A GENERALISED PRICE-SCORING MODEL FOR TENDER EVALUATION

019-026 rice scoring 9/20/05 12:12 PM Page 19 A GENERALISED PRICE-SCORING MODEL FOR TENDER EVALUATION Thum Peng Chew BE (Hons), M Eng Sc, FIEM, P. Eng, MIEEE ABSTRACT This aer rooses a generalised rice-scoring model derived from behavioural features of rosect theory for use in a tender evaluation rogram. It has the otential to overcome the limitations of existing rice models by allowing the withinrice attribute score variations to be adjusted by a reference factor. It imroves selectivity and incororates a reference function that emhasises the evaluator and decision maker s reference in the lowest rice by allowing a rice-scoring curve to be derived from the skewness of the distribution of tender rices and the tender articiation rate. It identifies rices that exceed the roject budget and enalises them. Statistical data from a survey of tender rices was used to secify the model for ractical use. Comarison between generic classes of models using rice difference and rice ratio functions as rice gain measures is made to illustrate the model s general alicability. With the choices made available in this generalised ricescoring model, it is ossible to comare the various methods of tender evaluation by comuter simulation. Keywords : Tender Prices, Price Comarison, Price Scoring Model, Tender Price Evaluation, Judgment, Decision Making INTRODUCTION The objective of tender rice scoring is to assign the right values to tender rices within a roject budget and to score them so that a fair evaluation that reflects the judgment and reference of the evaluator and decision maker, can be made. The rule of tender evaluation is to award a higher score to a lower rice tender because a lower rice has a higher value to the evaluator and decision maker. In manual evaluation methods, the rice values are assigned by human judgment on an ordinal scale which imlies a qualitative rice-value relationshi. If software decision-suort tools are used in integrated tender evaluation of rice and non-rice attributes, rice must be evaluated by a quantitative value function that converts rice in monetary unit to a score on an objective numerical scale. How rices are evaluated affects the final tender ranking. A survey of literature indicated that research on rice-value relationshi for the urose of tender evaluation is scarce. One simle quantitative value function that has been used by Karsak [1] for the evaluation of flexible manufacturing systems is a rice inverse model for scoring a cost-related attribute. The score synonymous with value, is derived from -1 the rice inverse, χ i = x i where x i is the rice of the i th tender. The model s weakness is its tendency towards indifference and not having the means to assign a degree of reference for low rices. Thus, there is no flexibility for it to fully exress the judgment of the evaluator and decision maker through a reference function. A wider search in the subject of judgment and decision making [7,8,9,10] revealed the existence of value functions in rosect theory roosed by Tversky and Kahneman [2,3] since 1979. The theory relaces the notion of utility with value which is defined in terms of gains and losses from a reference oint. They suggested that the value function defined over monetary gains is χ i = x i for x i above a reference rice(0) and over losses is λ( x i ) for x i below the reference rice. Determined from exerimental data, is 0.88 and λ is 2.25, indicating diminishing marginal value and asymmetry between gains and losses. The alication of the roosed value function has received attention recently [4,7,8,9] in a number of situations in which the norms and characteristics of decision makers are modelled. Variations in a tender s overall value are contributed by the within-attribute variations and the between-attribute variations. A value function deals only with the within-rice attribute variations while the weighting functions of the tender evaluation rocedures in [5,6] rovide the between-attribute variations. This aer resents a generalised rice-scoring model that determines the within-rice attribute variations from the rices of cometing tenders. Because all judgments and decisions are context-deendent [8], the credibility of a generalised rice-scoring model rests on its ability to cature an evaluator or decision maker s tendency for dominance or non-dominance and on its ability to exress mathematically the decision maker s judgment that is translated into a continuous continuum of strong to weak emhasis of tender rice. In a realistic model, the behavioural features in rosect theory have to combine with the cost objective to achieve a wide scoe of alication for situations in which the decision maker s behaviour varies. Prosect theory is helful u to this oint after which the reference factor is deendent on behaviour that has to be aroximated by information in a tender. To achieve this objective, it borrows from rosect theory some behavioural features that are not found in multi-attribute utility theory and alies it to tender rice judgment: Evaluators and decision makers are more sensitive to cost overrun than cost saving in the sense that they will reject tenders that have rices exceeding the roject budget. If there are sufficient tender roosals to select from without cost overrun, they will view the tender s lowest rice as indicative of the fair market value and the average rice lus margin as the budget s uer bound. Journal - The Institution of Engineers, Malaysia (Vol. 66, No. 1, March 2005) 19

019-026 rice scoring 9/20/05 12:12 PM Page 20 THUM PENG CHEW If the tender rices are significantly less than the roject budget, they become strongly gain-seeking so as to accrue larger cost savings and to comensate for the original overestimate. If the tender rices are close to but below the roject budget, their gain-seeking tendency is weaker because with lesser cost saving, attention is shifted to the non-rice attributes of the tender to accrue higher non-rice gains. This aer is organised as follows. First, a generalised rice-scoring function of a similar form as the value function of rosect theory is roosed. Next, the roject budget acting as the reference rice is defined and its imlications discussed. A reference function is develoed for strong to weak gainseeking, the strength of which is determined by the tender rice coefficient of skewness and tender articiation rate. A survey of tender rices elicits the two statistics which together with the evaluator s reference limits, are used to determine the constants of two generic classes of models. Within each generic class, comarison between rice difference and rice ratio functions as gain measures is made to illustrate the model s general alicability. Finally, aroriate alications of the two generic classes are suggested. PRICE VALUE FUNCTIONS In general, the mathematical function of a rice-scoring model that catures the essence of how a score, A relates to a rice gain variable, X must have the following roerties. (i) A monotonically increases with X (ii) 0 A 1 for 0 X 1 (iii) da 0 for 0 X 1 dx From rosect theory [2,3], the value/score, A i derived from the rice gain X i, of the i th tender rice out of m tender rices is suggested below for two generic classes of models. Un-normalised class: A i = X i Normalised class: A i = X i m X i i=1 (1a) (1b) where generally - < <. Equation 1b also ensures that m = 1 as a result of normalisation. When is negative, a Ai i=1 rice closer to the maximum rice will result in a higher score because the negative root of a small rice gain roduces a score close to unity. Because this is contrary to the tender evaluation rule of low rice-high score, the rice-scoring model must not consider values of that are negative. With this exclusion, the receding roerties are still satisfied but additional roerties are required to define the variable, for various behavioural states as follows. (i) A(X) is constant for = 0 (ii) A(X) is roortional to X for = 1 (iii) A(X 1) = 0 for = (iv) For Equation 1a, 0 1 The evaluator or decision maker s behaviour can be aroximated by the reference factor, which defines the gain-seeking tendency in the shae of the rice-scoring curve. With Equations 1a and 1b, there is no effect caused by rice gain on the scores when = 0 i.e. they all have the same unity score for the un-normalised class and 1 m for the normalised class. When increases from 0, the model starts to exhibit a lower rice-higher score characteristic, initially still having the tendency to be rice indifferent. When = 1, it scores linearly with rice gain and is not adjusted by reference (judgment). With Equation 1b, when =, the lowest rice attains the maximum score of 1 unit while the rest of the rices are scored zero irresective of their gain value. Thus, as moves from 0 to, the rice-scoring characteristic moves from one that is insensitive to rice gain, hence eliminating rice cometition, to one that exhibits the strongest reference for the lowest rice, hence creating the stiffest cometition that results in only one ossible contender. In other words, the rice-scoring model inherently allows for a range of effects to be accommodated, namely indifference, linear and non-linear deendence on rice gain through the secification of. THE PROJECT BUDGET AS A REFERENCE PRICE Before tenders are called, a value of the roject cost xˆ is estimated. Based on this value, a roject budget is given taking into account an assigned contingency cost which is rovided to mitigate situations of cost overrun caused by residual roject risk. If the contingency cost allocated is c, the roject budget, x B is xˆ + c. When tender submissions are received, their rices are comared with either the roject cost estimate or the budget allocation. Evaluators refer not to risk cost overrun and will be reluctant to recommend award of a contract whose rice exceeds the roject cost estimate or the budget. As a means of control, a rice-scoring model must assign the lowest rice score, usually zero, to rices exceeding the budget i.e. the value function for losses is set to zero for exclusion urose and is restricted to coding of rice gains. The budget is a reference rice on an objective scale and has the same meaning as that of rosect theory discussed in [2,3,4,7]. If cost overrun is limited by x B, then the rice gain has a range x B - x min. If the tender rices are ordered in ascending value in the sequence, x (1), x (2),.., x (j),.., x (m-1), x (m), then x min = x (1). Any rice, x (q) that exceeds x B should either be assigned an evaluation-adjusted value equal to x B or should be automatically excluded from the evaluation. If x (1) exceeds x B, it will be the only rice allowed to articiate in the evaluation with gain equal to zero. This requirement ensures that in the event that all tender rices exceed the budget value, the lowest rice tender will be the only alternative worthy of any consideration as far as tender rice evaluation is concerned. By accurate estimation, an under-budget situation can be avoided most of the time. PRICE-SCORING CURVES The exonent, in Equations 1a and 1b determines the shae of the rice-scoring curve. Strong gain-seeking means a 20 Journal - The Institution of Engineers, Malaysia (Vol. 66, No. 1, March 2005)

019-026 rice scoring 9/20/05 12:12 PM Page 21 A GENERALISED PRICE-SCORING MODEL FOR TENDER EVALUATION high reference ( > 1) for low rice. In zero (neutral) reference cases ( = 1), it would exhibit a straight line obtained by joining the lowest rice score with the score at x B over the rice difference range, x B - x min. Weak gain-seeking means a tendency to de-emhasise rice reference (0 < < 1). At the limit when = 0, all rices are treated equally and the scores are imartial to rice difference. To illustrate the rice-scoring curve, the highest score, Amax from Equation 1b is given by X A max = A (1) = (1) (1- P m = (1) ) (2) m X i (1- P i ) i=1 i=1 where the normalised rice is for simlicity, exressed as P i = 1- X i. If the highest score A (1), is set to unity, A (1) is exressed as follows: A (i) A max A (i) = = 1- P (i) 1- P (1) A continuous lot of A (i) against P i is shown in Figure 1 for various value of > 0 to illustrate the shae of the rice-scoring curves. Using the curves, the evaluator and decision maker can selected the degree of reference to match their judgment. (3) The coefficient of skewness, υ is roosed as a measure of the degree of gain-seeking. It qualifies by having the following roerties. (i) (ii) Its magnitude monotonically increases with skew For a ositive (right) asymmetry, it is ositive and it must reresent a strong gain-seeking characteristic that favours the lowest rice from the bunch of rices by making the scores reduce very quickly to zero for excetional rices that are located much higher than the lowest rice. (iii) For symmetry of rices, it is zero and it must not contribute to any reference by being neutral. (iv) For a negative (left) asymmetry, it is negative and it must reresent a weak gain-seeking characteristic that still favours the lowest rice but the scores of the bunch of rices are not reduced too quickly. The lowest rice is distant from the central bunch of rices because the ricing of the lowest tender is either made under a different condition or with a different strategy from those of the central bunch. The lowest tenderer may have a legitimate advantage over the rest because of technology, available caacity, resources and location which can be evaluated outside the rice domain as in [4,5]. Or in an attemt to win the contract, the lowest rice tenderer may resort to higher risk- Figure 1: Normalised score against normalised relative rice DISTRIBUTION OF PRICES Information about the tender can be extracted from the rice distribution statistics. In an ideally cometitive environment, the coefficient of variation and the variance are indicative of the degree of dissimilarity of the ricing characteristics of the tenderers. Price skew is an asymmetry arising out of either legitimate reasons or collusion [11,12]. It is not the urose here to determine its cause but to use it to determine reference. The existence of a low rice located to a distant left of a central bunch of rices (Figure 2a) gives rise to left asymmetry and a negative coefficient of skewness. If the rices are ideally symmetrical about their mean (Figure 2b), then the coefficient of skewness is zero. The existence of a high rice located to the distant right of a central bunch of rices (Figure 2c) gives rise to a right asymmetry and a ositive coefficient of skewness. Figure 2: Illustration of skewness in frequency distribution Journal - The Institution of Engineers, Malaysia (Vol. 66, No. 1, March 2005) 21

019-026 rice scoring 9/20/05 12:12 PM Page 22 THUM PENG CHEW taking and cutting rofit margin, a strategy that the others may not be willing to adot. If the determination of the roject budget is accurately done, such extreme rice deviation from the cost estimate must signal the need for extra caution during the evaluation and to uncover the hidden roject risks. While giving the highest score, the evaluator is risk-averse and must rudently limit the emhasis of the lowest rice. A similar effect may also be created from strategies that defeat fair cometition by many forms of cartel collusion. Non-cometing cartel rices are increased and the cartel-romoted tenderer s rice is raised to just below the next-lowest rice [12,13] as illustrated in the comarison between Figures 2a and 2b. If such strategies are revealed in the skew, they are enalised through exclusion by the budget limit as well as by rice deemhasis. When automatic action could not be taken effectively by the software rogram, re-assignment of (<1) is the way to counter this roblem. If behaviour is exressed in the reference factor, the distribution s coefficient of skewness υ, is a behaviour variable that induces a reference such that the relation between and υ meets the following: (i) > 0 for - < υ < (ii) = if υ (iii) = 1 if υ 0 (iv) = 0 if υ - A ossible general olynomial exression that satisfies these roerties is the skewness factor, k 1 given as follows: L1 k 1 = α2i-1υ2i-1 i =1 where α 2i-1 s are ositive constants to be determined for L 1 terms. υ is zero if the number of tenders evaluated is less than 3. TENDER PARTICIPATION RATE If there are sufficient tender roosals to select from without incurring cost overrun, the rice-scoring model should allow for increase in its rice selectivity under a situation of high tender articiation rate. When υ is very negative, is near zero. The scores tend to equalise and lose their discriminative ability. In Equation 1b, high tender articiation rate decreases the absolute scores by the effect of a larger denominator. The consequence is that when the rice scores are brought into the aggregation rocess with other non-rice factors, they become less contributory from their low score values and the lack of discrimination. This effect is counteracted by increasing the value of with tender articiation rate. One way is to include in, a searate tender articiation factor, k 2 which is a function of the number of tenderers, m being evaluated. It should have increasing monotonically with m i.e. d > 0 for m > 0. dm It is suggested that k 2 adots a general ositive olynomial function of m in the form L2 k 2 = β 0 + βjmj where β 0 is a constant and β j s are ositive constants to be j =1 (4) (5) determined for L 2 terms. When m = 1, there is no cometition and k 2 can be set to zero thus, β 0 = - βj. GENERAL PREFERENCE FUNCTION How k 1 and k 2 should be combined to obtain does not have a unique aroach. It must be alication deendent and satisfy ractical and intuitive requirements. One aroach is to add the effects of distribution skew, k 1 and the effects of tender articiation rate, k 2 i.e. k 1 + k 2. An additive effect has advantage over the multilicative one ( k 1 k 2 ) because it would not nullify the effect of tender articiation rate if the skewness were zero. However, addition alone cannot satisfy all the roerties of. A non-nullifying and roerty-comlying exression for as a function of k 1 and k 2 is suggested as follows: = ex(k 1 + k 2) (6) This exression restricts the value of to > 0 and thus, satisfies the required roerties of. k 1 contributes to by ensuring the ossibility of secifying a strong reference for low rice with ositive skew or a de-emhasis of reference for low rice with negative skew. It ensures that when υ = 0, zero (neutral) reference results from it. The tender articiation rate, m contributes ositively to through k 2. Exonentiation ensures a ositive all the time and it amlifies the combined effects, thus making the influence of stronger. Flexibility is given such that without the aid of these two factors, the evaluator and decision maker can still assign the value of indeendently. At this oint, it is seen that the rice-scoring model combines the ideas of rice gain measure, X and of the degree of reference, which is a function of the skewness of the rice distribution and the number of tenders evaluated, to calculate the rice scores, either un-normalised or normalised resectively. ANALYSIS OF TENDER PRICES If the urose of tender evaluation is to select an otimum tender, then the information that reveal rice relativities should be ut into good use for decision making. A survey was conducted to gather data on tender articiation rate and tender rices with the aim of obtaining statistics on tender articiation rate and on tender coefficients of variation and skewness. The range of values of m and υ can then be established for use in the rice-scoring model. From a total of 75 data sets from ast tender exercises, the coefficients of variation (deviation/mean) and skewness for each data set were calculated. From their distributions, the coefficients means, medians and deviations were obtained from statistical analysis and are summarised in Table 1. The coefficient of skewness is exected to vary from negative values to ositive values while the coefficient of variation is always ositive. 95% of the tenders has a articiation rate below 10 and the median is 4. The rice coefficient of variation s mean and median are close to each other at 0.1274 and 0.1221 resectively while its deviation is small at 0.0752. The survey data are indeed from tenderers who shared similar localised L2 j =1 22 Journal - The Institution of Engineers, Malaysia (Vol. 66, No. 1, March 2005)

019-026 rice scoring 9/20/05 12:12 PM Page 23 A GENERALISED PRICE-SCORING MODEL FOR TENDER EVALUATION Table 1: Analysis of tender articiation rate and tender rices Tender Price Tender Price Coefficient Price Coefficient Coefficients Particiation of Variation, of Skewness, Statistics Rate, m σ/µ υ Mean 4.933 0.1274 0.2828 Median 4 0.1221 0.4682 Deviation 2.124 0.0752 0.9764 5th Percentile 3 0.02637-1.5570 α = q min υ min β = q max υ min - q min υ max υ min (m max - 1) If β = β(m max-1), (9) (10a) 95th Percentile 9.3 0.2850 1.6301 characteristics [13] and quoted rices with small deviations from the mean rice. 95% of the coefficients of variation is below 0.2850. A budget u to 1.285 times the roject cost estimate will cature most tender rices for evaluation. The mean of the coefficient of skewness, µ υ is 0.2828 and its deviation, σ υ is larger at 0.9764. 90% of its variations lies within the range of 1.5570 and 1.6301 with a median of 0.4682. The large σ υ is an indication of the sensitivity of skewness to extreme values and υ can be suitably emloyed to vary the reference over a wide range. SPECIFIC PREFERENCE FUNCTION A secific reference function for the rice-scoring model adots the first order functions of k 1 = αυ from Equation 4 and of k 2 = β(m-1) from Equation 5, where α and β are the ositive constants in the exression of as follows: Thus, = ex[αυ + β(m-1)] (7a) or q = ln = αυ + β(m-1) (7b) The rice-scoring model is to work within the secified limits determined by: The maximum tender articiation rate, m max and the minimum, which is 1. The uer and lower ercentiles of the coefficient of skewness (υ max and υ min) of the tender distribution. The uer and lower limits ( max and min) of the reference factor The tender articiation rate can be estimated from the number of tender documents collected and thus a maximum can be set. A minimum of 1 is set to ensure that the effect of tender articiation rate could be felt at 2 onwards according to Equation 7a. Equation 7b shows that the variation of the model constants, α and β are deendent on the logarithm of the reference limits. They are less sensitive to the variations of and can accommodate the fuzziness of subjective judgments without large changes in value. Figure 1 rovides the evaluator and decision maker a means of secifying the limits by insection. Thus, q max = ln max = αυ max + β(m max - 1) q min = ln min = αυ min Solving for α and β, (8a) (8b) then, β = (10b) To ensure that α is ositive (> 0), then v min and q min must have the same sign. For β 0, (q max υ min q min υ max)/ υ min 0 (11a) (11b) max min κ (11c) min min κ (11d) The choice of is guided by the intuitive requirement that the effect of υ should be greater than that of m. λ Thus, α > β and max < min (12) 1 where λ = = κ + υ min. For a given min, combining Equations 11c and 12 gives the range of max. κ λ min < max < min q max υ min - q min υ max υmin ln ma x υ max = κ ln min υ min 1+υ max υ min 1 (13a) Similarly given max, λ max > min > κ max (13b) From the rice survey, υˆ min, υˆ max can be obtained for secified ercentile limits of its distribution. κ can be calculated. ˆmin is secified to calculate α. max is then selected so that β and β are ket ositive. In this aer, the range of the skewness coefficient is fixed at 1.5567 and at 1.6301 to give κ equals to 1.0472. To cater for at least 95% of tender cases, m max is fixed at 11. The corresonding values of α and β are shown in Table 2 for each air of min and max. The effect of tender articiation can be nullified (β = 0) by either min or max determined from Equations 11c and 11d. For a given min, the evaluator and decision maker can choose the strength of tender articiation rate from the range of values of max determined from Equation 13a. Nullification will not occur if max is larger than the lower extreme value. With 3 arameters fixed, the reference function, is determined from min and max for λ = -1.6896 as follows. Given min = 0.2, 5.395 < max < 15.167 and given max = 5.0, 0.3857 > min > 0.2151. 1 Journal - The Institution of Engineers, Malaysia (Vol. 66, No. 1, March 2005) 23

019-026 rice scoring 9/20/05 12:12 PM Page 24 THUM PENG CHEW PRICE GAIN FUNCTIONS Only rice gains are of interest. At this oint, the model excludes the coding of losses which is done in rosect theory. In an integrated tender evaluation rocedure, rice gain must be dimensionless. When rice is in monetary unit, it must be evaluated searately because it is not comatible with other dimensionless attributes. One roerty of a rice gain function is that for rices, x i > x j > x k, the rice gains derived from these rices must satisfy the condition X i < X j < X k. Transitivity of rice gains must be assured. The qualitative ordinal scale commonly used cannot be adoted here because it fails to reserve the numerical information of rice. Value scores must be made on quantitative scales. From exerience, both rice difference and rice ratio have been use in tender evaluation. Price difference is a comarison of rices in an interval scale whereas rice ratio reresents relative value comarison in a ratio scale. PRICE DIFFERENCE MODEL SUB-CLASS For the rice difference model, the rice gain variable is a measure of the distance of a rice from an uer rice limit and is exressed as follows. X i = x max - x i x (14) max - x min where x max is the reference rice on the interval scale and x min is the lower rice limit in the tender and x i is the tender rice of the i th tender alternative. The rice gain, 0 X i 1 is a normalised measure of rice difference. It is small if x i is close to the reference rice, x max and it is largest (unity) when x i = x min. With the roject budget x B, the rice gain in dimensionless unit is adjusted to X i = Table 2: Values of α and β for the reference function (υˆ max = 1.6301 and υˆ min = 1.5567) min max α x B - x (i) x B - x min such that x min = x (1) and if x min > x B, all X i s = 0. An un-normalised rice gain in monetary unit is given by X i = x B - x (i) β (m max = 11) 0.3 5.0 0.7733 0.3489 0.25 5.0 0.8904 0.1581 0.2151 5.0 0.9871 0.0000 0.2 7.5 1.0337 0.3300 0.2 5.395 1.0337 0.0000 (15a) (15b) This is a straightforward relation that equates rice score with rice difference. It is a useful measure for evaluation methods that comare marginal benefit with rice and evaluate rice searately from the other criteria. Although it does not satisfy the condition 0 X 1, it can be included in the generalised model as a secial case in which the rice gain value is equated with the rice difference. Corresondingly from Equation 15a and 15b, the ordered rice gain sequence, X (1), X (2),.., X (j),.., X (m-1), X (m) in descending rice difference, is obtained for calculating their scores, A (i). Thus, this method is a comarison based on ordered absolute values with reference to an artificial zero at x B. It also satisfies the transitivity roerty. PRICE RATIO MODEL SUB-CLASS If X is allowed to assume other non-linear functions of rice, -1 it is ossible to incororate X i = χ i = x i, Karsak s value function, into the generalised rice-scoring model. Although equals to unity, its scoring curve is strictly not linear. If x i = x min+ x i, where x i is a small ositive difference of the i th rice above the minimum rice, x min, then the value function is given by X i = 1 1 x 1- (16) min + x i x x i min x min If normalisation is done, the value function reresents a ratio as follows: X i = x min 1- x i (17) x min + x i x min Both Equations 16 and 17, when substituted into Equation 1b roduce identical effects. Both differ from the rice difference function. The generalised model accets both these two rice ratio functions but Equation 16 can only be used in the normalised class of models because on its own it is not dimensionless. Cost control is imlemented by excluding rices that exceed x B from evaluation. The largest rice does not necessarily have a low value because Equations 16 and 17 work in an ordered ratio scale with natural zero. Thus, a restriction to good discrimination is imosed. From the rice survey, the standard deviation of rice is small, making x i of Equations 16 and 17 small relative to x min. X i is thus closer to unity than to zero. These two rice gain functions satisfy the transitivity roerty but not the referencing roerty required in rosect theory. They reresent a secific sub-class that works on ratio comarison. COMPARISON BETWEEN PRICE DIFFERENCE AND PRICE RATIO MODELS Discrimination and cost control are the bases of comarison between the two model sub-classes. One set of tender rices for low-rice bunching and one set for high-rice bunching are used to illustrate the characteristics of the rice ratio and rice difference models. The two tender sets have the same minimum and maximum rices. Their statistics are tabulated in Tables 3 and 4 for both tenders. The higher rice tender naturally has the higher mean rice. When the large extreme rices are eliminated by cost control, the mean rices are reduced slightly but they remain relatively robust enough to continue to indicate the market norm. The behaviours of their standard deviations and coefficient of skewness are less redictable because they are deendent on the rice distribution after cost control action. A more reliable indication of the direction of change is seen in the reductions in the coefficients of variation after cost control. The tender with low-rice bunching has coefficient of skewness of 1.721 for 8 rices and decreases to 1.324 for 7 rices after cost control. The corresonding values of are 5.169 and 3.573 resectively. For the set with high-rice bunching the coefficient of skewness is 1.170 for 8 rices and increases to 1.274 for 7 rices after cost control. The skew has 24 Journal - The Institution of Engineers, Malaysia (Vol. 66, No. 1, March 2005)

019-026 rice scoring 9/20/05 12:12 PM Page 25 A GENERALISED PRICE-SCORING MODEL FOR TENDER EVALUATION Table 3: Statistics from low-rice bunching tender Tender Prices, Values RM Million Statistics (Cost-Controlled (Low-Price Bunching) Values in Brackets) 5.973 5.713 5.899 6.012 6.386 5.926 5.758 6.965 Mean 6.079 (5.952) Standard Deviation 0.4119 (0.2201) Coefficient of Variation 0.0678 (0.0370) Coefficient of Skewness 1.721 (1.324) Table 4: Statistics from high-rice bunching tender Tender Prices, Values RM Million Statistics (Cost-Controlled (Low-Price Bunching) Values in Brackets) 6.673 Mean 6.515 (6.451) 5.713 6.689 Standard Deviation 0.3998 (0.3846) 6.512 6.788 Coefficient of Variation 0.0614 (0.0596) 6.226 6.465 Coefficient of Skewness -1.170 (-1.274) 6.965 increased for high rice bunching because the extreme (lowest) rice is not eliminated by cost control in this case. The corresonding values of are 0.3942 and 0.3538 resectively. For a set of tender rices, 3 sets of normalised scores and 2 sets of un-normalised scores are calculated using a common reference function, whose coefficients α and β determined earlier were adoted. The rice scores from these 5 models are shown in Table 5 for low-rice bunching and in Table 6 for high-rice bunching. All 5 models show some degree of score bunching corresonding to rice bunching. In Karsak s model, equals to 1(α and β equal to zero) without cost control. Its scores are smaller comared with those of the rice ratio model in the next column. Tables 5 and 6 show that the rice ratio models do not convey significant score differences if the rice differences are small when comared to their absolute values. As illustrated in Table 6, they more effectively rovide information in terms of relative differences but are articularly weak in discriminating rices when there is highrice bunching. Comaratively, the rice difference models are more discriminating because they enhance the scores of the low rices and reduce the scores of the high rices. As a result, they will tend to strengthen the effect of rice when an overall tender evaluation is made together with the other non-rice attributes. By cost control, the over-riced scores of both the rice ratio and the rice difference models are set to zero, effectively eliminating over-riced tenders from consideration. To a small extent, cost control increases the normalised scores because the sum of scores is reduced by the elimination of the over-riced tender. The weakness of the rice ratio model in not being able to score the budget rice to zero is obvious here. Normalisation makes the sum of scores equals to 1 and the individual scores less than 1. An un-normalised model enhances all rice scores by making the score of the lowest rice equal to 1. The larger score magnitude in the un-normalised rice-scoring model has a greater ability to create rice dominance in the evaluation because it always starts with a score of 1 irresective of the number of tender rices. PRICE-SCORING MODELS From the generalised rice-scoring model, a number of ossible models for articular alications are derived as shown in Figure 3. The un-normalised class using the un-normalised rice-difference function with = 1 is erhas the simlest model and has attracted the widest alication, esecially in the 2-enveloe system of tender evaluation in which rice is searated from the non-rice attributes. This model suits direct rice comarison methods. Price difference in the mind of the evaluator is a straight forward and logical means of comaring rices in common monetary units. One would exect that a rocedure by rice difference is attractive because evaluators naturally judge and rank on the basis of rice difference. By assigning = 0.88, it becomes the original value function of rosect theory. Table 5: Comarison among the rice ratio models and the rice difference models in low-rice bunching case Normalised Price Scores Tender Prices, Karsak s Price Ratio Model Price Difference Model RM Million Model (xb = RM6.88million) (xb = RM6.88million) (α=0, β=0) (α=0.8904, β=0.01581) (α=0.8904, β=0.01581) 5.973 0.1267 0.1398 (0.8530) 0.1100 (0.4063) 5.713 0.1325 0.1639 (1.0000) 0.2708 (1.0000) 5.899 0.1283 0.1462 (0.8918) 0.1456 (0.5378) 6.012 0.1259 0.1366 (0.8334) 0.0940 (0.3473) 6.386 0.1185 0.1101 (0.6717) 0.0126 (0.0464) 5.926 0.1278 0.1438 (0.8774) 0.1318 (0.4867) 5.758 0.1315 0.1594 (0.9724) 0.2353 (0.8689) 6.965 0.1087 0.0000 (0.0000) 0.0000 (0.0000) Note: Price scores in bracketed italic are un-normalised scores using Equation 1a. Table 6: Comarison among the rice ratio models and the rice difference models in high-rice bunching case Normalised Price Scores Tender Prices, Karsak s Price Ratio Model Price Difference Model RM Million Model (xb =RM6.88million) (xb =RM6.88million) (α=0, β=0) (α=0.8904, β=0.01581) (α=0.8904, β=0.01581) 6.673 0.1216 0.1410 (0.9465) 0.1247 (0.5423) 5.713 0.1421 0.1490 (1.0000) 0.2300 (1.0000) 6.689 0.1213 0.1409 (0.9457) 0.1212 (0.5271) 6.512 0.1246 0.1423 (0.9547) 0.1529 (0.6648) 6.788 0.1180 0.1395 (0.9364) 0.0242 (0.1050) 6.226 0.1303 0.1446 (0.9700) 0.1874 (0.8147) 6.465 0.1255 0.1426 (0.9572) 0.1595 (0.6936) 6.965 0.1165 0.0000 (0.0000) 0.0000 (0.0000) Note: Price scores in bracketed italic are un-normalised scores using Equation 1a. Journal - The Institution of Engineers, Malaysia (Vol. 66, No. 1, March 2005) 25

019-026 rice scoring 9/20/05 12:12 PM Page 26 THUM PENG CHEW Figure 3: Particular rice-scoring models from the generalised model The un-normalised class using normalised rice gain functions are suitable for integrated evaluation of both rice and other nonrice attributes in a comuter rogram. Because of their larger score values comared with those of the normalised class, they have rice dominant tendencies. The rice-difference function can be adoted if dominance by a single rice is desired. On the other hand, the rice-ratio function will tend to move the scores away from singlerice dominance. When rice bunching occurs, collective dominance exists in the un-normalised function. The normalised class using normalised rice gain functions have different characteristics comared with those of the un-normalised class of models. They roduce an effect that tends not to emhasise on rice nor discriminate between rices thus, allow the other attributes to lay more influential roles in the evaluation. Their smaller scores are controlled by the number of tenders evaluated. The more tenders, the smaller the scores and the lesser the ability to dominate. The choice of any of the above models should be made by matching their characteristics with the evaluation objective. If the lowest rice is to dominate, then an un-normalised class of ricedifference scoring model is aroriate. If rice dominance is not intended, then a normalised class of rice-ratio scoring model is more effective. Both the un-normalised and normalised classes that use a dimensionless rice gain function based on either rice difference or rice ratio meets the requirement of dimensionless attribute scores in Thum s fuzzy tender evaluation model [5]. CONCLUSION A generalised rice-scoring model is develoed by incororating two rice gain functions, one based on rice difference on an interval scale and the other on rice ratio on a ratio scale. It attemts to revent cost overrun by eliminating over-riced tenders by comarison with the roject budget. The behaviour of evaluators and decision makers is modelled in the reference factor which is a function of the tender articiation rate and its rice distribution skew. In the resence of unusual influences or when fair cometition is threatened, the model will attemt to counteract the negative strategies adoted by tenderers, failing which a reassignment otion built into the rogram allows the evaluator to make judgment outside the model. The generalised rice-scoring model roduces two generic rice model classes. The un-normalised class tend to roduce rice dominant effects while the normalised class tends to allow non-rice attributes to have more influence in the overall evaluation. These two classes roduce their own rice ratio sub-class using a rice gain derived from the ratio of the minimum tender rice to a rice, and their rice difference sub-class derived from the rice gain measure of a rice with reference to the roject budget. A survey of tender rices rovides data for estimating the tender articiation rate and the range of variations of the rice distribution s coefficient of skewness to be used in the reference function. By aroriate choice of rice gain function and of the reference function, 5 articular rice-scoring models are derived. Price difference models are found to be better at discriminating rices than rice ratio models. Price ratio models are more suitable for matching tender evaluation objectives that do not emhasise rice. The combination of the value functions and the rice gain functions with built-in cost control roduces a diverse number of models for secific alications in a tender evaluation comuter rogram. Further investigation is required to test them with the nonrice attributes in order to define their alication in a tender evaluation rocedure. The generalised rice-scoring model hence, rovides a means of studying the existing methods of tender evaluation by comuter simulation. For examle, comarisons can be made between an integrated aroach in which rice and other attributes are combined in a 1-stage evaluation rocedure, and one in which rice is evaluated after the evaluation of other non-rice attributes is comleted in a 2-stage evaluation rocedure. REFERENCES [1] E. E. Karsak, Fuzzy MCDM Procedure for Evaluating Flexible Manufacturing System Alternatives, Proceedings of the 2000 IEEE Engineering Management Society,. 93-98, Albuquerque, New Mexico, USA, 2000. [2] D. Hahneman and A. Tversky, Prosect theory: An Analysis of Decision Under Risk, Econometrica Vol.74,.263-291, 1979. [3] A. Tversky and D. 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Hammond, Judgment and Decision Making: An Interdiscilinary Reader (Book) 2nd Edition, Cambridge University Press, 1999. [10] C. Starmer, Develoment in Non-Exected Utility Theory: The Hunt for a Descritive Theory of Choice under Risk, Journal of Economic Literature Vol. XXXVIII,.332-382, June 2000. [11] P. Bajari and L. Ye, Cometition Versus Collusion in Procurement Auctions: Identification and Testing, Stanford University Working Paer, 2001. [12] P. Bajari and G. Summers, Detecting Collusion in Procurement Auctions, Antitrust Law Journal Vol. 70,. 143-170, 2002. [13] L. M. Froeb and M. Shor, Auctions, Evidence and Antitrust in The Use of Econometrics In Antitrust, edited by John Harkrider, American Bar Association, 2003. 26 Journal - The Institution of Engineers, Malaysia (Vol. 66, No. 1, March 2005)