A study of cash flows in projects with fuzzy activity durations

icccbe 2010 Nottingham University Press Proceedings of the International Conference on Computing in Civil and Building Engineering W Tizani (Editor) A study of cash flows in projects with fuzzy activity durations Alexander Maravas National Technical University of Athens, Greece John Paris Pantouvakis National Technical University of Athens, Greece Abstract In order to deal with imprecision inherent in the planning and scheduling of projects, various extensions of CPM (Critical Path Method) and PERT (Program Evaluation and Review Technique) have been proposed in the literature. The main aim of these studies has mainly been the application of Fuzzy Set Theory in the computation of activity start and completion dates, and consequently in the calculation of the project completion time, and of the critical path. As a step forward, this paper studies cash flows in project networks whose activity durations are represented with fuzzy numbers. The paper aims at formulating a methodology for generating cash flows based upon the fuzzy dates calculated with the algorithms found in the fuzzy scheduling literature. The main finding of this research indicates that in the presence of uncertainty, the shape of the fuzzy project cash flow curve varies considerably from the well-established cumulative cost curve (S-curve). Furthermore, the application of the proposed methodology in decision support software may prove useful in both the evaluation of project proposals (during feasibility studies) and the assessment of working capital requirements (in project execution). Keywords: cost forecasting, cash flow, fuzzy project scheduling 1 Introduction The management of projects in an uncertain environment requires decisions that are based on inconsistent, vague and imprecise data. The shortcomings of traditional scheduling methods such as CPM and PERT, that employ a deterministic or a probabilistic view respectively, mandate the formulation of a new methodology to meet the real-world requirements. Probability theory and fuzzy set theory are two different approaches for the management of project uncertainty. Probability theory deals with random events by assigning probability distributions to the data that is necessary for making decisions. Alternatively, fuzzy set theory deals with the imprecision of data that must be processed in order to manage projects. Fuzzy Project Scheduling (FPS) is based on fuzzy set theory and is useful in dealing with circumstances that involve uncertainty, imprecision, vagueness and incomplete data. 1.1 Research in fuzzy project scheduling and fuzzy cash flows In terms of Fuzzy Project Scheduling, Prade (1979) was the first researcher to propose the application of fuzzy set theory in scheduling problems. Chanas and Kamburowski (1981) presented a fuzzy version of PERT which they named FPERT. Important research in Fuzzy Project Scheduling has been

conducted by McCahon and Lee (1988), Hapke et al. (1996), Lorterapong and Moselhi (1996), Dubois et al. (2003), Bonnal et al. (2004). Overall there have been about 40 papers published in major international journals addressing various aspects of this new scheduling approach. The necessity of applying fuzzy set theory or probability theory in the analysis of cash flow has been identified by several researchers. Boussabaine and Elhag (1999) studied the application of fuzzy techniques to cash flow management. Kumar et al. (2000) presented a methodology for the assessment of working capital requirements using fuzzy set theory. Lam et al. (2001) integrated the fuzzy reasoning technique with the fuzzy optimization technique to find an optimal path of corporate cash flow. Barazza et al. (2004) studied the application of stochastic S-curves in the probabilistic monitoring and forecasting of project performance. Yao et al. (2006) presented a fuzzy stochastic single-period model for cash management. In assessing existing research it is evident that FPS is aimed at the calculation of early/late start and finish dates as well as the determination of activity and path criticality and does not include the issue of cash flow generation. Also, the research in uncertainty in cash flows is not correlated to FPS. Therefore, the goal of this research is to combine research from FPS with important issues in cash flow generation. As a step forward, this paper studies cash flows in project networks whose activity durations are represented with fuzzy numbers. To this extent, fuzzy cash flows are generated based upon the fuzzy dates calculated from FPS algorithms. 1.2 Project cash flow significance The cash flow derived from an activity network is defined as the cumulative plot of project cost versus time as aggregated from each activity. To this extent, project management software such as Primavera Project Management and Microsoft Project facilitate the generation of project cash flows from activity network analysis. Furthermore, regression models based on actual construction cost data produce a cumulative project cash flow curve that usually has the shape of the letter S and is termed as the standard S-curve. The calculation of an accurate project cash flow is of utmost importance for the project managers and project owners. During feasibility studies an accurate cash flow is required for the Net Present Value Analysis, evaluation of the Internal Rate of Return, conducting cost-benefit analysis as well as the determination of project finance requirements. During project execution the cash flow is crucial in the assessment of working capital requirements. In effect, the difference between project expenditure (cash outflow) and project payments (cash inflow) determines the required capital reserves and is crucial for attaining financial viability. Overall, the study of the generated cash flow in conjunction with all other project constraints is crucial in project success. 1.3 Fuzzy set theory fundamentals Fuzzy set theory is used to characterize and quantify uncertainty and imprecision in data and functional relationships. Fuzzy set theory permits the gradual assessment of the membership of elements in a set in the real unit interval [0, 1]. Hence, a fuzzy set A of a universe X is characterized by a membership function μ Α : X[0,1] which associates with each element x of X a number μ A (x) in the interval [0,1] representing the grade of membership of x in A. In fuzzy set theory the triangular membership function which is defined by three numbers a, b, c is encountered very often. At b membership is 1, while a and c are the limits between zero and partial membership. The triangular fuzzy number ~ x = a, b, c has the following membership function:

0 x < a ( x a) /( b a) a x b μ A( x) = (1) ( c x) /( c b) b x c 0 x > c With any fuzzy set A we can associate a collection of crisp sets known as α-cuts (alpha-cuts) or α- level sets. A α-cut is a crisp set consisting of elements of A which belong to the fuzzy set at least to a degree of α. Therefore, if A is a subset of a universe U, then an α-level set of A is a non-fuzzy set denoted by Α α which comprises all elements of U whose grade membership in A is greater than or equal to α (Zadeh, 1975). In symbols, A α = { u μα ( u) α} (2) where: α is a parameter in the range 0 <α 1. 2 Fuzzy cash flow generation methodology 2.1 Fuzzy CPM algorithm The main goal of FPS is to apply fuzzy set theory concepts to real world projects. Thus, project activities can have durations that are fuzzy numbers instead of crisp numbers. Initially, the most appropriate membership function for the duration of every activity is selected through a process called fuzzification. Thereafter, the formulas for the forward and the backward recursion are applied using fuzzy arithmetic. The criticality of all the paths of the network can be calculated by various techniques. The fuzzy formulation of the CPM algorithm is based on the forward and backward pass of the data. The forward pass is performed by calculating the early start and early finish. ~ ~ ~ E = 0, ~ ~ ~ ~ ES = max( EF ) and EF = ES( + ) d S start p P p The backward pass is performed by calculating the late start and late finish. ~ ~ ~ ~ ~ ~ ~ LF end = T, LF = min( LS ) and LS = LF( ) d s S s where: p/s: is the preceding/succeeding activity, P/S: is the set of preceding/succeeding activities, E ~ S / E ~ F : is the fuzzy early start/finish time, L ~ S / L ~ F :is the fuzzy late start/finish time, d ~ : is the : is the fuzzy addition/subtraction operator, ~ ~ m ax / m i n : is the fuzzy maximum/minimum operator, T ~ :is the project completion time. fuzzy activity duration, ( + )/ ( ) 2.2 Activity duration The application of the FPS algorithm yields the fuzzy start and fuzzy completion dates of the activities (Fig. 1). The fuzzy start is due to the accumulation of uncertainty from preceding activities in the project network. The fuzzy completion date is the sum of the activity start with the activity duration. A fuzzy Gantt chart is the graphical representation of activity dates in networks with fuzzy activity durations (Hapke and Slowinski 1996; Slyepstov and Tsyshchuk 2003) since a standard Gantt chart is not applicable. Whereas the x-axis depicts time, the y-axis depicts the membership function of the activity duration. Figure 1 shows an activity with an early start of 7,8, 9, a duration of 8,10, 11 and an early finish of 15,18, 20. If in order to calculate the required cost per unit of time we need to divide the cost of this activity with its duration. However, the duration varies for different possibility measures and for optimistic and pessimistic scenarios. In the absolute best case (mind α ) the activity will start as early as possible and will last the minimum duration. In the absolute worst case (maxd α ) the activity will start as late as possible and will last the maximum duration. These intervals are defined as:

min D α = (inf Start, inf Start + inf Duration) (3) max D α = (supstart, supstart + supduration) (4) where: D α : is the interval of minimum/maximum duration at the respective α-cut, inf: Infimum (least), sup: supremum (greatest). In this specific example mind 0.5 = (7,5 16,5) and maxd 0.5 =(8,5 19) and hence the activity cost is distributed in these intervals. The overall project cash flow is simply the sum of the cost/unit of each activity. Because the resulting durations may not be integers they are rounded up/down accordingly. Figure 1, Fuzzy activity start and completion date. 2.3 Fuzzy time cost relation A very interesting issue is if there is a correlation between the uncertainty of the duration and the cost of the activity. To this extent there are several possibilities. 1 st case: Fixed Cost: The variation in duration is completely uncorrelated to the cost. This is true in instances where there is a negotiated fixed cost contract with a certain subcontractor. 2 nd case: Positive Time/Cost Correlation: Unexpected conditions increase both time and cost i.e. if adverse ground conditions are met there will be both an increase in the time and the cost to complete excavations. 3 rd case: Inverse Time/Cost Correlation: In this case the activity duration is decreased at the expense of a higher cost. This is done usually with the use of overtime labor or more expensive machinery. 4 th case: Fixed Time: The activity is performed in a crisp duration but the cost of activity may be fuzzy because of an uncertainty in the final price. It is the responsibility of the project manager to determine the type of time/cost relationship of each activity as well as the particular fuzzy functional relationship. This is done primarily based on the experience of project manager, the type of work and the form of payment of the specific activity. In all cases the underlying relationship between time and cost will determine this correlation in project environments with uncertainty. In this paper the cost of an activity is considered fixed (1 st case). 2.4 Methodology automation/computerization A computer program is developed to apply the methodology proposed herein by performing project network fuzzy arithmetic calculations, depicting the fuzzy-gantt schedule and generating the fuzzy cash flow curves. In networks with many activities the computer program performs calculations on membership functions, provides the graphical presentation of results and an advanced Graphical User Interface (GUI) can be used to zoom into activities of particular interest and their corresponding cost distribution. Furthermore, the computer program facilitates the extensive analysis of fuzzy cash flows by using multiple optimistic and pessimistic scenarios. The program applies the fuzzy CPM algorithm, calculates mind α and maxd α for several α-cuts (i.e. 1 0,9 0,1) and distributes costs

accordingly. As such, it can become a powerful decision support system for project cost forecasting and computer aided construction management. 3 Sample Network Analyses of Results To illustrate the application of the methodology a cash flow is generated for an activity network with the activities shown in Table 1. The project fuzzy 2 dimensional and 3 dimensional Gantt schedule of the project early dates is shown in Figure 2 and 3 respectively. Table 1. Sample activity network data. Activities Predecessor Lag Fuzzy Duration (days) Cost ( ) A 0 7,8, 9 40 B 0 4,6, 7 30 C A 0 18,20, 23 240 D A 2 12,15, 17 150 E A, B 0 8,10, 11 40 F C, D 0 9,10, 12 80 G E 0 8,10, 12 60 H F, G 0 3,4, 5 20 Figure 2, Fuzzy Gantt chart. Figure 3, 3D Fuzzy Gantt chart. Figure 4 shows five possible cash flows. At μ=1 the cash flow (black line) is equivalent to that generated from a deterministic analysis. At μ=0,5 (grey line) there are is an optimistic cash flow on the left and pessimistic one on the right. At μ=0 (grey dotted line) the optimistic and pessimistic scenario have a wider spread indicating and a higher level of uncertainty. In the best case the project will finish in 37 days whereas in the worst in 49 days. For every possibility there is a point of maximum uncertainty, the location of which is crucial for project managers. For μ=0,5 this point is on the 23 rd day in which case the cash flow varies from 404 to 489. Because of the assumption of fixed cost all cash flows add up to 660. Overall, instead of a single cash flow, there is an envelope of possible cash flows which becomes wider at increased risk levels. Thus, the fuzzy project cash flow varies from the well established S-curve. With the use of computer software the project manager can conduct optimistic and pessimistic scenarios at several possibility levels. Finally, at the project level the project inflows can be overlaid with the outflows and useful results can be deducted.

Figure 4, Fuzzy Cash Flow. 4 Conclusions This paper has contributed to the research of cash flows in networks with activities with fuzzy durations. Unlike other attempts at studying fuzzy project cash flows, the main advantage of this methodology is that the managers perception of uncertainty is entered at the activity level allowing the generation of several scenarios. Hence, cash flow curves are produced from detailed analysis and specific calculations at the activity level. The computerization of the methodology facilitates the execution of complex calculations, the presentation of high quality fuzzy Gantt schedules, the generation of cash flows and the location of the point with maximum cash flow uncertainty. Therefore its application in projects may prove useful in both the evaluation of project proposals (during feasibility studies) and the assessment of working capital requirements (during project execution) and, as such, it is an imperative tool for the practicing project manager. Ultimately, the proposed methodology could be incorporated in FPS commercial software packages providing practitioners a better perception of risk which is now hindered by the traditional software approach. References BARRAZA, G.A., BACK, W. E. and MATA, F., 2004. Probabilistic Forecasting of Project Performance Using Stochastic S Curves. Journal of Construction Engineering and Management, 130(1), 25-32. BONNAL, P., GOURCE, K. and LACOSTE, G., 2004. Where do we stand with Fuzzy Project Scheduling? Journal of Construction Engineering and Management, 130(1), 114-123. BOUSSABAINE, A.H. and ELHAG, T., 1999. Applying fuzzy techniques to cash flow analysis. Construction Management and Economics, 17, 745-755. CHANAS, S. and KAMBUROWSKI, J., 1981. The use of fuzzy variables in PERT, Fuzzy Sets and Systems, 5 (1), 11-19. DUBOIS, D., FARGIER, H. and GALVAGNON, V., 2003. On latest starting times and floats in activity networks with illknown durations, European Journal of Operational Research, 147, 266-280. HAPKE, M. and SLOWINSKI, R., 1996. Fuzzy priority heuristics for project scheduling. Fuzzy Sets and Sys., 83, 291-299. KUMAR, V.S.S., HANNA, A.S. and ADAMS, T., 2000. Assessment of working capital requirements by fuzzy set theory. Engineering Construction and Architectural Management, 7(1), 93-103. LAM, K.C., SO, A.T.P., HU, T., NG, T., YUEN,R.K.K., LO, S. M., CHEUNG, S.O. and YANG, H., 2001. An Integration of the fuzzy reasoning technique and the fuzzy optimization method in construction project management decision making. Construction Management and Economics, 19(1), 63-76. LORTERAPONG, P. and MOSELHI, O., 1996. Project-network analysis using fuzzy set theory. Journal of Construction Engineering and Management, 122, 308 318. McCAHON C.S. and LEE, E.S., 1988. Project network analysis with fuzzy activity times, Com. Mat. Ap., 15(10), 829-838. PRADE, H. (1979), Using Fuzzy Set Theory in a Scheduling Problem: A case study, Fuzzy Sets and Systems, 2, 153-165. SLYEPSTOV, A.I. and TSYSHCHUK, T. A., 2003. Fuzzy temporal characteristics of operations for project management on the network models basis. European Journal of Operational Research, 147, 253-265. YAO, J.S., CHEN, M.S. and LU, H.F., 2006. A fuzzy stochastic single-period model for cash management. European Journal of Operational Research, 170, 72-90. ZADEH, L.A., 1975. The Concept of a Linguistic Variable and its Application to Approximate Reasoning-I, Information Sciences, 8, 199-249.