The Application of Fuzzy VIKOR for the Design Scheme ...

Research ArticleThe Application of Fuzzy VIKOR for the Design SchemeSelection in Lean Management

Shuwei Jing Yao Tang and Junai Yan

School of Management Science and Engineering Shanxi University of Finance and Economics Taiyuan Shanxi China

Correspondence should be addressed to Shuwei Jing jingshuweiabcdaliyuncom

Received 28 February 2018 Accepted 15 August 2018 Published 10 September 2018

Academic Editor Ibrahim Zeid

Copyright copy 2018 Shuwei Jing et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Given that using the same evaluation criteria in group decision making (GDM) may cause solution bias this research proposesthe decision making model using improved fuzzy VIKOR and takes the lean management design scheme selection process in areal estate company as an example to study the way of decision making Research results are as follows (1) improved fuzzy VIKORallows different decision makers to evaluate all schemes with respect to different evaluation criteria systems and based on thisthis research proposes a decision making model which can effectively avoid making decision using the same criteria system (2)this decision making model is very scientific as it is not sensitive to the coefficient of selection mechanism the weight of decisionmaker and the weight of evaluation criterion so the decision results are very stable (3) the decision making model proposed inthis research can effectively obtain the optimal result and avoid suboptimal solution

1 Introduction

Lean management originates from lean production theprototype of lean production is Toyota production system(TPS) Lean production theory can work as the guideline ofenterprise strategic management [1]The concept of lean ideaproposed byWomack is the basis of leanmanagement whichmeans to manage various enterprise activities using lean idea[2] After that led by manufacturing multiple industries andsectors bring in lean management around the world Existingtheories and practices have verified that lean managementcan work in multiple industries to improve production effi-ciency reduce production cost bring enterprise considerableeconomic income and improve management innovationability The outstanding successes achieved by applying leanmanagement show that lean management is scientific andadvanced [3] During the past decades lean management haschanged the development track of enterprises and provided anew management mode [4]

Lean management is playing a more and more significantrole in enterprise management Multiple enterprises beginto learn lean management and regard lean management asa key way to improve enterprise efficiency When applyinglean management enterprises should design concrete lean

management schemes first At present enterprises mainlyuse the following three methods to design and select leanmanagement schemes (1) lean management experts designschemes and submit them to managers for decision (2)lean management experts and managers design schemesjointly (3) lean management experts guide managers todesign schemes Multiple enterprise managers lack leanmanagement knowledge so totally depending on managersto design lean management design schemes is quite unusualfor enterprises in China Actually lean management expertsplay main roles in lean management scheme design Whendesigning lean management schemes lean managementexperts usually design several lean management schemesand submit them to managers for decision or they presentonly one scheme and then seek for opinions of managers torevise the scheme time again In scheme selection processit takes a long time to reach a consensus because differentmanagers may apply different criteria to evaluate schemesSometimes scheme will be repeatedly revised in order toweigh the opinions of all managers which often leads to asuboptimal selection bringing great discount to the effect oflean management promotion Therefore researching on thedesign scheme selection method in lean management is thebasis of improving the effect of lean management promotion

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 9253643 15 pageshttpsdoiorg10115520189253643

2 Mathematical Problems in Engineering

Lacking scientific method of lean management designscheme selection is the main reason that leads to suboptimalsolution Currently decision makers mainly select lean man-agement design scheme through discussion which meanssubjective judgment is predominant Selecting lean man-agement design scheme is a typical group decision makingproblem which has threemain features (1) different schemeshave different advantages and they cannot be evaluated usingthe same evaluation criteria system (2) different decisionmakers may have different evaluation criteria systems (3)selecting lean management design scheme is a fuzzy processScientifically selecting leanmanagement design schememusttake the above three features into consideration

In order to solve the problems in selecting lean manage-ment design scheme this research applies fuzzy VIKOR tomake a selection from a quantitative perspective and con-structs a design scheme selectionmodel in leanmanagementMain contributions of this research include the following(1) the design scheme selection model constructed in thisresearch will save time in selecting scheme for enterpriseimprove the accuracy of lean management design schemeselection and provide a way of effectively avoiding subopti-mal scheme (2) this research expands the application rangeof fuzzy VIKOR method (3) this research creates a model ofdesign scheme selection in leanmanagement providing basisfor enterprises to select design scheme scientifically in leanmanagement practices

2 Literature Review

Designing schemes is the beginning of all managementactivities and the quality of management schemes reflectsthe management strategy of the enterprise Designing leanmanagement schemes is the basis of implementing leanmanagement Whether the schemes are splendid or notwill affect the result of lean management implementationTherefore scientifically selecting lean management designscheme is the basis of implementing lean management atthe same time the method of scheme selection is the basisof scientifically selecting lean management design schemeExisting researches have proposed the following six mainmethods of scheme selection

(1) Analytic Hierarchy Process (AHP) Since AHP wasproposed by American scholar Saaty in 1970s [5] it hasbeen applied in multiple sectors to solve decision makingproblems For example Yu et al [6] research on product lifecycle schemes selection applying AHP and establish hier-archical structure relation to select optimal design schemeoptimal scheme is obtained through pairwise comparisonand comprehensive evaluation Wang and Chin [7] pro-pose the new data envelopment analysis (DEA) methodfor priority determination in AHP helping decision makerschoose optimal scheme Zhu and Xu [8] apply AHP inhesitant group decision making (GDM) process to help withdecision making taking the water conservancy in China asan example which means to apply AHP in GDM processthat has hesitant judgments When applying AHP differentdecision makers hold different opinions on the importance

of various elements so the order of schemes obtained is alsodifferent Therefore Xie [9] combines covariance matrix andAHP to create aCov-AHPmethod thismethod can constructjudgmentmatrix using covariancematrix and obtain the onlyorder of scheme through calculation

(2) Multicriteria decision making (MCDM) MCDMincludes multiobjective decision making (MODM) and mul-tiattribute decision making (MADM) MCDM plays animportant role in many fields For example Medjoudj et al[10] apply MCDM to help decision makers improve profitand custom satisfaction in electric power company Giventhat the risk of credit guarantee products is complex He andWeng [11] use indeterminate language to describe the weightand attribute value of indexes in product risk evaluationindex system creating the group decision making model ofindeterminate language for credit guarantee products ChouandDing [12] constructMCDMmodel to evaluate the servicequality and the location choice of transshipment portsKhasanah et al [13] apply fuzzy MADM method in majorselection at senior high school research results show thatthis method can accommodate the presence of uncertaintyin decision making and the accuracy that decision makingresults are in accordance with the reality is as much as90 Buyukozkan et al [14] apply MCDMmethod includingintuitionistic fuzzy analytic network process intuitionisticfuzzy decision making trial and evaluation laboratory thismethod canmore effectively choose suitable business partnerWith the development of Internet technology many scholarsconstruct web based on intelligence framework to helpwith decision making For example combining case-basedreasoning system and MCDM technique to construct theweb based on intelligence framework can improve the speedand accuracy of decision making [15] constructing a webincluding all MCDM tools can help decision makers to selectschemes more quickly and reliably [16]

(3) Case-based decision theory (CBDT) The main ideaof CBDT is using previous method to solve current simi-lar decision making problems CBDT can avoid repetitivedecision making reducing time and cost Through collectingsimilar previous cases and calculating the similarity betweenthe target case and previous case the utility value of the effectbrought by previous case implementation can be calculatedand then integrated utility value of every scheme can becalculated [17] Guilfoos and Pape [18] use CBDT to analyzethe cooperative dynamics in prisonerrsquos dilemma proving thatCBDT model is more suitable when it comes to selectingschemes using cooperative dynamic analysis Studying therelationship between CBDT and ideal point model canreveal previous dependence point or provide reference pointwhich helps enterprise in product designing and productpositioning providing support for marketing decision [19]

(4) IF-TOPSIS Intuitionistic fuzzy sets (IFS) are mainlyapplied to deal with the indeterminacy in decision makingprocess Technique for order performance by similarity toideal solution (TOPSIS) can rank evaluation objects throughtesting the distance between evaluation objects and optimalsolution or worst solution an evaluation object will bethe best solution if it is closest to optimal solution andfarthest to worst solution IF-TOPSIS can be used to solve

Mathematical Problems in Engineering 3

complex decision making problems with high indetermi-nacy In complex decision making problem IF-TOPSIS caneliminate indeterminacy making the solution closer to thepreference of decisionmakers [20] Based on IF-TOPSIS Youet al [21] study evaluation information combining accuratenumbers and languages to helpwith decisionmaking throughsimulation study

(5) Prospect theory (PT) At the beginning PT is mainlyapplied in analyzing the decision making under risks [22]Hu et al [23] apply value function and decision weightingfunction to calculate the prospect value of scheme and obtainthe order of schemes according to prospect value Given thatthe coefficient of evaluation criterion in gray randomMCDMis not completely determined Wang and Zhou [24] proposea decision making method through analyzing examplesThrough comprehensively considering the risk preference ofdecision makers Hao et al [25] put forward the multistagerandom decision making method based on PT Li et al [26]combine IFS PT and gray correlation degree to solve randomdecision making problems

(6) VIKOR The process of decision making is alwaysfaced with problems such as lacking previous experiencelacking specific context of the organization and having lim-ited quantitative information of alternatives while VIKOR isable to help decision maker to select an alternative closest tothe ideal assuming compromising is acceptable [27] Awasthiand Kannan [28] combine fuzzy theory nominal grouptechnique and VIKOR and propose the decision makingmethod based on fuzzy NGT-VIKOR Aghajani et al [29]propose a method combining TOPSIS and VIKOR to helpdecision makers prioritize and select optimal alternative

Above all combining fuzzy VIKOR and other methodsto make decision is the most commonly used In previousstudy of fuzzy VIKOR application the process of rating withrespect to evaluation criteria is based on the condition thatall experts are using the same evaluation criteria systemto evaluate all schemes However in realistic process ofdecision making various experts may hold different opin-ions on criteria selection some criteria are considered tobe important by all decision makers but others are onlypartially favored In other words every decision maker tendsto adopt different criteria systems to evaluate all schemesConsequently in order to make fuzzy VIKOR better adaptedto realistic decision making process this research proposesthe decision making model using improved fuzzy VIKORin which different decision makers adopt different criteriasystems to evaluate all schemes At the fifth section of thisresearch sensitivity analysis is carried out to testify whetherthe decision making model is stable and scientific

3 Research Idea and Theoretical Basis

31 Research Idea When applying fuzzy VIKOR in schemeselection every decision maker is supposed to evaluateall alternatives with the same evaluation criteria systemHowever decision makers incline to evaluate schemes withdifferent evaluation criteria systems due to the differencesin professional background among them This research

proposes a scheme selection model using improved fuzzyVIKOR which allows different decision makers to applydifferent evaluation criteria systems in scheme selectionprocess The research idea is shown in Figure 1

32 Scheme Selection Model Using Fuzzy VIKOR

321 Evaluation Criteria Selection Before formulating theevaluation criteria system the objectives of decision makingshould be made clear first A comprehensive system ofevaluation criterion should referee previous experience andconsider the features of objectives at the same time

322 Linguistic Variables and Corresponding Fuzzy NumberIn scheme selection the understanding judgment intuitionand preference of decisionmakers are fuzzy which is difficultto bemeasured and representedwith precise numerical valueThus it would be more accurate to represent these linguisticvariables with fuzzy numbers This paper applies the fuzzyVIKOR method based on trapezoidal fuzzy numbers

When applying fuzzy VIKOR the weight of evaluationcriteria and the weight of alternatives with respect to eachcriterion are described using linguistic variables firstly Inthis research linguistic variables are divided into 7 levelswhich are very low (VL) low (L) fairly low (FL) medium(M) fairly high (FH) high (H) and very high (VH) [30]The trapezoidal fuzzy number corresponding to linguisticvariables is 119860 which can be defined as (1198991 1198992 1198993 1198994) |1198991 1198992 1198993 1198994 isin 119877 1198991 le 1198992 le 1198993 le 1198994 where 1198991 denotes thepossible minimum that each linguistic variable correspondswith 1198994 denotes the possiblemaximum and 1198992 1198993 denote thepossible values between 1198991 and 1198994 Themembership functionis defined using (1) [30]

120583119860 (119909) =

119909 minus 11989911198992 minus 1198991 119909 isin [1198991 1198992] 1 119909 isin [1198992 1198993] 1198994 minus 1199091198994 minus 1198993 119909 isin [1198993 1198994] 0 Otherwise

(1)

The details and values of 1198991 1198992 1198993 1198994 are shown inFigures 2 and 3 The linguistic variables and correspondingtrapezoidal fuzzy numbers are shown in Table 1 [30]

323 Data Collection Invite relevant experts or managersfrom internal decision unit to rate the selected criteria usinglinguistic variablesThen linguistic variables are transformedinto corresponding fuzzy set Let the fuzzy rating of scheme119860 119894 with respect to criteria 119885119895 by decision maker 119863119896 be 119877119894119895119896where 119877119894119895119896 = 1198771198941198951198961 1198771198941198951198962 1198771198941198951198963 1198771198941198951198964 and importance weightof evaluation criteria 119885119895 by decision maker119863119896 be119882119895119896119882119895119896 =1198821198951198961119882119895119896211988211989511989631198821198951198964


Decision maker

Need to construct a scientific design scheme selection model of lean management

Current situation1Selecting lean management design scheme is a typical GDM problem

2All decision makers apply the same evaluation criteria system in traditionalscheme selection process

3Decision compromisedtoleadsandtimemuchspendsprocessmakingsolution

Problem every decision maker hasdifferent preference on evaluation criteria

Sensitivity analysis

Create schemedesignmanagementleanselecttomodelscientifica

Check the anti-jamming ability of Decision making results

Solution

Commonevaluation criteria

Individual evaluation criteria

Problem the decision makingresult is not optimal

Based on

Fuzzy VIKOR

To obtain

Solution

D1

D2

D3

Dn

Alternatives A1 A2 A3 A4middotmiddotmiddot An

Zn

Z3

Z2

Z1Z Z＜ Z＝

Zh Zi Zj

Zu Zv Zw

Zo Zp Zq

Figure 1 Research idea

n1 n2 n3 n4

Figure 2 Trapezoidal fuzzy numbers 119860

324 Data Processing Aggregation Normalizationand Defuzzification

(1) AggregationThe aggregated fuzzy weight of each criterionis calculated in (2)ndash(6) [30 31]

119882119895 = 1198821198951119882119895211988211989531198821198954 (2)

0 0301 02 0604 05 0907 08 1

VL L FL M FH H VH

Figure 3 Corresponding trapezoidal fuzzy numbers

where

1198821198951 = min 1198821198951198961 (3)

1198821198952 = 1119896 sum1198821198951198962 (4)

1198821198953 = 1119896 sum1198821198951198963 (5)

1198821198954 = max 1198821198951198964 (6)


Table 1 Linguistic variables and corresponding fuzzy numbers

Linguistic variable Fuzzy numberVery low (VL) (00000102)Low (L) (01020203)Fairly low (FL) (02030405)Medium (M) (04050506)Fairly high (FH) (05060708)High (H) (07080809)Very high (VH) (08091010)

The aggregated fuzzy ratings of alternatives with respectto criterion are obtained using (7)ndash(11) [30 32]

119877119894119895 = 1198771198941198951 1198771198941198952 1198771198941198953 1198771198941198954 (7)

where

1198771198941198951 = min 1198771198941198951198961 (8)

1198771198941198952 = 1119896 sum1198771198941198951198962 (9)

1198771198941198953 = 1119896 sum1198771198941198951198963 (10)

1198771198941198954 = max 1198771198941198951198964 (11)

(2) Normalization In order to remove the dimensions of allcriteria linear normalization is applied to the fuzzy weight ofcriteria and weight of alternatives with respect to criteriaThecriterion whose higher value is desirable is called beneficialcriterion (119861) and its value should be divided by themaximumvalue of the decision matrix the criterion whose lower valueis desirable is called cost criterion (119862) and its value shouldbe divided by the minimum value of decision matrix Themethod is shown in (12)ndash(14) [30 33]

1198771015840119894119895 =

(1198771198941198951119877+1198941198954 1198771198941198952119877+1198941198954

1198771198941198953119877+1198941198954 1198771198941198954119877+1198941198954) 119885119895 isin 119861

(1198771198941198951119877minus1198941198951 1198771198941198952119877minus1198941198951

1198771198941198953119877minus1198941198951 1198771198941198954119877minus1198941198951) 119885119895 isin 119862 (12)

Here

119877+1198941198954 = max119894119889119890119888119894119904119894119900119899 119898119886119905119903119894119909

119885119895 isin 119861 (13)

119877minus1198941198951 = min119894119889119890119888119894119904119894119900119899 119898119886119905119903119894119909

119885119895 isin 119862 (14)

(3) Defuzzification The fuzzy weights of criteria and alterna-tives with respect to criteria are defuzzified through takingthe average of the normalized trapezoidal fuzzy numbers asis shown in (15) [30]

119891119894119895 = 119863119890119891119891119906119911 (1198771015840119894119895) = (11987710158401198941198951 + 11987710158401198941198952 + 11987710158401198941198953 + 119877101584011989411989544 ) (15)

325 Calculation of Utility Regret and VIKOR Index Utility(119878119894) regret (119866119894) and VIKOR index (119876119894) are calculated using(16)ndash(18) [30 32 34 35]

119878119894 = 119899sum119895=1

1198820119895 (119891lowast minus 119891119894119895)119891lowast minus 119891minus (16)

119866119894 = max119894

(1198820119895 (119891lowast minus 119891119894119895)119891lowast minus 119891minus ) (17)

119876119894 = V (119878119894 minus 119878lowast)119878minus minus 119878lowast + (1 minus V) (119866119894 minus 119866lowast)119866minus minus 119866lowast (18)

where 119878119894 119866119894 and 119876119894 respectively represent the util-ity regret and VIKOR index of the alternative 119860 119894 (119894 =1 2 sdot sdot sdot 119898) 119891lowastand 119891minus indicate the best and worst value of119891119894119895 and1198820119895 indicates the weight of criterion 119885119895 Here V is thecoefficient of decision making mechanism (weight of grouputility) and 1 minus V is the weight of individual regret 119878lowast =min119894119878119894 119878minus = max119894119878119894 119866lowast = min119894119866119894 119866minus = max119894119866119894

Rank all alternatives according to the values of 119878119894 119866119894 and119876119894 in increasing order the alternative which has the leastvalue of 119876119894 is quite likely to be the best solution However itstill needs further refining the detailed method for judgingthe best solution is shown in the following part of thisresearch

326 Proposing Compromise Solution If the alternative 119860(1)(119860(1) is the first position in the alternatives ranked by 119876)satisfies the following two conditions then it is the bestsolution [30]

Condition 1 119876(119860(2))minus119876(119860(1)) ge 1(119898minus1)119860(2) is the secondposition in the alternatives ranked by 119876Condition 2 Alternative119860(1)must also be at the first positionwhen ranked by 119878 orand 119877

If condition 1 is not satisfied then we could only get aset of compromise solutions [30] The compromise solutionsare composed of 119860(1) 119860(2) sdot sdot sdot 119860(119909) where 119860(119909) is obtainedaccording to 119876(119860(119909)) minus 119876(119860(1)) lt 1(119898 minus 1) for maximum 1199094 Case Study

This research brings several improvements to fuzzy VIKORmethod and proposes the lean management design schemeselection model based on previous research In order tomore clearly introduce the application of the design schemeselection model this research takes the decision makingprocess in W real estate development group Co Ltd (WGC)as a case

41 Case Situation WGC is an integrated enterprise group inChinawhich is based on real estate development and involvesbuilding constriction commercial logistics catering services


Given the problems which can be settled through leanmanagement

Lean management solutions

Select evaluation criterion system

Determine the weight of decision makers in decision making

Rate the evaluation criteria

Rate alternatives with respect to criteria

Data processing

Calculate Utility Regret measure and VIKOR index

Rank the alternatives

solution

alternativesofnumberfuzzywith respect to criteria should be

aggregated according to theweight of decision makers

Personal evaluation criteria

Common evaluation criteria

Lean management schemes

Propose

Design

Gain

Figure 4 Lean management design scheme selection model using improved fuzzy VIKOR

and culture media WGC has been rapidly developing sinceits establishment while at the same time some hiddenmanagement problems also showupwith the scale expansionExisting management problems are mainly in the followingaspects

(1)The responsibility and authority are divided unclearlyand performance appraisal system and stimulation measuresare incomplete

(2) The management is extensive lacking standard man-agement processes reasonable management systems andscientific management methods and the operation costs arehigh

(3) The management layer is aging badly and talentechelon construction is quite lagging

In order to improve management managers in WGPdecide to bring in lean management and invite an experi-enced lean management team Given the existing problemslean management team summarizes several improvementmeasures

(1) Management by objective (MBO) includes joband staff design objective design objective performanceappraisal and performance reward system

(2) Profit center management (PCM) includes profitcenter design and performance appraisal for executives

(3) Costmanagement (CM) includes designing budgetingand cost evaluation mechanism and setting research projectsof cost reduction

(4) Talents echelon building (TEB) includes lean ideapopularization and talents echelon bank construction

Based on these lean management experts and seniormanagers design five lean management schemes as is shownin Table 2

In order to select optimal lean management designscheme chairman of the board (1198634) CFO (1198635) and threelean management experts (1198631 1198632 1198633) form a decision mak-ing team The next part of this research will specificallyexplain how to use the lean management design schemeselection model to reach the optimal solution

42 Scheme Selection Model Using Improved Fuzzy VIKORThe model of design scheme selection in lean managementusing improved fuzzy VIKOR is shown in Figure 4

421 Evaluation Criteria and Alternatives Description Thecriteria considered in this research are shown in Table 3

The criteria considered in this research can be dividedinto common evaluation criteria and personal evaluation cri-teria Common evaluation criteria include capital investment(1198851) expected output (1198852) the reasonability of scheme (1198853)and long-term development of company (1198854) Personal eval-uation criteria include waste minimization (1198855) continuousimprovement (1198856) all staff participation (1198857) complexity ofscheme (1198858) difficulty of implementation (1198859) and valuemaximization (11988510)

Capital investment is directly linked with the costexpected output indicates the outcome of a scheme thereasonability of scheme is related to the coordination betweenall improvement plans in the scheme and the long-termdevelopment of company depends on whether the schemecan exert favorable and lasting influence on the developmentof enterprise Waste minimization is closely related to theutilization of various resources and materials continuousimprovement is required to ensure the perpetual optimiza-tion of the company and all staff participationmeans all staffs


Table 2 Lean management design schemes

Schemes Contents Costs(yen) Objectives1198601 MBO 400000 MBO 100 done1198602 MBO PCM 550000 MBO 100 done profits improve by 301198603 MBO CM 500000 MBO 100 done costs reduce by 101198604 MBO CM TEB 600000 MBO 100 done costs reduce by 10 TEB 100 done1198605 MBO PCM CM TEB 680000 MBO 100 done profits improve by 30 costs reduce by 10 TEB 100 done

Table 3 Evaluation criteria system

Common evaluationcriteria

Capital investment (1198851)Expected output (1198852)

Reasonability of scheme (1198853)Long-term development of company (1198854)1198631 1198632 1198633 1198634 1198635

Personal evaluationcriteria

Wasteminimization(1198855)

Continuousimprovement(1198856)





Complexity ofscheme (1198858)

All staffparticipation(1198857)

Difficulty ofimplementation(1198859)





Valuemaximization(11988510)


are involved in the scheme implementation The complexityof scheme includes the number of plans and the workloadof each plan the difficulty of implementation considersthe response of staff and the enthusiasm of managers andvalue maximization means to bring the company optimalimprovement under present condition

422 Data Collection Firstly this research invites five deci-sion makers to rate the weight of criteria using linguisticvariables the rating results are shown in Table 4 and thecorresponding fuzzy numbers are shown in Table 5

Then this research invites five decision makers to rateeach alternative with respect to criteria using linguisticvariables the rating results are shown in Table 6 and corre-sponding fuzzy numbers are shown in Table 7

423 Data Processing

(1) Aggregation Firstly weight of criterion is aggregated using(2)ndash(6) Taking the fuzzy ratings of 1198851 as an example theratings are (07080809) (07080809) (07080809)(08091010) (08091010)

11988211 = min 11988211198961 = 0711988212 = sum120579119896 ∙ 11988211198962

= 15 times (08 + 08 + 08 + 09 + 09) = 084

11988213 = sum120579119896 ∙ 11988211198963= 15 times (08 + 08 + 08 + 10 + 10) = 087

11988214 = max 11988211198964 = 10(19)

Therefore the aggregated fuzzy rating of 1198851 is (07 084 08710)

Then the weight of alternative with respect to criteriais aggregated In this research five decision makers applydifferent evaluation criteria systems to rate every alternativewhile for decision makers the influence of their decisionmaking is different Hence this research applies AHPmethodto determine the weights of decision makers in decisionmaking (shown in Table 8)

The weight of decision maker is calculated using the rootmethod (20)

120579119894 = ( 119899prod119895=1

119886119894119895)1119899

(20)

Taking the calculation of 1205791 as an example 1205791 =(prod119899119895=1119886119894119895)1119899 = (1 times 2 times 4 times 2 times 3)15 = 217The results are shown in Table 9The rating of alternatives with respect to criterion

assessed by different decision makers needs to be linearlynormalized the normalization results are shown in Table 10


Table 4 Weight of criteria assessed by decision makers (linguistic variable)

1198631 1198632 1198633 1198634 11986351198851 H H H VH VH1198852 VH VH VH VH VH1198853 VH VH VH H H1198854 H H H VH H1198855 H FH H M VH1198856 H H M H FH1198857 VH M VH FH H1198858 FH VH H M M1198859 M H M H M11988510 FH M FH VH VH

Table 5 Weight of criteria assessed by decision makers (fuzzy set)

1198631 1198632 1198633 1198634 11986351198851 (07080809) (07080809) (07080809) (08091010) (08091010)1198852 (08091010) (08091010) (08091010) (08091010) (08091010)1198853 (08091010) (08091010) (08091010) (07080809) (07080809)1198854 (07080809) (07080809) (07080809) (08091010) (07080809)1198855 (07080809) (05060708) (07080809) (04050506) (08091010)1198856 (07080809) (07080809) (04050506) (07080809) (05060708)1198857 (08091010) (04050506) (08091010) (05060708) (07080809)1198858 (05060708) (08091010) (07080809) (04050506) (04050506)1198859 (04050506) (07080809) (04050506) (07080809) (04050506)11988510 (05060708) (04050506) (05060708) (08091010) (08091010)

Table 6 Rating results (linguistic variables)

1198631 11986321198851 1198852 1198853 1198854 1198855 1198856 1198857 1198851 1198852 1198853 1198854 1198856 1198858 11988591198601 VH H VH M M H H H M H FH M FH VH1198602 FH VH FH M H M H FH H M H M VH H1198603 H VH H H H H VH H VH H M FH H H1198604 FH FH VH VH FH VH VH VH H VH H H FH H1198605 M H VH VH VH VH VH M VH VH VH VH M FH1198633 11986341198851 1198852 1198853 1198854 1198855 1198857 1198858 1198851 1198852 1198853 1198854 1198856 1198859 119885101198601 FH M FH H M FH H VH M FH H M VH M1198602 H H M FH FH FH H H FH H H H M FH1198603 H FH H FH H H H H VH H FH VH H H1198604 M FH VH VH FH VH FH FH FH VH FH VH FH FH1198605 VH VH VH VH VH VH M M H H VH H M H11986351198851 1198852 1198853 1198854 1198855 1198857 119885101198601 H M VH FH M M M1198602 VH H H H FH FH FH1198603 H H FH VH VH H H1198604 FH FH H FH FH H FH1198605 FH VH VH H FH VH VH


Table 7 Rating results (fuzzy set)

1198601 1198602 1198603 1198604 11986051198631 1198851 (08091010) (05060708) (07080809) (05060708) (05060708)1198852 (07080809) (08091010) (08091010) (05060708) (07080809)1198853 (08091010) (05060708) (07080809) (08091010) (08091010)1198854 (04050506) (04050506) (07080809) (08091010) (08091010)1198855 (04050506) (07080809) (07080809) (05060708) (08091010)1198856 (07080809) (04050506) (07080809) (08091010) (08091010)1198857 (07080809) (07080809) (08091010) (08091010) (08091010)1198632 1198851 (07080809) (07080809) (07080809) (08091010) (04050506)1198852 (04050506) (07080809) (07080809) (07080809) (08091010)1198853 (07080809) (04050506) (07080809) (08091010) (08091010)1198854 (05060708) (07080809) (04050506) (07080809) (08091010)1198856 (04050506) (04050506) (05060708) (07080809) (08091010)1198858 (05060708) (08091010) (07080809) (05060708) (04050506)1198859 (08091010) (07080809) (07080809) (07080809) (05060708)1198633 1198851 (05060708) (07080809) (07080809) (04050506) (08091010)1198852 (04050506) (07080809) (05060708) (05060708) (08091010)1198853 (05060708) (04050506) (07080809) (08091010) (08091010)1198854 (07080809) (05060708) (05060708) (08091010) (08091010)1198855 (04050506) (05060708) (07080809) (05060708) (08091010)1198857 (05060708) (05060708) (07080809) (08091010) (08091010)1198858 (07080809) (07080809) (07080809) (05060708) (04050506)1198634 1198851 (08091010) (07080809) (07080809) (05060708) (04050506)1198852 (04050506) (05060708) (08091010) (05060708) (07080809)1198853 (05060708) (07080809) (07080809) (08091010) (07080809)1198854 (07080809) (07080809) (05060708) (05060708) (08091010)1198856 (04050506) (07080809) (08091010) (08091010) (07080809)1198859 (08091010) (04050506) (07080809) (05060708) (04050506)11988510 (04050506) (05060708) (07080809) (05060708) (07080809)1198635 1198851 (07080809) (08091010) (07080809) (05060708) (05060708)1198852 (04050506) (07080809) (07080809) (05060708) (08091010)1198853 (08091010) (07080809) (05060708) (07080809) (08091010)1198854 (05060708) (07080809) (08091010) (05060708) (07080809)1198855 (04050506) (05060708) (08091010) (05060708) (05060708)1198857 (04050506) (05060708) (07080809) (07080809) (08091010)11988510 (04050506) (05060708) (07080809) (05060708) (08091010)

Table 8 The judgment results using a scale of 1-9

1198631 1198632 1198633 1198634 11986351198631 1 2 4 2 31198632 05 1 3 1 21198633 025 033 1 033 051198634 05 1 3 1 21198635 033 05 2 05 1

Table 9 Weight results of decision makers

1198631 1198632 1198633 1198634 11986351205791 217 125 042 125 070

Take the rating of1198601 with respect to1198851 as an example theratings are (08091010) (07080809) (05060708)(08091010) (07080809)

119877111 = min 119877111198961 = 05 (21)

119877112 = sum120579119896119877111198962= 037 times 09 + 022 times 08 + 006 times 06 + 022

times 09 + 013 times 08 = 085(22)


times 10 + 013 times 08 = 091(23)


Table 10 Weight of decision makers (normalized)

1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 119885101198631 037 037 037 037 066 046 0661198632 022 022 022 022 027 079 051198633 006 006 006 006 011 011 0211198634 022 022 022 022 027 05 0631198635 013 013 013 013 023 023 037

Table 11 Summary sheet of collected data

1198851 1198852 1198853 1198854 1198855119882119895 (0708408710) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0508509110) (0406106109) (0507908710) (0406206509) (04050506)1198602 (0507007710) (0508008510) (0406407009) (0407007109) (0507307709)1198603 (07080809) (0508709610) (07080809) (04080810) (0708008210)1198604 (0406607510) (050606509) (0708909710) (0507708510) (05060708)1198605 (0405406308) (0708408810) (0708809610) (0708909710) (0508409310)1198856 1198857 1198858 1198859 11988510119882119895 (070840910) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0406406409) (0407107209) (0506407209) (08091010) (04050506)1198602 (0405805809) (0507307409) (0708809610) (0406506509) (05060708)1198603 (0708308510) (0708709710) (07080809) (07080809) (07080809)1198604 (0708708910) (0708809510) (05060708) (050707509) (05060708)1198605 (0708809510) (08090910) (04050506) (040550608) (0708408710)

119877114 = max 119877111198964 = 10 (24)

Accordingly the aggregated fuzzy rating of 1198601 withrespect to 1198851 is (05 085 091 10) The aggregation resultsare shown in Table 11

Secondly using (12) to normalize the criterion weightsand alternative ratings with respect to criterion assessed bydecision makers in this research Z1 Z5 and Z7 are costcriteria and their values should be divided by the minimumvalue of decision matrix other criteria are benefit criteriaand their values should be divided by the maximum value ofthe decision matrix The normalization results are shown inTable 12

(2) Defuzzification The defuzzification of the data in Table 12is achieved using (15) the results of defuzzification are shownin Table 13

Take the defuzzification of the rating of A1 with respectto Z1 as an example the fuzzy rating is (125 213 228 25)

Deffuz (11990311) = (119903111 + 119903112 + 119903113 + 1199031144 )= 125 + 213 + 228 + 254 = 204 (25)

424 Arrangement and Analysis of Results

(1) Calculate the Best andWorst Values 119891lowastand 119891minusindicate thebest and worst values of alternative rating with respect to the

same criterion the best values (119891lowast) and worst values (119891minus) areshown in Table 14

(2) Calculation of Utility Regret and VIKOR Index Utility(119878119894) regret (119866119894) and VIKOR index (119876119894) are calculated using(16)ndash(18) take the calculation of 1198781 1198661 and1198761 as an example

1198781 = 119899sum119895=1

1198820119895 (119891lowast minus 1198911119895)119891lowast minus 119891minus= 215 times (143 minus 204)143 minus 204 + 093 times (063 minus 063)063 minus 086

+ 087 times (066 minus 082)066 minus 089 + 084 times (064 minus 066)064 minus 088+ 180 times (125 minus 125)125 minus 211 + 086 times (062 minus 064)062 minus 088+ 093 times (068 minus 068)068 minus 093 + 218 times (125 minus 173)125 minus 221+ 084 times (058 minus 093)058 minus 093 + 072 times (05 minus 05)05 minus 086

= 4761198661 = max

119894(1198820119895 (119891lowast minus 119891119894119895)119891lowast minus 119891minus ) = 215 times (143 minus 204)143 minus 204

= 215


Table 12 Normalization results

1198851 1198852 1198853 1198854119882119895 (175210225250) (080090100100) (070086092100) (070082084100)1198601 (125213228250) (040061061090) (050079087100) (040062065090)1198602 (125175193250) (050080085100) (040064070090) (040070071090)1198603 (175200200225) (050087096100) (070080080090) (040080080100)1198604 (100165213250) (050060065090) (070089097100) (050077085100)1198605 (100135158200) (070084088100) (070088096100) (070089097100)1198855 1198856 1198857 1198858119882119895 (100180190250) (070084090100) (080090100100) (175215230250)1198601 (100125125150) (040064064090) (040071072090) (125160180225)1198602 (125183193225) (040058058090) (050073074090) (175220240250)1198603 (175200210250) (070083085100) (070087097100) (175200200225)1198604 (100120140160) (070087089100) (070088095100) (125150175200)1198605 (125210233250) (070088095100) (080090090100) (100125125150)1198859 11988510119882119895 (070082084100) (040072076100)1198601 (080090100100) (040050050060)1198602 (040065065090) (050060070080)1198603 (070080080090) (070080080090)1198604 (050070075090) (050060070080)1198605 (040055060080) (070084087100)

Table 13 Defuzzification results

1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510Weight 215 093 087 084 180 086 093 218 084 0721198601 204 063 083 064 125 064 068 173 093 0501198602 185 079 066 067 181 062 072 221 065 0651198603 200 083 079 071 211 083 087 215 080 0801198604 176 069 089 078 130 088 088 163 071 0651198605 143 086 088 089 204 088 093 125 059 085

1198761 = V (1198781 minus 119878lowast)119878minus minus 119878lowast + (1 minus V) (1198661 minus 119866lowast)119866minus minus 119866lowast= 05 times (480 minus 480)480 minus 994 + 05 times (115 minus 215)115 minus 218= 049

(26)

The results are shown in Table 15In this research 1198604 is best ranked by 119876 and 119866 but it is

not best ranked by 119878 at the same time 119876(A5) minus 119876(A4) =044 minus 021 lt 1(5 minus 1) = 025 These facts mean that1198604 does not satisfy condition 1 in other words accordingto the judgment standard of traditional fuzzy VIKOR 1198604is not the best solution and 1198604 and 1198605 are components ofcompromise solutions However applying improved fuzzyVIKOR introduced in this research can effectively guarantee1198604 to be the best solution In order to prove that 1198604 is theoptimal solution this research proposes sensitivity analysis

5 Sensitivity Analysis

In order to prove the lean management design schemeselectionmodel using improved fuzzyVIKOR to be scientificthe antijamming ability of this model needs to be checkedExisting research has indicated that through testing whetherthe change in weights of various relevant factors will causegreat deviation to decision making result can prove thedecision making model to be scientific [36] Given thissensitivity analysis is approached from three aspects toanalyze the variation of decision making result the weight ofdecision maker 120579 the weight of evaluation criteria 120596 and theweight of mechanism coefficient ]

51 Analysis on the Coefficient of DecisionMakingMechanismLet the coefficient of decisionmakingmechanism ] change inthe interval [01 09] with a range of 01 each time the resultsare shown in Figure 5

Figure 5 shows that with the decisionmakingmechanismcoefficient changing from 01 to about 075 1198601 takes the


Table 14 The best and worst values

1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510119891lowast 143 063 066 065 125 062 068 125 059 05119891minus 204 086 089 089 211 088 093 221 093 085

Table 15 Utility regret and VIKOR index and their ranking

1198601 ranking 1198602 ranking 1198603 ranking 1198604 ranking 1198605 ranking119878 476 1 620 2 994 5 690 4 678 3119866 215 4 218 5 203 3 117 1 166 5119876(] = 05) 049 3 064 4 093 5 021 1 044 2

002040608

112

1 2 3 4 5 6 7 8 9

A1A2A3

A4A5

Figure 5 Sensitivity analysis on decision making mechanismcoefficient ]

optimized position when the coefficient changes from about075 to 09 1198601 comes to be the best solution

52 Analysis on the Weight of Decision Maker In thisresearch perturbation method [36] is applied to analysisthe sensitivity about the weight of decision maker Let theperturbation parameter be 120585 then 1205791015840119896 = 120585 ∙ 120579119896

For the five decision makers involved in this researchtheir weights are normalized which means

5sum119896=1

120579119896 = 1 (27)

1205791015840 + 5sum119898=1119898 =119896

1205791015840119898 = 1 (28)

120585 ∙ 120579119896 + 120582 5sum119898=1119898 =119896

120579119898 = 1 (29)

Combining (27) with (29) we can obtain that 120582 = (1 minus 120585 ∙120579119896)(1minus120579119896) whichmeans that when the weight of119863119896 changeswith a coefficient of 120585 then the weight of other decisionmakers will change with a coefficient of 120582 The sensitivityanalysis is taken making 120585 change within the interval [0812] with a range of 05 each time the results are shown inFigure 6

Figure 6 shows that in the 45 times experiments allschemes are not so sensitive to the change in the weight ofdecision makers and the priority order of schemes remains

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 450

010203040506070809

1

A1A2A3

A4A5

Figure 6 Sensitivity analysis on the weight of decision maker 120579

still except the 9th experiment The research results furthercertify that 1198604 is the best solution53 Analysis on the Weight of Evaluation Criteria To testthe sensitivity of decision making results affected by weightof evaluation criteria this experiment brings in a series ofrandom disturbances the method is as follows 200 sets ofrandom coefficients [1205721198861 1205721198862 sdot sdot sdot 12057211988610] (119886 = 1 2 sdot sdot sdot 200)that satisfy Gaussian distribution with mean 0 and variance06666 are provided usingMATLABAccording to the featureof Gaussian distribution the probability that values of theserandom coefficients are involved in the interval [01 09] is9974 Partial random coefficients are shown in Table 16the weight of evaluation criteria changed from1198821198950 to 1205721198861198951198821198950and this change will further affect VIKOR index119876 (shown inFigure 7)

Figure 7 shows that1198605 is very sensitive to the disturbanceand fluctuates severely However 1198604 is quite stable thatdisturbance exerts a negligible effect on1198604 Also1198604 takes theposition of optimal solution all the time (the VIKOR indexvalue of 1198604 is always the lowest) With no doubt 1198604 is theoptimal solution

6 Conclusions

Currently fuzzy VIKOR is a widely used decision makingmethod In decision making model using fuzzy VIKOR all


Table 16 Random coefficients that satisfy Gaussian distribution (partial)

Random coefficients that satisfy Gaussian distribution (partial)108 104 110 091 106 107 104 111 102 098106 106 106 106 086 100 095 102 098 095097 092 108 104 104 099 102 097 110 108096 096 109 095 089 093 095 102 100 105109 109 083 093 099 101 091 102 098 106098 104 103 098 091 104 095 108 096 101101 114 100 099 102 108 097 106 097 088108 098 100 100 096 095 105 107 099 114100 099 095 110 097 103 093 101 086 095094 099 089 110 112 105 104 102 106 104098 099 094 095 093 107 094 107 099 111108 108 102 105 106 099 107 101 111 101095 107 100 106 095 093 101 090 108 099108 127 104 106 097 097 099 101 092 105099 109 105 104 108 095 092 094 104 098094 086 093 103 103 100 092 099 093 095094 096 100 098 103 089 099 097 093 085106 102 103 096 108 096 108 095 098 099102 095 094 102 096 096 100 108 090 101

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12

Figure 7 Sensitivity analysis on the weight of evaluation criteria 120596

decisionmakers apply the same evaluation criteria to evaluateall schemes However in realistic decision making processdifferent decision makers may hold different opinions onevaluation criteria Under this situation applying the sameevaluation criteria to evaluate all schemes will spend muchtime in decision making and cannot always obtain optimalsolution In order toweigh the opinions of all decisionmakersand to obtain optimal solution in shorter time this researchproposes the decision making model using improved fuzzyVIKOR and implies this model in real lean managementdesign schememodel selection caseThis research also carriesout sensitivity analysis to test whether the solution is optimal

the analysis results confirm that this model is scientificThis research proposes the lean management design schemeselection model using improved fuzzy VIKOR This modelhas three advantages (1) the model can weigh the opinionsof all decision makers because the model allows differentdecisionmakers to evaluate all schemes with different evalua-tion criteria systems it is more applicable to realistic decisionmaking process in enterprise (2) solution obtained using thismodel has been tested to be quite stable so this model isscientific and can effectively avoid suboptimal solution Inconclusion the model is quite scientific and stable and itcan be widely applied in lean management design schemeselection process to obtain optimal solution

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this manuscript

Acknowledgments

This research work is supported by the Program for thePhilosophy and Social Sciences Research of Higher LearningInstitutions of ShanxiNo 2016 (39)No 2017326 theNationalNatural Science Foundation of China under Grant No


71071107 and the Program for the Outstanding InnovativeTeams ofHigher Learning Institutions of ShanxiNo 2016 (3)

References

[1] J PWomack and D T Jones ldquoFrom lean production to the leanenterpriserdquo Business Review vol 24 no 4 pp 38ndash46 1994

[2] A Taylor M Taylor and A Mcsweeney ldquoTowards greaterunderstanding of success and survival of lean systemsrdquo Interna-tional Journal of Production Research vol 51 no 22 pp 6607ndash6630 2013

[3] S Taj and C Morosan ldquoThe impact of lean operations on theChinese manufacturing performancerdquo Journal of Manufactur-ing Technology Management vol 22 no 2 pp 223ndash240 2011

[4] J Jayaram A Das and M Nicolae ldquoLooking beyond the obvi-ous Unraveling the Toyota production systemrdquo InternationalJournal of Production Economics vol 128 no 1 pp 280ndash2912010

[5] T L Saaty ldquoModeling unstructured decision problems - thetheory of analytical hierarchiesrdquoMathematics and Computers inSimulation vol 20 no 3 pp 147ndash158 1978

[6] S-R Yu J Tao and J-WWang ldquoMulti criteria decision analysisfor selecting product life cycle scenarios based on analyticalhierarchical process (AHP)rdquo Shanghai Jiaotong Daxue Xue-baoJournal of Shanghai Jiaotong University vol 41 no 4 pp520ndash531 2007

[7] Y-M Wang and K-S Chin ldquoA new data envelopment analysismethod for priority determination and group decision makingin the analytic hierarchy processrdquo European Journal of Opera-tional Research vol 195 no 1 pp 239ndash250 2009

[8] B Zhu and Z Xu ldquoAnalytic hierarchy process-hesitant groupdecision makingrdquo European Journal of Operational Researchvol 239 no 3 pp 794ndash801 2014

[9] Z Xie and S O Business ldquoCov-AHP an improvement ofanalytic hierarchy process methodrdquo Journal of Quantitative andTechnical Economics vol 8 pp 137ndash148 2015

[10] R Medjoudj D Aissani and K D Haim ldquoPower customer sat-isfaction and profitability analysis using multi-criteria decisionmaking methodsrdquo International Journal of Electrical Power ampEnergy Systems vol 45 no 1 pp 331ndash339 2013

[11] Y He and J X Weng ldquoMultiple attribute group decision basedon uncertain linguistic information for the credit guaranteeobjectsrdquo Systems Engineering vol 8 pp 25ndash29 2013

[12] C-C Chou and J-F Ding ldquoApplication of an integratedmodel with MCDM and IPA to evaluate the service quality oftransshipment portrdquoMathematical Problems in Engineering vol2013 Article ID 656757 7 pages 2013

[13] F N Khasanah A E Permanasari and S Suningkusumawar-dani ldquoFuzzy MADM for major selection at senior high schoolrdquoin Proceedings of the 2nd International Conference on Infor-mation Technology Computer and Electrical Engineering ICI-TACEE 2015 pp 41ndash45 Indonesia October 2015

[14] G Buyukozkan S Guleryuz and B Karpak ldquoA new combinedIF-DEMATEL and IF-ANP approach for CRM partner evalua-tionrdquo International Journal of Production Economics vol 191 pp194ndash206 2017

[15] G I Alptekin and G Buyukozkan ldquoAn integrated case-basedreasoning and MCDM system for Web based tourism destina-tion planningrdquo Expert Systems with Applications vol 38 no 3pp 2125ndash2132 2011

[16] M Hanine O Boutkhoum A Tikniouine and T AgoutildquoA new web-based framework development for fuzzy multi-criteria group decision-makingrdquo Springer Plus vol 5 no 1 p601 2016

[17] Y H Li X Chen Y Zhang and Zh P F ldquoMethods forselecting NPD project risk response alternatives using case-based decision theoryrdquo Journal of Industrial Engineering andEngineering Management vol 29 no 3 pp 257ndash264 2015

[18] T Guilfoos and A D Pape ldquoPredicting human cooperationin the Prisonerrsquos Dilemma using case-based decision theoryrdquoTheory and Decision vol 80 no 1 pp 1ndash32 2016

[19] K Kinjo and T Ebina ldquoCase-based decision model matchesideal point modelrdquo Journal of Intelligent Information Systemspp 1ndash22 2017

[20] G Buyukozkan and S Guleryuz ldquoMulti criteria group decisionmaking approach for smart phone selection using intuitionisticFuzzy TOPSISrdquo International Journal of Computational Intelli-gence Systems vol 9 no 4 pp 709ndash725 2016

[21] T H You B B Cao and B Xia ldquoAlternative selection methodconsidering air situation based on IFSST and TOPSISrdquo SoftScience vol 30 no 9 pp 131ndash135 2016

[22] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 pp 263ndash2911979

[23] J H Hu X H Chen and Y M Liu ldquoMulti-criteria decision-making method based on linguistic evaluation and prospecttheoryrdquo Control and Decision vol 24 no 10 pp 1477ndash14822009

[24] J Q Wang and Z Ling ldquoGrey-stochastic multi-criteriadecision-making approach based on prospect theoryrdquo SystemsEngineering-theory and Practice vol 30 no 9 pp 1658ndash16642010

[25] J J Hao J J Zhu and S F Liu ldquoA method for multi-stagestochastic multi-criteria decision-making concerning prospecttheoryrdquo Chinese Journal of Management Science vol 23 no 1pp 73ndash81 2015

[26] P Li Y Yang and C Wei ldquoAn intuitionistic fuzzy stochasticdecision-making method based on case-based reasoning andprospect theoryrdquo Mathematical Problems in Engineering vol2017 Article ID 2874954 13 pages 2017

[27] A Sanayei S F Mousavi and A Yazdankhah ldquoGroup decisionmaking process for supplier selection with VIKOR under fuzzyenvironmentrdquo Expert Systems with Applications vol 37 no 1pp 24ndash30 2010

[28] A Awasthi and G Kannan ldquoGreen supplier developmentprogram selection usingNGTandVIKORunder fuzzy environ-mentrdquo Computers amp Industrial Engineering vol 91 pp 100ndash1082016

[29] M M Aghajani G P Taherei N M N Sulaiman et alldquoApplication of TOPSIS and VIKOR improved versions in amulti criteria decision analysis to develop an optimized munic-ipal solid waste management modelrdquo Journal of EnvironmentalManagement vol 166 no 9 pp 109ndash115 2016

[30] S Vinodh S Sarangan and S C Vinoth ldquoApplication of fuzzycompromise solution method for fit concept selectionrdquo AppliedMathematical Modelling vol 38 no 3 pp 1052ndash1063 2014

[31] G Jung S M Kim S Y Kim and E S Jung ldquoEffects of designfactors of the instrument cluster panel on consumersrsquo affectionrdquoLectureNotes in Engineering andComputer Science vol 2182 no1 pp 2419ndash2446 2010


[32] R J Girubha and S Vinodh ldquoApplication of fuzzy VIKORand environmental impact analysis for material selection of anautomotive componentrdquo Materials and Corrosion vol 37 pp478ndash486 2012

[33] A Shemshadi H Shirazi M Toreihi andM J Tarokh ldquoA fuzzyVIKOR method for supplier selection based on entropy mea-sure for objective weightingrdquo Expert Systems with Applicationsvol 38 no 10 pp 12160ndash12167 2011

[34] S Vinodh A R Varadharajan and A Subramanian ldquoAppli-cation of fuzzy VIKOR for concept selection in an agile envi-ronmentrdquoThe International Journal of AdvancedManufacturingTechnology vol 65 no 5-8 pp 825ndash832 2013

[35] F B Cui X Y You H Shi and H C Liu ldquoOptimal sitingof electric vehicle charging stations using Pythagorean fuzzyVIKOR approachrdquo Mathematical Problems in Engineering vol2018 Article ID 9262067 12 pages 2018

[36] Y Yuan T Guan X-B Yan and Y-J Li ldquoBased on hybridVIKOR method decision making model for supplier selectionrdquoControl and Decision vol 29 no 3 pp 551ndash560 2014 (Chinese)

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Lacking scientific method of lean management designscheme selection is the main reason that leads to suboptimalsolution Currently decision makers mainly select lean man-agement design scheme through discussion which meanssubjective judgment is predominant Selecting lean man-agement design scheme is a typical group decision makingproblem which has threemain features (1) different schemeshave different advantages and they cannot be evaluated usingthe same evaluation criteria system (2) different decisionmakers may have different evaluation criteria systems (3)selecting lean management design scheme is a fuzzy processScientifically selecting leanmanagement design schememusttake the above three features into consideration

In order to solve the problems in selecting lean manage-ment design scheme this research applies fuzzy VIKOR tomake a selection from a quantitative perspective and con-structs a design scheme selectionmodel in leanmanagementMain contributions of this research include the following(1) the design scheme selection model constructed in thisresearch will save time in selecting scheme for enterpriseimprove the accuracy of lean management design schemeselection and provide a way of effectively avoiding subopti-mal scheme (2) this research expands the application rangeof fuzzy VIKOR method (3) this research creates a model ofdesign scheme selection in leanmanagement providing basisfor enterprises to select design scheme scientifically in leanmanagement practices

2 Literature Review

Designing schemes is the beginning of all managementactivities and the quality of management schemes reflectsthe management strategy of the enterprise Designing leanmanagement schemes is the basis of implementing leanmanagement Whether the schemes are splendid or notwill affect the result of lean management implementationTherefore scientifically selecting lean management designscheme is the basis of implementing lean management atthe same time the method of scheme selection is the basisof scientifically selecting lean management design schemeExisting researches have proposed the following six mainmethods of scheme selection

(1) Analytic Hierarchy Process (AHP) Since AHP wasproposed by American scholar Saaty in 1970s [5] it hasbeen applied in multiple sectors to solve decision makingproblems For example Yu et al [6] research on product lifecycle schemes selection applying AHP and establish hier-archical structure relation to select optimal design schemeoptimal scheme is obtained through pairwise comparisonand comprehensive evaluation Wang and Chin [7] pro-pose the new data envelopment analysis (DEA) methodfor priority determination in AHP helping decision makerschoose optimal scheme Zhu and Xu [8] apply AHP inhesitant group decision making (GDM) process to help withdecision making taking the water conservancy in China asan example which means to apply AHP in GDM processthat has hesitant judgments When applying AHP differentdecision makers hold different opinions on the importance

of various elements so the order of schemes obtained is alsodifferent Therefore Xie [9] combines covariance matrix andAHP to create aCov-AHPmethod thismethod can constructjudgmentmatrix using covariancematrix and obtain the onlyorder of scheme through calculation

(2) Multicriteria decision making (MCDM) MCDMincludes multiobjective decision making (MODM) and mul-tiattribute decision making (MADM) MCDM plays animportant role in many fields For example Medjoudj et al[10] apply MCDM to help decision makers improve profitand custom satisfaction in electric power company Giventhat the risk of credit guarantee products is complex He andWeng [11] use indeterminate language to describe the weightand attribute value of indexes in product risk evaluationindex system creating the group decision making model ofindeterminate language for credit guarantee products ChouandDing [12] constructMCDMmodel to evaluate the servicequality and the location choice of transshipment portsKhasanah et al [13] apply fuzzy MADM method in majorselection at senior high school research results show thatthis method can accommodate the presence of uncertaintyin decision making and the accuracy that decision makingresults are in accordance with the reality is as much as90 Buyukozkan et al [14] apply MCDMmethod includingintuitionistic fuzzy analytic network process intuitionisticfuzzy decision making trial and evaluation laboratory thismethod canmore effectively choose suitable business partnerWith the development of Internet technology many scholarsconstruct web based on intelligence framework to helpwith decision making For example combining case-basedreasoning system and MCDM technique to construct theweb based on intelligence framework can improve the speedand accuracy of decision making [15] constructing a webincluding all MCDM tools can help decision makers to selectschemes more quickly and reliably [16]

(3) Case-based decision theory (CBDT) The main ideaof CBDT is using previous method to solve current simi-lar decision making problems CBDT can avoid repetitivedecision making reducing time and cost Through collectingsimilar previous cases and calculating the similarity betweenthe target case and previous case the utility value of the effectbrought by previous case implementation can be calculatedand then integrated utility value of every scheme can becalculated [17] Guilfoos and Pape [18] use CBDT to analyzethe cooperative dynamics in prisonerrsquos dilemma proving thatCBDT model is more suitable when it comes to selectingschemes using cooperative dynamic analysis Studying therelationship between CBDT and ideal point model canreveal previous dependence point or provide reference pointwhich helps enterprise in product designing and productpositioning providing support for marketing decision [19]

(4) IF-TOPSIS Intuitionistic fuzzy sets (IFS) are mainlyapplied to deal with the indeterminacy in decision makingprocess Technique for order performance by similarity toideal solution (TOPSIS) can rank evaluation objects throughtesting the distance between evaluation objects and optimalsolution or worst solution an evaluation object will bethe best solution if it is closest to optimal solution andfarthest to worst solution IF-TOPSIS can be used to solve













120583119860 (119909) =


(1)




Decision maker









Solution




Based on

Fuzzy VIKOR

To obtain

Solution

D1

D2

D3

Dn


Zn

Z3

Z2

Z1Z Z＜ Z＝

Zh Zi Zj

Zu Zv Zw

Zo Zp Zq


n1 n2 n3 n4




119882119895 = 1198821198951119882119895211988211989531198821198954 (2)

0 0301 02 0604 05 0907 08 1

VL L FL M FH H VH


where

1198821198951 = min 1198821198951198961 (3)

1198821198952 = 1119896 sum1198821198951198962 (4)

1198821198953 = 1119896 sum1198821198951198963 (5)

1198821198954 = max 1198821198951198964 (6)





119877119894119895 = 1198771198941198951 1198771198941198952 1198771198941198953 1198771198941198954 (7)

where

1198771198941198951 = min 1198771198941198951198961 (8)

1198771198941198952 = 1119896 sum1198771198941198951198962 (9)

1198771198941198953 = 1119896 sum1198771198941198951198963 (10)

1198771198941198954 = max 1198771198941198951198964 (11)


1198771015840119894119895 =

(1198771198941198951119877+1198941198954 1198771198941198952119877+1198941198954

1198771198941198953119877+1198941198954 1198771198941198954119877+1198941198954) 119885119895 isin 119861

(1198771198941198951119877minus1198941198951 1198771198941198952119877minus1198941198951

1198771198941198953119877minus1198941198951 1198771198941198954119877minus1198941198951) 119885119895 isin 119862 (12)

Here

119877+1198941198954 = max119894119889119890119888119894119904119894119900119899 119898119886119905119903119894119909

119885119895 isin 119861 (13)

119877minus1198941198951 = min119894119889119890119888119894119904119894119900119899 119898119886119905119903119894119909

119885119895 isin 119862 (14)


119891119894119895 = 119863119890119891119891119906119911 (1198771015840119894119895) = (11987710158401198941198951 + 11987710158401198941198952 + 11987710158401198941198953 + 119877101584011989411989544 ) (15)


119878119894 = 119899sum119895=1


119866119894 = max119894

















Data processing



solution






Propose

Design

Gain











































423 Data Processing


11988211 = min 11988211198961 = 0711988212 = sum120579119896 ∙ 11988211198962

= 15 times (08 + 08 + 08 + 09 + 09) = 084

11988213 = sum120579119896 ∙ 11988211198963= 15 times (08 + 08 + 08 + 10 + 10) = 087

11988214 = max 11988211198964 = 10(19)




120579119894 = ( 119899prod119895=1

119886119894119895)1119899

(20)







1198631 1198632 1198633 1198634 11986351198851 (07080809) (07080809) (07080809) (08091010) (08091010)1198852 (08091010) (08091010) (08091010) (08091010) (08091010)1198853 (08091010) (08091010) (08091010) (07080809) (07080809)1198854 (07080809) (07080809) (07080809) (08091010) (07080809)1198855 (07080809) (05060708) (07080809) (04050506) (08091010)1198856 (07080809) (07080809) (04050506) (07080809) (05060708)1198857 (08091010) (04050506) (08091010) (05060708) (07080809)1198858 (05060708) (08091010) (07080809) (04050506) (04050506)1198859 (04050506) (07080809) (04050506) (07080809) (04050506)11988510 (05060708) (04050506) (05060708) (08091010) (08091010)





1198601 1198602 1198603 1198604 11986051198631 1198851 (08091010) (05060708) (07080809) (05060708) (05060708)1198852 (07080809) (08091010) (08091010) (05060708) (07080809)1198853 (08091010) (05060708) (07080809) (08091010) (08091010)1198854 (04050506) (04050506) (07080809) (08091010) (08091010)1198855 (04050506) (07080809) (07080809) (05060708) (08091010)1198856 (07080809) (04050506) (07080809) (08091010) (08091010)1198857 (07080809) (07080809) (08091010) (08091010) (08091010)1198632 1198851 (07080809) (07080809) (07080809) (08091010) (04050506)1198852 (04050506) (07080809) (07080809) (07080809) (08091010)1198853 (07080809) (04050506) (07080809) (08091010) (08091010)1198854 (05060708) (07080809) (04050506) (07080809) (08091010)1198856 (04050506) (04050506) (05060708) (07080809) (08091010)1198858 (05060708) (08091010) (07080809) (05060708) (04050506)1198859 (08091010) (07080809) (07080809) (07080809) (05060708)1198633 1198851 (05060708) (07080809) (07080809) (04050506) (08091010)1198852 (04050506) (07080809) (05060708) (05060708) (08091010)1198853 (05060708) (04050506) (07080809) (08091010) (08091010)1198854 (07080809) (05060708) (05060708) (08091010) (08091010)1198855 (04050506) (05060708) (07080809) (05060708) (08091010)1198857 (05060708) (05060708) (07080809) (08091010) (08091010)1198858 (07080809) (07080809) (07080809) (05060708) (04050506)1198634 1198851 (08091010) (07080809) (07080809) (05060708) (04050506)1198852 (04050506) (05060708) (08091010) (05060708) (07080809)1198853 (05060708) (07080809) (07080809) (08091010) (07080809)1198854 (07080809) (07080809) (05060708) (05060708) (08091010)1198856 (04050506) (07080809) (08091010) (08091010) (07080809)1198859 (08091010) (04050506) (07080809) (05060708) (04050506)11988510 (04050506) (05060708) (07080809) (05060708) (07080809)1198635 1198851 (07080809) (08091010) (07080809) (05060708) (05060708)1198852 (04050506) (07080809) (07080809) (05060708) (08091010)1198853 (08091010) (07080809) (05060708) (07080809) (08091010)1198854 (05060708) (07080809) (08091010) (05060708) (07080809)1198855 (04050506) (05060708) (08091010) (05060708) (05060708)1198857 (04050506) (05060708) (07080809) (07080809) (08091010)11988510 (04050506) (05060708) (07080809) (05060708) (08091010)


1198631 1198632 1198633 1198634 11986351198631 1 2 4 2 31198632 05 1 3 1 21198633 025 033 1 033 051198634 05 1 3 1 21198635 033 05 2 05 1


1198631 1198632 1198633 1198634 11986351205791 217 125 042 125 070


119877111 = min 119877111198961 = 05 (21)


times 09 + 013 times 08 = 085(22)


times 10 + 013 times 08 = 091(23)



1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 119885101198631 037 037 037 037 066 046 0661198632 022 022 022 022 027 079 051198633 006 006 006 006 011 011 0211198634 022 022 022 022 027 05 0631198635 013 013 013 013 023 023 037


1198851 1198852 1198853 1198854 1198855119882119895 (0708408710) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0508509110) (0406106109) (0507908710) (0406206509) (04050506)1198602 (0507007710) (0508008510) (0406407009) (0407007109) (0507307709)1198603 (07080809) (0508709610) (07080809) (04080810) (0708008210)1198604 (0406607510) (050606509) (0708909710) (0507708510) (05060708)1198605 (0405406308) (0708408810) (0708809610) (0708909710) (0508409310)1198856 1198857 1198858 1198859 11988510119882119895 (070840910) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0406406409) (0407107209) (0506407209) (08091010) (04050506)1198602 (0405805809) (0507307409) (0708809610) (0406506509) (05060708)1198603 (0708308510) (0708709710) (07080809) (07080809) (07080809)1198604 (0708708910) (0708809510) (05060708) (050707509) (05060708)1198605 (0708809510) (08090910) (04050506) (040550608) (0708408710)

119877114 = max 119877111198964 = 10 (24)





Deffuz (11990311) = (119903111 + 119903112 + 119903113 + 1199031144 )= 125 + 213 + 228 + 254 = 204 (25)





1198781 = 119899sum119895=1



= 4761198661 = max


= 215



1198851 1198852 1198853 1198854119882119895 (175210225250) (080090100100) (070086092100) (070082084100)1198601 (125213228250) (040061061090) (050079087100) (040062065090)1198602 (125175193250) (050080085100) (040064070090) (040070071090)1198603 (175200200225) (050087096100) (070080080090) (040080080100)1198604 (100165213250) (050060065090) (070089097100) (050077085100)1198605 (100135158200) (070084088100) (070088096100) (070089097100)1198855 1198856 1198857 1198858119882119895 (100180190250) (070084090100) (080090100100) (175215230250)1198601 (100125125150) (040064064090) (040071072090) (125160180225)1198602 (125183193225) (040058058090) (050073074090) (175220240250)1198603 (175200210250) (070083085100) (070087097100) (175200200225)1198604 (100120140160) (070087089100) (070088095100) (125150175200)1198605 (125210233250) (070088095100) (080090090100) (100125125150)1198859 11988510119882119895 (070082084100) (040072076100)1198601 (080090100100) (040050050060)1198602 (040065065090) (050060070080)1198603 (070080080090) (070080080090)1198604 (050070075090) (050060070080)1198605 (040055060080) (070084087100)


1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510Weight 215 093 087 084 180 086 093 218 084 0721198601 204 063 083 064 125 064 068 173 093 0501198602 185 079 066 067 181 062 072 221 065 0651198603 200 083 079 071 211 083 087 215 080 0801198604 176 069 089 078 130 088 088 163 071 0651198605 143 086 088 089 204 088 093 125 059 085


(26)









1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510119891lowast 143 063 066 065 125 062 068 125 059 05119891minus 204 086 089 089 211 088 093 221 093 085



002040608

112

1 2 3 4 5 6 7 8 9

A1A2A3

A4A5





5sum119896=1

120579119896 = 1 (27)

1205791015840 + 5sum119898=1119898 =119896

1205791015840119898 = 1 (28)

120585 ∙ 120579119896 + 120582 5sum119898=1119898 =119896

120579119898 = 1 (29)



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 450

010203040506070809

1

A1A2A3

A4A5




6 Conclusions





1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12




Data Availability




Acknowledgments




References













































Journal of












Journal of







Volume 2018






Volume 2018




















120583119860 (119909) =


(1)




Decision maker









Solution




Based on

Fuzzy VIKOR

To obtain

Solution

D1

D2

D3

Dn


Zn

Z3

Z2

Z1Z Z＜ Z＝

Zh Zi Zj

Zu Zv Zw

Zo Zp Zq


n1 n2 n3 n4




119882119895 = 1198821198951119882119895211988211989531198821198954 (2)

0 0301 02 0604 05 0907 08 1

VL L FL M FH H VH


where

1198821198951 = min 1198821198951198961 (3)

1198821198952 = 1119896 sum1198821198951198962 (4)

1198821198953 = 1119896 sum1198821198951198963 (5)

1198821198954 = max 1198821198951198964 (6)





119877119894119895 = 1198771198941198951 1198771198941198952 1198771198941198953 1198771198941198954 (7)

where

1198771198941198951 = min 1198771198941198951198961 (8)

1198771198941198952 = 1119896 sum1198771198941198951198962 (9)

1198771198941198953 = 1119896 sum1198771198941198951198963 (10)

1198771198941198954 = max 1198771198941198951198964 (11)


1198771015840119894119895 =

(1198771198941198951119877+1198941198954 1198771198941198952119877+1198941198954

1198771198941198953119877+1198941198954 1198771198941198954119877+1198941198954) 119885119895 isin 119861

(1198771198941198951119877minus1198941198951 1198771198941198952119877minus1198941198951

1198771198941198953119877minus1198941198951 1198771198941198954119877minus1198941198951) 119885119895 isin 119862 (12)

Here

119877+1198941198954 = max119894119889119890119888119894119904119894119900119899 119898119886119905119903119894119909

119885119895 isin 119861 (13)

119877minus1198941198951 = min119894119889119890119888119894119904119894119900119899 119898119886119905119903119894119909

119885119895 isin 119862 (14)


119891119894119895 = 119863119890119891119891119906119911 (1198771015840119894119895) = (11987710158401198941198951 + 11987710158401198941198952 + 11987710158401198941198953 + 119877101584011989411989544 ) (15)


119878119894 = 119899sum119895=1


119866119894 = max119894

















Data processing



solution






Propose

Design

Gain











































423 Data Processing


11988211 = min 11988211198961 = 0711988212 = sum120579119896 ∙ 11988211198962

= 15 times (08 + 08 + 08 + 09 + 09) = 084

11988213 = sum120579119896 ∙ 11988211198963= 15 times (08 + 08 + 08 + 10 + 10) = 087

11988214 = max 11988211198964 = 10(19)




120579119894 = ( 119899prod119895=1

119886119894119895)1119899

(20)







1198631 1198632 1198633 1198634 11986351198851 (07080809) (07080809) (07080809) (08091010) (08091010)1198852 (08091010) (08091010) (08091010) (08091010) (08091010)1198853 (08091010) (08091010) (08091010) (07080809) (07080809)1198854 (07080809) (07080809) (07080809) (08091010) (07080809)1198855 (07080809) (05060708) (07080809) (04050506) (08091010)1198856 (07080809) (07080809) (04050506) (07080809) (05060708)1198857 (08091010) (04050506) (08091010) (05060708) (07080809)1198858 (05060708) (08091010) (07080809) (04050506) (04050506)1198859 (04050506) (07080809) (04050506) (07080809) (04050506)11988510 (05060708) (04050506) (05060708) (08091010) (08091010)





1198601 1198602 1198603 1198604 11986051198631 1198851 (08091010) (05060708) (07080809) (05060708) (05060708)1198852 (07080809) (08091010) (08091010) (05060708) (07080809)1198853 (08091010) (05060708) (07080809) (08091010) (08091010)1198854 (04050506) (04050506) (07080809) (08091010) (08091010)1198855 (04050506) (07080809) (07080809) (05060708) (08091010)1198856 (07080809) (04050506) (07080809) (08091010) (08091010)1198857 (07080809) (07080809) (08091010) (08091010) (08091010)1198632 1198851 (07080809) (07080809) (07080809) (08091010) (04050506)1198852 (04050506) (07080809) (07080809) (07080809) (08091010)1198853 (07080809) (04050506) (07080809) (08091010) (08091010)1198854 (05060708) (07080809) (04050506) (07080809) (08091010)1198856 (04050506) (04050506) (05060708) (07080809) (08091010)1198858 (05060708) (08091010) (07080809) (05060708) (04050506)1198859 (08091010) (07080809) (07080809) (07080809) (05060708)1198633 1198851 (05060708) (07080809) (07080809) (04050506) (08091010)1198852 (04050506) (07080809) (05060708) (05060708) (08091010)1198853 (05060708) (04050506) (07080809) (08091010) (08091010)1198854 (07080809) (05060708) (05060708) (08091010) (08091010)1198855 (04050506) (05060708) (07080809) (05060708) (08091010)1198857 (05060708) (05060708) (07080809) (08091010) (08091010)1198858 (07080809) (07080809) (07080809) (05060708) (04050506)1198634 1198851 (08091010) (07080809) (07080809) (05060708) (04050506)1198852 (04050506) (05060708) (08091010) (05060708) (07080809)1198853 (05060708) (07080809) (07080809) (08091010) (07080809)1198854 (07080809) (07080809) (05060708) (05060708) (08091010)1198856 (04050506) (07080809) (08091010) (08091010) (07080809)1198859 (08091010) (04050506) (07080809) (05060708) (04050506)11988510 (04050506) (05060708) (07080809) (05060708) (07080809)1198635 1198851 (07080809) (08091010) (07080809) (05060708) (05060708)1198852 (04050506) (07080809) (07080809) (05060708) (08091010)1198853 (08091010) (07080809) (05060708) (07080809) (08091010)1198854 (05060708) (07080809) (08091010) (05060708) (07080809)1198855 (04050506) (05060708) (08091010) (05060708) (05060708)1198857 (04050506) (05060708) (07080809) (07080809) (08091010)11988510 (04050506) (05060708) (07080809) (05060708) (08091010)


1198631 1198632 1198633 1198634 11986351198631 1 2 4 2 31198632 05 1 3 1 21198633 025 033 1 033 051198634 05 1 3 1 21198635 033 05 2 05 1


1198631 1198632 1198633 1198634 11986351205791 217 125 042 125 070


119877111 = min 119877111198961 = 05 (21)


times 09 + 013 times 08 = 085(22)


times 10 + 013 times 08 = 091(23)



1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 119885101198631 037 037 037 037 066 046 0661198632 022 022 022 022 027 079 051198633 006 006 006 006 011 011 0211198634 022 022 022 022 027 05 0631198635 013 013 013 013 023 023 037


1198851 1198852 1198853 1198854 1198855119882119895 (0708408710) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0508509110) (0406106109) (0507908710) (0406206509) (04050506)1198602 (0507007710) (0508008510) (0406407009) (0407007109) (0507307709)1198603 (07080809) (0508709610) (07080809) (04080810) (0708008210)1198604 (0406607510) (050606509) (0708909710) (0507708510) (05060708)1198605 (0405406308) (0708408810) (0708809610) (0708909710) (0508409310)1198856 1198857 1198858 1198859 11988510119882119895 (070840910) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0406406409) (0407107209) (0506407209) (08091010) (04050506)1198602 (0405805809) (0507307409) (0708809610) (0406506509) (05060708)1198603 (0708308510) (0708709710) (07080809) (07080809) (07080809)1198604 (0708708910) (0708809510) (05060708) (050707509) (05060708)1198605 (0708809510) (08090910) (04050506) (040550608) (0708408710)

119877114 = max 119877111198964 = 10 (24)





Deffuz (11990311) = (119903111 + 119903112 + 119903113 + 1199031144 )= 125 + 213 + 228 + 254 = 204 (25)





1198781 = 119899sum119895=1



= 4761198661 = max


= 215



1198851 1198852 1198853 1198854119882119895 (175210225250) (080090100100) (070086092100) (070082084100)1198601 (125213228250) (040061061090) (050079087100) (040062065090)1198602 (125175193250) (050080085100) (040064070090) (040070071090)1198603 (175200200225) (050087096100) (070080080090) (040080080100)1198604 (100165213250) (050060065090) (070089097100) (050077085100)1198605 (100135158200) (070084088100) (070088096100) (070089097100)1198855 1198856 1198857 1198858119882119895 (100180190250) (070084090100) (080090100100) (175215230250)1198601 (100125125150) (040064064090) (040071072090) (125160180225)1198602 (125183193225) (040058058090) (050073074090) (175220240250)1198603 (175200210250) (070083085100) (070087097100) (175200200225)1198604 (100120140160) (070087089100) (070088095100) (125150175200)1198605 (125210233250) (070088095100) (080090090100) (100125125150)1198859 11988510119882119895 (070082084100) (040072076100)1198601 (080090100100) (040050050060)1198602 (040065065090) (050060070080)1198603 (070080080090) (070080080090)1198604 (050070075090) (050060070080)1198605 (040055060080) (070084087100)


1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510Weight 215 093 087 084 180 086 093 218 084 0721198601 204 063 083 064 125 064 068 173 093 0501198602 185 079 066 067 181 062 072 221 065 0651198603 200 083 079 071 211 083 087 215 080 0801198604 176 069 089 078 130 088 088 163 071 0651198605 143 086 088 089 204 088 093 125 059 085


(26)









1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510119891lowast 143 063 066 065 125 062 068 125 059 05119891minus 204 086 089 089 211 088 093 221 093 085



002040608

112

1 2 3 4 5 6 7 8 9

A1A2A3

A4A5





5sum119896=1

120579119896 = 1 (27)

1205791015840 + 5sum119898=1119898 =119896

1205791015840119898 = 1 (28)

120585 ∙ 120579119896 + 120582 5sum119898=1119898 =119896

120579119898 = 1 (29)



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 450

010203040506070809

1

A1A2A3

A4A5




6 Conclusions





1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12




Data Availability




Acknowledgments




References













































Journal of












Journal of







Volume 2018






Volume 2018









Decision maker









Solution




Based on

Fuzzy VIKOR

To obtain

Solution

D1

D2

D3

Dn


Zn

Z3

Z2

Z1Z Z＜ Z＝

Zh Zi Zj

Zu Zv Zw

Zo Zp Zq


n1 n2 n3 n4




119882119895 = 1198821198951119882119895211988211989531198821198954 (2)

0 0301 02 0604 05 0907 08 1

VL L FL M FH H VH


where

1198821198951 = min 1198821198951198961 (3)

1198821198952 = 1119896 sum1198821198951198962 (4)

1198821198953 = 1119896 sum1198821198951198963 (5)

1198821198954 = max 1198821198951198964 (6)





119877119894119895 = 1198771198941198951 1198771198941198952 1198771198941198953 1198771198941198954 (7)

where

1198771198941198951 = min 1198771198941198951198961 (8)

1198771198941198952 = 1119896 sum1198771198941198951198962 (9)

1198771198941198953 = 1119896 sum1198771198941198951198963 (10)

1198771198941198954 = max 1198771198941198951198964 (11)


1198771015840119894119895 =

(1198771198941198951119877+1198941198954 1198771198941198952119877+1198941198954

1198771198941198953119877+1198941198954 1198771198941198954119877+1198941198954) 119885119895 isin 119861

(1198771198941198951119877minus1198941198951 1198771198941198952119877minus1198941198951

1198771198941198953119877minus1198941198951 1198771198941198954119877minus1198941198951) 119885119895 isin 119862 (12)

Here

119877+1198941198954 = max119894119889119890119888119894119904119894119900119899 119898119886119905119903119894119909

119885119895 isin 119861 (13)

119877minus1198941198951 = min119894119889119890119888119894119904119894119900119899 119898119886119905119903119894119909

119885119895 isin 119862 (14)


119891119894119895 = 119863119890119891119891119906119911 (1198771015840119894119895) = (11987710158401198941198951 + 11987710158401198941198952 + 11987710158401198941198953 + 119877101584011989411989544 ) (15)


119878119894 = 119899sum119895=1


119866119894 = max119894

















Data processing



solution






Propose

Design

Gain











































423 Data Processing


11988211 = min 11988211198961 = 0711988212 = sum120579119896 ∙ 11988211198962

= 15 times (08 + 08 + 08 + 09 + 09) = 084

11988213 = sum120579119896 ∙ 11988211198963= 15 times (08 + 08 + 08 + 10 + 10) = 087

11988214 = max 11988211198964 = 10(19)




120579119894 = ( 119899prod119895=1

119886119894119895)1119899

(20)







1198631 1198632 1198633 1198634 11986351198851 (07080809) (07080809) (07080809) (08091010) (08091010)1198852 (08091010) (08091010) (08091010) (08091010) (08091010)1198853 (08091010) (08091010) (08091010) (07080809) (07080809)1198854 (07080809) (07080809) (07080809) (08091010) (07080809)1198855 (07080809) (05060708) (07080809) (04050506) (08091010)1198856 (07080809) (07080809) (04050506) (07080809) (05060708)1198857 (08091010) (04050506) (08091010) (05060708) (07080809)1198858 (05060708) (08091010) (07080809) (04050506) (04050506)1198859 (04050506) (07080809) (04050506) (07080809) (04050506)11988510 (05060708) (04050506) (05060708) (08091010) (08091010)





1198601 1198602 1198603 1198604 11986051198631 1198851 (08091010) (05060708) (07080809) (05060708) (05060708)1198852 (07080809) (08091010) (08091010) (05060708) (07080809)1198853 (08091010) (05060708) (07080809) (08091010) (08091010)1198854 (04050506) (04050506) (07080809) (08091010) (08091010)1198855 (04050506) (07080809) (07080809) (05060708) (08091010)1198856 (07080809) (04050506) (07080809) (08091010) (08091010)1198857 (07080809) (07080809) (08091010) (08091010) (08091010)1198632 1198851 (07080809) (07080809) (07080809) (08091010) (04050506)1198852 (04050506) (07080809) (07080809) (07080809) (08091010)1198853 (07080809) (04050506) (07080809) (08091010) (08091010)1198854 (05060708) (07080809) (04050506) (07080809) (08091010)1198856 (04050506) (04050506) (05060708) (07080809) (08091010)1198858 (05060708) (08091010) (07080809) (05060708) (04050506)1198859 (08091010) (07080809) (07080809) (07080809) (05060708)1198633 1198851 (05060708) (07080809) (07080809) (04050506) (08091010)1198852 (04050506) (07080809) (05060708) (05060708) (08091010)1198853 (05060708) (04050506) (07080809) (08091010) (08091010)1198854 (07080809) (05060708) (05060708) (08091010) (08091010)1198855 (04050506) (05060708) (07080809) (05060708) (08091010)1198857 (05060708) (05060708) (07080809) (08091010) (08091010)1198858 (07080809) (07080809) (07080809) (05060708) (04050506)1198634 1198851 (08091010) (07080809) (07080809) (05060708) (04050506)1198852 (04050506) (05060708) (08091010) (05060708) (07080809)1198853 (05060708) (07080809) (07080809) (08091010) (07080809)1198854 (07080809) (07080809) (05060708) (05060708) (08091010)1198856 (04050506) (07080809) (08091010) (08091010) (07080809)1198859 (08091010) (04050506) (07080809) (05060708) (04050506)11988510 (04050506) (05060708) (07080809) (05060708) (07080809)1198635 1198851 (07080809) (08091010) (07080809) (05060708) (05060708)1198852 (04050506) (07080809) (07080809) (05060708) (08091010)1198853 (08091010) (07080809) (05060708) (07080809) (08091010)1198854 (05060708) (07080809) (08091010) (05060708) (07080809)1198855 (04050506) (05060708) (08091010) (05060708) (05060708)1198857 (04050506) (05060708) (07080809) (07080809) (08091010)11988510 (04050506) (05060708) (07080809) (05060708) (08091010)


1198631 1198632 1198633 1198634 11986351198631 1 2 4 2 31198632 05 1 3 1 21198633 025 033 1 033 051198634 05 1 3 1 21198635 033 05 2 05 1


1198631 1198632 1198633 1198634 11986351205791 217 125 042 125 070


119877111 = min 119877111198961 = 05 (21)


times 09 + 013 times 08 = 085(22)


times 10 + 013 times 08 = 091(23)



1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 119885101198631 037 037 037 037 066 046 0661198632 022 022 022 022 027 079 051198633 006 006 006 006 011 011 0211198634 022 022 022 022 027 05 0631198635 013 013 013 013 023 023 037


1198851 1198852 1198853 1198854 1198855119882119895 (0708408710) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0508509110) (0406106109) (0507908710) (0406206509) (04050506)1198602 (0507007710) (0508008510) (0406407009) (0407007109) (0507307709)1198603 (07080809) (0508709610) (07080809) (04080810) (0708008210)1198604 (0406607510) (050606509) (0708909710) (0507708510) (05060708)1198605 (0405406308) (0708408810) (0708809610) (0708909710) (0508409310)1198856 1198857 1198858 1198859 11988510119882119895 (070840910) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0406406409) (0407107209) (0506407209) (08091010) (04050506)1198602 (0405805809) (0507307409) (0708809610) (0406506509) (05060708)1198603 (0708308510) (0708709710) (07080809) (07080809) (07080809)1198604 (0708708910) (0708809510) (05060708) (050707509) (05060708)1198605 (0708809510) (08090910) (04050506) (040550608) (0708408710)

119877114 = max 119877111198964 = 10 (24)





Deffuz (11990311) = (119903111 + 119903112 + 119903113 + 1199031144 )= 125 + 213 + 228 + 254 = 204 (25)





1198781 = 119899sum119895=1



= 4761198661 = max


= 215



1198851 1198852 1198853 1198854119882119895 (175210225250) (080090100100) (070086092100) (070082084100)1198601 (125213228250) (040061061090) (050079087100) (040062065090)1198602 (125175193250) (050080085100) (040064070090) (040070071090)1198603 (175200200225) (050087096100) (070080080090) (040080080100)1198604 (100165213250) (050060065090) (070089097100) (050077085100)1198605 (100135158200) (070084088100) (070088096100) (070089097100)1198855 1198856 1198857 1198858119882119895 (100180190250) (070084090100) (080090100100) (175215230250)1198601 (100125125150) (040064064090) (040071072090) (125160180225)1198602 (125183193225) (040058058090) (050073074090) (175220240250)1198603 (175200210250) (070083085100) (070087097100) (175200200225)1198604 (100120140160) (070087089100) (070088095100) (125150175200)1198605 (125210233250) (070088095100) (080090090100) (100125125150)1198859 11988510119882119895 (070082084100) (040072076100)1198601 (080090100100) (040050050060)1198602 (040065065090) (050060070080)1198603 (070080080090) (070080080090)1198604 (050070075090) (050060070080)1198605 (040055060080) (070084087100)


1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510Weight 215 093 087 084 180 086 093 218 084 0721198601 204 063 083 064 125 064 068 173 093 0501198602 185 079 066 067 181 062 072 221 065 0651198603 200 083 079 071 211 083 087 215 080 0801198604 176 069 089 078 130 088 088 163 071 0651198605 143 086 088 089 204 088 093 125 059 085


(26)









1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510119891lowast 143 063 066 065 125 062 068 125 059 05119891minus 204 086 089 089 211 088 093 221 093 085



002040608

112

1 2 3 4 5 6 7 8 9

A1A2A3

A4A5





5sum119896=1

120579119896 = 1 (27)

1205791015840 + 5sum119898=1119898 =119896

1205791015840119898 = 1 (28)

120585 ∙ 120579119896 + 120582 5sum119898=1119898 =119896

120579119898 = 1 (29)



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 450

010203040506070809

1

A1A2A3

A4A5




6 Conclusions





1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12




Data Availability




Acknowledgments




References













































Journal of












Journal of







Volume 2018






Volume 2018












119877119894119895 = 1198771198941198951 1198771198941198952 1198771198941198953 1198771198941198954 (7)

where

1198771198941198951 = min 1198771198941198951198961 (8)

1198771198941198952 = 1119896 sum1198771198941198951198962 (9)

1198771198941198953 = 1119896 sum1198771198941198951198963 (10)

1198771198941198954 = max 1198771198941198951198964 (11)


1198771015840119894119895 =

(1198771198941198951119877+1198941198954 1198771198941198952119877+1198941198954

1198771198941198953119877+1198941198954 1198771198941198954119877+1198941198954) 119885119895 isin 119861

(1198771198941198951119877minus1198941198951 1198771198941198952119877minus1198941198951

1198771198941198953119877minus1198941198951 1198771198941198954119877minus1198941198951) 119885119895 isin 119862 (12)

Here

119877+1198941198954 = max119894119889119890119888119894119904119894119900119899 119898119886119905119903119894119909

119885119895 isin 119861 (13)

119877minus1198941198951 = min119894119889119890119888119894119904119894119900119899 119898119886119905119903119894119909

119885119895 isin 119862 (14)


119891119894119895 = 119863119890119891119891119906119911 (1198771015840119894119895) = (11987710158401198941198951 + 11987710158401198941198952 + 11987710158401198941198953 + 119877101584011989411989544 ) (15)


119878119894 = 119899sum119895=1


119866119894 = max119894

















Data processing



solution






Propose

Design

Gain











































423 Data Processing


11988211 = min 11988211198961 = 0711988212 = sum120579119896 ∙ 11988211198962

= 15 times (08 + 08 + 08 + 09 + 09) = 084

11988213 = sum120579119896 ∙ 11988211198963= 15 times (08 + 08 + 08 + 10 + 10) = 087

11988214 = max 11988211198964 = 10(19)




120579119894 = ( 119899prod119895=1

119886119894119895)1119899

(20)







1198631 1198632 1198633 1198634 11986351198851 (07080809) (07080809) (07080809) (08091010) (08091010)1198852 (08091010) (08091010) (08091010) (08091010) (08091010)1198853 (08091010) (08091010) (08091010) (07080809) (07080809)1198854 (07080809) (07080809) (07080809) (08091010) (07080809)1198855 (07080809) (05060708) (07080809) (04050506) (08091010)1198856 (07080809) (07080809) (04050506) (07080809) (05060708)1198857 (08091010) (04050506) (08091010) (05060708) (07080809)1198858 (05060708) (08091010) (07080809) (04050506) (04050506)1198859 (04050506) (07080809) (04050506) (07080809) (04050506)11988510 (05060708) (04050506) (05060708) (08091010) (08091010)





1198601 1198602 1198603 1198604 11986051198631 1198851 (08091010) (05060708) (07080809) (05060708) (05060708)1198852 (07080809) (08091010) (08091010) (05060708) (07080809)1198853 (08091010) (05060708) (07080809) (08091010) (08091010)1198854 (04050506) (04050506) (07080809) (08091010) (08091010)1198855 (04050506) (07080809) (07080809) (05060708) (08091010)1198856 (07080809) (04050506) (07080809) (08091010) (08091010)1198857 (07080809) (07080809) (08091010) (08091010) (08091010)1198632 1198851 (07080809) (07080809) (07080809) (08091010) (04050506)1198852 (04050506) (07080809) (07080809) (07080809) (08091010)1198853 (07080809) (04050506) (07080809) (08091010) (08091010)1198854 (05060708) (07080809) (04050506) (07080809) (08091010)1198856 (04050506) (04050506) (05060708) (07080809) (08091010)1198858 (05060708) (08091010) (07080809) (05060708) (04050506)1198859 (08091010) (07080809) (07080809) (07080809) (05060708)1198633 1198851 (05060708) (07080809) (07080809) (04050506) (08091010)1198852 (04050506) (07080809) (05060708) (05060708) (08091010)1198853 (05060708) (04050506) (07080809) (08091010) (08091010)1198854 (07080809) (05060708) (05060708) (08091010) (08091010)1198855 (04050506) (05060708) (07080809) (05060708) (08091010)1198857 (05060708) (05060708) (07080809) (08091010) (08091010)1198858 (07080809) (07080809) (07080809) (05060708) (04050506)1198634 1198851 (08091010) (07080809) (07080809) (05060708) (04050506)1198852 (04050506) (05060708) (08091010) (05060708) (07080809)1198853 (05060708) (07080809) (07080809) (08091010) (07080809)1198854 (07080809) (07080809) (05060708) (05060708) (08091010)1198856 (04050506) (07080809) (08091010) (08091010) (07080809)1198859 (08091010) (04050506) (07080809) (05060708) (04050506)11988510 (04050506) (05060708) (07080809) (05060708) (07080809)1198635 1198851 (07080809) (08091010) (07080809) (05060708) (05060708)1198852 (04050506) (07080809) (07080809) (05060708) (08091010)1198853 (08091010) (07080809) (05060708) (07080809) (08091010)1198854 (05060708) (07080809) (08091010) (05060708) (07080809)1198855 (04050506) (05060708) (08091010) (05060708) (05060708)1198857 (04050506) (05060708) (07080809) (07080809) (08091010)11988510 (04050506) (05060708) (07080809) (05060708) (08091010)


1198631 1198632 1198633 1198634 11986351198631 1 2 4 2 31198632 05 1 3 1 21198633 025 033 1 033 051198634 05 1 3 1 21198635 033 05 2 05 1


1198631 1198632 1198633 1198634 11986351205791 217 125 042 125 070


119877111 = min 119877111198961 = 05 (21)


times 09 + 013 times 08 = 085(22)


times 10 + 013 times 08 = 091(23)



1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 119885101198631 037 037 037 037 066 046 0661198632 022 022 022 022 027 079 051198633 006 006 006 006 011 011 0211198634 022 022 022 022 027 05 0631198635 013 013 013 013 023 023 037


1198851 1198852 1198853 1198854 1198855119882119895 (0708408710) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0508509110) (0406106109) (0507908710) (0406206509) (04050506)1198602 (0507007710) (0508008510) (0406407009) (0407007109) (0507307709)1198603 (07080809) (0508709610) (07080809) (04080810) (0708008210)1198604 (0406607510) (050606509) (0708909710) (0507708510) (05060708)1198605 (0405406308) (0708408810) (0708809610) (0708909710) (0508409310)1198856 1198857 1198858 1198859 11988510119882119895 (070840910) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0406406409) (0407107209) (0506407209) (08091010) (04050506)1198602 (0405805809) (0507307409) (0708809610) (0406506509) (05060708)1198603 (0708308510) (0708709710) (07080809) (07080809) (07080809)1198604 (0708708910) (0708809510) (05060708) (050707509) (05060708)1198605 (0708809510) (08090910) (04050506) (040550608) (0708408710)

119877114 = max 119877111198964 = 10 (24)





Deffuz (11990311) = (119903111 + 119903112 + 119903113 + 1199031144 )= 125 + 213 + 228 + 254 = 204 (25)





1198781 = 119899sum119895=1



= 4761198661 = max


= 215



1198851 1198852 1198853 1198854119882119895 (175210225250) (080090100100) (070086092100) (070082084100)1198601 (125213228250) (040061061090) (050079087100) (040062065090)1198602 (125175193250) (050080085100) (040064070090) (040070071090)1198603 (175200200225) (050087096100) (070080080090) (040080080100)1198604 (100165213250) (050060065090) (070089097100) (050077085100)1198605 (100135158200) (070084088100) (070088096100) (070089097100)1198855 1198856 1198857 1198858119882119895 (100180190250) (070084090100) (080090100100) (175215230250)1198601 (100125125150) (040064064090) (040071072090) (125160180225)1198602 (125183193225) (040058058090) (050073074090) (175220240250)1198603 (175200210250) (070083085100) (070087097100) (175200200225)1198604 (100120140160) (070087089100) (070088095100) (125150175200)1198605 (125210233250) (070088095100) (080090090100) (100125125150)1198859 11988510119882119895 (070082084100) (040072076100)1198601 (080090100100) (040050050060)1198602 (040065065090) (050060070080)1198603 (070080080090) (070080080090)1198604 (050070075090) (050060070080)1198605 (040055060080) (070084087100)


1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510Weight 215 093 087 084 180 086 093 218 084 0721198601 204 063 083 064 125 064 068 173 093 0501198602 185 079 066 067 181 062 072 221 065 0651198603 200 083 079 071 211 083 087 215 080 0801198604 176 069 089 078 130 088 088 163 071 0651198605 143 086 088 089 204 088 093 125 059 085


(26)









1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510119891lowast 143 063 066 065 125 062 068 125 059 05119891minus 204 086 089 089 211 088 093 221 093 085



002040608

112

1 2 3 4 5 6 7 8 9

A1A2A3

A4A5





5sum119896=1

120579119896 = 1 (27)

1205791015840 + 5sum119898=1119898 =119896

1205791015840119898 = 1 (28)

120585 ∙ 120579119896 + 120582 5sum119898=1119898 =119896

120579119898 = 1 (29)



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 450

010203040506070809

1

A1A2A3

A4A5




6 Conclusions





1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12




Data Availability




Acknowledgments




References













































Journal of












Journal of







Volume 2018






Volume 2018















Data processing



solution






Propose

Design

Gain











































423 Data Processing


11988211 = min 11988211198961 = 0711988212 = sum120579119896 ∙ 11988211198962

= 15 times (08 + 08 + 08 + 09 + 09) = 084

11988213 = sum120579119896 ∙ 11988211198963= 15 times (08 + 08 + 08 + 10 + 10) = 087

11988214 = max 11988211198964 = 10(19)




120579119894 = ( 119899prod119895=1

119886119894119895)1119899

(20)







1198631 1198632 1198633 1198634 11986351198851 (07080809) (07080809) (07080809) (08091010) (08091010)1198852 (08091010) (08091010) (08091010) (08091010) (08091010)1198853 (08091010) (08091010) (08091010) (07080809) (07080809)1198854 (07080809) (07080809) (07080809) (08091010) (07080809)1198855 (07080809) (05060708) (07080809) (04050506) (08091010)1198856 (07080809) (07080809) (04050506) (07080809) (05060708)1198857 (08091010) (04050506) (08091010) (05060708) (07080809)1198858 (05060708) (08091010) (07080809) (04050506) (04050506)1198859 (04050506) (07080809) (04050506) (07080809) (04050506)11988510 (05060708) (04050506) (05060708) (08091010) (08091010)





1198601 1198602 1198603 1198604 11986051198631 1198851 (08091010) (05060708) (07080809) (05060708) (05060708)1198852 (07080809) (08091010) (08091010) (05060708) (07080809)1198853 (08091010) (05060708) (07080809) (08091010) (08091010)1198854 (04050506) (04050506) (07080809) (08091010) (08091010)1198855 (04050506) (07080809) (07080809) (05060708) (08091010)1198856 (07080809) (04050506) (07080809) (08091010) (08091010)1198857 (07080809) (07080809) (08091010) (08091010) (08091010)1198632 1198851 (07080809) (07080809) (07080809) (08091010) (04050506)1198852 (04050506) (07080809) (07080809) (07080809) (08091010)1198853 (07080809) (04050506) (07080809) (08091010) (08091010)1198854 (05060708) (07080809) (04050506) (07080809) (08091010)1198856 (04050506) (04050506) (05060708) (07080809) (08091010)1198858 (05060708) (08091010) (07080809) (05060708) (04050506)1198859 (08091010) (07080809) (07080809) (07080809) (05060708)1198633 1198851 (05060708) (07080809) (07080809) (04050506) (08091010)1198852 (04050506) (07080809) (05060708) (05060708) (08091010)1198853 (05060708) (04050506) (07080809) (08091010) (08091010)1198854 (07080809) (05060708) (05060708) (08091010) (08091010)1198855 (04050506) (05060708) (07080809) (05060708) (08091010)1198857 (05060708) (05060708) (07080809) (08091010) (08091010)1198858 (07080809) (07080809) (07080809) (05060708) (04050506)1198634 1198851 (08091010) (07080809) (07080809) (05060708) (04050506)1198852 (04050506) (05060708) (08091010) (05060708) (07080809)1198853 (05060708) (07080809) (07080809) (08091010) (07080809)1198854 (07080809) (07080809) (05060708) (05060708) (08091010)1198856 (04050506) (07080809) (08091010) (08091010) (07080809)1198859 (08091010) (04050506) (07080809) (05060708) (04050506)11988510 (04050506) (05060708) (07080809) (05060708) (07080809)1198635 1198851 (07080809) (08091010) (07080809) (05060708) (05060708)1198852 (04050506) (07080809) (07080809) (05060708) (08091010)1198853 (08091010) (07080809) (05060708) (07080809) (08091010)1198854 (05060708) (07080809) (08091010) (05060708) (07080809)1198855 (04050506) (05060708) (08091010) (05060708) (05060708)1198857 (04050506) (05060708) (07080809) (07080809) (08091010)11988510 (04050506) (05060708) (07080809) (05060708) (08091010)


1198631 1198632 1198633 1198634 11986351198631 1 2 4 2 31198632 05 1 3 1 21198633 025 033 1 033 051198634 05 1 3 1 21198635 033 05 2 05 1


1198631 1198632 1198633 1198634 11986351205791 217 125 042 125 070


119877111 = min 119877111198961 = 05 (21)


times 09 + 013 times 08 = 085(22)


times 10 + 013 times 08 = 091(23)



1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 119885101198631 037 037 037 037 066 046 0661198632 022 022 022 022 027 079 051198633 006 006 006 006 011 011 0211198634 022 022 022 022 027 05 0631198635 013 013 013 013 023 023 037


1198851 1198852 1198853 1198854 1198855119882119895 (0708408710) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0508509110) (0406106109) (0507908710) (0406206509) (04050506)1198602 (0507007710) (0508008510) (0406407009) (0407007109) (0507307709)1198603 (07080809) (0508709610) (07080809) (04080810) (0708008210)1198604 (0406607510) (050606509) (0708909710) (0507708510) (05060708)1198605 (0405406308) (0708408810) (0708809610) (0708909710) (0508409310)1198856 1198857 1198858 1198859 11988510119882119895 (070840910) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0406406409) (0407107209) (0506407209) (08091010) (04050506)1198602 (0405805809) (0507307409) (0708809610) (0406506509) (05060708)1198603 (0708308510) (0708709710) (07080809) (07080809) (07080809)1198604 (0708708910) (0708809510) (05060708) (050707509) (05060708)1198605 (0708809510) (08090910) (04050506) (040550608) (0708408710)

119877114 = max 119877111198964 = 10 (24)





Deffuz (11990311) = (119903111 + 119903112 + 119903113 + 1199031144 )= 125 + 213 + 228 + 254 = 204 (25)





1198781 = 119899sum119895=1



= 4761198661 = max


= 215



1198851 1198852 1198853 1198854119882119895 (175210225250) (080090100100) (070086092100) (070082084100)1198601 (125213228250) (040061061090) (050079087100) (040062065090)1198602 (125175193250) (050080085100) (040064070090) (040070071090)1198603 (175200200225) (050087096100) (070080080090) (040080080100)1198604 (100165213250) (050060065090) (070089097100) (050077085100)1198605 (100135158200) (070084088100) (070088096100) (070089097100)1198855 1198856 1198857 1198858119882119895 (100180190250) (070084090100) (080090100100) (175215230250)1198601 (100125125150) (040064064090) (040071072090) (125160180225)1198602 (125183193225) (040058058090) (050073074090) (175220240250)1198603 (175200210250) (070083085100) (070087097100) (175200200225)1198604 (100120140160) (070087089100) (070088095100) (125150175200)1198605 (125210233250) (070088095100) (080090090100) (100125125150)1198859 11988510119882119895 (070082084100) (040072076100)1198601 (080090100100) (040050050060)1198602 (040065065090) (050060070080)1198603 (070080080090) (070080080090)1198604 (050070075090) (050060070080)1198605 (040055060080) (070084087100)


1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510Weight 215 093 087 084 180 086 093 218 084 0721198601 204 063 083 064 125 064 068 173 093 0501198602 185 079 066 067 181 062 072 221 065 0651198603 200 083 079 071 211 083 087 215 080 0801198604 176 069 089 078 130 088 088 163 071 0651198605 143 086 088 089 204 088 093 125 059 085


(26)









1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510119891lowast 143 063 066 065 125 062 068 125 059 05119891minus 204 086 089 089 211 088 093 221 093 085



002040608

112

1 2 3 4 5 6 7 8 9

A1A2A3

A4A5





5sum119896=1

120579119896 = 1 (27)

1205791015840 + 5sum119898=1119898 =119896

1205791015840119898 = 1 (28)

120585 ∙ 120579119896 + 120582 5sum119898=1119898 =119896

120579119898 = 1 (29)



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 450

010203040506070809

1

A1A2A3

A4A5




6 Conclusions





1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12




Data Availability




Acknowledgments




References













































Journal of












Journal of







Volume 2018






Volume 2018


































423 Data Processing


11988211 = min 11988211198961 = 0711988212 = sum120579119896 ∙ 11988211198962

= 15 times (08 + 08 + 08 + 09 + 09) = 084

11988213 = sum120579119896 ∙ 11988211198963= 15 times (08 + 08 + 08 + 10 + 10) = 087

11988214 = max 11988211198964 = 10(19)




120579119894 = ( 119899prod119895=1

119886119894119895)1119899

(20)







1198631 1198632 1198633 1198634 11986351198851 (07080809) (07080809) (07080809) (08091010) (08091010)1198852 (08091010) (08091010) (08091010) (08091010) (08091010)1198853 (08091010) (08091010) (08091010) (07080809) (07080809)1198854 (07080809) (07080809) (07080809) (08091010) (07080809)1198855 (07080809) (05060708) (07080809) (04050506) (08091010)1198856 (07080809) (07080809) (04050506) (07080809) (05060708)1198857 (08091010) (04050506) (08091010) (05060708) (07080809)1198858 (05060708) (08091010) (07080809) (04050506) (04050506)1198859 (04050506) (07080809) (04050506) (07080809) (04050506)11988510 (05060708) (04050506) (05060708) (08091010) (08091010)





1198601 1198602 1198603 1198604 11986051198631 1198851 (08091010) (05060708) (07080809) (05060708) (05060708)1198852 (07080809) (08091010) (08091010) (05060708) (07080809)1198853 (08091010) (05060708) (07080809) (08091010) (08091010)1198854 (04050506) (04050506) (07080809) (08091010) (08091010)1198855 (04050506) (07080809) (07080809) (05060708) (08091010)1198856 (07080809) (04050506) (07080809) (08091010) (08091010)1198857 (07080809) (07080809) (08091010) (08091010) (08091010)1198632 1198851 (07080809) (07080809) (07080809) (08091010) (04050506)1198852 (04050506) (07080809) (07080809) (07080809) (08091010)1198853 (07080809) (04050506) (07080809) (08091010) (08091010)1198854 (05060708) (07080809) (04050506) (07080809) (08091010)1198856 (04050506) (04050506) (05060708) (07080809) (08091010)1198858 (05060708) (08091010) (07080809) (05060708) (04050506)1198859 (08091010) (07080809) (07080809) (07080809) (05060708)1198633 1198851 (05060708) (07080809) (07080809) (04050506) (08091010)1198852 (04050506) (07080809) (05060708) (05060708) (08091010)1198853 (05060708) (04050506) (07080809) (08091010) (08091010)1198854 (07080809) (05060708) (05060708) (08091010) (08091010)1198855 (04050506) (05060708) (07080809) (05060708) (08091010)1198857 (05060708) (05060708) (07080809) (08091010) (08091010)1198858 (07080809) (07080809) (07080809) (05060708) (04050506)1198634 1198851 (08091010) (07080809) (07080809) (05060708) (04050506)1198852 (04050506) (05060708) (08091010) (05060708) (07080809)1198853 (05060708) (07080809) (07080809) (08091010) (07080809)1198854 (07080809) (07080809) (05060708) (05060708) (08091010)1198856 (04050506) (07080809) (08091010) (08091010) (07080809)1198859 (08091010) (04050506) (07080809) (05060708) (04050506)11988510 (04050506) (05060708) (07080809) (05060708) (07080809)1198635 1198851 (07080809) (08091010) (07080809) (05060708) (05060708)1198852 (04050506) (07080809) (07080809) (05060708) (08091010)1198853 (08091010) (07080809) (05060708) (07080809) (08091010)1198854 (05060708) (07080809) (08091010) (05060708) (07080809)1198855 (04050506) (05060708) (08091010) (05060708) (05060708)1198857 (04050506) (05060708) (07080809) (07080809) (08091010)11988510 (04050506) (05060708) (07080809) (05060708) (08091010)


1198631 1198632 1198633 1198634 11986351198631 1 2 4 2 31198632 05 1 3 1 21198633 025 033 1 033 051198634 05 1 3 1 21198635 033 05 2 05 1


1198631 1198632 1198633 1198634 11986351205791 217 125 042 125 070


119877111 = min 119877111198961 = 05 (21)


times 09 + 013 times 08 = 085(22)


times 10 + 013 times 08 = 091(23)



1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 119885101198631 037 037 037 037 066 046 0661198632 022 022 022 022 027 079 051198633 006 006 006 006 011 011 0211198634 022 022 022 022 027 05 0631198635 013 013 013 013 023 023 037


1198851 1198852 1198853 1198854 1198855119882119895 (0708408710) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0508509110) (0406106109) (0507908710) (0406206509) (04050506)1198602 (0507007710) (0508008510) (0406407009) (0407007109) (0507307709)1198603 (07080809) (0508709610) (07080809) (04080810) (0708008210)1198604 (0406607510) (050606509) (0708909710) (0507708510) (05060708)1198605 (0405406308) (0708408810) (0708809610) (0708909710) (0508409310)1198856 1198857 1198858 1198859 11988510119882119895 (070840910) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0406406409) (0407107209) (0506407209) (08091010) (04050506)1198602 (0405805809) (0507307409) (0708809610) (0406506509) (05060708)1198603 (0708308510) (0708709710) (07080809) (07080809) (07080809)1198604 (0708708910) (0708809510) (05060708) (050707509) (05060708)1198605 (0708809510) (08090910) (04050506) (040550608) (0708408710)

119877114 = max 119877111198964 = 10 (24)





Deffuz (11990311) = (119903111 + 119903112 + 119903113 + 1199031144 )= 125 + 213 + 228 + 254 = 204 (25)





1198781 = 119899sum119895=1



= 4761198661 = max


= 215



1198851 1198852 1198853 1198854119882119895 (175210225250) (080090100100) (070086092100) (070082084100)1198601 (125213228250) (040061061090) (050079087100) (040062065090)1198602 (125175193250) (050080085100) (040064070090) (040070071090)1198603 (175200200225) (050087096100) (070080080090) (040080080100)1198604 (100165213250) (050060065090) (070089097100) (050077085100)1198605 (100135158200) (070084088100) (070088096100) (070089097100)1198855 1198856 1198857 1198858119882119895 (100180190250) (070084090100) (080090100100) (175215230250)1198601 (100125125150) (040064064090) (040071072090) (125160180225)1198602 (125183193225) (040058058090) (050073074090) (175220240250)1198603 (175200210250) (070083085100) (070087097100) (175200200225)1198604 (100120140160) (070087089100) (070088095100) (125150175200)1198605 (125210233250) (070088095100) (080090090100) (100125125150)1198859 11988510119882119895 (070082084100) (040072076100)1198601 (080090100100) (040050050060)1198602 (040065065090) (050060070080)1198603 (070080080090) (070080080090)1198604 (050070075090) (050060070080)1198605 (040055060080) (070084087100)


1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510Weight 215 093 087 084 180 086 093 218 084 0721198601 204 063 083 064 125 064 068 173 093 0501198602 185 079 066 067 181 062 072 221 065 0651198603 200 083 079 071 211 083 087 215 080 0801198604 176 069 089 078 130 088 088 163 071 0651198605 143 086 088 089 204 088 093 125 059 085


(26)









1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510119891lowast 143 063 066 065 125 062 068 125 059 05119891minus 204 086 089 089 211 088 093 221 093 085



002040608

112

1 2 3 4 5 6 7 8 9

A1A2A3

A4A5





5sum119896=1

120579119896 = 1 (27)

1205791015840 + 5sum119898=1119898 =119896

1205791015840119898 = 1 (28)

120585 ∙ 120579119896 + 120582 5sum119898=1119898 =119896

120579119898 = 1 (29)



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 450

010203040506070809

1

A1A2A3

A4A5




6 Conclusions





1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12




Data Availability




Acknowledgments




References













































Journal of












Journal of







Volume 2018






Volume 2018












1198631 1198632 1198633 1198634 11986351198851 (07080809) (07080809) (07080809) (08091010) (08091010)1198852 (08091010) (08091010) (08091010) (08091010) (08091010)1198853 (08091010) (08091010) (08091010) (07080809) (07080809)1198854 (07080809) (07080809) (07080809) (08091010) (07080809)1198855 (07080809) (05060708) (07080809) (04050506) (08091010)1198856 (07080809) (07080809) (04050506) (07080809) (05060708)1198857 (08091010) (04050506) (08091010) (05060708) (07080809)1198858 (05060708) (08091010) (07080809) (04050506) (04050506)1198859 (04050506) (07080809) (04050506) (07080809) (04050506)11988510 (05060708) (04050506) (05060708) (08091010) (08091010)





1198601 1198602 1198603 1198604 11986051198631 1198851 (08091010) (05060708) (07080809) (05060708) (05060708)1198852 (07080809) (08091010) (08091010) (05060708) (07080809)1198853 (08091010) (05060708) (07080809) (08091010) (08091010)1198854 (04050506) (04050506) (07080809) (08091010) (08091010)1198855 (04050506) (07080809) (07080809) (05060708) (08091010)1198856 (07080809) (04050506) (07080809) (08091010) (08091010)1198857 (07080809) (07080809) (08091010) (08091010) (08091010)1198632 1198851 (07080809) (07080809) (07080809) (08091010) (04050506)1198852 (04050506) (07080809) (07080809) (07080809) (08091010)1198853 (07080809) (04050506) (07080809) (08091010) (08091010)1198854 (05060708) (07080809) (04050506) (07080809) (08091010)1198856 (04050506) (04050506) (05060708) (07080809) (08091010)1198858 (05060708) (08091010) (07080809) (05060708) (04050506)1198859 (08091010) (07080809) (07080809) (07080809) (05060708)1198633 1198851 (05060708) (07080809) (07080809) (04050506) (08091010)1198852 (04050506) (07080809) (05060708) (05060708) (08091010)1198853 (05060708) (04050506) (07080809) (08091010) (08091010)1198854 (07080809) (05060708) (05060708) (08091010) (08091010)1198855 (04050506) (05060708) (07080809) (05060708) (08091010)1198857 (05060708) (05060708) (07080809) (08091010) (08091010)1198858 (07080809) (07080809) (07080809) (05060708) (04050506)1198634 1198851 (08091010) (07080809) (07080809) (05060708) (04050506)1198852 (04050506) (05060708) (08091010) (05060708) (07080809)1198853 (05060708) (07080809) (07080809) (08091010) (07080809)1198854 (07080809) (07080809) (05060708) (05060708) (08091010)1198856 (04050506) (07080809) (08091010) (08091010) (07080809)1198859 (08091010) (04050506) (07080809) (05060708) (04050506)11988510 (04050506) (05060708) (07080809) (05060708) (07080809)1198635 1198851 (07080809) (08091010) (07080809) (05060708) (05060708)1198852 (04050506) (07080809) (07080809) (05060708) (08091010)1198853 (08091010) (07080809) (05060708) (07080809) (08091010)1198854 (05060708) (07080809) (08091010) (05060708) (07080809)1198855 (04050506) (05060708) (08091010) (05060708) (05060708)1198857 (04050506) (05060708) (07080809) (07080809) (08091010)11988510 (04050506) (05060708) (07080809) (05060708) (08091010)


1198631 1198632 1198633 1198634 11986351198631 1 2 4 2 31198632 05 1 3 1 21198633 025 033 1 033 051198634 05 1 3 1 21198635 033 05 2 05 1


1198631 1198632 1198633 1198634 11986351205791 217 125 042 125 070


119877111 = min 119877111198961 = 05 (21)


times 09 + 013 times 08 = 085(22)


times 10 + 013 times 08 = 091(23)



1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 119885101198631 037 037 037 037 066 046 0661198632 022 022 022 022 027 079 051198633 006 006 006 006 011 011 0211198634 022 022 022 022 027 05 0631198635 013 013 013 013 023 023 037


1198851 1198852 1198853 1198854 1198855119882119895 (0708408710) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0508509110) (0406106109) (0507908710) (0406206509) (04050506)1198602 (0507007710) (0508008510) (0406407009) (0407007109) (0507307709)1198603 (07080809) (0508709610) (07080809) (04080810) (0708008210)1198604 (0406607510) (050606509) (0708909710) (0507708510) (05060708)1198605 (0405406308) (0708408810) (0708809610) (0708909710) (0508409310)1198856 1198857 1198858 1198859 11988510119882119895 (070840910) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0406406409) (0407107209) (0506407209) (08091010) (04050506)1198602 (0405805809) (0507307409) (0708809610) (0406506509) (05060708)1198603 (0708308510) (0708709710) (07080809) (07080809) (07080809)1198604 (0708708910) (0708809510) (05060708) (050707509) (05060708)1198605 (0708809510) (08090910) (04050506) (040550608) (0708408710)

119877114 = max 119877111198964 = 10 (24)





Deffuz (11990311) = (119903111 + 119903112 + 119903113 + 1199031144 )= 125 + 213 + 228 + 254 = 204 (25)





1198781 = 119899sum119895=1



= 4761198661 = max


= 215



1198851 1198852 1198853 1198854119882119895 (175210225250) (080090100100) (070086092100) (070082084100)1198601 (125213228250) (040061061090) (050079087100) (040062065090)1198602 (125175193250) (050080085100) (040064070090) (040070071090)1198603 (175200200225) (050087096100) (070080080090) (040080080100)1198604 (100165213250) (050060065090) (070089097100) (050077085100)1198605 (100135158200) (070084088100) (070088096100) (070089097100)1198855 1198856 1198857 1198858119882119895 (100180190250) (070084090100) (080090100100) (175215230250)1198601 (100125125150) (040064064090) (040071072090) (125160180225)1198602 (125183193225) (040058058090) (050073074090) (175220240250)1198603 (175200210250) (070083085100) (070087097100) (175200200225)1198604 (100120140160) (070087089100) (070088095100) (125150175200)1198605 (125210233250) (070088095100) (080090090100) (100125125150)1198859 11988510119882119895 (070082084100) (040072076100)1198601 (080090100100) (040050050060)1198602 (040065065090) (050060070080)1198603 (070080080090) (070080080090)1198604 (050070075090) (050060070080)1198605 (040055060080) (070084087100)


1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510Weight 215 093 087 084 180 086 093 218 084 0721198601 204 063 083 064 125 064 068 173 093 0501198602 185 079 066 067 181 062 072 221 065 0651198603 200 083 079 071 211 083 087 215 080 0801198604 176 069 089 078 130 088 088 163 071 0651198605 143 086 088 089 204 088 093 125 059 085


(26)









1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510119891lowast 143 063 066 065 125 062 068 125 059 05119891minus 204 086 089 089 211 088 093 221 093 085



002040608

112

1 2 3 4 5 6 7 8 9

A1A2A3

A4A5





5sum119896=1

120579119896 = 1 (27)

1205791015840 + 5sum119898=1119898 =119896

1205791015840119898 = 1 (28)

120585 ∙ 120579119896 + 120582 5sum119898=1119898 =119896

120579119898 = 1 (29)



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 450

010203040506070809

1

A1A2A3

A4A5




6 Conclusions





1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12




Data Availability




Acknowledgments




References













































Journal of












Journal of







Volume 2018






Volume 2018










1198601 1198602 1198603 1198604 11986051198631 1198851 (08091010) (05060708) (07080809) (05060708) (05060708)1198852 (07080809) (08091010) (08091010) (05060708) (07080809)1198853 (08091010) (05060708) (07080809) (08091010) (08091010)1198854 (04050506) (04050506) (07080809) (08091010) (08091010)1198855 (04050506) (07080809) (07080809) (05060708) (08091010)1198856 (07080809) (04050506) (07080809) (08091010) (08091010)1198857 (07080809) (07080809) (08091010) (08091010) (08091010)1198632 1198851 (07080809) (07080809) (07080809) (08091010) (04050506)1198852 (04050506) (07080809) (07080809) (07080809) (08091010)1198853 (07080809) (04050506) (07080809) (08091010) (08091010)1198854 (05060708) (07080809) (04050506) (07080809) (08091010)1198856 (04050506) (04050506) (05060708) (07080809) (08091010)1198858 (05060708) (08091010) (07080809) (05060708) (04050506)1198859 (08091010) (07080809) (07080809) (07080809) (05060708)1198633 1198851 (05060708) (07080809) (07080809) (04050506) (08091010)1198852 (04050506) (07080809) (05060708) (05060708) (08091010)1198853 (05060708) (04050506) (07080809) (08091010) (08091010)1198854 (07080809) (05060708) (05060708) (08091010) (08091010)1198855 (04050506) (05060708) (07080809) (05060708) (08091010)1198857 (05060708) (05060708) (07080809) (08091010) (08091010)1198858 (07080809) (07080809) (07080809) (05060708) (04050506)1198634 1198851 (08091010) (07080809) (07080809) (05060708) (04050506)1198852 (04050506) (05060708) (08091010) (05060708) (07080809)1198853 (05060708) (07080809) (07080809) (08091010) (07080809)1198854 (07080809) (07080809) (05060708) (05060708) (08091010)1198856 (04050506) (07080809) (08091010) (08091010) (07080809)1198859 (08091010) (04050506) (07080809) (05060708) (04050506)11988510 (04050506) (05060708) (07080809) (05060708) (07080809)1198635 1198851 (07080809) (08091010) (07080809) (05060708) (05060708)1198852 (04050506) (07080809) (07080809) (05060708) (08091010)1198853 (08091010) (07080809) (05060708) (07080809) (08091010)1198854 (05060708) (07080809) (08091010) (05060708) (07080809)1198855 (04050506) (05060708) (08091010) (05060708) (05060708)1198857 (04050506) (05060708) (07080809) (07080809) (08091010)11988510 (04050506) (05060708) (07080809) (05060708) (08091010)


1198631 1198632 1198633 1198634 11986351198631 1 2 4 2 31198632 05 1 3 1 21198633 025 033 1 033 051198634 05 1 3 1 21198635 033 05 2 05 1


1198631 1198632 1198633 1198634 11986351205791 217 125 042 125 070


119877111 = min 119877111198961 = 05 (21)


times 09 + 013 times 08 = 085(22)


times 10 + 013 times 08 = 091(23)



1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 119885101198631 037 037 037 037 066 046 0661198632 022 022 022 022 027 079 051198633 006 006 006 006 011 011 0211198634 022 022 022 022 027 05 0631198635 013 013 013 013 023 023 037


1198851 1198852 1198853 1198854 1198855119882119895 (0708408710) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0508509110) (0406106109) (0507908710) (0406206509) (04050506)1198602 (0507007710) (0508008510) (0406407009) (0407007109) (0507307709)1198603 (07080809) (0508709610) (07080809) (04080810) (0708008210)1198604 (0406607510) (050606509) (0708909710) (0507708510) (05060708)1198605 (0405406308) (0708408810) (0708809610) (0708909710) (0508409310)1198856 1198857 1198858 1198859 11988510119882119895 (070840910) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0406406409) (0407107209) (0506407209) (08091010) (04050506)1198602 (0405805809) (0507307409) (0708809610) (0406506509) (05060708)1198603 (0708308510) (0708709710) (07080809) (07080809) (07080809)1198604 (0708708910) (0708809510) (05060708) (050707509) (05060708)1198605 (0708809510) (08090910) (04050506) (040550608) (0708408710)

119877114 = max 119877111198964 = 10 (24)





Deffuz (11990311) = (119903111 + 119903112 + 119903113 + 1199031144 )= 125 + 213 + 228 + 254 = 204 (25)





1198781 = 119899sum119895=1



= 4761198661 = max


= 215



1198851 1198852 1198853 1198854119882119895 (175210225250) (080090100100) (070086092100) (070082084100)1198601 (125213228250) (040061061090) (050079087100) (040062065090)1198602 (125175193250) (050080085100) (040064070090) (040070071090)1198603 (175200200225) (050087096100) (070080080090) (040080080100)1198604 (100165213250) (050060065090) (070089097100) (050077085100)1198605 (100135158200) (070084088100) (070088096100) (070089097100)1198855 1198856 1198857 1198858119882119895 (100180190250) (070084090100) (080090100100) (175215230250)1198601 (100125125150) (040064064090) (040071072090) (125160180225)1198602 (125183193225) (040058058090) (050073074090) (175220240250)1198603 (175200210250) (070083085100) (070087097100) (175200200225)1198604 (100120140160) (070087089100) (070088095100) (125150175200)1198605 (125210233250) (070088095100) (080090090100) (100125125150)1198859 11988510119882119895 (070082084100) (040072076100)1198601 (080090100100) (040050050060)1198602 (040065065090) (050060070080)1198603 (070080080090) (070080080090)1198604 (050070075090) (050060070080)1198605 (040055060080) (070084087100)


1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510Weight 215 093 087 084 180 086 093 218 084 0721198601 204 063 083 064 125 064 068 173 093 0501198602 185 079 066 067 181 062 072 221 065 0651198603 200 083 079 071 211 083 087 215 080 0801198604 176 069 089 078 130 088 088 163 071 0651198605 143 086 088 089 204 088 093 125 059 085


(26)









1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510119891lowast 143 063 066 065 125 062 068 125 059 05119891minus 204 086 089 089 211 088 093 221 093 085



002040608

112

1 2 3 4 5 6 7 8 9

A1A2A3

A4A5





5sum119896=1

120579119896 = 1 (27)

1205791015840 + 5sum119898=1119898 =119896

1205791015840119898 = 1 (28)

120585 ∙ 120579119896 + 120582 5sum119898=1119898 =119896

120579119898 = 1 (29)



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 450

010203040506070809

1

A1A2A3

A4A5




6 Conclusions





1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12




Data Availability




Acknowledgments




References













































Journal of












Journal of







Volume 2018






Volume 2018










1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 119885101198631 037 037 037 037 066 046 0661198632 022 022 022 022 027 079 051198633 006 006 006 006 011 011 0211198634 022 022 022 022 027 05 0631198635 013 013 013 013 023 023 037


1198851 1198852 1198853 1198854 1198855119882119895 (0708408710) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0508509110) (0406106109) (0507908710) (0406206509) (04050506)1198602 (0507007710) (0508008510) (0406407009) (0407007109) (0507307709)1198603 (07080809) (0508709610) (07080809) (04080810) (0708008210)1198604 (0406607510) (050606509) (0708909710) (0507708510) (05060708)1198605 (0405406308) (0708408810) (0708809610) (0708909710) (0508409310)1198856 1198857 1198858 1198859 11988510119882119895 (070840910) (08091010) (0708609210) (0708208410) (0407207610)1198601 (0406406409) (0407107209) (0506407209) (08091010) (04050506)1198602 (0405805809) (0507307409) (0708809610) (0406506509) (05060708)1198603 (0708308510) (0708709710) (07080809) (07080809) (07080809)1198604 (0708708910) (0708809510) (05060708) (050707509) (05060708)1198605 (0708809510) (08090910) (04050506) (040550608) (0708408710)

119877114 = max 119877111198964 = 10 (24)





Deffuz (11990311) = (119903111 + 119903112 + 119903113 + 1199031144 )= 125 + 213 + 228 + 254 = 204 (25)





1198781 = 119899sum119895=1



= 4761198661 = max


= 215



1198851 1198852 1198853 1198854119882119895 (175210225250) (080090100100) (070086092100) (070082084100)1198601 (125213228250) (040061061090) (050079087100) (040062065090)1198602 (125175193250) (050080085100) (040064070090) (040070071090)1198603 (175200200225) (050087096100) (070080080090) (040080080100)1198604 (100165213250) (050060065090) (070089097100) (050077085100)1198605 (100135158200) (070084088100) (070088096100) (070089097100)1198855 1198856 1198857 1198858119882119895 (100180190250) (070084090100) (080090100100) (175215230250)1198601 (100125125150) (040064064090) (040071072090) (125160180225)1198602 (125183193225) (040058058090) (050073074090) (175220240250)1198603 (175200210250) (070083085100) (070087097100) (175200200225)1198604 (100120140160) (070087089100) (070088095100) (125150175200)1198605 (125210233250) (070088095100) (080090090100) (100125125150)1198859 11988510119882119895 (070082084100) (040072076100)1198601 (080090100100) (040050050060)1198602 (040065065090) (050060070080)1198603 (070080080090) (070080080090)1198604 (050070075090) (050060070080)1198605 (040055060080) (070084087100)


1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510Weight 215 093 087 084 180 086 093 218 084 0721198601 204 063 083 064 125 064 068 173 093 0501198602 185 079 066 067 181 062 072 221 065 0651198603 200 083 079 071 211 083 087 215 080 0801198604 176 069 089 078 130 088 088 163 071 0651198605 143 086 088 089 204 088 093 125 059 085


(26)









1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510119891lowast 143 063 066 065 125 062 068 125 059 05119891minus 204 086 089 089 211 088 093 221 093 085



002040608

112

1 2 3 4 5 6 7 8 9

A1A2A3

A4A5





5sum119896=1

120579119896 = 1 (27)

1205791015840 + 5sum119898=1119898 =119896

1205791015840119898 = 1 (28)

120585 ∙ 120579119896 + 120582 5sum119898=1119898 =119896

120579119898 = 1 (29)



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 450

010203040506070809

1

A1A2A3

A4A5




6 Conclusions





1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12




Data Availability




Acknowledgments




References













































Journal of












Journal of







Volume 2018






Volume 2018










1198851 1198852 1198853 1198854119882119895 (175210225250) (080090100100) (070086092100) (070082084100)1198601 (125213228250) (040061061090) (050079087100) (040062065090)1198602 (125175193250) (050080085100) (040064070090) (040070071090)1198603 (175200200225) (050087096100) (070080080090) (040080080100)1198604 (100165213250) (050060065090) (070089097100) (050077085100)1198605 (100135158200) (070084088100) (070088096100) (070089097100)1198855 1198856 1198857 1198858119882119895 (100180190250) (070084090100) (080090100100) (175215230250)1198601 (100125125150) (040064064090) (040071072090) (125160180225)1198602 (125183193225) (040058058090) (050073074090) (175220240250)1198603 (175200210250) (070083085100) (070087097100) (175200200225)1198604 (100120140160) (070087089100) (070088095100) (125150175200)1198605 (125210233250) (070088095100) (080090090100) (100125125150)1198859 11988510119882119895 (070082084100) (040072076100)1198601 (080090100100) (040050050060)1198602 (040065065090) (050060070080)1198603 (070080080090) (070080080090)1198604 (050070075090) (050060070080)1198605 (040055060080) (070084087100)


1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510Weight 215 093 087 084 180 086 093 218 084 0721198601 204 063 083 064 125 064 068 173 093 0501198602 185 079 066 067 181 062 072 221 065 0651198603 200 083 079 071 211 083 087 215 080 0801198604 176 069 089 078 130 088 088 163 071 0651198605 143 086 088 089 204 088 093 125 059 085


(26)









1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510119891lowast 143 063 066 065 125 062 068 125 059 05119891minus 204 086 089 089 211 088 093 221 093 085



002040608

112

1 2 3 4 5 6 7 8 9

A1A2A3

A4A5





5sum119896=1

120579119896 = 1 (27)

1205791015840 + 5sum119898=1119898 =119896

1205791015840119898 = 1 (28)

120585 ∙ 120579119896 + 120582 5sum119898=1119898 =119896

120579119898 = 1 (29)



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 450

010203040506070809

1

A1A2A3

A4A5




6 Conclusions





1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12




Data Availability




Acknowledgments




References













































Journal of












Journal of







Volume 2018






Volume 2018










1198851 1198852 1198853 1198854 1198855 1198856 1198857 1198858 1198859 11988510119891lowast 143 063 066 065 125 062 068 125 059 05119891minus 204 086 089 089 211 088 093 221 093 085



002040608

112

1 2 3 4 5 6 7 8 9

A1A2A3

A4A5





5sum119896=1

120579119896 = 1 (27)

1205791015840 + 5sum119898=1119898 =119896

1205791015840119898 = 1 (28)

120585 ∙ 120579119896 + 120582 5sum119898=1119898 =119896

120579119898 = 1 (29)



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 450

010203040506070809

1

A1A2A3

A4A5




6 Conclusions





1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12




Data Availability




Acknowledgments




References













































Journal of












Journal of







Volume 2018






Volume 2018











1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103

109

115

121

127

133

139

145

151

157

163

169

175

181

187

193

199

A1A2A3

A4A5

0

02

04

06

08

1

12




Data Availability




Acknowledgments




References













































Journal of












Journal of







Volume 2018






Volume 2018










References













































Journal of












Journal of







Volume 2018






Volume 2018





















Journal of












Journal of







Volume 2018






Volume 2018








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