Structural Safety 33 (2011) 232–241

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Structural Safety

journal homepage: www.elsevier .com/ locate/s t rusafe

Cost-effectiveness fuzzy analysis for an efficient reduction of uncertainty

Uwe Reuter a,⇑, Ulrike Schirwitz b

a Faculty of Civil Engineering, Technische Universität Dresden, 01062 Dresden, Germanyb Institute of Construction Informatics, Technische Universität Dresden, 01062 Dresden, Germany

a r t i c l e i n f o

Article history:Received 1 November 2010Received in revised form 10 March 2011Accepted 11 March 2011Available online 23 April 2011

Keywords:Cost-effectiveness fuzzy analysisCost-effectiveness analysisFuzzy analysisEpistemic uncertainty

0167-4730/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.strusafe.2011.03.005

⇑ Corresponding author.E-mail addresses: Uwe.Reuter@tu-dresden.de (U

tu-dresden.de (U. Schirwitz).

a b s t r a c t

Many planning and production processes are characterized by uncertain data and uncertain information.For realistic modeling of such processes these uncertainties have to be considered. The new approachpresented in this paper takes epistemic uncertainty into account, for which fuzzy set theory is applicable.In some cases it is possible and useful to reduce epistemic uncertainty by additional monetary invest-ments. It is postulated that uncertain forecast values, e.g. expected safety, quality, or the completion dateof a structure, can be improved or scheduled more precisely by a higher investment. Aim of the presentedcost-effectiveness fuzzy analysis is the evaluation of the effectiveness of monetary investments on thereduction of uncertainty of the analyzed forecast values.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Quality management requires organizational measures forassurance, control, and improvement of the quality of products,processes, or services. In this context quality assurance refers toall measures for ensuring that standards of quality are being metat costs as low as possible. An important instrument of qualityassurance is thus the cost-effectiveness analysis. With the help ofsuitable criteria, a cost-effectiveness analysis evaluates the relativecosts and effects of different courses of action within an invest-ment project. Effects to be evaluated are usually the safety, thequality, or the completion date of the intended results. Such crite-ria are for example statements about the certainty, that the givenquality or time intervals are adhered to.

In general input data and information of a cost-effectivenessanalysis are uncertain. The evaluation of the aforementioned crite-ria thus yields also only uncertain forecast values. Uncertainty ofdata can occur in aleatoric as well as in epistemic form [1]. Modeluncertainty has to be distinguished from data uncertainty. It existsdue to the abstraction of reality to a model. That means there isuncertainty in the mapping [2].

Aleatoric uncertainty means random variability. Methods ofstochastics can be used to deal with such type of uncertainty.Hence variable data and information are modelled as random vari-ables whose distribution functions are estimated. These methodsare applied to engineering problems e.g. in context of investments

ll rights reserved.

. Reuter), Ulrike.Schirwitz@

in seismic upgrades of buildings [3] or bridges [4]. The approachpresented in this paper takes epistemic uncertainty in terms ofimprecision into account. Due to the absence of complete knowl-edge in the case of imprecision, it is impossible or at least costlyto describe a single data or information exactly [5]. Fuzzy set the-ory is applicable for modeling this kind of epistemic uncertainty[2,6]. Uncertain input data are modelled as fuzzy variables. Thefuzzy input variables are mapped onto fuzzy forecast variables(e.g. forecasts like expected safety or completion date of a struc-ture) with the help of the fuzzy analysis, which is presented in Sec-tion 2. It is postulated that the uncertain forecast values can beimproved or scheduled more precisely by a higher monetaryinvestment. That is, a higher investment yields a more precise pre-diction with a lower uncertainty of the forecast value. Aim of thecost-effectiveness fuzzy analysis presented in Section 3 is the eval-uation of the effectiveness of monetary investments on the reduc-tion of uncertainty of the analyzed forecast variable.

Example 1. The idea of the cost-effectiveness fuzzy analysis can beexemplarily pointed out by means of an investment project of theautomotive industry. The thickness of deep-drawing steel sheetcan only be specified in uncertain terms, i.e. lower and upperboundaries are normally given. This uncertainty can be reduced bya higher monetary investment, i.e. a higher investment allows thereduction of the possible range of the thickness. The uncertainsheet thicknesses have a significant impact on the crash behaviorof the automobiles [7]. Crash test is an essential measure of qualityassurance. The forecast of the crash behavior should thus becharacterized by minimum uncertainty. Amongst others, theuncertainty of the impact force is an important criterion for

U. Reuter, U. Schirwitz / Structural Safety 33 (2011) 232–241 233

evaluation of the crash behavior [8]. Aim of a cost-effectivenessfuzzy analysis is thus the evaluation of the effectiveness ofinvestments for reduction of the uncertainty of the sheet thick-nesses on the uncertainty of the impact force.

Example 2. A building site which installs prefabricated columnscan serve as a second example. The delivery of the columns is madeby trucks and the unloading as well as the installation is carriedout using a single crane. The average velocity of the trucks is char-acterized by uncertainty, e.g. because of the unknown traffic situ-ation or the possibility of checks at national borders, buildingsite entrances, or by the police on motorways. No statisticallyfirmed data are available. Instead, only the expert knowledge ofthe truck drivers is usable. They can roughly estimate the approx-imated velocity on the route and the velocity they will not exceedor fall below. Because of the uncertain average velocity of thetrucks the total duration of the delivery process of the columns isalso uncertain. By investing determined amounts of money, differ-ent possibilities for the reduction of uncertainty of the velocity ofthe trucks exist. The most expensive variant is to close the wholedelivery route to public traffic for the duration of the transport.As cheaper alternative a smaller section of the route can be closed.These additional investments can reduce the uncertainty of thetotal duration of the delivery process and therefore a more pre-cisely planning is possible. By means of this scenario the applica-tion of cost-effectiveness fuzzy analysis is explained exemplarilyin Section 4.1. An extension of the scenario to a multi-dimensionalproblem is dealt with in Section 4.2.

2. Fuzzy analysis

Fuzzy analysis is a method which enables uncertain terms to behandled in form of fuzzy variables. For this purpose fuzzy inputvariables ~x1; ~x2; . . . ; ~xl are mapped onto fuzzy forecast variables~z1;~z2; . . . ;~zm [2].

~z1;~z2; . . . ;~zmð Þ ¼ ~f ~x1; ~x2; . . . ; ~xlð Þ ð1Þ

A fuzzy variable ~x is characterized by its normalized member-ship function l~xðxÞ.

0 6 l~xðxÞ 6 1 8 x 2 R ð2Þ9 xl; xr with l~xðxÞ ¼ 1 8 x 2 ½xl;xr� ð3Þ

A fuzzy variable ~x is referred to as convex if its membershipfunction l~xðxÞ monotonically decreases on each side of the maxi-mum value, i.e. if

l~xðx2ÞP min½l~xðx1Þ; l~xðx3Þ� 8 x1; x2; x3 2 R with x1 6 x2 6 x3

ð4Þ

Fig. 1. Fuzzy variable ~x dis

applies, see Fig. 1.Eq. (1) represents a mapping problem which can be solved by

means of the extension principle. Under the constraint of convexfuzzy input variables a-level optimization according to [2] or [9]is numerically more efficient. For this, a-level sets of the fuzzy vari-ables are used. A convex fuzzy variable ~x is characterized by a fam-ily of a-level sets Xa.

~x ¼ ðXa ¼ ½xal; xar �ja 2 ½0; 1�Þ ð5Þ

Each a-level set Xa is a connected interval [xal; xar]. The bound-aries xal and xar for a – 0 are given by

xal ¼min½x 2 Rjl~xðxÞP a� ð6Þxar ¼ max½x 2 Rjl~xðxÞP a� ð7Þ

The set

S~x ¼ fx 2 Rjl~xðxÞ > 0g ð8Þ

of a fuzzy variable ~x is referred to as the support. The support S~x of afuzzy variable ~x is referred to as an a-level set Xa with a = 0 not-withstanding Eqs. (6) and (7). The interval boundaries xa=0l andxa=0r of the a-level set Xa=0 are then given by

xa¼0l ¼ lima0!þ0

min x 2 Rjl~xðxÞ > a0� �� �

ð9Þ

xa¼0r ¼ lima0!þ0

max x 2 Rjl~xðxÞ > a0� �� �

ð10Þ

If the number of a-level sets is denoted by n P 2 the followingholds between the a-level sets Xa.

0 6 ai 6 aiþ1 6 1 with i ¼ 1;2; . . . ;n� 1 ð11Þa1 ¼ 0 and an ¼ 1 ð12ÞXaiþ1

# Xaið13Þ

In Fig. 1 a convex fuzzy variable ~x discretized by n = 4 a-levelsets Xa is shown.

The a-level optimization is based on discretization of all fuzzyinput variables ~x1; ~x2; . . . ; ~xl and fuzzy forecast variables~z1;~z2; . . . ;~zm with the same number of a-levels a1, a2, . . . , an. Thea-level set Xk;ai

on the a-level ai, i = 1, 2, . . . , n, is assigned to eachfuzzy input variable ~xk; k ¼ 1;2; . . . ; l. All a-level sets Xk;ai

formthe l-dimensional crisp subspace Xai

. Mutual dependencies orrather interactions between fuzzy variables can exist. Withoutinteractions, the subspace Xai

forms an l-dimensional hypercuboid.Otherwise, the hypercuboid forms the envelope. On the same a-le-vel the crisp subspace Xai

is assigned to the crisp subspace Zai,

which is formed by the a-level sets Zj;ai, j = 1, 2, . . . , m, of the fuzzy

forecast variables ~zj. Each point of Xaican be described by its coor-

dinates x1, x2, . . . , xl. The computation of each ~zj can be carried outby means of any mapping

zj ¼ fjðx1; x2; . . . ; xlÞ ð14Þ

cretized by 4 a-levels.

234 U. Reuter, U. Schirwitz / Structural Safety 33 (2011) 232–241

where fj(�) is referred to as the corresponding deterministic funda-mental solution for the deterministic result variable zj describingthe a-level sets Zj;ai

. Under the condition of convex fuzzy variablesit is sufficient to calculate the smallest and the largest element ofZj;ai

for a sufficient number of a-levels. Therewith, the membershipfunction l~zj

ðzjÞ is given in a discretized form. Finding the minimumand maximum elements is an optimization problem with the objec-tive functions

zj ¼ fjðx1; x2; . . . ; xlÞ )max ð15Þzj ¼ fjðx1; x2; . . . ; xlÞ )min ð16Þ

subject to

ðx1; x2; . . . ; xlÞ 2 Xaið17Þ

Fig. 2 illustrates the procedure exemplarily for l = 2 fuzzy inputvariables ~xk and m = 1 fuzzy forecast variable ~zj.

3. Cost-effectiveness fuzzy analysis

The cost-effectiveness analysis is a method for evaluating sev-eral courses of action and for supporting decision making. In con-trast to the related cost-benefit analysis, the cost-effectivenessanalysis does not determine the benefits as monetary values. In-stead, any other unit is applicable. Hence the cost-effectivenessanalysis is used when the effects are not monetarily measurableand therefore primarily in medical and health care [10], as wellas in educational policy [11]. In this paper, a different approachis presented to extend the application of this method to investmentprojects under uncertainty.

Principally, the intended objective when using the cost-effec-tiveness analysis is to minimize the cost-effectiveness ratio, i.e.to have the lowest costs per effective unit. Additionally, this canbe limited by a maximum budget and/or a minimum effect. Fur-thermore, the maximum effectiveness by a given amount of moneyor the minimum cost for an intended effect can be analyzed.

A cost-effectiveness fuzzy analysis under consideration of epi-stemic uncertainty of input data succeeds in three steps providedthat uncertain data and information, respectively, are modelledas fuzzy variables. Each forecast value of the investment project

Fig. 2. a-Level o

is thus also a fuzzy variable. Defining the membership functionof the fuzzy input variables requires that a higher monetary invest-ment yields a reduction of the uncertainty of the input data.

In the following Section 3.1, the cost-effectiveness fuzzy analy-sis is presented for the case of a single fuzzy input variable. Themulti-dimensional case is provided in Section 3.2. In view of thecomputational effort a straightforward local as well as a globalmethod is introduced. All variants are considered for the case ofa single forecast variable. The extension to multiple forecast vari-ables succeeds analogously.

3.1. One-dimensional case

First step: Possible investment costs are interrelated to the a-levels of the fuzzy input variable ~x. Each potential expense ci

is assigned to an a-level ai, i = 1, 2, . . . , n, using a monotonicincreasing inverse cost function C�1(c). The inverse cost func-tion C�1(c) maps monetary investments c onto the interval[0; 1] and thus assigns the possible investment costs ci to thea-levels of the fuzzy input variable. Provided that a higher mon-etary investment allows a more precise input value, higherinvestment costs are thus assigned to ‘lesser’ a-level sets.Therefore, the requirement according to Eq. (13) is fulfilled, thatis, a-level sets Xai� are contained in subjacent a-level sets Xai

forai; ai� 2 ½0; 1�; ai 6 ai� . Fig. 3 shows exemplarily the mappingof investment costs onto the a-levels of a fuzzy input variable.Second step: The fuzzy input variable ~x modelled under consid-eration of possible investment costs is mapped onto the fuzzyforecast variable ~z using the fuzzy analysis presented in Sec-tion 2. The deterministic model for computation of the deter-ministic forecast value z represents the deterministicfundamental solution of the fuzzy analysis.Third step: The effectiveness of the possible investment costs isevaluated by means of uncertainty measures for the fuzzy fore-cast variable and is illustrated in a cost-effectiveness diagram.For evaluation of the effectiveness of investments on the reduc-tion of uncertainty of the forecast variable ~z and thus on thepreciseness of the forecast, different investment costs and theassociated a-levels of ~z are analyzed. For each a-level the reduc-tion of uncertainty is evaluated with the normalized measure

ptimizat

ion.

Eð

CE

Fig. 3. Inverse cost function C�1(c) for mapping of investment costs onto the a-levels of a fuzzy input variable ~x.

U. Reuter, U. Schirwitz / Structural Safety 33 (2011) 232–241 235

aiÞ ¼1� zai r�zai l

za1 r�za1 lfor za1r – za1 l

0 for za1r ¼ za1 l

(ð18Þ

Fig. 4. Cost-effectiveness diagram.

that is, each investment level is assigned to a measure of effec-tiveness.The computation according to Eq. (18) for all analyzed a-levelsyields the cost-effectiveness diagram describing the interrela-tion between possible investment costs ci and the effectivenessE(ai) = E(C�1(ci)) regarding the reduction of uncertainty, seeFig. 4. With the help of the cost-effectiveness diagram, the effec-tiveness of the investments can be analyzed by evaluating theratio of the change in investment costs to the change in effectsof uncertainty reduction. In other words, those costs which havethe best effect on the reduction of uncertainty of the forecast va-lue are definable.These facts can also be expressed by a cost-effectiveness ratioCER(ai), according to Eq. (19), which can be interpreted as thecosts required to obtain one unit of effectiveness. Therewith, itbecomes clear how much the ratio between costs and effec-tiveness differs. The fundamental objective is to find the min-imum CER(ai).

RðaiÞ ¼CðaiÞEðaiÞ

)min for EðaiÞ > 0 ð19Þ

E½k

3.2. Multi-dimensional case

In case of multiple fuzzy input variables ~x1; ~x2; . . . ; ~xl, a local(Section 3.2.1) and a global variant (Section 3.2.2) of analysis ispresented. Naturally, with the global analysis better results areattained, but the computational time increases. The local analysisserves as rough estimation of the impact of the individual fuzzyinput variables on the forecast variable. However, interactionsbetween the fuzzy input variables are not considered withinthe local analysis, but it enables an orientation for distributingfunds.

CE

3.2.1. Local analysis

First step: According to the first step of the one-dimensionalcase (Section 3.1) the investment costs are interrelated to thea-levels of each fuzzy input variable ~xk; k ¼ 1;2; . . . ; l, by meansof an inverse cost function C[k]�1(c).Second step: Altogether l fuzzy analyses are performed like pre-sented in Section 2. The input variables for each fuzzy analysis kare defined as follows: From all fuzzy input variables~xp; p ¼ 1;2; . . . ; l; p – k, merely the a-level sets Xp;a1¼0 are inputvariables of the fuzzy analysis k, solely ~xk remains as fuzzy inputvariable (see Fig. 5). The fuzzy forecast variable obtained for

each fuzzy input variable ~xk is denoted by ~z½k�, which can be usedas an indicator of the influence of ~xk on the system.

Remark 1. When breaking down the fuzzy analyses to the optimi-zation problems according to Eqs. (15) and (16), decrease in calcu-lation time is possible. Theoretically the complexity ofoptimization problems is 2 � (l � n), but numerically this can bereduced to 2 � (l � n � (l � 1)). However, the results have to berejoined to all individual ~z½k� additionally.

Third step: The procedure is similar to the third step of the one-dimensional case (Section 3.1), but it is done for each singlefuzzy input variable ~xk and forecast variable ~z½k�, respectively.Analogous to Eq. (18), the reduction of uncertainty for each a-level of ~z½k� is evaluated by8

�ðaiÞ ¼1�

z½k�ai r�z½k�ai l

z½k�a1r�z½k�a1 l

for z½k�a1r – z½k�a1 l

0 for z½k�a1r ¼ z½k�a1 l

><>: ð20Þ

The cost-effectiveness ratio CER[k](ai) is thus obtained for eachpossible investment on fuzzy input variable ~xk analogous to Eq.(19):

R½k�ðaiÞ ¼C ½k�ðaiÞ½k� )min for E½k�ðaiÞ > 0 ð21Þ

E ðaiÞ

As all fuzzy input variables except for ~xk are fixed on a-level a1 = 0,no additional cost are incurred, only costs for reducing ~xk are con-sidered.

After that, the cost-effectiveness curves of the cost-effec-tiveness diagram can be analyzed. The cost-effectiveness ratioscan be compared within one fuzzy input variable but also be-tween all fuzzy input variables. It becomes clear which fuzzyinput variables separately have the most impact on the fuzzyforecast variable, but also which investments are most cost-effective.

Fig. 5. Local cost-effectiveness fuzzy analysis with l fuzzy input variables and corresponding input intervals Xp;a1¼0.

236 U. Reuter, U. Schirwitz / Structural Safety 33 (2011) 232–241

3.2.2. Global analysis

First step: According to the first step of the local analysis (Sec-tion 3.2.1), the investment costs are interrelated to the a-levelsof each fuzzy input variable ~xk; k ¼ 1;2; . . . ; l, by means of theinverse cost function C[k]�1(c).

Fig. 6. Sequence of fuzzy analyses for glo

Second step: For each fuzzy input variable ~xknl�1 fuzzy analysesare carried out. The corresponding input variables for eachfuzzy analysis are defined as follows:(1) For the fuzzy input variables ~xp; p ¼ 1;2; . . . ; l p – k, all

combinations of a-level sets Xp;aip; ip ¼ 1;2; . . . ;n are con-

sidered, so that a single combination consists of one

bal cost-effectiveness fuzzy analysis.

Fig. 7. Global analysis with l = 2 fuzzy input variables and corresponding input intervals Xp;aipfor n = 3 a-levels.

Fig. 8. Inverse cost function C�1(c) for mapping of investment costs ci onto the a-levels of the fuzzy input variable ~v .

U. Reuter, U. Schirwitz / Structural Safety 33 (2011) 232–241 237

238 U. Reuter, U. Schirwitz / Structural Safety 33 (2011) 232–241

concrete a-level set Xp;aipof each ~xp. The a-level sets Xp;aip

aswell as the corresponding a-levels aip , which are necessaryfor the costing in the third step, are summarized in matri-ces B[k,u] of size (l � 1) � 2, u = 1, 2, . . . , nl�1.

CE

C ½

B½k;u� ¼

b½k;u�1;1 b½k;u�1;2

..

. ...

b½k;u�k�1;1 b½k;u�k�1;2

b½k;u�k;1 b½k;u�k;2

b½k;u�kþ1;1 b½k;u�kþ1;2

..

. ...

b½k;u�l�1;1 b½k;u�l�1;2

266666666666666664

377777777777777775

¼

X1;ai1ai1

..

. ...

Xk�1;aik�1aik�1

Xkþ1;aikþ1aikþ1

Xkþ2;aikþ2aikþ2

..

. ...

Xl;ailail

26666666666666664

37777777777777775

ð22Þ

(2) The intervals Xp;aip; p ¼ 1;2; . . . ; l; p – k, of each matrix

B[k,u] – supplemented with the fuzzy input variable ~xk eachtime – are the input variables for each fuzzy analysis.

Sub-steps (1) and (2) are repeated for each fuzzy variable~xk. This sequence is illustrated in Fig. 6. The fuzzy forecastvariable obtained for each fuzzy input variable ~xk and foreach cycle u is denoted by ~z½k;u�.

For the case of l = 2 fuzzy input variables and n = 3 a-levels, the

input variables for all required fuzzy analyses are shown in Fig. 7.

Remark 2. Fig. 6 represents the theoretical view of the procedureof global cost-effectiveness fuzzy analysis. From the numericalperspective, the computation can be stopped after k = 1. At thispoint in time all results of the necessary optimization problemsaccording to Eqs. (15) and (16) are available, but have to berejoined for all individual ~z½k;u�. That means, instead of l � nl�1 fuzzyanalyses, nl�1 are sufficient.

analysis (Section 3.2.1), but is done for each l � nl�1 calculated½k;u�

Fig. 9. Fuzzy forecast variable ~t.

Third step: The procedure is similar to the third step of the local

forecast variable ~z . The reduction of uncertainty of the ana-lyzed forecast variable ~z½k;u� subjected to the possible combina-tions of investments is evaluated by

E½k;u�ðaiÞ ¼1�

z½k;u�ai r �z½k;u�ai l

z½k�a1 r�z½k�a1 l

for z½k�a1r – z½k�a1 l

0 for z½k�a1r ¼ z½k�a1 l

8><>: ð23Þ

Analogous to the local analysis, z½k�a1 l and z½k�a1r are the intervalboundaries of the forecast variable ~z½k� obtained on condition thatip = 1 for all p = 1, 2, . . . , l, p – k.The cost-effectiveness ratio CER[k,u](ai) is obtained for eachinvestment on fuzzy input variable ~xk analogous to Eq. (21):

R½k;u�ðaiÞ ¼C ½k;u�ðaiÞ½k;u� )min for E½k;u�ðaiÞ > 0 ð24Þ

Fig. 10. Cost-effectiveness diagram for one-dimensional case.

E ðaiÞ

For this, the costs C[k,u](ai) corresponding to a particular a-le-vel ai of the fuzzy input variable ~xk in combination with theintervals of the first columns of matrix B[k,u] are defined as fol-lows:

Table 1Cost-effectiveness ratios CER(ai) for one-dimensional case.

ai C(ai) (€) E(ai) (–) CER(ai) (€)

k;u�ðaiÞ ¼ C ½k�ðaiÞ þX

p2f1;2;...;ljp–kg

C ½p� b½k;u�p;2

� �for p < k

C ½p� b½k;u�p�1;2

� �for p > k

8><>:

0 0 0 –0.25 100 000 0.31 322 5810.5 200 000 0.57 350 8770.75 300 000 0.65 461 5391 400 000 0.65 615 385

ð25ÞAfter that, the l � nl cost-effectiveness ratios can be analyzed

and compared within one fuzzy input variable but also betweenall fuzzy input variables. It becomes clear which fuzzy input

variables globally have the most impact on the fuzzy forecastvariable, but also which investments are most cost-effective.

Remark 3. According to Remark 2, from numerical point of view itis sufficient to consider only the cost-effectiveness ratios CER[1,-

u](ai), because all combinations of investments are contained inthem. In case of a large amount of fuzzy input variables, the com-putation and evaluation effort can be reduced by carrying out localcost-effectiveness fuzzy analysis as a kind of sensitivity analysisand then to apply global cost-effectiveness fuzzy analysis onlyfor the most significant fuzzy input variables. Thereby, less influen-tial fuzzy input variables can be fixed at their support.

4. Example

4.1. One-dimensional case

The presented method is demonstrated in the following byExample 2 mentioned in Section 1. The deterministic fundamentalsolution is calculated by simulating the process using the simula-tion program AnyLogic�. The a-level optimization is implementedadditionally within this program by the authors. The total durationof the delivery process of the columns is to be analyzed. The mod-elled scenario is as follows: A factory produces columns and sev-eral trucks deliver them to the building site. A crane lifts thecolumns from the trucks to a storage place. Additionally, the crane

Fig. 11. Inverse cost functions for mapping of investment costs ci onto the a-levels of the fuzzy input variables ~v1 and ~v2.

Table 2Input variables of each fuzzy analysis k for local analysis.

k Fuzzy inputvariable (km/h)

Input interval(km/h)

Fuzzy forecastvariable

1 ~v1 ¼ h50; 69:2; 74; 74i V2;a1 ¼ ½8;10� ~t½1�

2 ~v2 ¼ h8; 9:6; 10; 10i V1;a1 ¼ ½50;74� ~t½2�

U. Reuter, U. Schirwitz / Structural Safety 33 (2011) 232–241 239

is needed to transport columns from the storage place to the build-ing. The average velocity of the trucks is modelled as fuzzy inputvariable ~v . It is defined as a normalized trapezoid fuzzy interval~v ¼ hv l; vml; vmr ; vri (see Fig. 8), where vl and vr are the velocitieswhich will not be exceeded or not be fallen below, i.e. the left andright boundaries of the support S~v . The values within the interval[vml; vmr] have a degree of membership l~v ðvÞ ¼ 1, which corre-sponds to the velocities estimated as accurately as possible bythe drivers.

The number of trucks notruck and the number of columns nocol-

umn are deterministic input parameters and known in advance.The implemented algorithm allows to choose the amount ofa-level sets, which are defined equally spaced within the interval

Fig. 12. Fuzzy forecast variables

[0; 1] according to Eqs. (11)–(13). The fuzzy forecast variable ~t ofthe a-level optimization is the total duration of the delivery. Dueto the discretization ~t is obtained as polygonal fuzzy variable.

First step: It is assumed, that four possibilities (close wholedelivery route to public traffic or specific sections of it for theduration of the transport) exists to reduce the uncertainty ofthe velocity of the trucks by investing a determined amountof money. The membership function of the velocity of the trucks~v ¼ h50; 69:2; 74; 74i km

h and the examined inverse cost func-tion C�1(ci) are shown in Fig. 8. The amounts to be investedare assumed to be between 100 000 € and 400 000 € and areassigned to the a-levels a2 = 0.25, a3 = 0.5, a4 = 0.75 and a5 = 1.Second step: The a-level optimization had been carried outusing five a-levels (including a-level a1 = 0), because four possi-bilities to reduce the uncertainty of the fuzzy input variable areassumed. The result of the a-level optimization is the fuzzyforecast variable ~t, see Fig. 9.Third step: Based on the calculated results, a cost-effectivenessdiagram is obtained, see Fig. 10. It is obvious that the invest-ment of 100 000 € has the steepest incline and therefore thebest ratio between cost and effectiveness. However, the differ-

~t½1� and ~t½2� of local analysis.

Fig. 13. Cost-effectiveness diagram of local analysis.

Table 4Input variables of fuzzy analyses for global analysis.

k Fuzzy input variable(km/h)

u Input interval(km/h)

Fuzzy forecastvariable

1 ~v1 ¼ h50; 69:2; 74; 74i 1 V2;a1 ¼ ½8;10� ~t½1;1�

2 V2;a2 ¼ ½8:8;10� ~t½1;2�

3 V2;a3 ¼ ½9:6;10� ~t½1;3�

2 ~v2 ¼ h8; 9; 10; 10i 1 V1;a1 ¼ ½50;74� ~t½2;1�

2 V1;a2 ¼ ½59:6;74� ~t½2;2�

3 V1;a3 ¼ ½69:2;74� ~t½2;3�

Table 5Cost-effectiveness ratios CER[k,u](ai) of global analysis.

k Fuzzy inputvariable

u ai C[k, u] (ai) (€) E[k,u](ai) CER[k, u](ai) (€)

1 ~v1 1 0 0 0 –0.5 200 000 0.433 461 6231 400 000 0.666 600 609

2 0 2000 0.048 41 4710.5 202 000 0.436 463 4551 402 000 0.693 580 413

3 0 4000 0.057 70 135

240 U. Reuter, U. Schirwitz / Structural Safety 33 (2011) 232–241

ence of this ratio between the investments of 100 000 € and200 000 € is only minimal. That is, the uncertainty of the deliv-ery process can be reduced efficiently by investments in theamount of 100 000 € or 200 000 €. An expense of more than200 000 € seems less recommendable, because the curvebecomes significantly shallower. Definitely, 300 000 € shouldbe the maximum investment, because an additional expendi-ture is without any effect.The cost-effectiveness ratios CER(ai) for all a-levels according toEq. (19) are presented in Table 1 and confirm the foregoingstatements.

4.2. Multi-dimensional case

For the multi-dimensional case a second fuzzy input variable isconsidered within the scenario of Section 4.1. Beside the averagevelocity of the trucks ~v truck, also the average velocity of the crane~vcrane is only given as uncertain value. Both variables are definedas normalized trapezoidal fuzzy intervals (see Fig. 11). To simplifythe calculation for this example the number of a-levels is reducedto three in contrast to the one-dimensional case.

4.2.1. Local analysis

First step: It is assumed, that two possibilities (close wholedelivery route to public traffic or a specific section of it for theduration of the transport) exists to reduce the uncertainty ofthe velocity of the trucks by investing a determined amountof money. The membership function of the velocity of the trucks~v trucks ¼ ~v1 ¼ h50; 69:2; 74; 74i km

h and the examined inversecost function C[trucks]�1(c) = C[1]�1(c) are shown in Fig. 11. Theamounts to be invested are assumed to be 200 000 € or400 000 € and are assigned to the a-levels a2 = 0.5 and a3 = 1.Furthermore it is possible to reduce the uncertainty of thevelocity of the crane by additional staff which supports the pro-cess of loading and unloading. The membership function of~vcrane ¼ ~v2 ¼ h8; 9:6; 10; 10i km

h and the examined inverse costfunction C[crane]�1 = C[2]�1 = (c) are shown in Fig. 11. Theamounts to be invested are assumed to be 2000 € or 4000 €

and are assigned to the a-levels a2 = 0.5 and a3 = 1.Second step: Two fuzzy analyses had been carried out usingthree a-levels (including a-level a1 = 0), because two fuzzyinput variables are given (see Table 2). The results of the fuzzyanalyses in the form of the fuzzy forecast variables ~t½1� and ~t½2�

are shown in Fig. 12.Third step: Table 3 contains all cost-effectiveness ratiosaccording to Eq. (21). The cost-effectiveness diagram is shownin Fig. 13. Due to the low cost, the investment of 2000 € hasthe best cost-effectiveness ratio, although there is only asmall reduction of uncertainty of the forecast variable. Toget a more significant reduction of uncertainty, the invest-ment of 200 000 € is preferred over the investment of400 000 €.

Table 3Cost-effectiveness ratios CER[k](ai) of local analysis.

k Fuzzy input variable ai C[k](ai) (€) E[k](ai) (–) CER[k](ai) (€)

1 ~v1 0 0 0 –0.5 200 000 0.43 461 6231 400 000 0.67 600 609

2 ~v2 0 0 0 –0.5 2000 0.05 41 4711 4000 0.06 70 135

4.2.2. Global analysis

First step: This step corresponds to the first step of the localanalysis (Section 4.2.1).Second step: Altogether six fuzzy analyses are performed (seeTable 4), because two fuzzy input variables are given and twopossibilities to reduce uncertainty are known each.Third step: Table 5 contains all cost-effectiveness ratios accord-ing to Eq. (24). As mentioned in Remark 3, only the calculationsfor k = 1 are necessary, but the results are shown completely inorder to point out the repetition.

The best cost-effectiveness ratio is obtained when reducingthe uncertainty of ~v2 to a-level a2 = 0.5, which means aninvestment of 2000 €. As in local analysis, the lowest cost-effec-tiveness ratio for a more significant reduction of uncertainty is

0.5 204 000 0.436 468 0431 404 000 0.739 547 030

2 ~v2 1 0 0 0 –0.5 2000 0.048 41 4711 4000 0.057 70 135

2 0 200 000 0.433 461 6230.5 202 000 0.436 463 4551 204 000 0.436 468 043

3 0 400 000 0.666 600 6090.5 402 000 0.693 580 4131 404 000 0.739 547 030

U. Reuter, U. Schirwitz / Structural Safety 33 (2011) 232–241 241

achieved by the investment of 200 000 €. However, a combina-tion of both investments, i.e. the reduction of the uncertainty ofboth fuzzy input variables, yields no improvement. This iscaused by the nonlinearity of the deterministic fundamentalsolution, that is the construction process model, and is onlyidentifiable using global cost-effectiveness fuzzy analysis.

5. Conclusion

In this paper, a new approach is presented for evaluating differ-ent monetary investment possibilities for reducing the uncertaintyof process variables. Uncertain process variables are modelled asfuzzy variables and mapped onto fuzzy forecast variables usingthe fuzzy analysis. For evaluation of the effectiveness of monetaryinvestments on the reduction of uncertainty of the fuzzy forecastvariables, the new cost-effectiveness fuzzy analysis is introduced.The cost-effectiveness fuzzy analysis is applicable for multi-dimensional problems in order to support decision making. Theapproach is successfully applied by means of a constructionprocess example using the simulation program AnyLogic�.

Acknowledgments

The authors gratefully acknowledge the financial support of theEuropean Union (EFRE) and the Free State of Saxony as well as of

the German Federal Ministry of Education and Research (BMBF)for the project MEFISTO.

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