HANDLING UNCERTAIN INFORMATION IN WHOLE LIFE COSTING - A COMPARATIVE STUDY Mohammed Kishk, Assem Al-Hajj and Robert Pollock Scott Sutherland School, The Robert Gordon University, Aberdeen AB10 7QB, UK. ABSTRACT A number of recently developed algorithms to handle uncertain information in whole life costing (WLC) are explained and validated in the context of two example applications. In the first example application, the proposed methodology is compared to the sensitivity analysis technique. The break-even point has been correctly identified in almost all cases. Furthermore, it has been shown how the employed fuzzy methodology may be seen as a generalised sensitivity approach to which has been added a measure of the precision with which input variables are known to the decision-maker. In the second example application, the proposed methodology is compared to two probabilistic techniques: the confidence index method and the Monte Carlo simulation (MCS) technique. The proposed methodology correctly identified the most uncertain cost items and portrayed well the confidence in ranking. Besides, all predicted net present values were in close agreement with those obtained by the MCS technique. This showed once more the robustness of various measures and concepts employed in the developed algorithms. INTRODUCTION Whole life costing (WLC) is a technique that is used to facilitate the effective choice between a number of competing alternatives that differ not only in their initial costs but in their subsequent maintenance and operational costs as well. The method consists of two processes. First, all costs and revenues associated with the acquisition, use and maintenance and disposal of various alternatives are discounted to the present time. Then, these discounted costs of each alternative are summed up to calculate the net present value (NPV) of that alternative. In the second process, various alternatives are ranked in order of ascendant magnitude according to their NPVs and the best alternative is selected such that it has the lowest NPV. For results that are deterministic, there is no ambiguity in ranking the alternatives and the decision is straightforward. WLC, however, deals with the future and the future is unknown. Thus, it is crucial to risk assess the results before the final decision is made. In doing so, either a sensitivity analysis (SA) or a probabilistic risk assessment technique, usually the Monte Carlo simulation (MCS), is employed. The SA is used to identify the impact of a change in the value of a single risky independent parameter on the dependent variable, i.e. the NPV in a WLC exercise. The objective is usually to determine the break-even point defined as the value of the input-data element that causes the NPV of the least-cost alternative to equal that of the next-lowest-cost alternative (Kirk and Dell Isola, 1995). The major advantage of the SA is that it explicitly shows the robustness of the ranking of alternatives (Flanagan and Norman, 1993, Woodward, 1995). However, it has two limitations. First, it is a univariate approach, i.e., only one parameter can be varied at a time. Thus, it is effective only when the uncertainty in one input-data element is predominant (Kirk and Dell Isola, 1995). Secondly, it does not aim to quantify risk but rather to identify factors that are risk sensitive (Flanagan et al., 1989). Thus, it does not provide a definitive method of making the decision.
Kishk, Al-Hajj and Pollock The MCS has been used in WLC modelling by many authors including Flanagan et al. (1987, 1989), Ko et al. (1998) and Goumas et al. (1999). In a MCS, every uncertain variable is represented by a probability distribution function (PDF). The resulting NPVs become random variables represented by PDFs. As noted by Flanagan et al. (1989), this provides the decision-maker with a wider view in the final choice between alternatives but will not remove the need for the decision-maker to apply judgement and there will be, inevitably, a degree of subjectivity in this judgement. Moreover, many researchers (Woodward, 1995; Chau, 1997; Byrne, 1997; Edwards and Bowen, 1998; among others) have criticized simulation techniques for their complexity and their expense in terms of computation time and expertise required to extract the knowledge. The confidence index (CI) is a simplified probabilistic approach. It is based on two assumptions (Kirk and Dell Isola, 1995): (1) the uncertainties in all cost data are normally distributed; and (2) the high and low 90% estimates for each cost do in fact correspond to the true 90% points of the normal PDF for that cost. Obviously, the CI approach tackles some of the difficulties of MCS. However, the necessary assumption of normally distributed data and the above two restrictions limit its generality. The main assumption in probabilistic risk assessment techniques is that all uncertainties follow the characteristics of random uncertainty. This implies that all uncertainties are due to stochastic variability or to measurement or sampling error; and consequently are expressible by means of PDFs that are best derived from significant data. However, historic data for WLC within the construction industry is sparse (Bull, 1993; Ashworth, 1996, 1999; Wilkinson, 1996; Sterner, 2000, among others). Besides, facets of uncertainty in WLC data are not only random but also of a judgmental nature (Kishk, 2001). Due to the above limitations of existing risk assessment models in handling uncertain data in WLC modelling, the authors suggested that the fuzzy set theory (FST) might be more appropriate. This is mainly because it is easier to define fuzzy variables than random variables when no or limited information is available (Kaufmann and Gupta, 1985). Furthermore, mathematical concepts and operations within the framework of FST are much simpler than those within the probability theory (Ferrari and Savoia, 1998). In a series of papers (Kishk and Al-Hajj, 2000a, 2000b, 2000c, 2000d), the authors employed the FST to develop three models and algorithms to handle subjective assessments of input variables. In a subsequent paper (Kishk and Al-Hajj, 2001a), another algorithm has been developed to handle stochastic data and expert assessments as represented by probability density functions (PDFs) and fuzzy numbers (FNs), respectively, within the same model calculation. Recently, a WLC-based decision support algorithm has also been proposed (Kishk and Al-Hajj, 2001b). This algorithm systematically analyses uncertain input data and provides the decision-maker with a better impression of their validity and usability by the employment of two sets of measures. The first set includes two confidence measures CI 1 and CI 2, to evaluate the rank ordering of various competing alternatives. These factors may be interpreted as measures of the confidence in the two statements: A is better than B and A is at least as better as B, respectively, where A and B are two competing alternatives. The second set includes two uncertainty measures U and F to identify the significance of various costs regarding the ambiguity of the decision. In this paper, these algorithms are explained and compared to existing assessment techniques in the context of two example applications. This is done to highlight some of their unique features and to identify their relation to the SA, CI and MCS techniques.
Handling uncertain information in whole life costing 2.0 EXAMPLE (1) In this section, an example problem given in Kirk and Dell Isola (1995) is solved. A student recreation centre including a gymnasium was to be designed and it was required to undertake a feasibility study for the use of a solar-assisted domestic hot water system. Cost data of the two design alternatives are summarised in Table (1). Because the uncertainty of the economic life of the solar-energy system, T, a SA of varying its value from 10 to 25 with a best estimate of 18 years using a discount rate of 7% has been requested. Kirk and Dell Isola (1995) solved the problem by calculating the present worth savings from using the solar energy system and a break-even point of 16 years was obtained. To ease the comparison with the proposed algorithms, the problem is resolved by calculating the net present values of both alternatives. Again, the break-even point is 16 years and the corresponding net present value is $144,723 as shown in Fig. (1). Table (1): Cost data for example (1). Cost Fuel System Solar System Initial construction cost Baseline $ 129,000 Annual Maintenance and repair costs Baseline $ 2,720 Annual fuel cost $ 15,320 0 Salvage value 0 $ 29,440 180 Net Present Value ($1000s) 160 140 120 144.723 Solar. Fuel. 100 8 12 16 20 24 28 T Figure (1): The sensitivity test using the net present values. The net present value (NPV) algorithm (Kishk and Al-Hajj, 2000b) was also employed to solve this example problem. Low, best and high estimates were used to define the membership function of the life cycle, T. Five different MFs were considered: a triangular fuzzy number (TFN), a trapezoidal fuzzy number (TrFN) and three normal PDFs. Figure (2) depicts the resulting NPVs of both alternatives for the TFN and TrFN cases. For the TFN case, the same break-even point is calculated by the algorithm at net present value of $144,723 with a membership value of 0.75. This value corresponds to an 3
Kishk, Al-Hajj and Pollock economic life of 16 years (Fig. 3) as previously calculated by the sensitivity approach. For the TrFN case, the resulting MFs of the net present values are two crisp sets (intervals) whose boundaries are defined by the NPVs of both alternatives at the corresponding boundaries of the TrFN, i.e. [ 10, 25] years. Thus, a break-even interval, [$ 133,138, $155,274], was obtained. The true break-even point of $144,723 is included in this interval as shown in Fig. (2). Obviously, a single break-even point could not be obtained because the use of a TrFN implies that the low, best and high estimates were given the same membership value, i.e. = 1. 0.750 Solar Fuel TFN TrFN 100 120 140 144.723 160 180 Figure (2): The net present values for the TFN and TrFN cases (example 1). 0.750 8 12 16 20 24 28 T, years Figure (3): Break-even point for the TFN case (example 1).
Handling uncertain information in whole life costing To further investigate the effect of the shape of the membership function, three normal PDFs have been used to model the economic life of the solar system with the best estimate of 18 years as the mean value and standard deviations, σ, of 1, 1.5 and 2. Figure (4) shows the resulting NPVs for these three cases. In all cases, the correct break-even point of $144,723 was predicted with associated membership values of 11, 74 and 04 for σ = 1,1.5 and 2, respectively. Again, these membership values correspond to the same economic life of 16 years as shown in Figure (5). This shows the robustness of the transformation algorithm. 04 74 Solar Fuel PDF, σ = PDF, σ = 1.5 PDF, = 2.0 σ 11 100 120 144.723 140 160 180 Figure (4): MFs for the net present values for PDF cases (example 1). 04 74 σ = σ = 1.5 σ = 2.0 11 8 12 16 20 24 28 T, years Figure (5): Break-even points for PDF cases (example 1). 5
Kishk, Al-Hajj and Pollock Figure (4) shows also how the uncertainty in information is reflected in the predicted solution as the spread of the calculated NPVs increases with the uncertainty of data, i.e. as the value of the standard deviation increases. As expected, the spread is zero for the certain case, σ = 0, shown with solid lines in the figure. In the TrFN case, the fuel system was ranked first, while the solar-assisted system was recommended in all other cases. The confidence measure in ranking for all studied cases are summarised in Table (2). All these measures are in general agreement with common sense which shows the effectiveness of the ranking algorithm and the employed confidence measures. Table (2): Confidence measures of ranking (example 1). Case CI 1 CI 2 Crisp 00 00 Normal PDF, σ = 0.550 0.743 Normal PDF, σ = 1.5 34 56 Normal PDF, σ = 2.0 0.369 00 TFN 0.342 0.572 TrFN 74 0.512 3.0 EXAMPLE (2) Cost estimates of two alternative floor finishes for an administrative facility are summarised in Table (3). It is required to choose the best option for an analysis period of 25 years and a discount rate of 10%. Kirk and Dell Isola (1995) used the confidence index approach to solve this example. Alternative L was ranked first with a confidence index of 0.1498 which indicates low confidence in the choice of alternative L for implementation. Besides, their computations indicated also that the high uncertainty of annual M&R are the main cause to this low confidence in ranking. Because the CI approach does not give detailed NPVs, this problem is re-solved using MCS using the Crystal Ball 2000 simulation software (Decisioneering, 2000) as shown in Fig. (6). The fuzzy NPV algorithm was also used and its results are depicted with the thick curves in Fig. (7). Results of the MCS and fuzzy solutions are summarised in Table (4). To allow for a clearer comparison between the MCS and fuzzy results, the relative frequencies in Fig. (6) were transformed to possibility distributions (Kishk and Al-Hajj, 2001a) and are plotted in Fig. (7) with thin curves. As shown, fuzzy results agreed well with those obtained from MCS. Table (3): Cost estimates for example (2). Alternatives Estimates High Best Low 1. Alternative (L): Initial construction costs $ 7,600 $ 7,200 $ 6,800 Maintenance and repair annual cost. $ 2,700 $ 1,800 $ 700 2. Alternative (H): Initial construction costs $ 14,500 $ 14,500 $ 13,700 Maintenance and repair annual cost. $ 2,000 $ 1,200 $ 400 Alternative L was ranked first and the confidence measures in this ranking are summarised in Table (5). These relatively low measures reflect low confidence in this ranking. Besides, annual M&R costs were identified to have the dominant contribution to the ambiguity of ranking as clearly indicated by the calculated measures of uncertainty in Table (6). These results are in general agreement with those obtained by the confidence index approach.
Handling uncertain information in whole life costing 4 Alternative L. Alternative H. 8000 Relative frequency 3 2 1 6000 4000 2000 Frequency (200,000 trials) 0 0 12 16 20 24 28 32 36 Figure (6): Simulation results (example 2). Alternative L. Alternative H. Alternative 12 16 20 24 28 32 36 Figure (7): MFs of Net present values (example 2 ) Table (4): Summary of results (example 2). MCS 7 Fuzzy Minimum Maximum Mean Minimum Maximum Removal L $14,992 $32,095 $23,535 $14,967 $32,105 $23,536 H $17,345 $33,407 $25,407 $17,331 $33,454 $25,392
Kishk, Al-Hajj and Pollock Table (5): Measures of confidence (example 2). Rank Alternatives Alternative L Alternative H CI 1 CI 2 CI 1 CI 2 1 Alternative L --- --- 0.132 0.566 2 Alternative H 00 34 --- --- Table (6): Measures of uncertainty (example 2). Cost and value items Alternative H Alternative L (discounted & normalised) U F U F Initial cost 00 00 07 18 Maintenance and repair annual costs 0.560 0.307 0.598 0.335 Kirk and Dell Isola re-solved this example problem using closer estimates for annual M&R costs (Table 7). Alternative L was ranked first with a confidence index of 59 indicating high confidence in the choice of alternative L for implementation. Again, both the Crystal Ball 2000 simulation software (Decisioneering, 2000) and the NPV algorithms were used to obtain the revised NPVs of both alternatives. These revised results are summarised in Table (8). Figure (8) shows the results of the simulation exercise (200,000 trials) and their equivalent MFs are depicted with the thin lines in Fig. (9). As shown in Fig. (9) and Table (8), the revised results obtained from the proposed fuzzy algorithms agreed well with those obtained from MCS and are slightly more conservative. Again, alternative L was ranked first and the confidence measures in this ranking are summarised in Table (9). These relatively high measures reflect higher confidence in this ranking as previously indicated by the confidence index technique. These results indicate again the robustness of various measures employed in the proposed algorithms. Table (7): Revised cost estimates (example 2). Alternatives Estimates High Best Low 1. Alternative (L): Initial construction costs 7,600 7,200 6,800 Maintenance and repair annual cost. 2,300 1,800 1,300 2. Alternative (H): Initial construction costs 14,500 14,500 13,700 Maintenance and repair annual cost. 1,800 1,200 600 Table (8): Summary of revised results (example 2). Alternative MCS Fuzzy Minimum Maximum Mean Minimum Maximum Removal L $18,612 $28,464 $23,533 $18,597 $28,474 $23,536 H $19,151 $31,630 $25,394 $19,146 $31,639 $25,392 Table (9): Revised measures of confidence (example 2). Rank Alternatives Alternative L Alternative H CI 1 CI 2 CI 1 CI 2 1 Alternative L --- --- 0.196 0.598 2 Alternative H 003 02 --- ---
4 Alternative L. Alternative H. Handling uncertain information in whole life costing 8000 Relative frequency 3 2 1 6000 4000 2000 Frequency (200,000 trials) 0 0 16 20 24 28 32 Figure (8): Simulation results (example 2, revised case). Alternative L. Alternative H. 16 20 24 28 32 Figure (9): MFs of NPVs (example 2, revised case). 4.0 CONCLUSIONS AND FUTURE RESEARCH A number of recently developed algorithms have been illustrated using two case studies. Typical results showed a well agreement with traditional risk assessment techniques. Besides, smooth output MFs have been obtained despite the relatively small number of intervals employed, illustrating the robustness and computational efficiency of the implemented algorithms. Furthermore, these algorithms not only have the desirable features of existing techniques but also have more advantages as follows. 9
Kishk, Al-Hajj and Pollock Compared to the SA technique, they have the desirable property of transparency. Besides, they can deal with the uncertainty of multiple variables simultaneously rather than being a univariate approach. Compared to the MCS technique, they have the desirable property of producing the distribution of all possible values of the output variable but in a less complex and a more effective way. Compared to the CI approach, they have the desirable property of ranking competing alternatives and giving confidence measures in this ranking but they are not limited to a specific type of distributions of input variables. Because of these unique features, these algorithms will be employed to develop IT applications of whole life costing to support life-cycle decision-making in the design and management of construction assets. Part of this objective will be achieved through an ongoing EPSRC funded research project (Al-Hajj et al., 2001; Aouad et al., 2001) undertaken by a joint collaboration with another team from the University of Salford. 5.0 REFERENCES Al-Hajj, A., Pollock, R., Kishk, M., Aouad, G., Sun, M. and Bakis, N. (2001) On the requirements for effective information management in whole life costing within an integrated environment. Proceedings of the Annual Conference of the RICS Research Foundation (COBRA 2001), Glasgow Caledonian University, 3-5 September, 2, 402-413. Aouad, G., Bakis, N., Amaratunga, D., Osbaldiston, S., Sun, M., Kishk, M., Al-Hajj, A. and Pollock, R. (2001) An integrated life cycle costing database a conceptual framework. Proceedings of the 17th Annual Conference of the Association of Researchers in Construction Management (ARCOM 2001), University of Salford, 5-7 September, 1, 421-431. Ashworth, A. (1996) Estimating the life expectancies of building components in life cycle costing calculations, Structural Survey, 14 (2), 4-8. Ashworth, A. (1999) Cost studies of buildings, Longman. Bull, J. W. (1993) The way ahead for life cycle costing in the construction industry. In Bull, J. W. (ed.) Life Cycle Costing for Construction. Blackie Academic & Professional, Glasgow, UK. Byrne, P. (1997) Fuzzy DCF: a contradiction in terms, or a way to better investment appraisal? Proceedings of Cutting Edge 97, RICS. Chau, K. W. (1997) Monte Carlo simulation of construction costs using subjective data: response, Construction Management and Economics, 15, 109-115. Decisioneering (2000) Crystall Ball 2000 User Manual. Decisioneering, Denver, USA. Edwards, P.J. and Bowen, P. A. (1998) Practices, barriers and benefits of risk management process in building services cost estimation: comment, Construction Management and Economics, 16, 105-108. Goumas, M. G., Lygerou, V. A. and Papayannakis, L. E. (1999) Computational methods for planning and evaluating geothermal energy projects, Energy Policy, 27, 147-154. Flanagan, R., Kendell, A., Norman, G. and Robinson, G. (1987) Life Cycle Costing and Risk Management, Construction Management and Economics, 5, 53-71. Flanagan, R., Norman, G., Meadows, J. and Robinson, G. (1989) Life Cycle Costing - Theory and Practice. BSP Professional Books.
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