 4469 4421 34308 12Niger 3743 374 3889 3995 4327 4496 4691 4763 33643 15Nigeria 4446 4416 4283 4265 4393 437 4372 4733 35278 9Senegal 3679 4248 4189 4208 4394 4427 4609 4937 34691 11Sierra Leone 3545 4253 3987 4431 4839 4918 5278 5458 36709 6Togo 356 3573 3858 3927 4162 4361 4703 5129 33273 16

1st

3rd2nd

4th

6th5th

7th

9th8th

10th

12th11th

13th

15th14th

16th17th18th19th

Rank

ings

West African countries

Beni

nBu

rkin

a Fas

oCa

mer

oon

Cape

Ver

deCh

ad

Equa

toria

l Gui

nea

The G

ambi

aG

hana

Gui

nea

Gui

nea-

Biss

auLi

beria

Mal

iM

aurit

ania

Nig

erN

iger

iaSe

nega

lSi

erra

Leo

neTo

go

DBPGWSM-grey weightsGWSM-equal weights

Cocirct

e drsquoIv

oire

Figure 1 Rankings comparison between the DBP and the GWSM

decision-making The DBP uses equal weights for the eval-uation of business environment For a balanced comparisonbetween the DBP method and GWSM we also present theGWSM using equal weight Equal weights are real numbers

Following the GWSM steps using equals weights for all thefirst-level criteria and their second-level criteria we have

1198821015840

= (0025 0025 0025 0025 00333 00333

00333 00333 00333 00333 00333 00333

00333 0025 0025 0025 0025 00333

00333 00333 002 002 002 002 002

00167 00167 00167 00167 00167 00167

00333 00333 00333 00333 00333

00333)119879

(27)

The ranking of West African countries from the 1st positionis as follows the Gambia gtGhana gtCape Verde gt EquatorialGuinea gtNiger gtMali gtGuinea gtMauritania gt Togo gtCotedrsquoIvoire gt Cameroon gt Benin gt Liberia gt Senegal gt BurkinaFasogtChadgtGuinea-BissaugtNigeriagt Sierra Leone Basedon equal weights the Gambia is still ranked the 1st positionusing the GWSM and Ghana is the 1st using the cumulativeresult of the DBP But after considering uncertainty over theperiod of the year 2008 to 2015 Ghana is ranked the 2nd posi-tion with Cape Verde the 2nd position using DBP methodAlso Cape Verde is a rank lower the 3rd position using theGWSM and theGambia is then ranked the 3rd position usingthe DBP method Chad which is ranked the 19th positionusing the DBP method is ranked the 16th position usingGWSM Although Sierra Leone is ranked the 4th position bytheDBP GWSM ranks Sierra Leone the 19th position becauseof the highest degree of uncertainty with a value of 02814


measured as the aggregated boundary distances The degreeof uncertainty of Sierra Leone is more than twice that ofthe Gambia that has a value of 01093 Likewise Nigeriais ranked the 18th position using the GWSM which has adegree of uncertainty of 02633 A corresponding moderatedegree of uncertainty of Guinea and Niger has improved itranking from the 17th position and the 15th position usingthe cumulative DBP method to the 7th position and the 5thposition respectively

Lastly we compare equal weights and grey weights usingthe GWSM Liberia Mali and Niger are majorly affected byour investorrsquos preferencesMali that is ranked the 6th positionusing equal weights becomes the 13th position using greyweights and Liberia that is the 13th position using equalweights becomes the 7th position using grey weights Theranking of Niger from the 5th position using equal weightsbecomes 10th position using grey weights The weights of theDMs did not affect the rankings of the Gambia EquatorialGuinea Nigeria and Sierra Leone as the 1st 4th 18th and19th positions respectively The outcome of making a long-term investment in countries with lower uncertainty valuesis highly predictable It is not unusual for different rankingmethods to produce different rankings most especially underdifferent conditions [78] The degree of uncertainty theboundary distance plays a significant role in ranking theGambia as the 1st country Also uncertainty and the weightsof the criteria are the two factors for the difference in rankingsusing the DBP and GWSMmethods

5 Sensitivity Analysis

Sensitivity analysis is performed to show the robustness andthe degree of uncertainty covered by the GWSM resultsThrough the sensitivity analysis ranges of values for the inputparameters and coefficients that do not affect the rankings canbe obtained

51 Whitenization Sensitivity The white value is dependenton the whitenization coefficient (120582)Thewhitenization coeffi-cient helps to determine theminimum andmaximum overallperformances of the countries The whitenization coefficientchanges with an increment of 02 while keeping other inputparameters unchanged The results of the whitenizationsensitivity when 120582 = 0 02 04 06 08 1 are given inTable 5 showing the effect of a change in 120582 on the rankingsCape Verde Cote drsquoIvoire the Gambia and Guinea-Bissauare unaffected by the changes in whitenization coefficient As120582 increases the rankings for Cameroon and Chad rankingsbecome better while the rankings for Benin Ghana andMauritania become worse

52 Period Sensitivity Interval grey numbers are used torepresent the values of the criteria from the year 2008 tothe year 2015 a period of 8 years However the effects ofrepresenting a period of 6 years (2010ndash2015) and a period of 3years (2012ndash2015) with grey numbers are shown in Table 6The Gambia is ranked the 1st position over a long period(2008ndash2015) Ghana is ranked the 2nd position and CapeVerde is ranked the 3rd position However for a short period

Table 5 Whitenization sensitivity on rankings

Countries120582 0 02 04 06 08 1Benin 10 10 10 11 11 12Burkina Faso 15 14 14 14 14 14Cameroon 13 12 12 12 12 11Cape Verde 2 2 2 2 2 2Chad 19 18 17 17 17 17Cote drsquoIvoire 9 9 9 9 9 9Equatorial Guinea 4 4 4 4 3 3The Gambia 1 1 1 1 1 1Ghana 3 3 3 3 4 5Guinea 5 5 6 6 6 6Guinea-Bissau 16 16 16 16 16 16Liberia 8 7 7 7 7 7Mali 12 13 13 13 13 13Mauritania 6 6 5 5 5 4Niger 11 11 11 10 10 10Nigeria 17 17 18 18 18 18Senegal 14 15 15 15 15 15Sierra Leone 18 19 19 19 19 19Togo 7 8 8 8 8 8

(2010ndash2015 and 2012ndash2015) Cape Verde is ranked the 1stposition and the Gambia is ranked the 2nd position Ghana isranked the 3rd position for the preriod of 2010 to 2015 whileMali is ranked the 3rd position for the period of 2012 to 2015Burkina Faso made a very significant change from the 15thposition in the period of the year 2008 to 2015 and the 10th inthe period of 2010ndash2015 to the 7th position in a shorter period(2012ndash2015) Niger moves from the 5th position in 2008ndash2015to the 13th position in the year 2010 to 2015 and then 16thposition in 2012ndash2015

53 Distance Sensitivity The distance measurement of thelower and upper bounds of grey number has a directimpact on the degree of uncertainty Using different distancemeasurement values of 119901 in (20) the effects of distancemeasurement considering the Manhattan (119901 = 1) Euclidian(119901 = 2) and Minkowski (119901 = 3) distances on the ranking areobtained and shown in Table 7 FromTable 7 it is evident thatthe distances measurement has an influence on the assess-ment of decision alternatives because as 119901 trends to infinitythe boundary distance approaches the upper bound of theweighted aggregated sum of the criteria as shown in (28) For119910119894gt 119910119894

119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

=119901rarrinfin

119910119894 (28)

TheManhattan distancemeasurement reflects the true degreeof uncertainty in the business environment However usingthe Euclidian and the Minkowski (119901 = 3) distances Equato-rial Guinea and Mauritania are ranked the 1st and 2nd posi-tions respectively When 119901 = 2 Guinea-Bissau is ranked the3rd position using Euclidian distance while Chad is rankedthe 3rd when 119901 = 3


Table 6 Period sensitivity on rankings

Countriesperiod 2008ndash2015 2010ndash2015 2012ndash2015Benin 12 12 12Burkina Faso 15 10 7Cameroon 11 15 11Cape Verde 3 1 1Chad 16 19 19Cote drsquoIvoire 10 14 15Equatorial Guinea 4 5 6The Gambia 1 2 2Ghana 2 3 9Guinea 7 9 10Guinea-Bissau 17 17 17Liberia 13 4 4Mali 6 7 3Mauritania 8 6 5Niger 5 13 16Nigeria 18 18 18Senegal 14 8 8Sierra Leone 19 16 13Togo 9 11 14

Table 7 Distance measurement sensitivity on rankings

Countriesdistance Manhattan(p= 1)

Euclidian(p = 2)

Minkowski(p =3)

Benin 11 14 13Burkina Faso 14 11 9Cameroon 12 7 7Cape Verde 2 13 15Chad 17 9 3Cote drsquoIvoire 9 6 10Equatorial Guinea 4 1 1The Gambia 1 10 12Ghana 3 16 19Guinea 6 3 4Guinea-Bissau 16 12 8Liberia 7 4 5Mali 13 15 14Mauritania 5 2 2Niger 10 5 6Nigeria 18 18 16Senegal 15 17 17Sierra Leone 19 19 18Togo 8 8 11

6 Conclusion

All investors have their methods and criteria for decidingwhich country to invest The World Bank has done a greatwork by providing the annual data on countryrsquos performancesto help investors and researchers However care should be

taken in making use of these data because the DBP approachcannot provide an evaluation over a period of years Inthis regard we have assessed the performances of the WestAfrican countries using a new method called GWSM Inevaluating the business environment values of the criteria forall the alternatives are transformed to grey numbers based ontheir performances over the period of the year 2008 to theyear 2015 A decision matrix is constructed and normalizedand then the weighted grey decision matrix is obtained anduncertainties are considered in the evaluationThemain con-tribution of this paper is that GWSMcan evaluate alternativesover a period of years and it can open new possibilities formultiple hybrid evaluation methods

The ranking provided by the DBP presents a generalisedranking by using equal weights of the criteria in evaluatingthese countries It is almost impossible for an investor to havean equal preference for all measurable criteria that is allcriteria having equal weights Grey weights of the criteria areobtained from investors measured as grey linguistic valuesThe GWSM provides a better evaluation result over a periodof 8 years The results obtained from the research ranked theGambia as the 1st position inWest Africa based on the degreeof uncertainty Cape Verde and Ghana are highly recom-mended There should be deeper thought investing in SierraLeone and Nigeria Be sure these countries are in your line ofbusiness for example Nigeria is the oil giant in West Africaproducing over 25 million barrels of crude oil per day [79]

Future studies on reducing the numbers of criteria andfurther work on evaluating the DMrsquos weights can be donebecause not all experts should be rated equally Since someDMs have longer years of experiences and some have workedin different countries all these account for their preferencesThe GWSM is a framework for decision-making underuncertain decision environment Other hybrid methods canbe derived when GWSM is combined with existing MCDMweighing techniques which may provide new methods toevaluate alternatives under uncertain decision-making envi-ronment

Competing Interests

The authors declare that they have no competing interests

References

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[2] K Chelst and Y B Canbolat Value-Added Decision Making forManagers CRC Press New York NY USA 2011

[3] D R Karanki Uncertainty Management in Reliability Assess-ment Lambert Academic Publishing 2010

[4] The World Bank ldquoAbout Doing Business Measuring forimpactrdquo 2014 httpelibraryworldbankorgdoiabs101596978-0-8213-9984-2 About Doing Business

[5] G-H Tzeng and J-J Huang Multiple Attribute Decision Mak-ing Methods and Applications CRC Press Boca Raton FlaUSA 2011


[6] E K Zavadskas Z Turskis and S Kildiene ldquoState of art surveysof overviews on MCDMMADM methodsrdquo Technological andEconomic Development of Economy vol 20 no 1 pp 165ndash1792014

[7] D Jato-Espino E Castillo-Lopez J Rodriguez-Hernandezand J C Canteras-Jordana ldquoA review of application of multi-criteria decisionmaking methods in constructionrdquoAutomationin Construction vol 45 pp 151ndash162 2014

[8] T L Saaty ldquoDecision making with the analytic hierarchy pro-cessrdquo International Journal of Services Sciences vol 1 no 1 p 832008

[9] T L Saaty ldquoDecision makingmdashthe Analytic Hierarchy andNetwork Processes (AHPANP)rdquo Journal of Systems Science andSystems Engineering vol 13 no 1 pp 1ndash35 2004

[10] J Antucheviciene A Zakarevicius and E K ZavadskasldquoMeasuring congruence of ranking results applying particularMCDMmethodsrdquo Informatica vol 22 no 3 pp 319ndash338 2011

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[12] HG ShakouriMNabaee and S Aliakbarisani ldquoA quantitativediscussion on the assessment of power supply technologiesDEA (data envelopment analysis) and SAW (simple additiveweighting) as complementary methods for the lsquoGrammarrsquordquoEnergy vol 64 pp 640ndash647 2014

[13] B Roy ldquoClassement et choix en presence de points de vuemultiplesrdquoRAIROmdashRecherche Operationnelle vol 2 no V1 pp57ndash75 1968

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[16] G Qian and X Qian ldquoThe evidential reasoning approach formultiple attribute decision analysis using intuitionistic fuzzyinformationrdquo in Proceedings of the 4th International Conferenceon Wireless Communications Networking and Mobile Comput-ing (WiCOM rsquo08) pp 1ndash5 Dalian China October 2008

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[18] C Hamzacebi and M Pekkaya ldquoDetermining of stock invest-ments with grey relational analysisrdquo Expert Systems with Appli-cations vol 38 no 8 pp 9186ndash9195 2011

[19] J Wallenius J S Dyer P C Fishburn R E Steuer S Ziontsand K Deb ldquoMultiple criteria decision making multiattributeutility theory recent accomplishments and what lies aheadrdquoManagement Science vol 54 no 7 pp 1336ndash1349 2008

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[21] X Xu ldquoThe SIR method a superiority and inferiority rankingmethod for multiple criteria decision makingrdquo European Jour-nal of Operational Research vol 131 no 3 pp 587ndash602 2001

[22] W Edwards and F H Barron ldquoSMARTS and SMARTERimproved simple methods for multiattribute utility measure-mentrdquo Organizational Behavior and Human Decision Processesvol 60 no 3 pp 306ndash325 1994

[23] F H Barron and B E Barrett ldquoThe efficacy of SMARTERmdashsimple multi-attribute rating technique extended to rankingrdquoActa Psychologica vol 93 no 1ndash3 pp 23ndash36 1996

[24] J-J Wang Y-Y Jing C-F Zhang and J-H Zhao ldquoReviewon multi-criteria decision analysis aid in sustainable energydecision-makingrdquo Renewable and Sustainable Energy Reviewsvol 13 no 9 pp 2263ndash2278 2009

[25] L A Zadeh ldquoThe concept of a linguistic variable and its applica-tion to approximate reasoningmdashIrdquo Information Sciences vol 8no 3 pp 199ndash249 1975

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[27] D Wang ldquoExtension of TOPSIS method for RampD personnelselection problem with interval grey numberrdquo in Proceedings ofthe International Conference onManagement and Service Science(MASS rsquo09) pp 1ndash4 Wuhan China September 2009

[28] C Bai and J Sarkis ldquoA grey-based DEMATEL model for eval-uating business process management critical success factorsrdquoInternational Journal of Production Economics vol 146 no 1pp 281ndash292 2013

[29] E Kose M Kabak and H Aplak ldquoGrey theory based MCDMprocedure for sniper selection problemrdquo Grey Systems Theoryand Application vol 3 no 1 pp 35ndash45 2013

[30] H Kuang KWHipel andDM Kilgour ldquoEvaluation of sourcewater protection strategies in Waterloo Region based on GreySystems Theory and PROMETHEE IIrdquo in Proceedings of theIEEE International Conference on SystemsMan andCybernetics(SMC rsquo12) pp 2775ndash2779 IEEE Seoul Republic of KoreaOctober 2012

[31] Y Z Mehrjerdi ldquoStrategic system selection with linguisticpreferences and grey information using MCDMrdquo Applied SoftComputing vol 18 pp 323ndash337 2014

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[34] N Zhang ldquoMethod for aggregating correlated interval greylinguistic variables and its application to decision makingrdquoTechnological and Economic Development of Economy vol 19no 2 pp 189ndash202 2013

[35] F Jin P Liu and X Zhang ldquoThemulti-attribute group decisionmaking method based on the interval grey linguistic variablesweighted harmonic aggregation operatorsrdquo Technological andEconomic Development of Economy vol 19 no 3 pp 409ndash4302013

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change in the weight of one attribute on the final ranking ofalternativesrdquo Journal of Industrial Engineering vol 4 pp 13ndash182009

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[40] S-Y Chou Y-H Chang and C-Y Shen ldquoA fuzzy simple addi-tive weighting system under group decision-making for facilitylocation selection with objectivesubjective attributesrdquo Euro-pean Journal of Operational Research vol 189 no 1 pp 132ndash1452008

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[44] C-H Goh Y-C A Tung and C-H Cheng ldquoA revisedweighted sum decisionmodel for robot selectionrdquoComputers ampIndustrial Engineering vol 30 no 2 pp 193ndash199 1996

[45] E Triantaphyllou and C-T Lin ldquoDevelopment and evaluationof five fuzzy multiattribute decision-making methodsrdquo Interna-tional Journal of Approximate Reasoning vol 14 no 4 pp 281ndash310 1996

[46] M Modarres and S Sadi-Nezhad ldquoFuzzy simple additiveweightingmethod by preference ratiordquo Intelligent AutomationmdashSoft Computing vol 11 no 4 pp 235ndash244 2005

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[48] E K Zavadskas J Antucheviciene S H Razavi Hajiagha andS S Hashemi ldquoExtension of weighted aggregated sum productassessment with interval-valued intuitionistic fuzzy numbers(WASPAS-IVIF)rdquo Applied Soft Computing Journal vol 24 pp1013ndash1021 2014

[49] E K Zavadskas Z Turskis and J Antuceviciene ldquoSelecting acontractor by using a novel method for multiple attribute anal-ysis weighted aggregated sum product assessment with greyvalues (WASPAS-G)rdquo Studies in Informatics andControl vol 24no 2 pp 141ndash150 2015

[50] D Stanujkic and E K Zavadskas ldquoA modified weighted summethod based on the decision-makerrsquos preferred levels ofperformancesrdquo Studies in Informatics and Control vol 24 no4 pp 461ndash469 2015

[51] T-Y Chen ldquoAn interactive signed distance approach for mul-tiple criteria group decision-making based on simple additiveweighting method with incomplete preference informationdefined by interval type-2 fuzzy setsrdquo International Journal ofInformation Technology and Decision Making vol 13 no 5 pp979ndash1012 2014

[52] Y-J Wang ldquoA fuzzy multi-criteria decision-making modelbased on simple additive weighting method and relative prefer-ence relationrdquo Applied Soft Computing Journal vol 30 pp 412ndash420 2015

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approach under interval type-2 fuzzy set concepts for MCDMproblemrdquo in Advanced Computer and Communication Engi-neering Technology H A Sulaiman M A Othman M F IOthman YA Rahim andNC Pee Eds pp 833ndash842 SpringerInternational Berlin Germany 2015

[54] Y Xu W Zhang and H Wang ldquoA conflict-eliminatingapproach for emergency group decision of unconventionalincidentsrdquo Knowledge-Based Systems vol 83 no 1 pp 92ndash1042015

[55] J-L Deng ldquoControl problems of grey systemsrdquo Systems amp Con-trol Letters vol 1 no 5 pp 288ndash294 1982

[56] M-S Yin ldquoFifteen years of grey system theory research ahistorical review and bibliometric analysisrdquo Expert Systems withApplications vol 40 no 7 pp 2767ndash2775 2013

[57] P Delcea Camelia and D Camelia ldquoGrey systems theory ineconomicsmdasha historical applications reviewrdquo Grey SystemsTheory and Application vol 5 no 2 pp 263ndash276 2015

[58] J Xu and M Sasaki ldquoTechnique of order preference by sim-ilarity for multiple attribute decision making based on greymembersrdquo IEEJ Transactions on Electronics Information andSystems vol 124 no 10 pp 1999ndash2005 2004

[59] G-D Li D Yamaguchi and M Nagai ldquoA grey-based decision-making approach to the supplier selection problemrdquoMathemat-ical and Computer Modelling vol 46 no 3-4 pp 573ndash581 2007

[60] Z Turskis and E K Zavadskas ldquoA novel method for multiplecriteria analysis grey additive ratio assessment (ARAS-G)methodrdquo Informatica vol 21 no 4 pp 597ndash610 2010

[61] L Liu J-H Chen G-M Wang and D-Z Lao ldquoMulti-attributed decision making for mining methods based on greysystem and interval numbersrdquo Journal of Central South Univer-sity vol 20 no 4 pp 1029ndash1033 2013

[62] S M Mousavi B Vahdani R Tavakkoli-Moghaddam and NTajik ldquoSoft computing based on a fuzzy grey group compromisesolution approach with an application to the selection problemof material handling equipmentrdquo International Journal of Com-puter IntegratedManufacturing vol 27 no 6 pp 547ndash569 2014

[63] B Oztaysi ldquoA decisionmodel for information technology selec-tion using AHP integrated TOPSIS-Grey the case of contentmanagement systemsrdquo Knowledge-Based Systems vol 70 pp44ndash54 2014

[64] E K Zavadskas T Vilutiene Z Turskis and J TamosaitieneldquoContractor selection for construction works by applying saw-gand topsis grey techniquesrdquo Journal of Business Economics andManagement vol 11 no 1 pp 34ndash55 2010

[65] S H Zolfani M Sedaghat and E K Zavadskas ldquoPerformanceevaluating of rural ICT centers (telecenters) applying fuzzyAHP SAW-G and TOPSIS Grey a case study in Iranrdquo Techno-logical and Economic Development of Economy vol 18 no 2 pp364ndash387 2012

[66] S Datta N Sahu and S Mahapatra ldquoRobot selection basedon greyminusMULTIMOORA approachrdquo Grey Systems Theory andApplication vol 3 no 2 pp 201ndash232 2013

[67] Q Li and N Zhao ldquoStochastic interval-grey number VIKORmethod based on prospect theoryrdquo Grey Systems Theory andApplication vol 5 no 1 pp 105ndash116 2015

[68] S Liu and Y Lin Grey Information Theory and Practical Appli-cations Springer 2005

[69] S Liu and Y Lin ldquoIntroduction to grey systems theoryrdquo inGreySystems chapter 1 Springer New York NY USA 2011

[70] T Aven and E Zio ldquoSome considerations on the treatment ofuncertainties in risk assessment for practical decision makingrdquo


Reliability Engineering and System Safety vol 96 no 1 pp 64ndash74 2011

[71] Y-H Lin P-C Lee and H-I Ting ldquoDynamic multi-attributedecision making model with grey number evaluationsrdquo ExpertSystems with Applications vol 35 no 4 pp 1638ndash1644 2008

[72] G R Pophali A B Chelani and R S Dhodapkar ldquoOptimalselection of full scale tannery effluent treatment alternativeusing integrated AHP and GRA approachrdquo Expert Systems withApplications vol 38 no 9 pp 10889ndash10895 2011

[73] M Kabak S Burmaoglu and Y Kazancoglu ldquoA fuzzy hybridMCDM approach for professional selectionrdquo Expert Systemswith Applications vol 39 no 3 pp 3516ndash3525 2012

[74] Y Beikkhakhian M Javanmardi M Karbasian and BKhayambashi ldquoThe application of ISM model in evaluatingagile suppliers selection criteria and ranking suppliers usingfuzzyTOPSIS-AHPmethodsrdquoExpert SystemswithApplicationsvol 42 no 15-16 pp 6224ndash6236 2015

[75] J D Fage ldquoWestern Africamdashregion Africardquo httpglobalbritannicacomEBcheckedtopic640491western-Africa

[76] Doing Business ldquoDoing Business 2015 Going Beyond Effi-ciencyrdquo 2014

[77] Doing Business ldquoHistorical Data Sets and Trendsrdquo World BankGroup httpwwwdoingbusinessorgcustom-query

[78] K J Arrow ldquoA difficulty in the concept of social welfarerdquo Journalof Political Economy vol 58 no 4 pp 328ndash346 1950

[79] Nigerian National Petroleum Corporation ldquoOil Productionrdquohttpwwwnnpcgroupcomnnpcbusinessupstreamventuresoilproductionaspx

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


related alternatives Future data cannot be known from onlycurrent data but it is most likely that future data may bewithin the ranges of already known past and current datawhich are represented as interval grey numbers Althoughthe uncertainties of the future are absolute grey numbersare used to model the uncertainties of the future caused byincomplete and insufficient information inmaking an invest-ment decision This paper proposes an original approachcalled Grey Weighted Sum Model (GWSM) in evaluatingthe business environment in West Africa by accounting foruncertainties over a period of yearsThe contributions in thispaper are twofold Firstly we address a major problem notcovered by theDBP that is ranking countries over a period ofyears considering the degree of uncertainty in the country aswell as investors preferences represented as criteria weightsSecondly we extend the traditional Weighted Sum Model(WSM) by using grey numbers to represent performances ofevaluation criteria that may vary from time to time

This work is presented as follows Section 2 is LiteratureReview and Section 3 is GreyWeighted SumModel (GWSM)Section 4 presents the results and some discussions Thesensitivity analysis using GWSM is given in Section 5 Finallya conclusion is drawn in Section 6

2 Literature Review

There are several methods for solving and evaluating selec-tion problems in MCDM [5ndash7] Some of these methodsinclude the Analytical Hierarchy Process (AHP) developedbyThomas Saaty for evaluating alternatives by comparing allthe criteria of the alternatives [8] The Analytical NetworkProcess (ANP) was later introduced by Thomas Saaty as amethod to evaluate alternatives with interdependent decisioncriteria and alternatives [9] Zavadskas et al developed theCOmplex PRoportional ASsessment (COPRAS) for eval-uating alternatives which measure the complex efficiencyof an alternative as a proportion of the minimum andmaximum criteria values [10 11] Data Envelopment Analysis(DEA) is a method for evaluating the effective efficiencyof decision-making unit Shakouri et al [12] used the DEAin conjunction with Simple Additive Weighting (SAW) toassess power supply technologies Elimination and ChoiceExpressing Reality (ELECTRE) was developed as early as inthe 1960s [13] and has had more than four major iterativeimprovements to address the problem of not being ableto rank alternatives Several improvements that have beenintroduced to ELECTRE include ELECTRE I-IV ElECTRE-IS and ELECTREndashTRI [14] Technique for Order Preferencesby Similarity to an Ideal Solution (TOPSIS) was introducedby Hwang and Yoon [15] in selecting the best alternative as acompromise solution TOPSIS method selects the alternativewith the shortest distance to the ideal solution and farthestdistance to the negative ideal solution these distances areusually measured as Euclidean distances Evidential Reason-ing Approach (ERA) [16] evaluates alternatives with qualita-tive and quantitative criteria based on Dempster-Shafer (D-S) theory of evidence [17] with some level of ignorance GreyRelational Analysis (GRA) [18] is based on the Grey SystemTheory (GST) and uses the relative relational coefficient of

evaluation alternatives to the ideal alternative in determiningthe best alternative Multiattribute Utility Theory (MAUT)[19] is a classical method for evaluating alternatives by aggre-gating the satisfaction obtainable from every criterion mea-sured as utils Preference Ranking Organization Method forEnrichment of Evaluation (PROMETHEE) [20] was devel-oped by Jean-Pierre Brans which is an outranking methodResearchers have widely accepted PROMETHEE and theyhave developed it to a fully ranking technique Superiorityand Inferiority Ranking (SIR) [21] is a combination of theWSM and PROMETHEE where WSM is used for aggrega-tion of procedure and PROMETHEE for ranking proceduresSimple Multiattribute Rating Technique (SMART) [22] is anMCDM method in which the DM assigns points to everycriterion for all alternatives as well as the relative criteria forestimating the weights for evaluation SMART Extended toRanking (SMARTER) [23] is an improvement of SMART byusing rankings of the criteria for weight estimation

The relative importance of evaluation criteria for anMCDM problem is expressed using weights by the DM(s)With clear goal and alternatives the method of assigningweights to the evaluation criteria must be determined bymaking sure that it can produce themost accurate ratio of thecriteria based on the relative levels of importance Subjectiveor objective weighting methods [24] for MCDM problemscan be used in determining the weights For instance groupdecision-making process can be used in estimating theweights as well as the introduction of linguistic value becauseof the way humans think [25] Li et al [26] applied greyarithmetic mean weighting method in aggregating the groupweights of DMs in software selection Wang [27] appliedthis same technique to the selection of personnel using greylinguistic values for measuring the preferences of the groupDMs Similarly Bai and Sarkis Kose et al Kuang et aland Mehrjerdi [28ndash31] all used grey arithmetic mean incomputing the weights of criteria in a group decision-makingenvironment Kang et al [32] applied grey geometric mean inaggregating the weights of the DMs to determine the suitablemeeting location based on the service offered and the prefer-ences of the DMs Grey correlation operators can be used tocompute the relationships among all theDMspreferences andassign weights to the criteria based on the dependences in therelationship of the preferences [33] Other grey correlationoperators are Interval Grey Linguistic Variables OrderedWeighted Aggregation (IGLOWA) Interval Grey Linguis-tic Correlated Ordered Arithmetic Aggregation (IGLCOA)and Interval Grey Linguistic Correlated Ordered GeometricAggregation (IGLCOGA) [34] Jin et al [35] estimated theweights of the criteria using grey harmonic operators whichis based on the number of occurrences of theweights assignedby the group of DMs Esangbedo and Che [36] proposedthe Grey Rank Order Centroid (GROC) weights for groupdecision-making and applied it to evaluate African businessenvironment in a specific year rather than over a period ofseveral years addressed in this study based on the GRAmethod

The main techniques closely related to this paper areWSM and GST WSM can be described as a weighted linearcombination or scoring method [37 38] The WSM is also



119860lowast

WSM = max1⩽119894⩽119898

119899

sum

119895=1

119886119894119895119908119895 (1)








1= [119894 7119894] and otimes119868

2=





otimes119866 + otimes119867 = [119892 + ℎ 119892 + ℎ]



(2)


119892 times otimes119867 = [119892ℎ 119892ℎ] (3)



119883 =(

otimes11990911

otimes11990912


otimes11990921

otimes11990922


d

otimes1199091198981

otimes1199091198982


) (4)



119883119894= (otimes119909

1198941 otimes1199091198942 otimes119909

119894119899) (5)


otimes119909lowast

119894119895= [

119909119894119895minusmin

1le119894le119898119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

119909119894119895minusmin

1le119894le119898119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

]

(6)


otimes119909lowast

119894119895= [

max1le119894le119898

119909119894119895minus 119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

max1le119894le119898

119909119894119895minus 119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

]

(7)


119883lowast

=(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

) (8)


119883lowast

119894= (otimes119909

lowast


1198942 otimes119909

lowast

119894119899) (9)


For grey weights


119899)119879

(10)


where otimes119882119895= [119908119895 119908119895] and 119908

119895and 119908



otimes1198821015840

119895=

otimes119882119895

sum119899

119895=1119908119895

(11)

where otimes1198821015840119895= [1199081015840

119895 1199081015840119895] and sum119899

119895=11199081015840119895= 1


1198821015840

= (otimes1198821015840

1 otimes1198821015840

2 otimes119882

1015840

119899)119879

(12)


119882 = (1199081 1199082 119908

119899)119879

(13)


we have

1198821015840

= (1199081015840

1 1199081015840

2 119908

1015840

119899)119879

(14)

where 1199081015840119895= 119908119895sum119899

119895=1119908119895and sum119899

119895=11199081015840

119895= 1


119884 = 119883lowast

times1198821015840

(15)

For grey weights

119884 =(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

)(

(

otimes1198821015840

1

otimes1198821015840

2

otimes1198821015840

119899

)

)

(16)


119884 =(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

)(

(

1199081015840

1

1199081015840

2

1199081015840

119899

)

)

(17)

Therefore

119884 =(

otimes1199101

otimes1199102

otimes119910119898

) (18)

where otimes119910119894= [119910119894




119881119894= 119910119894

(1 minus 120582) + 119910119894120582 (19)



119894 for the


119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

(20)





119911119894= 119881119894(1 minus 119878

119894) (21)


119885119894= [

119911119894

max (119911119894)] times 100 (22)










otimes119909119894119895= [119909119894119895 119909119894119895] (23)

where

119909119894119895= min (V2008

119894119895 V2009119894119895

V2015119894119895

)

119909119894119895= max (V2008

119894119895 V2009119894119895

V2015119894119895

)

(24)




119894= (otimes119909

1198941 otimes1199091198942 otimes119909

11989437) For example 119883

1=


1is otimes11990911


119894have similar







1198821015840

= ([00144 00256] [00151 00265]

[00123 00229] [00123 00229] [00123 00229]

[00129 00237] [00123 00229] [0011 0021]

[00131 0024] [00104 00203] [00158 00274]

[00188 00313] [0015 00264] [00201 00331]

[0018 00303] [00144 00256] [00144 00256]

[00219 00354] [00238 00377] [00219 00354]

[00144 00256] [00115 00218] [00165 00284]

[00172 00294] [00144 00256] [00123 00229]





Starting a Business




Time 6Cost 7


Time 9Cost 10


Time 12Cost 13

Getting Credit





Paying Taxes

Payments 21Time 22


Other Taxes 25





Procedures 34


Recovery Rate 37



[00178 00302] [00185 00311] [00123 00229]

[00171 00293] [00171 00293] [0018 00303]

[00171 00293] [0018 00303] [00112 00212]

[00137 00246] [00155 00272])119879

(25)






119884 = ([03222 0738] [02841 06751]

[03692 07732] [03045 06857] [02471 06091]

[03204 07043] [03298 06677] [03757 07671]

[03688 07927] [03307 06977] [02645 06564]

[03211 06929] [06929 03142] [03249 06658]

[031 06878] [02643 07138] [0295 07349]

[0263 07164] [03232 07098])119879

(26)


1= 05301 119881

2= 04796 119881

3= 05712 119881

19=



1= 04157 119878

2= 03910 119878

3= 04040 119878

19=

03866


lated using (21) 1199111=03097 119911

2=02921 119911

3= 03405 119911

19=


(22)1198851=8906119885

2=8400119885

3=9791 119885

19=9111





2 BurkinaFaso 04796 03910 02921 84 14





11 Guinea-Bissau 04604 03920 028 8051 16








1st

3rd2nd

4th

6th5th

7th

9th8th

10th

12th11th

13th

15th14th

16th17th18th19th

Rank

ings


Beni

nBu

rkin

a Fas

oCa

mer

oon

Cape

Ver

deCh

ad

Equa

toria

l Gui

nea

The G

ambi

aG

hana

Gui

nea

Gui

nea-

Biss

auLi

beria

Mal

iM

aurit

ania

Nig

erN

iger

iaSe

nega

lSi

erra

Leo

neTo

go


Cocirct

e drsquoIv

oire




1198821015840

= (0025 0025 0025 0025 00333 00333

00333 00333 00333 00333 00333 00333

00333 0025 0025 0025 0025 00333

00333 00333 002 002 002 002 002

00167 00167 00167 00167 00167 00167

00333 00333 00333 00333 00333

00333)119879

(27)













119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

=119901rarrinfin

119910119894 (28)







Euclidian(p = 2)

Minkowski(p =3)


6 Conclusion





Competing Interests


References





























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119860lowast

WSM = max1⩽119894⩽119898

119899

sum

119895=1

119886119894119895119908119895 (1)








1= [119894 7119894] and otimes119868

2=





otimes119866 + otimes119867 = [119892 + ℎ 119892 + ℎ]



(2)


119892 times otimes119867 = [119892ℎ 119892ℎ] (3)



119883 =(

otimes11990911

otimes11990912


otimes11990921

otimes11990922


d

otimes1199091198981

otimes1199091198982


) (4)



119883119894= (otimes119909

1198941 otimes1199091198942 otimes119909

119894119899) (5)


otimes119909lowast

119894119895= [

119909119894119895minusmin

1le119894le119898119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

119909119894119895minusmin

1le119894le119898119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

]

(6)


otimes119909lowast

119894119895= [

max1le119894le119898

119909119894119895minus 119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

max1le119894le119898

119909119894119895minus 119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

]

(7)


119883lowast

=(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

) (8)


119883lowast

119894= (otimes119909

lowast


1198942 otimes119909

lowast

119894119899) (9)


For grey weights


119899)119879

(10)


where otimes119882119895= [119908119895 119908119895] and 119908

119895and 119908



otimes1198821015840

119895=

otimes119882119895

sum119899

119895=1119908119895

(11)

where otimes1198821015840119895= [1199081015840

119895 1199081015840119895] and sum119899

119895=11199081015840119895= 1


1198821015840

= (otimes1198821015840

1 otimes1198821015840

2 otimes119882

1015840

119899)119879

(12)


119882 = (1199081 1199082 119908

119899)119879

(13)


we have

1198821015840

= (1199081015840

1 1199081015840

2 119908

1015840

119899)119879

(14)

where 1199081015840119895= 119908119895sum119899

119895=1119908119895and sum119899

119895=11199081015840

119895= 1


119884 = 119883lowast

times1198821015840

(15)

For grey weights

119884 =(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

)(

(

otimes1198821015840

1

otimes1198821015840

2

otimes1198821015840

119899

)

)

(16)


119884 =(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

)(

(

1199081015840

1

1199081015840

2

1199081015840

119899

)

)

(17)

Therefore

119884 =(

otimes1199101

otimes1199102

otimes119910119898

) (18)

where otimes119910119894= [119910119894




119881119894= 119910119894

(1 minus 120582) + 119910119894120582 (19)



119894 for the


119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

(20)





119911119894= 119881119894(1 minus 119878

119894) (21)


119885119894= [

119911119894

max (119911119894)] times 100 (22)










otimes119909119894119895= [119909119894119895 119909119894119895] (23)

where

119909119894119895= min (V2008

119894119895 V2009119894119895

V2015119894119895

)

119909119894119895= max (V2008

119894119895 V2009119894119895

V2015119894119895

)

(24)




119894= (otimes119909

1198941 otimes1199091198942 otimes119909

11989437) For example 119883

1=


1is otimes11990911


119894have similar







1198821015840

= ([00144 00256] [00151 00265]

[00123 00229] [00123 00229] [00123 00229]

[00129 00237] [00123 00229] [0011 0021]

[00131 0024] [00104 00203] [00158 00274]

[00188 00313] [0015 00264] [00201 00331]

[0018 00303] [00144 00256] [00144 00256]

[00219 00354] [00238 00377] [00219 00354]

[00144 00256] [00115 00218] [00165 00284]

[00172 00294] [00144 00256] [00123 00229]





Starting a Business




Time 6Cost 7


Time 9Cost 10


Time 12Cost 13

Getting Credit





Paying Taxes

Payments 21Time 22


Other Taxes 25





Procedures 34


Recovery Rate 37



[00178 00302] [00185 00311] [00123 00229]

[00171 00293] [00171 00293] [0018 00303]

[00171 00293] [0018 00303] [00112 00212]

[00137 00246] [00155 00272])119879

(25)






119884 = ([03222 0738] [02841 06751]

[03692 07732] [03045 06857] [02471 06091]

[03204 07043] [03298 06677] [03757 07671]

[03688 07927] [03307 06977] [02645 06564]

[03211 06929] [06929 03142] [03249 06658]

[031 06878] [02643 07138] [0295 07349]

[0263 07164] [03232 07098])119879

(26)


1= 05301 119881

2= 04796 119881

3= 05712 119881

19=



1= 04157 119878

2= 03910 119878

3= 04040 119878

19=

03866


lated using (21) 1199111=03097 119911

2=02921 119911

3= 03405 119911

19=


(22)1198851=8906119885

2=8400119885

3=9791 119885

19=9111





2 BurkinaFaso 04796 03910 02921 84 14





11 Guinea-Bissau 04604 03920 028 8051 16








1st

3rd2nd

4th

6th5th

7th

9th8th

10th

12th11th

13th

15th14th

16th17th18th19th

Rank

ings


Beni

nBu

rkin

a Fas

oCa

mer

oon

Cape

Ver

deCh

ad

Equa

toria

l Gui

nea

The G

ambi

aG

hana

Gui

nea

Gui

nea-

Biss

auLi

beria

Mal

iM

aurit

ania

Nig

erN

iger

iaSe

nega

lSi

erra

Leo

neTo

go


Cocirct

e drsquoIv

oire




1198821015840

= (0025 0025 0025 0025 00333 00333

00333 00333 00333 00333 00333 00333

00333 0025 0025 0025 0025 00333

00333 00333 002 002 002 002 002

00167 00167 00167 00167 00167 00167

00333 00333 00333 00333 00333

00333)119879

(27)













119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

=119901rarrinfin

119910119894 (28)







Euclidian(p = 2)

Minkowski(p =3)


6 Conclusion





Competing Interests


References





























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












otimes119866 + otimes119867 = [119892 + ℎ 119892 + ℎ]



(2)


119892 times otimes119867 = [119892ℎ 119892ℎ] (3)



119883 =(

otimes11990911

otimes11990912


otimes11990921

otimes11990922


d

otimes1199091198981

otimes1199091198982


) (4)



119883119894= (otimes119909

1198941 otimes1199091198942 otimes119909

119894119899) (5)


otimes119909lowast

119894119895= [

119909119894119895minusmin

1le119894le119898119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

119909119894119895minusmin

1le119894le119898119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

]

(6)


otimes119909lowast

119894119895= [

max1le119894le119898

119909119894119895minus 119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

max1le119894le119898

119909119894119895minus 119909119894119895

max1le119894le119898

119909119894119895minusmin

1le119894le119898119909119894119895

]

(7)


119883lowast

=(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

) (8)


119883lowast

119894= (otimes119909

lowast


1198942 otimes119909

lowast

119894119899) (9)


For grey weights


119899)119879

(10)


where otimes119882119895= [119908119895 119908119895] and 119908

119895and 119908



otimes1198821015840

119895=

otimes119882119895

sum119899

119895=1119908119895

(11)

where otimes1198821015840119895= [1199081015840

119895 1199081015840119895] and sum119899

119895=11199081015840119895= 1


1198821015840

= (otimes1198821015840

1 otimes1198821015840

2 otimes119882

1015840

119899)119879

(12)


119882 = (1199081 1199082 119908

119899)119879

(13)


we have

1198821015840

= (1199081015840

1 1199081015840

2 119908

1015840

119899)119879

(14)

where 1199081015840119895= 119908119895sum119899

119895=1119908119895and sum119899

119895=11199081015840

119895= 1


119884 = 119883lowast

times1198821015840

(15)

For grey weights

119884 =(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

)(

(

otimes1198821015840

1

otimes1198821015840

2

otimes1198821015840

119899

)

)

(16)


119884 =(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

)(

(

1199081015840

1

1199081015840

2

1199081015840

119899

)

)

(17)

Therefore

119884 =(

otimes1199101

otimes1199102

otimes119910119898

) (18)

where otimes119910119894= [119910119894




119881119894= 119910119894

(1 minus 120582) + 119910119894120582 (19)



119894 for the


119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

(20)





119911119894= 119881119894(1 minus 119878

119894) (21)


119885119894= [

119911119894

max (119911119894)] times 100 (22)










otimes119909119894119895= [119909119894119895 119909119894119895] (23)

where

119909119894119895= min (V2008

119894119895 V2009119894119895

V2015119894119895

)

119909119894119895= max (V2008

119894119895 V2009119894119895

V2015119894119895

)

(24)




119894= (otimes119909

1198941 otimes1199091198942 otimes119909

11989437) For example 119883

1=


1is otimes11990911


119894have similar







1198821015840

= ([00144 00256] [00151 00265]

[00123 00229] [00123 00229] [00123 00229]

[00129 00237] [00123 00229] [0011 0021]

[00131 0024] [00104 00203] [00158 00274]

[00188 00313] [0015 00264] [00201 00331]

[0018 00303] [00144 00256] [00144 00256]

[00219 00354] [00238 00377] [00219 00354]

[00144 00256] [00115 00218] [00165 00284]

[00172 00294] [00144 00256] [00123 00229]





Starting a Business




Time 6Cost 7


Time 9Cost 10


Time 12Cost 13

Getting Credit





Paying Taxes

Payments 21Time 22


Other Taxes 25





Procedures 34


Recovery Rate 37



[00178 00302] [00185 00311] [00123 00229]

[00171 00293] [00171 00293] [0018 00303]

[00171 00293] [0018 00303] [00112 00212]

[00137 00246] [00155 00272])119879

(25)






119884 = ([03222 0738] [02841 06751]

[03692 07732] [03045 06857] [02471 06091]

[03204 07043] [03298 06677] [03757 07671]

[03688 07927] [03307 06977] [02645 06564]

[03211 06929] [06929 03142] [03249 06658]

[031 06878] [02643 07138] [0295 07349]

[0263 07164] [03232 07098])119879

(26)


1= 05301 119881

2= 04796 119881

3= 05712 119881

19=



1= 04157 119878

2= 03910 119878

3= 04040 119878

19=

03866


lated using (21) 1199111=03097 119911

2=02921 119911

3= 03405 119911

19=


(22)1198851=8906119885

2=8400119885

3=9791 119885

19=9111





2 BurkinaFaso 04796 03910 02921 84 14





11 Guinea-Bissau 04604 03920 028 8051 16








1st

3rd2nd

4th

6th5th

7th

9th8th

10th

12th11th

13th

15th14th

16th17th18th19th

Rank

ings


Beni

nBu

rkin

a Fas

oCa

mer

oon

Cape

Ver

deCh

ad

Equa

toria

l Gui

nea

The G

ambi

aG

hana

Gui

nea

Gui

nea-

Biss

auLi

beria

Mal

iM

aurit

ania

Nig

erN

iger

iaSe

nega

lSi

erra

Leo

neTo

go


Cocirct

e drsquoIv

oire




1198821015840

= (0025 0025 0025 0025 00333 00333

00333 00333 00333 00333 00333 00333

00333 0025 0025 0025 0025 00333

00333 00333 002 002 002 002 002

00167 00167 00167 00167 00167 00167

00333 00333 00333 00333 00333

00333)119879

(27)













119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

=119901rarrinfin

119910119894 (28)







Euclidian(p = 2)

Minkowski(p =3)


6 Conclusion





Competing Interests


References





























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where otimes119882119895= [119908119895 119908119895] and 119908

119895and 119908



otimes1198821015840

119895=

otimes119882119895

sum119899

119895=1119908119895

(11)

where otimes1198821015840119895= [1199081015840

119895 1199081015840119895] and sum119899

119895=11199081015840119895= 1


1198821015840

= (otimes1198821015840

1 otimes1198821015840

2 otimes119882

1015840

119899)119879

(12)


119882 = (1199081 1199082 119908

119899)119879

(13)


we have

1198821015840

= (1199081015840

1 1199081015840

2 119908

1015840

119899)119879

(14)

where 1199081015840119895= 119908119895sum119899

119895=1119908119895and sum119899

119895=11199081015840

119895= 1


119884 = 119883lowast

times1198821015840

(15)

For grey weights

119884 =(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

)(

(

otimes1198821015840

1

otimes1198821015840

2

otimes1198821015840

119899

)

)

(16)


119884 =(

otimes119909lowast



lowast

1119899

otimes119909lowast



lowast

2119899

d

otimes119909lowast



lowast

119898119899

)(

(

1199081015840

1

1199081015840

2

1199081015840

119899

)

)

(17)

Therefore

119884 =(

otimes1199101

otimes1199102

otimes119910119898

) (18)

where otimes119910119894= [119910119894




119881119894= 119910119894

(1 minus 120582) + 119910119894120582 (19)



119894 for the


119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

(20)





119911119894= 119881119894(1 minus 119878

119894) (21)


119885119894= [

119911119894

max (119911119894)] times 100 (22)










otimes119909119894119895= [119909119894119895 119909119894119895] (23)

where

119909119894119895= min (V2008

119894119895 V2009119894119895

V2015119894119895

)

119909119894119895= max (V2008

119894119895 V2009119894119895

V2015119894119895

)

(24)




119894= (otimes119909

1198941 otimes1199091198942 otimes119909

11989437) For example 119883

1=


1is otimes11990911


119894have similar







1198821015840

= ([00144 00256] [00151 00265]

[00123 00229] [00123 00229] [00123 00229]

[00129 00237] [00123 00229] [0011 0021]

[00131 0024] [00104 00203] [00158 00274]

[00188 00313] [0015 00264] [00201 00331]

[0018 00303] [00144 00256] [00144 00256]

[00219 00354] [00238 00377] [00219 00354]

[00144 00256] [00115 00218] [00165 00284]

[00172 00294] [00144 00256] [00123 00229]





Starting a Business




Time 6Cost 7


Time 9Cost 10


Time 12Cost 13

Getting Credit





Paying Taxes

Payments 21Time 22


Other Taxes 25





Procedures 34


Recovery Rate 37



[00178 00302] [00185 00311] [00123 00229]

[00171 00293] [00171 00293] [0018 00303]

[00171 00293] [0018 00303] [00112 00212]

[00137 00246] [00155 00272])119879

(25)






119884 = ([03222 0738] [02841 06751]

[03692 07732] [03045 06857] [02471 06091]

[03204 07043] [03298 06677] [03757 07671]

[03688 07927] [03307 06977] [02645 06564]

[03211 06929] [06929 03142] [03249 06658]

[031 06878] [02643 07138] [0295 07349]

[0263 07164] [03232 07098])119879

(26)


1= 05301 119881

2= 04796 119881

3= 05712 119881

19=



1= 04157 119878

2= 03910 119878

3= 04040 119878

19=

03866


lated using (21) 1199111=03097 119911

2=02921 119911

3= 03405 119911

19=


(22)1198851=8906119885

2=8400119885

3=9791 119885

19=9111





2 BurkinaFaso 04796 03910 02921 84 14





11 Guinea-Bissau 04604 03920 028 8051 16








1st

3rd2nd

4th

6th5th

7th

9th8th

10th

12th11th

13th

15th14th

16th17th18th19th

Rank

ings


Beni

nBu

rkin

a Fas

oCa

mer

oon

Cape

Ver

deCh

ad

Equa

toria

l Gui

nea

The G

ambi

aG

hana

Gui

nea

Gui

nea-

Biss

auLi

beria

Mal

iM

aurit

ania

Nig

erN

iger

iaSe

nega

lSi

erra

Leo

neTo

go


Cocirct

e drsquoIv

oire




1198821015840

= (0025 0025 0025 0025 00333 00333

00333 00333 00333 00333 00333 00333

00333 0025 0025 0025 0025 00333

00333 00333 002 002 002 002 002

00167 00167 00167 00167 00167 00167

00333 00333 00333 00333 00333

00333)119879

(27)













119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

=119901rarrinfin

119910119894 (28)







Euclidian(p = 2)

Minkowski(p =3)


6 Conclusion





Competing Interests


References





























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















otimes119909119894119895= [119909119894119895 119909119894119895] (23)

where

119909119894119895= min (V2008

119894119895 V2009119894119895

V2015119894119895

)

119909119894119895= max (V2008

119894119895 V2009119894119895

V2015119894119895

)

(24)




119894= (otimes119909

1198941 otimes1199091198942 otimes119909

11989437) For example 119883

1=


1is otimes11990911


119894have similar







1198821015840

= ([00144 00256] [00151 00265]

[00123 00229] [00123 00229] [00123 00229]

[00129 00237] [00123 00229] [0011 0021]

[00131 0024] [00104 00203] [00158 00274]

[00188 00313] [0015 00264] [00201 00331]

[0018 00303] [00144 00256] [00144 00256]

[00219 00354] [00238 00377] [00219 00354]

[00144 00256] [00115 00218] [00165 00284]

[00172 00294] [00144 00256] [00123 00229]





Starting a Business




Time 6Cost 7


Time 9Cost 10


Time 12Cost 13

Getting Credit





Paying Taxes

Payments 21Time 22


Other Taxes 25





Procedures 34


Recovery Rate 37



[00178 00302] [00185 00311] [00123 00229]

[00171 00293] [00171 00293] [0018 00303]

[00171 00293] [0018 00303] [00112 00212]

[00137 00246] [00155 00272])119879

(25)






119884 = ([03222 0738] [02841 06751]

[03692 07732] [03045 06857] [02471 06091]

[03204 07043] [03298 06677] [03757 07671]

[03688 07927] [03307 06977] [02645 06564]

[03211 06929] [06929 03142] [03249 06658]

[031 06878] [02643 07138] [0295 07349]

[0263 07164] [03232 07098])119879

(26)


1= 05301 119881

2= 04796 119881

3= 05712 119881

19=



1= 04157 119878

2= 03910 119878

3= 04040 119878

19=

03866


lated using (21) 1199111=03097 119911

2=02921 119911

3= 03405 119911

19=


(22)1198851=8906119885

2=8400119885

3=9791 119885

19=9111





2 BurkinaFaso 04796 03910 02921 84 14





11 Guinea-Bissau 04604 03920 028 8051 16








1st

3rd2nd

4th

6th5th

7th

9th8th

10th

12th11th

13th

15th14th

16th17th18th19th

Rank

ings


Beni

nBu

rkin

a Fas

oCa

mer

oon

Cape

Ver

deCh

ad

Equa

toria

l Gui

nea

The G

ambi

aG

hana

Gui

nea

Gui

nea-

Biss

auLi

beria

Mal

iM

aurit

ania

Nig

erN

iger

iaSe

nega

lSi

erra

Leo

neTo

go


Cocirct

e drsquoIv

oire




1198821015840

= (0025 0025 0025 0025 00333 00333

00333 00333 00333 00333 00333 00333

00333 0025 0025 0025 0025 00333

00333 00333 002 002 002 002 002

00167 00167 00167 00167 00167 00167

00333 00333 00333 00333 00333

00333)119879

(27)













119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

=119901rarrinfin

119910119894 (28)







Euclidian(p = 2)

Minkowski(p =3)


6 Conclusion





Competing Interests


References





























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Starting a Business




Time 6Cost 7


Time 9Cost 10


Time 12Cost 13

Getting Credit





Paying Taxes

Payments 21Time 22


Other Taxes 25





Procedures 34


Recovery Rate 37



[00178 00302] [00185 00311] [00123 00229]

[00171 00293] [00171 00293] [0018 00303]

[00171 00293] [0018 00303] [00112 00212]

[00137 00246] [00155 00272])119879

(25)






119884 = ([03222 0738] [02841 06751]

[03692 07732] [03045 06857] [02471 06091]

[03204 07043] [03298 06677] [03757 07671]

[03688 07927] [03307 06977] [02645 06564]

[03211 06929] [06929 03142] [03249 06658]

[031 06878] [02643 07138] [0295 07349]

[0263 07164] [03232 07098])119879

(26)


1= 05301 119881

2= 04796 119881

3= 05712 119881

19=



1= 04157 119878

2= 03910 119878

3= 04040 119878

19=

03866


lated using (21) 1199111=03097 119911

2=02921 119911

3= 03405 119911

19=


(22)1198851=8906119885

2=8400119885

3=9791 119885

19=9111





2 BurkinaFaso 04796 03910 02921 84 14





11 Guinea-Bissau 04604 03920 028 8051 16








1st

3rd2nd

4th

6th5th

7th

9th8th

10th

12th11th

13th

15th14th

16th17th18th19th

Rank

ings


Beni

nBu

rkin

a Fas

oCa

mer

oon

Cape

Ver

deCh

ad

Equa

toria

l Gui

nea

The G

ambi

aG

hana

Gui

nea

Gui

nea-

Biss

auLi

beria

Mal

iM

aurit

ania

Nig

erN

iger

iaSe

nega

lSi

erra

Leo

neTo

go


Cocirct

e drsquoIv

oire




1198821015840

= (0025 0025 0025 0025 00333 00333

00333 00333 00333 00333 00333 00333

00333 0025 0025 0025 0025 00333

00333 00333 002 002 002 002 002

00167 00167 00167 00167 00167 00167

00333 00333 00333 00333 00333

00333)119879

(27)













119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

=119901rarrinfin

119910119894 (28)







Euclidian(p = 2)

Minkowski(p =3)


6 Conclusion





Competing Interests


References





























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119884 = ([03222 0738] [02841 06751]

[03692 07732] [03045 06857] [02471 06091]

[03204 07043] [03298 06677] [03757 07671]

[03688 07927] [03307 06977] [02645 06564]

[03211 06929] [06929 03142] [03249 06658]

[031 06878] [02643 07138] [0295 07349]

[0263 07164] [03232 07098])119879

(26)


1= 05301 119881

2= 04796 119881

3= 05712 119881

19=



1= 04157 119878

2= 03910 119878

3= 04040 119878

19=

03866


lated using (21) 1199111=03097 119911

2=02921 119911

3= 03405 119911

19=


(22)1198851=8906119885

2=8400119885

3=9791 119885

19=9111





2 BurkinaFaso 04796 03910 02921 84 14





11 Guinea-Bissau 04604 03920 028 8051 16








1st

3rd2nd

4th

6th5th

7th

9th8th

10th

12th11th

13th

15th14th

16th17th18th19th

Rank

ings


Beni

nBu

rkin

a Fas

oCa

mer

oon

Cape

Ver

deCh

ad

Equa

toria

l Gui

nea

The G

ambi

aG

hana

Gui

nea

Gui

nea-

Biss

auLi

beria

Mal

iM

aurit

ania

Nig

erN

iger

iaSe

nega

lSi

erra

Leo

neTo

go


Cocirct

e drsquoIv

oire




1198821015840

= (0025 0025 0025 0025 00333 00333

00333 00333 00333 00333 00333 00333

00333 0025 0025 0025 0025 00333

00333 00333 002 002 002 002 002

00167 00167 00167 00167 00167 00167

00333 00333 00333 00333 00333

00333)119879

(27)













119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

=119901rarrinfin

119910119894 (28)







Euclidian(p = 2)

Minkowski(p =3)


6 Conclusion





Competing Interests


References





























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1st

3rd2nd

4th

6th5th

7th

9th8th

10th

12th11th

13th

15th14th

16th17th18th19th

Rank

ings


Beni

nBu

rkin

a Fas

oCa

mer

oon

Cape

Ver

deCh

ad

Equa

toria

l Gui

nea

The G

ambi

aG

hana

Gui

nea

Gui

nea-

Biss

auLi

beria

Mal

iM

aurit

ania

Nig

erN

iger

iaSe

nega

lSi

erra

Leo

neTo

go


Cocirct

e drsquoIv

oire




1198821015840

= (0025 0025 0025 0025 00333 00333

00333 00333 00333 00333 00333 00333

00333 0025 0025 0025 0025 00333

00333 00333 002 002 002 002 002

00167 00167 00167 00167 00167 00167

00333 00333 00333 00333 00333

00333)119879

(27)













119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

=119901rarrinfin

119910119894 (28)







Euclidian(p = 2)

Minkowski(p =3)


6 Conclusion





Competing Interests


References





























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















119878119894=119901

radic(119910119894)119901

minus (119910119894

)

119901

=119901rarrinfin

119910119894 (28)







Euclidian(p = 2)

Minkowski(p =3)


6 Conclusion





Competing Interests


References





























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Euclidian(p = 2)

Minkowski(p =3)


6 Conclusion





Competing Interests


References





























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







