A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression

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1 Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty, No. 38, Wu-chuan L Chung-l, Taoyuan, Tawan 320, R.O.C. bstract In the contractng busness, constructon frms are generally more concerned wth short-term fnancal strateges than the long-term ones. Workng captal management s the central ssue of all short-term fnancal concerns. More mportantly, cash management s ultmate goal for achevng hgh lqudty and proftablty. Ths study focuses on the cash porton of workng captal management by usng the concept of target cash balance. The am s to develop a practcal model for constructon frms n Tawan for ratonalzng the amount of cash and current assets, whch should be possessed n any pont of tme. The Fuzzy S-curve regresson s ntroduced here for understandng the ssues nvolved. Based on the project cash flow and progress payment records of an example project taken from Housng and Urban Development Bureau, Tawan Provncal Government, ths model s demonstrated and tentatve conclusons concernng the model are gven. Key words: Fuzzy S-curve regresson, workng captal management, cash flow. Introducton In the present-day research of complex systems, such as engneerng technology, envronment and socetal economy, become large n dmenson and complexty so that the exact numercal data can not be obtaned for the nformaton of systems. To solve the problems arsng from complex systems may become very neffcent or even mpossble f usng the tradtonal uthor to whom all correspondence should be addressed

2 mathematcal tools that are not constructed for dealng wth hgh dmensonalty models. Smlarly, the tradtonal least square regresson may not be applcable when dealng wth curve fttng problems. In the past twenty years, some approaches contanng fuzzy nformaton have been attractng ncreasngly attenton, as proposed n the lterature [-4]. Tanaka et al. [] develop a fuzzy lnear regresson model by usng lnear programmng technques n 982. In 988, Damond [2] resembled tradtonal least squares regresson to establsh fuzzy lnear least squares models. Ruonng [3-4] consdered the ratonalty of metrc defnton and dscussed the problem for least squares fttng of fuzzy-value data, whch are expressed as fuzzy numbers, and to develop an S-shaped curve regresson model for fttng ths type of data. The characterstcs of large publc constructons are large-scaled, tme-lmted, hgh cost, and complex-techncal etc., and there are many uncertan factors n whch. Therefore, to perform ths knda project s dffcult, especally for program and dspatch of workng captal. In order to overcome the problem of controllng projects, the S-curves are wdely used. They are valuable to project management n reportng current status and predcton the future of projects. Therefore, the S-type dstrbuton s very sutable n regresson on constructon management and socal economy etc. However, as far as we know, the workng captal management for large publc constructons by usng fuzzy S-curve regresson remans unresolved yet. Ths study s organzed as follows. Frst, classc S-curve theory s recalled. Then, based on fuzzy set theory and fuzzy nference engne as well as center of gravty defuzzfcaton, a fuzzy S-curve s obtaned for curve fttng problems. Fnally, a numercal example wth smulatons s gven to demonstrate the methodology, and the conclusons are drawn. Classc S-Curve Theory In bology and socal economy, an S-shaped cure s often used to reflect the phenomena. It means that the trend of growth gets slow frst and fnally saturaton rapdly. In practcal problem 2

3 of constructons, contractors budgets are often performed on an overall bass. Changes n strateges and mx of contracts are ver dffcult to evaluate on such a bass [5]. Therefore, the prncple of smulaton wth tools of computer was proposed to generate possble scenaros based on the specfed strateges and the expected envronment. The relatonshp between budgets and tme lmt for a project can be represented va S-curve fttng. typcal S-curve fgure s shown n Fg.. The x-axs and y-axs denote project duraton and complete progress, respectvely. Mskaw [6] proposed an S-curve equaton whch can be used n a varety of applcatons related to project control. The S-curve model s of the followng form: T 3 π ( T ) T + (.5 Tp ) 3 2 P = sn sn( T ) log 2T + 3T 2 2 π () Tp T + where P denotes percentage completon of a project or an actvty; T denotes tme at any pont of the duraton of a project or an actvty; T P s shape factor. Fg. 2 s plotted wth varous values of T P between T = 0 and T = 00% duraton and the envelope of curves for T P = 0 and T P = 00% n Eq. (). Here we suppose we can exactly get all observed data takng part n the problems, but, actually, we may not know exact values, rather some approxmaton. For ths reason, the tradtonal fttng method may not be qute sutable. Before ntroducng fuzzy S-curve regresson, we gve some relatve defntons and conclusons n the followng. Fuzzy Set Theory Defnton : Let R s a real number set. fuzzy set ~ on R s sad to be a fuzzy number f the followng condtons are satsfed: () x 0 R, such that µ ~ x ) = ; and membershp functon µ ~ ( x) s pecewse contnuous; and (2) (0,] ( 0, { x µ ~ ( x), x R} s a convex set on R. 3

4 Where x s the mean value of 0 ~ and s a crsp set. The convex set means that x x, x ], f x) mn( f ( x ), f ( )). [ 2 ( x2 Defnton 2: fuzzy number ~ s sad to be bounded f supp( ~ ) { x ~ ( x) > 0} set. Where supp( ~ ) s a crsp set. µ s a bounded Evdently for any ( 0, ] the -level set, ~, wll be expressed as a closed nterval [ p, q]. Based on the fuzzy extenson prncple [7], lnear operatons about closed ntervals are obtaned as follows: Lemma : Let [a, b], [d, e] be closed ntervals of real number. Then [a, b] + [d, e] = [ a + d, b + e] ; [a, b] [d, e] = [ a e, b d], (2) [a, b] [d, e] = [mn( ad, ae, b d,be), max( ad, ae, b d,be)] ; (3) [a, b] / [ d, e] = [a, b] [ / e, /d] = [mn( a / d, a / e, b / d,b / e), max( a / d, a / e, b / d,b / e) ]. (4) Remark : Gven any operatons whch have commutatve and assocatve characterstcs, the operatons of extenson stll have these characters. From the theory of -level descrbed above and decomposton theorem [8], we have ( B) B (5) U (0,] B ( B) (6) where denotes any arthmetc operaton; and B are fuzzy numbers and B wll be a fuzzy number. Remark 2: Wang et al. [9] proposed that the resultant fuzzy number s the same type as the orgnal fuzzy numbers after the operaton of addton or subtracton. Defnton 3: Extended Operatons for -Representaton of Fuzzy Sets fuzzy number ~ s -type, f there exsts postve constants β >0, γ >0 and 4

5 m x L for x m β µ ~ ( x) = (7) x m R for x m γ where m, a real number, s mean value of ~ ; β, γ denote left spread and rght spread, respectvely; moreover, ~ could be represented as ( m, β, γ ). Lemma 2: Gven two -type fuzzy numbers ~ and B ~, we have ( m, β, γ ) ( m, β, γ ) ( n, δ, η) = ( m n, β δ, γ + η) (8) ( n, δ, η) = ( m n, β + η, γ + δ ) (9) S-curve va Fuzzy Inference Engne fuzzy nference s descrbed by a set of fuzzy IF-THEN rules n the followng form: Rule : IF x ~ ~ s, s and y B L and x ng s ~ ng, y ng s ~ B n g THEN Y = a x + b (0) k k k where n ponts x, y ) ~ ( x n, y ) and k order curve fttng s adopted and r s the number of ( n IF-THEN rules for =,2, L, r ; ~ np and B ~ np ( p =, 2, L, g ) are the -type fuzzy sets, and x ~ x ng as well as ng y ~ y are the premse varables. Usng the center of gravty defuzzfcaton, product nference, and sngle fuzzfer, the fnal output s nferred as follows: r k w ( t)[ ak x + bk ] r = k x& ( t) = = r h ( t)( ak x + bk ) = w ( t) = () It s assumed that w ( t) 0, =,2, L, r; w ( t) > 0. Therefore, h ( t) 0 and r = h ( t) = for all t. r = 5

6 Example To llustrate the procedure of ths fuzzy regresson model, consder the followng example project, whch s taken from Housng and Urban Development Bureau, Tawan Provncal Government. The data nclude seven metro bds of valuaton and the mean scale and the mean.of tme lmt for a project are 2.7 bllons and 6 years or so, respectvely. From the data whch are normalzed and represented by percentage frst, t s easly known that the frst tme of evaluaton 4.5% of total work tme. The observed data are gven n table, where the data M = (X, Y, u, v) are all trangular fuzzy numbers, and u =0 %, v = 0 % are the left and rght spreads respectvely. Fg.3 s the valuaton data of seven metro bds and Fg. 4 s plotted by the proposed fuzzy regresson. Moreover, from Fg.4, the smulaton results show that C = T T 49.49T T.98T T by fuzzy regresson approach, where C denotes the percentage of completeness (%), and T s tme (%). Conclusons The least-squares method can usually be used to the problems of curve fttng, but when the observed data are not obtaned exactly t may not be sutable. Therefore, we propose here a Fuzzy S-curve regresson method here for understandng the ssues nvolved. The am fnally s to develop a practcal model for constructon frms n Tawan for ratonalzng the amount of cash and current assets, whch should be possessed n any pont of tme. Furthermore, from the smulaton results shown s Fgs. 3-4, there are somehow dscrete and delayed stuaton concernng the data. The progress of the former 30 % of tme lmt for a project s a bt slow. In addton, the frst evaluaton tme of total work s 4.5% when a contractor proposes. It mples that we have to notce the delayed phenomenon of cash flow and mantan the lqudty of cash n some degree n the control and management of cash to verfy the project can be fnshed smoothly and successfully. 6

7 References [] H. Tanaka, S Uejma and K. sa, Lnear regresson analyss wth fuzzy model, IEEE Trans. SMCB. 2 (982) [2] P. Damond, Fuzzy least squares, Inform. Sc. 46 (988) [3] Xu Ruonng, lnear regresson model n fuzzy envronment, dv. Modellng Smulaton 27 (99) [4] Xu Ruonng, S curve regresson model n fuzzy envronment, Fuzzy sets and Syst. 90 (997) [5] Kaka,., P., Contractors fnancal budgetng usng computer smulaton, Constructon Management and Economcs, 2, pp.3-24, 994. [6] Mskaw, Z., n S-curve equaton for project control, Constructon Management and Economcs, 7, pp.5-24, 989. [7] L.. Zadeh, Fuzzy sets, Inform. and Control 8 (965) [8] G. J. Klr, B. Yuan, Fuzzy sets and fuzzy logc theory and applcatons, Prentc-Hall, Englewood Clffs, NJ, 995. [9] Wen-June Wang and C.H. Chu, Entropy varaton on the fuzzy numbers wth arthmetc operatons, Fuzzy Sets and Systems, vol.03, no.3, pp , May

8 Fg.. Typcal S-curve fgure Fg.2. Mskaw S-curve model 8

9 Fg.3 The valuaton data of seven metro bds 00% y = x x x x x x R 2 = % 60% 40% 20% 0% 0% 20% 40% 60% 80% 00% Fg.4 n S-curve by fuzzy regresson method. 9