Jorge J Maté-García

José M Rodríguez-Fernández

Methods: Single Equation Models < C -Mathematical and Quantitative Methods, O32 -Management of Technological Innovation and R&D < O3 -Technological Change|Research and Development < O -Economic Development, Technological Change, and Growth, L60

Keywords: Productivity, R&D expenditures, Firm panel data, Spanish manufacturing firms Productivity, R&D expenditures, firm panel data, Spanish manufacturing firms. JEL Classification: C23, D24, L60, O32

This paper analyses the relationship between productivity growth and R&D investments of Spanish manufacturing firms during the 1990s. The theoretical model is a version of the Cobb-Douglas production function in its growth rate form. The purpose is to estimate the rate of return to R&D expenditures. The econometric specification is a distributed lag model. The estimation applies the GMM method. The main empirical finding is that a positive and significant role is played by R&D expenditures on productivity growth. The rate of return to R&D expenditures is 26.598 per cent.

These expenditures contribute to technological improvements and the latter expedite the growth of the economy. So the investment in R&D is considered to have a relevant impact on productivity growth.

The empirical analysis of the relationship between R&D and productivity is made usually through the estimation of a production function in which the technological capital is included as an explanatory variable. 2 This input makes easy into firms the existence of new ideas, which could end up by being to the advantage of a major productivity. Papers that use this approach have a common characteristic. Starting out from a production function, they determine the output elasticity with regard to the technological capital. The underlying problem is that this kind of analyses needs data about research capital stock, which are not available. Thus, it is necessary to make an estimation of the technological capital stock. In order to do so, they usually use the perpetual inventory method. That is, the capital of each period is calculated from the capital of the previous period minus the depreciation and plus the investment in this capital in the period. This process requires to make a hypothesis about the value of capital depreciation rate and to use an initial value of this capital. 3These problems could be avoided if what we estimate is some transformation of the production function that only needs to know the R&D expenditures in each F o r P e e r R e v i e w period, that is, a flow variable. This is the aim of this paper. Instead of estimating production functions in which research capital stock is an additional factor, we relate productivity growth directly to R&D investment over a sample of Spanish manufacturing firms during the 1990s. The goal of the estimation will be to determine the rate of return to the mentioned capital, instead of its elasticity. It implies a certain degree of newness in relation to previous researches made about the Spanish manufacturing sector. According to our knowledge, there are no studies made with Spanish firm-level data in this line of research. This paper is organised as follows. Section 2 describes previous researches about productivity growth and technological capital. Section 3 describes the used theoretical model. A description of data, variables, and empirical methodology applied in the econometric analysis is presented in section 4. Section 5 contains the estimation results of the Spanish manufacturing industry. The last Section includes the summary and the most important conclusions of the research.

Background and previous literature.

According to the preceding section, there are two approaches to deal with the relationship between productivity and technological capital. On the one hand, we can estimate the output elasticity with regard to the technological capital; on the other hand, a rate of return to R&D capital could be estimated. Till now, the former is the approach applied in the Spanish papers concerning this topic. Thus, [START_REF] Lafuente | Productividad, capital tecnológico e investigación en la economía española[END_REF] [START_REF] Fluviá | Capital tecnológico y externalidades: un análisis de panel[END_REF] and [START_REF] Grandón | Capital tecnológico e incrementos de productividad en la industria española 1975-1981[END_REF] However, we use the second mentioned approach. That is, this paper studies the relationship between labour productivity growth and investment in R&D capital for the Spanish manufacturing industry in order to estimate the rate of return to R&D expenditures. We use microdata from the Business Strategy Survey (ESEE). Using these panel data allows having a large number of observations, and doing a tracking of the behaviour of individual firms for several periods. The individual firm is the relevant unit of decision relating to R&D investment and, therefore, the dynamic aspects are better studied to this micro level. Everything There are many studies in other countries that try to estimate the rate of return to R&D expenditures by using firm-level data. In what follows, we will highlight some of the most relevant ones. [START_REF] Griliches | Comparing productivity growth: An exploration of French and U.S. industrial and firm data[END_REF] analyse the influence of R&D expenditures on productivity by using firm-level data for the United States and France between 1973 and 1978. They only consider firms that realise R&D investment in intensive R&D capital industries. For their part, [START_REF] Clark | Productivity growth and R&D at the business level: Results of the PIMS Data Base[END_REF] offer results about productivity growth and R&D expenditures for the period 1970-1980. Their sample includes 924 U.S. manufacturing firms. In the work of [START_REF] Lichtenberg | The impact of R&D investment on productivity: New evidence using linked R&D-LRD data[END_REF], panel data were used to estimate the relationship between R&D and productivity growth in the U.S. manufacturing industry from 1972 to 1985. [START_REF] Bessen | Adoption costs and the rate of return to research and development[END_REF] uses a sample of 471 U.S. firms between 1983 and 1989 in order to obtain results about the relationship between productivity and R&D expenditures, although the principal aim of his research is measuring the costs of adopting new technology associated with the firm's R&D expenditures.

In what related to Japan, [START_REF] Odagiri | The impact of R&D on productivity increase in Japanese manufacturing companies[END_REF] estimate the impact of R&D expenditures on productivity growth rate by using firm-level data in two different periods : 1966: to 1973: and 1974: to 1982[START_REF] Fecher | Effects directs et indirects de la R&D sur la productivité: une analyse de l´industrie manufacturière belge[END_REF] [START_REF] Wakelin | Productivity growth and R&D expenditure in UK manufacturing firms[END_REF] analyses the relationship between productivity growth and R&D intensity in the United Kingdom by using information of 170 firms between 1988 and 1996. Finally, [START_REF] Parisi | Productivity, innovation creation and absorption, and R&D: micro evidence for Italy[END_REF] present an empirical evidence of the effects of both productive process innovation and product innovation on productivity in Italy. Likewise, they study the effect of R&D investment over the probability of introducing innovations. The data are extracted, initially, from 941 firms between 1992 and 1997. After carrying out a selection in the sample to eliminate firms with missing values, inconsistencies or extreme values for the variables, authors work with a panel data where are over-represented firms operating in hightechnological industries.

The estimates of the rate of return offered by these works are different. It should be noted that the rate of return depends on both, the unit values of the different variables included in the estimations and the data sources used. Later, we will compare the results achieved here for the Spanish case with those of other countries.

The theoretical model.

The empirical analysis of the relationship between productivity and R&D expenditures is mainly based on [START_REF] Griliches | Issues in assessing the contribution of research and development to productivity growth[END_REF][START_REF] Griliches | Productivity puzzles and R&D: Another non-explanation[END_REF] In this paper, the starting point in building our model is a Cobb-Douglas production function with three productive factors:

Q it = A e t C it L it K it e µi e it [1]
where the subscripts i and t denote the firm and the time, respectively; Q is the output; L represents the labour factor; C measures the physical capital stock; K measures the research capital stock; A is a constant; , , and are output elasticities with regard to physical capital, labour and R&D capital, respectively;

is the rate of disembodied technical change (exogenous changes in the productive technology along the time which cause variations in the productivity growth rate that are common to all firms); µ represents a firm-specific unobserved effect, which is constant over time; is a random error term.

The use of a Cobb-Douglas function separable in R&D factor allows estimating it as a linear model in first differences. We take equation [1], we make a logarithmic transformation and apply first differences, and then we have the following equations: If we subtract labour logarithm from both sides of expression [1a], we will obtain the following equation:

q it = a + t + c it + l it + k it + µ i + it [1a] q it = + c it + l it + k it + It [1b]
q it -l it = a + t + c it + l it + k it -l it + µ i + it
By replacing this value of by 1 --and making some operations we get the following result:

q it -l it = a + t + c it + l it -l it -l it + k it -l it + µ i + it
Re-ordering terms:

F o r P e e r R e v i e w q it -l it = a + t + (c -l) it -l it + k it + µ i + it [2a]
By applying first differences to this expression we get:

(q -l) it = + (c -l) it + k it -l it + it [2b]
We appointed out earlier that is the output elasticity with regard to R&D capital, which could be estimated from the following expression:

= ( Q/ K) it (K/Q) it [3a]
On the other hand, the growth rate of this kind of capital could be estimated by the expression:

k it = ( K/K) it [3b]
If we call to the marginal productivity of research capital -that is, =( Q/ K) it -, we obtain from the expressions [3a] and [3b], the following result:

k it = ( Q/ K) it (K/Q) it ( K/K) it = (R/Q) it [4]
where R it denotes expenditures on R&D of the firm i in the year t net of depreciation of the previously accumulated R&D capital. Therefore, R it is a proxy of the net investment in R&D capital.

(q-l) it = + (c -l) it -l it + (R/Q) it + it [2c]
In this expression it = it ; (q-l) it is the labour productivity growth rate; (c -l) it is the rate of growth of the capital-labour ratio; l it is the rate of growth of the employment; and the ratio (R/Q) it is the R&D intensity or the technological effort.

In the estimation of the expression [2c] we only require data of R&D expenditures, instead of R&D capital stock.

If the market is characterized by competitive conditions, matches with the rate of return to R&D expenditures 4 . It is important, however, to apply some controls on this rate.

First, this rate is the gross rate of return. The net rate is obtained removing the (unknown) depreciation rate of R&D capital. 5 Therefore, although the problem of measuring the K variable is avoided, some difficulties arise to determine the correct value of the depreciation rate. However, [START_REF] Mairesse | R&D and productivity: A survey of econometric studies at the firm level, Working Paper 3666[END_REF] show that, if the R&D capital depreciation rate is small with respect to R&D expenditures growth rate, then -estimates are enough close to their real value, even though without differentiating between net and gross rate.

4 See [START_REF] Clark | Productivity growth and R&D at the business level: Results of the PIMS Data Base[END_REF] and [START_REF] Griliches | R&D and productivity growth at the industry level: Is there still a relationship?[END_REF]. 5 See [START_REF] Hall | Exploring the relationship between R&D and productivity in French manufacturing firms[END_REF], for example. They use a depreciation rate of 15 per cent. the disagreement between the private profits and social profits of the activity. The whole economy will be better off with the positive externalities generated by R&D investment of particular firms (spillover effects) because the knowledge can be transferred and utilised by others firms. For instance, it can not be prevented acquired qualifications by research personnel of some firms from passing on to others firms when there is labour mobility between companies. If this spillover effect is taken into account in the estimations, it would be possible, at first, to estimate separately the private and the social rate of return. If this effect is not considered separately, the estimates based on data of individual firms could be reflecting-to an uncertain degree-both the social and the private rate of return.7

Some interesting overviews and papers about spillover effects are those done by [START_REF] Griliches | The search for R&D spillovers[END_REF], [START_REF] Nadiri | Innovation and technological spillovers[END_REF], [START_REF] Mairesse | R&D productivity: A survey of the econometric literature[END_REF], [START_REF] Aiello | R&D spillovers and productivity growth: Evidence from Italian manufacturing microdata[END_REF] and [START_REF] Chen | Technological knowledge, spillover and productivity: evidence from Taiwanese firm level panel data[END_REF]. These effects have also been studied in Spain by [START_REF] Fluviá | Capital tecnológico y externalidades: un análisis de panel[END_REF], López Pueyo and Sanaú Villarroya (1998) and [START_REF] Beneito | R&D productivity and spillovers at the firm level: Evidence from Spanish panel data[END_REF].

6 See, for example, [START_REF] Bessen | Adoption costs and the rate of return to research and development[END_REF]. 7 See [START_REF] Mairesse | R&D and productivity: A survey of econometric studies at the firm level, Working Paper 3666[END_REF]. variable is the result of the product of two variables: the average total number of employees and the number of worked hours per full-time employee and year. The average total number of employees is calculated as the sum of the following items: full-time em ployees, ½ of the part -time employees (both items at December 31), and the average number of temporary employees during the year.

The labour variable includes all kind of employees (those who work in R&D activities or other activities). This implies a double-counting problem, which also affects other variables, as argued by [START_REF] Mairesse | Estimating the productivity of research and development: An exploration of GMM methods using data on French and United States manufacturing firms[END_REF]. In accordance with these researchers, value added, physical capital, and labour should be corrected by taking into account the cost of R&D materials, physical capital used in R&D laboratories, and personnel dedicated to R&D activities, respectively, since these inputs are already included in R&D expenditures. Nonetheless, except for the labour variable, there is a lack of information to carry out these adjustments. The ESEE offers details about the number of employees dedicated to R&D. Thus, it might be possible, at first, to deduct these R&D employees from the total number of employees. However, firms offer information about the R&D labour variable once each four years. As a consequence, we decided to use the variable of the average total number of employees in the estimate of the number of labour hours.

Despite of not applying the adjustment for the double-counting, the obtained estimates for the rate of return to R&D expenditures could be considered fair. 10 10 However, the value of the rate of return is somewhat lower than that obtained by the corrected values. Nonetheless, the difference in the estimates for adjusted and non-adjusted data is practically cancelled if the corrections are only made for the labour variable but not for the rest of variables with a similar problem. For more information, see [START_REF] Hall | Exploring the relationship between R&D and productivity in French manufacturing firms[END_REF] or Smith el al. (2004). On the part of R/Q (the technological effort or the R&D intensity), it is the most interesting explanatory variable for the aim of this paper. The R variable is defined as the total R&D expenditures during the year in real terms. 12 The Q variable is measured as indicated previously, and it is the firm's real value added.

Finally, other variables that do not appear explicitly in equation [2c] were also included in the econometric estimation. On the one hand, the specification represented by such equation is mainly based on a long-term perspective by considering R&D expenditures and disembodied technical change, among others, as productivity determinants. However, it is important to control also for shortterm phenomena associated with demand fluctuations. Firms partially face temporal demand fluctuations by changing the intensity of using their physical capital stock. This phenomenon involves changes in the productivity of the firms.

One way to include these factors in the model is to add as an explanatory variable 11 The use of the book value of the equipment in real terms as a proxy of the physical capital stock is frequent in the economic literature. See, for example, [START_REF] Clark | Productivity growth and R&D at the business level: Results of the PIMS Data Base[END_REF], [START_REF] Hall | Exploring the relationship between R&D and productivity in French manufacturing firms[END_REF], [START_REF] Mairesse | Estimating the productivity of research and development: An exploration of GMM methods using data on French and United States manufacturing firms[END_REF], [START_REF] Beneito | R&D productivity and spillovers at the firm level: Evidence from Spanish panel data[END_REF], [START_REF] Wakelin | Productivity growth and R&D expenditure in UK manufacturing firms[END_REF] and [START_REF] Parisi | Productivity, innovation creation and absorption, and R&D: micro evidence for Italy[END_REF]. 12 The firms' total R&D expenditures refer to those made by both private funds and public sector subsidies. They are deflated by the aggregate Industrial Price Index provided by the INE. the rate of growth of capacity utilisation ( lnCU = cu). 13 The ESEE provides information about the capacity utilisation (the CU variable). 14 On the other hand, for the estimations that we make by using equation [2c] it must be taken into account that changes in the firms' productivity might depend on the specific characteristics of the industry to which firms belong. Therefore, dummy variables are included in the model in order to reflect the industry to which each firm belongs. In this way the bias due to sector-specific unobservable heterogeneity is reduced. The 18 industries used in the ESEE are summarised in Appendix, as we indicated previously.

The theoretical model presented in equation [2c] must be properly specified in econometric terms in order to estimate it. The possible individual effects were already eliminated because this theoretical model is deduced by applying first differences. Thus, the most important point to take into account is that the impact of investment in R&D over the increase of labour productivity is not used to being immediate one. Meanwhile, such impact, once produced, might not be limited to only one period. As a consequence, these effects will be distributed over time.

The causes of all this might be several. On the one hand, a determined R&D project might have a life of more than a year; thus, their final effects will not be appreciated till the project has reached its end. Besides, in initial steps of the project, the investigation staff will produce ideas, that they do not become immediately into bigger production. On the other hand, even if the project is 13 See [START_REF] Clark | Productivity growth and R&D at the business level: Results of the PIMS Data Base[END_REF]. totally developed, certain time is needed to take the decision of applying it in the productive process; probably, the innovation will be implanted in a gradual way in order not to change traumatically the firm's cost structure. Add to this the learning process at the time of applying in practice the introduced innovation, which surely will be completed in specific details little by little. 15 Considering the behavior of R/Q variable and the arguments mentioned in previous paragraphs, we have decided to specify equation [2c] econometrically through a distributed lag model. However, the difficulty that entails determining the exact structure of lags with which the variable R&D expenditure operates over the productivity growth rate must be emphasized. To well understand this structure, many (unavailable) data about R/Q variable are needed along time. In addition, the value of the mentioned variable must be independent between several periods, which is not used to being the case: generally, R&D expenditures in a determined period are correlated with those of preceding periods. 16 Based on all these remarks, in this paper we use the following econometric specification for equation [2c]:

(q-l) it = + (c -l) it -l it + 1 (R/Q) i,t-1 + 2 (R/Q) i,t-2 +… ... + n (R/Q) i,t-n + cu it + industry effect + it [2d]
14 It is defined in the ESEE as average percentage of utilisation-during the year-of the firm's standard capacity. 15 Econometric arguments also support the introduction of lags to the technological effort variable. On the one hand, due to the correlation that might exist between the actual investment in R&D and the value added of the period. On the other hand, the presence of the value added in both sides of the equation might generate biases in the coefficient of the technological effort variable. See [START_REF] Mairesse | Estimating the productivity of research and development: An exploration of GMM methods using data on French and United States manufacturing firms[END_REF]. 16 See, for example, [START_REF] Griliches | R&D and productivity growth at the industry level: Is there still a relationship?[END_REF] and [START_REF] Rouvinen | R&D-productivity dynamics: causality, lags, and 'dry holes[END_REF]. In this equation all the comments made in the previous paragraphs are already taken into account: we have written the short-term phenomena ( cu); R/Q variable is lagged for the periods t-1,t-2,...,t-n; and we have added the industry effect defined as 2 IC 2 + 3 IC 3 + ...+ 18 IC 18 ; where 's are parameters and IC's represent industry dummy variables.

In order to estimate the mentioned [2d] model, generalized method of moments (GMM) is applied. This is a robust approach to heteroskedasticity across firms and to correlation of disturbances within firms over time; thus, GMM can be efficient without making very restrictive assumptions. 17

The used sample in the model estimation is extracted from the ESEE panel data between 1993 and 1999. Estimations were done including all firms that provide information for these variables. 18 The firms that fulfil this requirement are 1,312, an unbalanced data panel with a total of 8,636 observations. The panel is unbalanced because of entry and exit of firms. A filter for mergers was also applied. 19 Most existing models that concern the relationship between productivity and investment in technology have used data for only those firms that make this kind of expenditure at least for a few years. However, from an econometric perspective, it seems reasonable to include also the data offered by remainder 17 See, for example, [START_REF] Mairesse | Estimating the productivity of research and development: An exploration of GMM methods using data on French and United States manufacturing firms[END_REF] and Wooldridge (2002, especially Chs. 8, 11, and 14). 18 Significant problems relative to outliers were not found. 19 A balanced panel would contain an econometrically insufficient number of firms. firms of the sample (those firms with annual R&D investment that always equals zero). These firms act as a control group and they allow having more complete information as they pick up the changes of labour productivity if no technological effort is made.

Empirical results.

In this section we present the results of the estimated rate of return to R&D expenditures. The previous equation [2d] is considered as the starting point of the analysis.

The obtained empirical results are presented in Table 1. Only the results of the model with one lag for R/Q variable are presented, because other lags for this variable are statistically insignificant. 20 The usual tests (C statistic or differencein-Sargan statistic) allow for the acceptance of the exogeneity hypothesis for the variables (c-l), l, and uc, but not for the R/Q variable. As a consequence, this latter one has been instrumented. The instruments used in the econometric estimations are enumerated in the mentioned above Table. 21 The overidentifying restrictions are tested by Hansen J statistic, which is consistent in the presence of heteroskedasticity. The results of this test do not reject the validity of the instruments, accepting by that the null hypothesis with p-value of 0.379. 20 This is similar to the published results of [START_REF] Clark | Productivity growth and R&D at the business level: Results of the PIMS Data Base[END_REF], [START_REF] Hall | Exploring the relationship between R&D and productivity in French manufacturing firms[END_REF], and [START_REF] Bessen | Adoption costs and the rate of return to research and development[END_REF], for example. The empirical results with other lags are available from the authors by request. 21 Several instruments were analysed and those with the best econometric results were chosen. They are similar to those applied in other studies, which also use three or four lags for some explanatory variables of the model. For instance, see [START_REF] Mairesse | Estimating the productivity of research and development: An exploration of GMM methods using data on French and United States manufacturing firms[END_REF]. and regressors are jointly significant (the null hypothesis is accepted: without the constant term, 2 (4)=43.10, p-value=0.0000; with the constant term, 2 (5)=45.49, p-value=0.0000).

The coefficient of R&D intensity variable is statistically significant and has a positive sign. That means that the technological effort has a positive effect over productivity. The other variables used in the estimation are significant from the statistical viewpoint for some of the usual levels.

In accordance with the estimated econometric model and by taking into account what we pointed out in the previous section, we observe that the rate of return to R&D investment ( ) is situated in 0.26598. Thus, for each additional monetary unit invested in R&D, the output would increase by 1.26598 monetary units, ceteris paribus.

With regard to the capital-labour ratio, its influence over productivity growth is positive (a positive value of the coefficient in the equation [2d]). The coefficient of capacity utilisation variable ( in equation [2d]) also has positive sign, which means that productivity growth rate is also influenced by factors associated with temporary demand fluctuations that firms face. The variable number of labour hours has a coefficient ( in equation [2d]) that is significantly 22 The inclusion of the capacity utilisation variable might be capturing the different characteristics of each industry and, as a consequence, reducing the significance of dummy variables. The model was also estimated by imposing the assumption of constant returns to scale for capital and labour (zero value for the coefficient ). This constraint does not very substantially change the empirical results, because a similar return rate is obtained (0.26120). In consequence, at the Spanish manufacturing industry the existence of constant returns to scale can be admitted, even though also a light bias toward diminishing returns to scale can be admitted. However, empirical results without the constraint of constant returns to scale were chosen to be presented because they have better econometric properties.

In short, productivity growth in the Spanish manufacturing industry could be explained by innovative effort of firms, capacity utilisation, and capital-labour ratio, in addition to other non-modeled factors.

These empirical results for the Spanish firms could be compared with similar estimates made in other countries. Nonetheless, it must be clarified that, in the economic literature, the estimated values of the rate of return to R&D expenditures ( ) change in a wide interval, depending on the used sample, on how the variables have been defined and on the econometric specification. The estimate of the rate of return to R&D expenditures obtained in this paper is situated in most usual interval. In most empirical researches existing till now, the estimates of move in the interval 0.2-0.4, where the values between 0.2 and 0.3 are specially frequent. This is the case of [START_REF] Griliches | Comparing productivity growth: An exploration of French and U.S. industrial and firm data[END_REF] for the United States and France, as well as [START_REF] Griliches | Productivity and R&D at the firm level[END_REF], [START_REF] Griliches | Productivity, R&D and basic research at the firm level in the 1970s[END_REF], [START_REF] Jaffe | Demand and supply influences in R&D intensity and productivity growth[END_REF], [START_REF] Jones | Measuring the social return to R&D[END_REF][START_REF] Jones | Measuring the social return to R&D[END_REF][START_REF] Bessen | Adoption costs and the rate of return to research and development[END_REF]; these five later papers refer to the United States. The same holds for the estimates of [START_REF] Odagiri | R&D expenditures, royalty payments and sales growth in Japanese manufacturing corporations[END_REF] and [START_REF] Sassenou | Recherche-développement et productivité dans les entreprises japonaises: une étude économétrique sur données de panel[END_REF] for Japan, [START_REF] Griliches | R&D and productivity growth: Comparing Japanese and U.S. manufacturing firms[END_REF] for the United States and Japan, and [START_REF] Wakelin | Productivity growth and R&D expenditure in UK manufacturing firms[END_REF] for the United Kingdom.

Nonetheless, we must notice that these works have used different econometric models and estimation techniques, in addition to several forms of defining the relevant variables. For instance, rates of return below 0.2 appear in studies such as those of [START_REF] Clark | Productivity growth and R&D at the business level: Results of the PIMS Data Base[END_REF], [START_REF] Schankerman | A test of static equilibrium models and rates of return to quasi-fixed factors with an application to the Bell System[END_REF], [START_REF] Bernstein | Rates of return on physical and R&D capital and structure of the production process: Cross section and time series evidence[END_REF],

and [START_REF] Lichtenberg | The impact of R&D investment on productivity: New evidence using linked R&D-LRD data[END_REF]; all of them have used U.S. data. Similar thing holds for papers by [START_REF] Odagiri | The impact of R&D on productivity increase in Japanese manufacturing companies[END_REF] for Japan and [START_REF] Hall | Exploring the relationship between R&D and productivity in French manufacturing firms[END_REF] for France.

Even by comparing values obtained by models whose explanatory variables are defined by a similar way, we still observe differences between the estimated values. This fact could be due, among others, to several factors. The first factor is that each research concerns different territorial and temporal ambit. Second, the quality of the data used is different. Third, each researcher considers a different functional specification for the empirical estimation (the lag structure for R/Q variable is different). Forth, some analyses use data of the privately financed R&D expenditures, while others include also the R&D investment financed by the public sector. Finally, some studies include the role of spillover effect in productivity growth.

Conclusions.

This paper presents a theoretical model that associates labour productivity growth with R&D expenditures. In order to estimate the econometric model we used the generalized method of moments (GMM). The empirical results were showed to be consistent to the theoretical perspectives. They indicated that R&D investment by Spanish firms has a positive and statistically significant effect on labour productivity rate of growth, with one lag (the additional lags do not seem to be statistically significant). More specifically, the rate of return of this kind of investment is about 26.598 per cent. En general, this result is in the line of the rate estimated by studies made for other countries.

On the other hand, a positive relationship between labour productivity growth and capital-labour ratio was found. Similar relationship was also found between this productivity growth and capacity utilisation by firms. This indicates that productivity changes are related to short-term factors associated with temporary demand fluctuations that firms face. Lastly, we detected a production function for the Spanish industrial firms with diminishing returns to scale on capital and labour, which is in accordance with results of previous researches. However, the relationship between productivity growth and technological effort does not very substantially change when the constraint of constant returns to scale is imposed.

Of course, the model estimation might be affected by econometric limitations. [START_REF] Griliches | Production functions: The economic search for identification[END_REF] pointed out that estimations of production functions by using micro data show some problems; these problems also appear when a transformed production function is estimated. 25 As a consequence, the empirical results achieved in this paper, even though they are enough reasonable, they must be interpreted with some caution. But, in spite of this, studies of this kind are useful.

Finally, it is important to draw the attention to the weakness of R&D investment in the Spanish firms. At the first glance, the rate of return in the Spanish manufacturing industry seems to be high. This rate of return should enhance the Spanish investment in technological capital. However, in the practice it is shown that this does not occur. The high risk associated with R&D projects and the difficulty of appropriating exclusively all profits derived from innovation can withdraw the firms (specially the small and medium-sized firms) from carrying out this kind of activities, in spite of the high-expected return. 26 Moreover, the Spanish business structure is very focused in the services sector that does not require large efforts in R&D; also the Spanish firms might probably face special problems to finance their R&D investments, mainly because firms 25 See [START_REF] Mairesse | R&D and productivity: A survey of econometric studies at the firm level, Working Paper 3666[END_REF] and [START_REF] Hall | Exploring the relationship between R&D and productivity in French manufacturing firms[END_REF]. 26 See [START_REF] Nadiri | Innovation and technological spillovers[END_REF]. At this moment, in which the Spanish economic conditions are relatively stable, a step forward could be taken in order to solve the mentioned problems. Generalized method of moments (GMM). Instruments: (c-l) t , l t , cu t, (c-l) t-1 , l t-1, (R/Q) t-2 , (R/Q) t-3, (R/Q) t-4,, cu t-1 . Heteroskedasticity-autocorrelation robust standard errors.

  centre their attention on the analysis of R&D expenditures.

  , using panel data from a survey of large Spanish firms (Encuesta de Grandes Empresas Españolas, published by the Ministry of Industry and Energy) for the period 1975-1981, offer evidence on R&D elasticity. García et al. (1998) estimate a production function using a panel from the Business Strategy Survey (Encuesta sobre Estrategias Empresariales, ESEE) provided by Fundación SEPI. Their main goal is the estimation of the direct elasticity of employment with respect to innovation. López Pueyo and Sanaú Villarroya (1998), using panel data of thirteen industrial sectors for the years 1986-1992, offer own R&D elasticities. Beneito (2001), using panel data from the ESEE, estimates a production function and analyses the effect of R&D capital on productivity, for the period 1990-1996. Finally, López Pueyo and Sanaú Villarroya (2001) estimate production functions for ten countries (nine belonging to the EU and the USA) using panel data for the period 1982-1993. They obtain the own R&D elasticities in each country, taking into account the technological externalities coming from other countries.

  if compared with other publications that use industry-level data.

  analyses the influence of R&D expenditures over productivity by using individual data of Belgian firms between 1981 and 1983. For the case of France, Hall and Mairesse have updated some results of earlier works that study the relationship between productivity and R&D in the French economy. The sample period extends from 1980 to 1987 and it includes information of 351 individual firms.

  as an additional production factor, together with usual productive factors.

  letter symbol denote the logarithm of the corresponding variable and represents the first difference of the pertinent variable. Firm effects, µ, are eliminated when first differences are applied.The main obstacle of an estimation of this kind is the need to have a proper measurement of R&D capital stock. In order to overcome this obstacle, some transformations could be made to the Cobb-Douglas function. Let us assume that in this function the representative parameter of returns to scale (only considered for the standard inputs: physical capital and labour) it differs from the unitary value in a quantity equal to . That is, = 1--. If the coefficient takes a zero value in the estimation of the empirical model, then it indicates a production function with constant returns on capital and labour.

  Taking into account [4], the expression [2b] could be rewritten by the following way:

  (c-l) of the equation[2c] represents the rate of growth of the capital-labour ratio. In order to measure the physical capital stock in real terms (C), we used the gross book value of tangible fixed assets adjusted by the deflator for equipment. 11 This deflator is extracted from the Industrial Price Indexes provided by the INE. The labour variable is measured as indicated in the previous paragraph (average total number of labour hours).

  known tests, industry dummy variables are jointly insignificant (the null hypothesis is accepted: 2 (17)=14.67, p-value=0.6192);22

  and with negative sign. This indicates production function with diminishing returns on capital and labour.23

  other studies the estimates of the rate of return to R&D investment are positioned in values somewhat lower than those mentioned above.

  problems and because government support to the technological innovation is low.The inadequate stock of R&D capital in Spain is burdening the labour productivity growth. The lack of investment might cause Spanish firms a loss of competitiveness and a slow growth of potentiality of the economy in a context of opening markets where goods with high technological content are exchanged.Therefore, it is worthwhile to stand up and to signal the need for applying some measurements of industrial policy that solve the Spanish economy deficiencies in this scope. In this sense, the company-financed R&D investments should be properly backed up by the public sector because, in Spain, the share of the public sector in R&D activities is majority, in contrast to what occurs in other developed countries. Also the activity of Technology Transference Centres should be enhanced, among other measurements, in order to improve the relationship between firms and universities. Likewise, it would be useful to increase the resources channelled toward research made by government agencies. This one should be accompanied by controls in the allocation of funds and appropriate mechanisms for evaluation of results.

  estimate the own R&D elasticity using a time series of aggregate data for the period 1966-1981. They calculate the stock of

	research capital.
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  Second, some researchers 6 argue that the costs of adoptin g a new technology must be taken into account in order to obtain the true R&D investment rate of return. If each monetary unit spent in R&D requires, for example, another additional monetary unit to implement the technology, then the investment' return equals only half of what could be estimated.Finally, it is important to differentiate between private and social rate of return. This is because the incomplete appropriation of the research effects causes

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Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

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  The estimation of equation [2c], which allows the approximation of the value of the technological capital rate of return, , is the main objective of this paper.

	Submitted Manuscript
	Through this specification, which does not take into account the spillover effect, a
	gross rate of return is obtained and the costs of adopting a new technology are not
	considered.
	4. Data sources, variable definitions, and empirical methodology.
	F o r F o r
	P P
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	The denominator of
	this productivity variable is the labour variable (L), defined as the number of
	labour hours that, in average, the firm had during the corresponding year. This
	13 14

Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK The estimation of the proposed equation will be done using the data included in the Business Strategy Survey (ESEE). This survey includes a sample of firms' panel, which is considered representative to the Spanish manufacturing industry. The firms are distributed into 18 industries or branches (energy and mining related activities are excluded). 8 Firms included in the ESEE are those with 10 or more employees. In what follows, we describe the variables used in the econometric specification of the model [2c].

The dependent variable is the rate of growth of labour productivity. The numerator of this productivity variable is the real value added, which is used as a form of measuring the output (Q) 9 . The value added has been deflated using Industrial Price Indexes at two-digit level, which are published by the Spanish Statistics Institute (Instituto Nacional de Estadística, INE).

Page 17 of 32 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

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  22estimates of the rate of return, but not necessarily the real value of the mentioned rate. In this way, the estimates are different according to how the variable of technological effort is defined, because R/Q ratio may present in its denominator the value added or the value of sales. There are also some differences depending on, first, if productivity is referred to total factor productivity or labour productivity; second, if representative dummy variables for each industry are or are not included; third, if a lag structure or another is proposed, etc. Therefore, the results presented in this paper could be directly compared, up to a point, to only those results achieved by researches that define the relevant variables by the same way.

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	differences are completely expected because the empirical analysis only provides

  The aim is verify in Spain if a growing investment in R&D generates an increase in the productivity of firms. The model to R&D capital stock.

	allows the estimation of the rate of return to R&D investments by using a flow
	variable (R&D expenses) without the need to build the (unavailable) research
	capital stock variable.
	The theoretical model was later specified through an econometric model with
	distributed lags over time. Subsequently, the econometric model has been
	estimated by means of a panel data extracted from the Business Strategy Survey
	(Encuesta sobre Estrategias Empresariales, ESEE) for the period 1993-1999. The F data belong to firms of 18 Spanish manufacturing industries, in which we observe o r obvious differences with regard to the elements examined here.
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Page 31 of 32 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  All these measures would be able to create a stable framework for the achievement of technological investment, what would revert in a greater return for this type of investment.

	Submitted Manuscript
	APPENDIX
	Manufacturing industries
	Industry 1: Primary Metals
	Industry 2: Non-Metallic Mineral Manufacturing
	Industry 3: Chemical Products
	Industry 4: Metal Manufacturing
	F o r F o r F o r F o r F o r Industry 5: Machinery for Agriculture and Industry Industry 6: Office and Data Machinery F o r
	P e P e P e P e P e Industry 7: Electrical and Electronic Machinery & Accessories P Industry 8: Autos and Motors e
	e r e r e r e r e r Industry 9: Other Transport Material Industry 10: Meat and by -products e r
	R e v i e w R e v i e w R e v i e w R e v i e w R e v i e w R e v i e Industry 15: Wood and Wood Furniture Industry 11: Food and Tobacco Industry 12: Beverages Industry 13: Textiles and Clothing Industry 14: Leather and Footwear w
	Industry 16: Paper and Printing
	Industry 17: Rubber and Plastics
	Industry 18: Other Manufacturing
	31 32

Table 1 . Estimation results of empirical model Dependent variable: (q-l) t

 1

	Variable	Coefficient	Stand. Error	Z	Significance
	Constant ( )	0.03241	0.01771	1.83	0.067
	(c -l) t	0.05228	0.03082	1.70	0.090
	l t	-0.31735	0.06273	-5.06	0.000
	(R/Q) t-1	0.26598	0.12295	2.16	0.031
	cu t	0.14354	0.6869	2.09	0.037
	No. of firms	1,312	Period	1993-1999
	Hansen J statistic	5.312	[ 2 (5)]	Significance	0.37933

In this way, see the technological capital model from[START_REF] Griliches | Issues in assessing the contribution of research and development to productivity growth[END_REF], which, in addition to the usual productive factors, provides in the production function another differentiated productive factor that could be called research capital, technological capital or R&D capital.

In this approach, researchers usually make different estimates by assuming several values for this depreciation rate.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

See in the Appendix the industry classification of the two-digit level used by the ESEE.

Measuring the output through the value added is frequent in the economic literature. See, for example,[START_REF] Odagiri | The impact of R&D on productivity increase in Japanese manufacturing companies[END_REF],[START_REF] Hall | Exploring the relationship between R&D and productivity in French manufacturing firms[END_REF],[START_REF] Mairesse | Estimating the productivity of research and development: An exploration of GMM methods using data on French and United States manufacturing firms[END_REF] and[START_REF] Rouvinen | R&D-productivity dynamics: causality, lags, and 'dry holes[END_REF].

This result was also presented by[START_REF] Smith | R&D and productivity in Danish firms: some empirical evidence[END_REF] for Denmark and[START_REF] Wakelin | Productivity growth and R&D expenditure in UK manufacturing firms[END_REF] for the United Kingdom. As it is argued by the latter, one might attribute that result to the exclusion of raw materials and intermediate products from the production function.

See Mairesse and Sassenou (1991) concerning papers about rate of return to R&D and[START_REF] Atella | Do R&D expenditures really matter for TFP[END_REF] in relation to papers about elasticity of total factor productivity with respect

Acknowledgements

Authors thank the Fundación SEPI from Spain for the access to used empirical data.

The authors are also grateful for the comments received from the participants at the 6 th Annual EUNIP Conference (Turku, Finland, December 2002); V Jornadas de Economía Laboral (Tarragona, Spain, July 2003); 7 th ISINI International Congress (Lille, France, August 2003); and XIX Jornadas de Economía Industrial (Castellón, Spain, September 2003). Nonetheless, results and conclusions of this paper are exclusive responsibility of the authors. .

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript


Giacomo ; Galante

Antonella Certa

Mario Enea

Concetta Manuela

La Fata

E Ing

Gestionale Certa

Antonella Enea

Mario La Fata

Multi-objective Human Resources Allocation in R&D Projects Planning

Keywords: PROJECT MANAGEMENT, DECISION SUPPORT Human resources allocation, Mullti-objective optimization Human resources allocation, Multi-objective optimization, ELECTRE III, ANP

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The assignment of human resources is a very important phase of the decision-making process, especially with respect to research and development (R&D) projects where performance strongly depends on human resources capabilities.

R&D projects are an important function of any business' activity in particular for high-tech companies in electronics, software development, etc., whose primary function is product development and innovation. In a R&D department, several projects may have to be implemented simultaneously within a certain period of time by a limited number of human resources with different skills, making the process of assigning tasks difficult. The purpose of this paper is to propose a multi-objective methodology to allocate human resources with different skill levels to projects that require several competencies. The model focuses on three fundamental aspects of human resources: the different skill levels, the learning process undertaken and the social relationships existing in working teams.

Regarding the first aspect, many studies show how it is possible to improve the efficiency of the assignment process by taking into account the resource capability level. Koshijima and Umeda (2001) formulated a method to allocate workers with different skill levels to tasks in order to minimize the project execution cost.

Authors highlight how the traditional approach of project management doesn't give explicit consideration to human factors. Based on an equation representing the dynamic working process, the results show the improvement in the economic performance and in the project management process. The authors propose a linear programming model in which certain levels of skill are required to perform a task by multiple engineers who are classified on the basis of their experiences and capabilities. Also, Gomar et al (2000) developed a linear programming model to optimize the assignment process of a partially multi-skilled workforce. Human resources have a range of skills that allow them to participate in one or more projects. In particular, the model considers four objective functions: minimizing the total number of workers used on a project, increasing employment on the construction phase, minimizing human resource hires and fires and minimizing the number of human resource switches between crews. Bassett (2000) proposed a model to optimally assign people to roles in order to minimize outside contracts by better utilization of the available expertise. A one-month assignment is proposed and the scheduling problem is solved by mathematical programming and heuristic approaches.

Significant studies have examined the learning process, the second aspect considered herein, and stated that it has significant potential to improve productivity in organizations. Wu and Sun (2006) developed a mixed nonlinear program for multi-project R&D scheduling and staff allocation problems, which considers the staff learning effect. The goal is to minimize outsourcing costs, assuming the need to turn to outsourcing strategy whenever a task cannot be completed before its due date. A genetic algorithm (GA) is proposed to solve the problem. The learning process is also considered by Nembhard (2001), who suggested a heuristic approach for assigning workers to tasks based on the learning rate of the individual workers. The author shows by simulation that worker assignment based on empirical learning parameters has significant potential to increase the overall productivity. 3 Some studies highlight the importance of the third aspect, the social relationships between workers. In particular, Yaakob and Kawata (1999) studied the problem of placement of workers in a production line. By using linguistic variables and triangular fuzzy numbers, they express the worker's suitability for each job.

Since human resources operate in team, the relationships among workers are included as input data in order to make a more accurate decision. The team relationship function is calculated as the sum of all the relationships between pair of workers. The objective is the maximization of a composite function obtained by summing the workers suitability and the relationship among the team members for each job. Shen et al (2002) also proposed a multi-criteria assessment model that considers the relationships between human resources. They underline the fact that several workers, with different skills and expertises, may share the same role in the organization but the selection of appropriate individuals, based merely on the relationship between a role and a task, is not very effective. They use linguistic variables and fuzzy numbers to evaluate worker suitability for each task. In particular, the task assignment considers three factors. First, the comparison between the workers' skill sets and the skills required to perform a task determines the worker capability in performing the task. The other two factors take into account the social relationships among workers and the relation among tasks, respectively, the latter being either conflicting or complementary. In order to prevent fraud, conflicting tasks must be performed by different workers. The authors also take into consideration the learning process to evaluate worker performance.

Extensive research has been done on handling the multi-objective human resource assignment. However, for the most part, the goal is to optimize a single utility function determined as the weighted sum of the score functions. The difficulty of normalizing the objective functions and of quantifying the weights, however, is well known.

In this paper, the search for the optimal solution is carried out in two steps. Firstly, a set of non-dominated solutions is obtained by exploring the optimal Pareto frontier and subsequently, based on further information, the decision maker can select the best compromise among the objectives, without a priori introduction of arbitrary weights. Yoshimura et al (2006) developed a project selection algorithm for choosing the set of projects that maximizes the total estimated profit and then presented a human resources allocation algorithm for optimally placing human resources among the selected projects. A genetic algorithm is used to obtain a Pareto solution set from which the optimal solution is selected. The process for discarding unsuitable solutions is based on setting lower limit constraints for each objective considered; therefore the authors do not suggest any formalized tool for this phase of the optimization procedure. Conversely, the present work uses the ELECTRE III multi-criteria method as a tool to support the decision maker's process of choosing the optimal solution. By using proper thresholds, this method allows the decision maker to take into account the uncertainty and ambiguity that affect the performance evaluation.

The uncertainty in estimating some input parameters of the model is expressed by the use of fuzzy numbers, thus the value of each objective function corresponding to a certain solution is also a fuzzy number. These values are used to calculate the ELECTRE III thresholds. A set of weights for the decision criteria is introduced in order to apply ELECTRE III method. Its evaluation is difficult in our case since the decision criteria are not independent. In order to support the decision maker regarding this aspect, an adaptation of the Analytic Network Process (ANP) technique is proposed.

Model description

The problem here is to allocate the available human resources to a fixed number of projects that have to be carried out totally or partially within a certain scheduling period. The projects to be planned are those previously selected by the company according to the resources availability. In this context, no explicit mention to project's activities is made, but the competencies needed for their execution and the total time required for each competency are considered.

In R&D projects, each human resource often participates in more than one project simultaneously and works in a team within a specific project. Furthermore, each activity is usually allowed to stop and then restart during project execution. Therefore, human resource flexibility in project implementation, at least in the planning phase considered here, can lead to ignoring or disregarding of existing relationships between activities.

The challenge here is not a classical resource-constrained project-scheduling problem (RCPSP), in which the objective is generally the minimization of project completion time, considering the precedence constraints among activities and the maximum availability of the resources. In RCPSP, the human resource allocation problem consists of determining the number of people assigned to each task. The resources are considered as uniform entities and their particular abilities and personalities are not taken into account.

On the contrary, the proposed model takes into account these factors explicitly and it aims to maximize the expected quality in project execution, allocating human resources to activities by considering their owns abilities and the importance that each skill brings to the project. In R&D organizations, human resources are so fundamental that management must optimize their utilization. A success of a company can be enhanced when management encourages a positive and collaborative work environment, allowing workers to grow and mature. These two aspects are also considered in the model, which is actually a multi-objective problem. Described below are the objective functions that maximize the satisfaction of skills required by projects, the increase of skills and the human resource satisfaction in team working. For the sake of simplicity, the scheduling interval, whose duration usually depends on the environment in which companies work, is assumed unitary. The projects that need to be scheduled are just those beginning in this interval, while for those already started and not completed, it is assumed to maintain the previous planned resource assignments. x jki time that resource j works on project i dispensing skill k measured as a fraction of the planning period;

Nomenclature

I
y jki binary variable that takes value 1 if resource j works on project i dispensing skill k, 0 otherwise; n ji total number of skills supplied by resource j working on project i;

H ji binary variable that takes value 1 if resource j works on project i, 0 otherwise.

Objectives functions

The three objectives here considered are the maximization of the project quality, the increase of the resources skill and the satisfaction in working team. They are expressed by the following objective functions.

Project quality

Project quality mainly depends on the satisfaction of skills required by each project. Therefore the first objective function to maximize is:

F o r P e e r R e v i e w O n l y 6 1 , , i jk ki jki j i k F p c p x = ⋅ ⋅ ⋅ ∑ (1)
where:

i = 1,…,I is the generic project among those in business portfolio I; j = 1,…,J represents the human resource among those available; k = 1,…,K is the generic skill among those K needed to complete the project set I.

The decision variable x jki is continuous and defined in [0,1] range. It represents a fraction of the unitary planning time that resource j employs to bring the skill k to the project i. Furthermore, every skill has a different importance in reaching the stated goals for each project, so a weight p ki is introduced to explain such priority. For every project, the sum of these weights is equal to 1:

∑ = k ki p 1 i ∀ (2)
Each project is also characterized by a coefficient p i that expresses how the considered project is strategically important among all those in the business portfolio. Also in this case, the sum of the aforementioned coefficients is equal to 1:

∑ 1 = i i p (3)
The generic worker j has an initial skill level c jk that represents, for each skill k, the worker's ability in performing those activities that require k. All parameters c jk are obtained from qualitative judgments expressed by a team of experts by using the procedure described in next section.

The objective function is subject to the following constraints:

, ,

jki j max k i x a ≤ ∑ j ∀ (4) with 0 jki x ≥ , , j k i ∀ (5)
and a j,max is the resource time availability within the planning period. Since this period is supposed to be unitary, the value of a j,max is equal to 1 for workers not yet hired in projects already started and still being executed. On the contrary, the availability of human resources already hired is less than the planning period given that the engagements scheduled have to be carried out.

The respect of each skill k required by the project i is expressed by

F o r P e e r R e v i e w O n l y 7 ∑ = j ki jki o x i k, ∀ (6)
where the o ki is the temporal request of skill k by the project i.

Considering that it could be unproductive to employ a resource for only a small fraction of time, it is here supposed that each worker's employment must be not less than a fixed percentage r ki of demand o ki :

( )

jki ki ki jki jki jki x a r o y x y ≥ + ⋅ ⋅ ≤ i k j , , ∀ (7)
where y jki is a Boolean variable equal to 1 if the human resource j works on project i dispensing skill k and 0 otherwise. Moreover, the constraint assures that y jki takes 1 if x jki has a value not equal to 0, becomes 0 if x jki has a value equal to 0. The parameter a is a small arbitrary number that assures the respect of the previous condition when o ki is equal to 0.

In order to focus the attention of each resource on few goals, it is assumed that the employer can at most participate in a stated number N of projects.

The introduction of two dummy variables n ji and H ji is necessary to express this condition.

The first one, n ji , is an integer variable representing the number of competencies totally performed by human resource j in completing project i:

ji jki k n y = ∑ i j, ∀ (8)
The second dummy variable, H ji , is Boolean and takes 1 if resource j works on project i, regardless of the specific competency, and 0 otherwise. The relationship between the variables just defined is expressed by:

ji bn ji ji ji n U H H n ≤ ⋅ ≤ , j i ∀ (9)
where U bn is the upper bound of n ji stated as the number of skills required by each project.

Constraint (9) implies that H ji is 0 if n ji is 0 and 1 if n ji is not equal to zero.
It is now possible to express the constraint concerning the maximum number of projects on which the generic resource j can work as:

N H i ji ≤ ∑ j ∀ (10)
For a given skill k, in order to avoid an over splitting of workloads between resources employed on the same project, it is convenient to fix the maximum number of team members. In this context, considering that R&D projects are taken into account, the number of team members is at most 2.

2 jki j y ≤ ∑ i k, ∀ (11)

Increase of the resources skill

The second objective function aims to maximize the minimum amount of skill available for business activities by learning process. For each skill k, the total availability is calculated by summing the amount of the same skill globally held by all resources.

The initial value c jk arises from the last evaluation made by a team of experts stating their judgments by linguistic variables connected to fuzzy numbers, as described in section 3. The value of c jk is the crisp value obtained by defuzzificating the corresponding fuzzy number.

To evaluate the resource skill improvement, the following learning curve is used:

( ) 1 jk t f t e -β = - (12)
where β jk is the specific parameter for resource j and skill k and t is the learning time.

In order to simplify the model, the improvement is calculated by a linear approximation of the learning curve.

A growth factor m jk is calculated as the derivative of the learning curve in correspondence to c jk and the improvement is therefore expressed by:

( jk jki i m x ⋅ ∑ ) (13)
The linear approximation is valid because the scheduling period is significantly smaller than the one in which the learning curve is defined.

Resource ability at the end of the planning period is therefore:

jk jk jki i c m x + ⋅ ∑ (14)
while the generic skill k globally available at the end of same period is:

( ) k jk jk jki j i c c m x = + ⋅ ∑ ∑ (15)
The second objective function is to maximize the minimum value of skill availability:

2 k k F min c = (16)

Satisfaction in work team

A function that takes into account the preference expressed by resources j and l in working together within the same team is here formulated. This preference is defined by a coefficient α jl in the so-called social Zadeh (1975) proposed a fuzzy approach to numerically represent the uncertainty associated to qualitative evaluations. Fuzzy numbers are standard fuzzy sets defined on a set of real numbers. In this paper, triangular and trapezoidal fuzzy numbers are used. Triangular fuzzy numbers are widely used due to their simplicity and solid theoretical basis (Pedrycz , 1994).

The membership function of a triangular fuzzy number A is µ A : R → [0,1] and can be represented by the equations:

x d m d - - when x ∈ [d, m] = ) (x A µ u x u m - - when x ∈ [m ,u] (18
10 x d m d - - when x ∈ [d, m] 1 when x ∈ [m, n] = ) (x µ B u x u n - - when x ∈ [n ,u] (19) 0 otherwise
where d<m<n<u.

Similarly, a trapezoidal fuzzy number is fully characterized by four real numbers (d, m, n, u). The parameters m and n give the maximum grade of µ B (x).

Figure 1 shows the linguistic terms used to express the judgments about c jk and α jl and the corresponding fuzzy numbers.

The MIN fuzzy operator is used in this context in order to aggregate the judgments expressed by each expert, so α jl represents the minimum affinity level declared from each pair of resources. Concerning the evaluation of c jk , the same operator is applied because the procedure guarantees that the estimated skill level is really transferred to the project each resource works on.

The values of c jk and α jl in the objective functions are the crisp values obtained by defuzzificating the corresponding fuzzy number using the centroid method. The information about the uncertainty of the parameters will be used in the second step of the optimization procedure (section 4.2.2).

Insert Fig. 1 4

. Resolution approach

As mentioned in the introduction, the resolution approach proposed for the multi-objective optimization problem consists of two steps. In the first, various trade-off solutions best approximating the Pareto-optimal frontier are determined. Next, in order to select the optimal solution among those obtained in the previous step, the ELECTRE III multi-criteria method is applied. Further information about the problem is then used to compare the solutions. In this phase the solutions that best describe the Pareto optimal frontier are individuated and the number of those to be considered in the next step of selection process is reduced, by eliminating similar solutions . In this way, maintaining a complete description of the non-dominated solution space, the selection procedure is made easier.

The Lexicographic Goal Programming Method (LGP), the Weighted Sum Method and the ε-constraint

Method are used to determine those solutions. To reduce the number of solutions being submitted to ELECTRE III and maintain a complete description of the Pareto-optimal frontier, the concept of crowding distance is then applied.

Pareto-optimal solutions

The LGP is initially used to individuate the extreme Pareto-optimal solutions. This method considers the three objective functions separately, thereby reducing a multi-objective problem to a mono-objective one. For example, these sequential steps determine one extreme solution:

• maximizing F 1 as a single objective problem;

• maximizing F 2 considering the previous value of F 1 as a further constraint;

• maximizing F 3 considering the previous values of F 1 and F 2 as further constraints.

The procedure is analogously applied changing the objectives hierarchy to find the other five solutions, normalized using the expression:

min , max min - ( 1,2,3) - i i i norm i i F F F i F F = = (20)
The next step is to apply the Weighted Sum Method to reduce the objectives set into a single objective by multiplying each objective function by a weight, resulting in the following composite objective function: subject to constraints (4), ( 5), ( 6), ( 7), ( 8), ( 9), ( 10) and ( 11).

F = w 1 •F 1,norm + w 2 •F 2,norm + w 3 •F 3,norm (21)
Since our model is non-linear, the solution space may be non-convex and the Weighted Sum Approach (Deb, 2001) may not be able to determine all Pareto-optimal solutions.

Anyway, ignoring the shape of the objective space, the setting of the weights vector to obtain a complete description of the optimal Pareto frontier is really difficult: a uniformly distributed set of weights vectors does not correspond to a uniformly distributed set of Pareto-optimal solutions. Moreover, different weights vectors do not necessarily lead to different Pareto-optimal solutions. For these reasons, in order to find other solutions in the region where few solutions have been found by using the Weighted Sum Approach, the ε-constraint

Method is applied. This method is able to find solutions also in the non-convex region. As Haimes et al.

(1971) suggested, the problem can be modified keeping one of the objectives and restricting the others within user-specified values.

The problem becomes:

Max F µ (x) Subject to F i (x) ≥ ε i, i =1 to 3 and i ≠ µ (24)
and constraints (4), ( 5), ( 6), ( 7),( 8), ( 9), ( 10) and ( 11)

Hence, different Pareto-optimal solutions can be found by using different sets of ε i that must be chosen within the minimum and maximum values of the objective functions found by using the LGP. The choice of a particular set of values for ε i restricts the possible location of the solution to a desired region of the Paretooptimal frontier.

Reduction of the number of solutions

This procedure reduces the solutions obtained in the previous step by ranking them on the basis of the crowding distance. The crowding distance provides an estimate of the density of solutions around each solution belonging to the frontier. Considering a generic objective i, the n solutions are sorted in descending order. Fixing d i 1 = d i n = ∞, the crowding distance of the generic solution z is expressed by:

1 -1 max min - ( 1, 2,3; 2,..., -1) - z z z i i i i i F F d i z n F F + = = = (25)
z z i i d d z n = = = ∑ (26)
The solutions selected for the next step are the first m, with the higher values of the parameter.

Second step: ELECTRE III

ELECTRE III (Roy, 1996) is a multi-criteria decision making procedure that can be applied when a set of alternatives must be ranked according to a set of criteria reflecting the decision maker's preferences. In a multi-criteria decision problem, although logical and mathematical conditions required to determine an optimum solution do not exist, a solution representing a good compromise according to the conflicting criteria established can be individuated. The ELECTRE III method is based upon pseudo-criteria. Using proper thresholds, a pseudo-criterion takes into account the uncertainty and ambiguity that can affect the evaluation of the performance, so that, if the difference in the performance of two alternatives is minimal, according to a certain criterion, such alternatives can be considered indifferent according to that criterion. Another peculiarity that differentiates ELECTRE III from other methodologies is that it is not compensative, which means that a very bad score in one objective function is not compensated by good scores in other objectives.

In other words, the decision maker will not choose an alternative if it is very bad compared to another one, even on a single criterion. This occurs if the difference between the values of an attribute of two alternatives is greater than a fixed veto threshold. ELECTRE III is based upon outranking relations: an alternative outranks another alternative if sufficient reasons exist to assert that the first is as good as the second and good reasons to reject such assertions do not exist. Outranking is therefore based upon a concordance/discordance principle.

This principle consists of the verification of the existence of a concordance of criteria in favour of the assertion that one alternative is as good as another and that a verifiably strong discordance among the score values that may reject the previous assertion does not exist.

The following thresholds are introduced: q -indifference threshold; p -preference threshold; v -veto threshold;

where q ≤ p ≤ v.

Objective weights

The weights associated to each objective function represent the relative importance of each goal compared to the others. In this case, the objectives are not independent: reaching one goal positively affects the other goals.

Since it is impossible to translate these influences between goals into mathematical expressions, the model does not take them into account.

In particular, let Ob 1 , Ob 2 and Ob 3 represent the three goals: project quality, increase of the resources skill level and satisfaction in working teams respectively. The following relationships may be considered:

• Ob 1 →Ob 3 : a good development of project implies an improvement in social relations between human resources who work together;

• Ob 3 →Ob 1 : a good affinity among resources working together positively influences project development, since it puts the resources in a position to entirely bring their own skills to the project;

• Ob 1 →Ob 2 : a good development of project implies a faster learning process;

• Ob 2 →Ob 1 : improvement in skill level during project execution implies improvement in expected project quality;

• Ob 2 →Ob 3 : the satisfaction for the individual skill level increase implies an improvement of teamwork relationship;

• Ob 3 →Ob 2 : a higher level of teamwork relations leads to more professional growth in human resources.

The Analytic Network Process (ANP) is a valid and useful approach in selecting a particular solution among all those found when the business objectives are not independent (Saaty, 1996). The ANP method, as the more famous AHP, is based on the comparison between couple of alternatives, so it may handle only a restricted number of alternatives. Saaty has shown that to maintain a reasonable consistency when deriving priorities from paired comparisons, the number of elements being considered must be less or equal to nine (Saaty, 1980).

In our case, a complete description of the Pareto optimal frontier is presented to the decision maker, implying a number of solutions that makes ANP unusable. For this reason and for the ones mentioned in section 4.2, the ELECTRE III method is chosen in order to select the best solution. However, the dependence among the objectives cannot be disregarded and here is considered in determining the weights. Such dependence implies For such reasons, an adaptation of the Analytic Network Process (ANP) to the particular considered problem is proposed.

The ANP extends the hierarchy structure of the AHP method to a network structure, whose nodes consist of clusters of elements (criteria, alternatives, etc.) and arcs represent the link among them. In particular, ANP deals with the dependence of elements within a cluster (inner dependence) and among elements belonging to different clusters (outer dependence). The influence that a set of elements belonging to the same cluster has on each element of the problem is represented by a priority vector, obtained by paired comparisons in the usual way of AHP. These priority vectors are grouped and arranged into a matrix called unweighted supermatrix.

The ANP calculation process employs other two matrices, namely the weighted supermatrix and the limit supermatrix. The weighted supermatrix is obtained by multiplying all the elements of the unweighted supermatrix by the corresponding cluster weight. The limit supermatrix is obtained by multiplying the weighted supermatrix by itself until the values in each row are equal. These values constitute the final priority of the elements. The simple network structure of our decisional problem is represented in figure 2.

Insert Fig. 2 The loop in figure 2 represents the inner dependence. Therefore, it is possible to calculate the relative importance of each criterion with respect to company's satisfaction and the influence that each one has on the others. The relative importance of the objectives is the weight introduced in ELECTRE III. The indifference threshold q is the minimum difference between values taken from a goal that the decision maker considers meaningful to state that one solution is preferable over another. It is due to a real indifference of the decision maker about two similar solutions but it should take into account the uncertainty associated to each solution. Since available information is usually not sufficient, the last issue is not properly taken into account.

In this study, the indifference threshold is associated to the uncertainty described by the fuzzy number corresponding to the generic objective function calculated in a solution. It is useful to keep in mind that judgements about skill levels and affinity are the fuzzy numbers c jk and α jk .

To solve the model described in section 2, these numbers have been defuzzified using the Centroid Method to find the corresponding crisp value. Nevertheless, once the model is solved, it is possible to express the value of each objective function by a fuzzy number obtained executing the operations on the fuzzy number of the correspondent function. For a given goal and solution, the indifference threshold is stated as the difference between the upper bound u of the defined interval and the central value m of the fuzzy number. If the crisp value of the objective function of a solution is greater than the corresponding value of another solution but this gap is lower than the interval threshold, this difference could be due to the uncertainty rather to an effective prevalence of the first solution on the second. The preference threshold p represents, instead, the minimum gap between values taken by the objective function which the decision maker thinks as important to strictly state that a solution is preferable over another one.

A deep knowledge of the problem is required to fix p; the only possible comment is that p is significantly greater than q.

Finally, if an alternative is worse than another based on a certain criterion for a value greater than the so called veto threshold v and the same solution is strictly better based on other criteria, this solution is declared outranked by the other. Using ELECTRE III, it is not necessary to introduce the veto threshold. In this study, it is used only for the first objective function, as showed in the numerical example in the following section.

Numerical example

The proposed approach has been applied to a simulated case including 8 human resources, 5 projects and 5 skills. The input data are reported in Tables 1234567.

The data are generated on the basis of the following considerations to make the model as realistic as possible:

• the sum of time required in Table 1 matches the constraint on maximum availability of the resources; • each resource has unevenly distributed skill levels;

• a value of o ik =0 implies p ik =0;

• a judgment value smaller than "low" (see figure 1) about c jk implies that no possibility of growth is given to resource j with relation to skill k;

• the values of β k have been created assuming that the skill level improvement, from a minimum value of 0.3 (centroid of the fuzzy number associated with the linguistic variable "low") to a maximum value of 0.99, takes place in 5-10 times the scheduling period, randomly generated .

For the sake of simplicity, it is assumed that, for each competence, all resources are on the same learning path during the project's execution. thus table 6 shows β k and not β jk . It is thus considered a learning curve for each skill, regardless of the particular human resource taken into account.

Parameter r ik, appearing in constraint ( 7), is equal to 0.2 without taking into account the project or the skill.

Each worker may works on a number N of projects, which are at most 3.

Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Table 7 By using the procedure described in section 4.1.1 and Lingo 10 software, 50 solutions are found. Among those, 15 solutions are selected by the procedure described in section 4.1.2 (see table 8). The second column of the table below shows the method used to find the solution.

Table10

Table 11 Table 12 Table 13 Since the network is characterized by one cluster, the unweighted supermatrix in table 14 matches the weighted one. In table 15 the global weights of the limit matrix are reported. To evaluate the indifference threshold, the procedure described in 4.2.2 is applied. Figure 3 shows, as example, the fuzzy number that matches the first objective function calculated in solution 11 by using equation (1). The corresponding threshold is the difference between the upper bound u of the definition interval and the central value m.

Fig.3

For the sake of simplicity, the preference threshold is fixed at twice the indifference threshold. The veto threshold, used for the first objective only, is three times the indifference threshold.

The indifference thresholds for each objective function and for each solution are in table 16. III has not been able to select between these two solutions, the choose of the solution to implement has to be carried out by the decision maker on the basis of further aspects not considered in the present model.

Tables 17 and18 show the workers assigned to the projects and the skills dispensed for the two selected solutions.

Table 17 Table 18 Tables 17 and18 highlight that worker 7 is not involved during the considered planning period, that confirms the accuracy of the results obtained. In fact, since the global resources availability is greater than the request needed in projects execution, the most unsuitable resource in matching the business goals is not hired. In particular, looking at table 7, it is evident that the resource 7 has the lowest affinity with the others and so he/she is penalized in hiring.

Conclusions

The developed multi-objectives model allows to find the best allocation of human resources to projects considering worker features. The resolution approach of the problem consists in two steps: first, a set of nondominated solutions that describes the optimal Pareto frontier is obtained and subsequently, based on further information, the ELECTRE III method is used to select the solution that represents the best compromise according to the considered objectives. The advantages in applying the described procedure compared to the classical one, which optimizes a utility function, are already aforementioned. Two further aspects of this method have to be here highlighted as innovative: the computation of the indifference threshold and the specification of the weights utilized for the ELECTRE III. The choice of the indifference threshold is subjective: it depends on the decision maker who expresses the equivalence judgment between two generic solutions according to a specific goal. However, in the problem handled, the most important aspect that needs to be considered is the uncertainty of values taken by the objective functions. In this paper a fuzzy approach is used to numerically translate the qualitative judgments of the different aspects of human resources. The uncertainty described by the fuzzy number corresponding to the generic objective function is used as value of the indifference threshold. Regarding the second aspect, the AHP method is usually utilized in literature to calculate the weights associated to each goal. In evaluating the relative importance between two objectives, it is hard for the decision maker to take into account the dependences between them. In order to overcome this • Nembhard, D., A., Heuristic approach for assigning workers to tasks based on individual learning rates, International Journal of Production Research, 2001Research, , 39(9), 1955Research, -1968.

• Pedrycz, W., Why Triangular Membership Functions?, Fuzzy Sets and Systems, 1994, 64(1), 21-30.

• Roy, B., Multicriteria methodology for decision aiding, Kluwer, Dordrecht, The Netherlands, 1996.

• Saaty, T., L., Decision making with dependence and feedback: The analytic network process, RWS Publications, Pittsburgh, 1996.

  on the base of judgments expressed by human resources. These judgments are linguistic variables, which are linked to fuzzy numbers. The social relationships matrix is updated after each collaboration teamwork experience.Therefore, the objective function to maximize is: of c jk and α α α α jl parametersThe judgments about the individual skill level and the affinity grade of each couple are expressed by linguistic variables.

  triangular fuzzy number is fully characterized by three real numbers(d, m, u). The parameter m gives the maximum grade of µ A (x) that is equal to 1; d and u are the lower and upper bounds of the definition interval.Analogously, the membership function of a trapezoidal fuzzy number B is µ B : R → [0,1] and can be represented by the equations:

  of the Pareto-frontier is obtained by varying the weights and solving the problem:

  max and F i min are the maximum and minimum values of the objective function i of the solutions belonging to the frontier. The overall crowding distance of the solution z, considering all objectives, is:

  decision maker is involved to supply his/her preferences by choosing the objective function weights and the indifference, preference and veto thresholds for each objective function.

  the objectives importance based only on the fulfilment of the company expectations. For example, as already mentioned before, a good affinity among resources in working team positively influences the expected project quality and therefore the importance of objective 3 increases.

  Fig.1 Fuzzy scale

Fig. 2

 2 Fig.2 Network structure for objective functions weights assessment

Table 8

 8 As highlighted in section 4.2.1, the weights associated to the objective functions and required by ELECTRE III are found using the ANP method and Super Decisions Software.

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	18

Table 10

 10 shows the comparison between couples of objectives with respect to the company satisfaction, while table 11, 12 and 13 represent the evaluation of mutual dependences. The pair wise comparisons are expressed by Saaty's semantics scale (table9).

Table 9

 9

Table 14 Table 15

 1415

Table 16

 16 Applying ELECTRE III to alternatives in table 8, normalizing the values of the objective functions by equation (20), solutions 9 and 10 are selected as the best compromise among those supplied. Since ELECTRE

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	19

  modified version of ANP method is proposed. The ANP is generally used in alternatives selection phase and it is capable of handling interdependence among elements. Instead, in this specific case, the aim is not the choose of the best alternative, carried out by ELECTRE III, but the calculation of the weights of the objectives considered as elements belonging to a cluster.

	References • Bassett, M., Assigning projects to optimize the utilization of employees' time and expertise, Computers & Chemical Engineering, 2000, 24, 1013-1021. • Deb, K., Multi-Objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, New problem, a Further developments can focus on the exploration of the Pareto frontier that could be improved by using a York, 2001.
	costrained multi-objective genetic algorithm. • Gomar, J., E., Terrien, R., K., Haas, C., T., Morton, D., P., Tucker, R., L., Assignment and allocation
	F o r optimization of a partially multiskilled workforce, Internal Report of Center for Construction Industry F Studies, University of Texas at Austin, 2000. o • Haimes, Y., Y., Lasdon, L. S., Wismer, D., A., On a bicriterion formulation of the problems of r integrated system identification and system optimization, IEEE Transactions on Systems, Man and
	P P Cybernetics, 1971, 1(3), 296-297.
	e e r e e r
	R e v i e w R e v i e w
	O n O n
	l y l y
	20 21

• Koshijima, I., Umeda, T., Human Resource Allocation in Project Management -Management Science Approach, Internal report of Chiba Institute of Technology Tsudanuma, Narashino, Chiba, 2001.

Table 1 .

 1 Zadeh, L., A., The Concept of a Linguistic Variable and Its Application to Approximate Reasoning, Time required by any project for each skill o ki

	Proceedings of the 36th Hawaii International Conference on System Sciences, Big Island, Hawaii,
	USA, January 6-9 2003, 202-210 .
	• Wu, M-C., Sun S-H., A project scheduling and staff assignment model considering learning effect,
	International Journal of Advanced Manufacturing Technology, 2006, 28, 1190-1195.
	• Yaakob, S., B., Kawata, S., Workers'placement in an industrial environment, Fuzzy sets and system,
	1999, 106 (3), 289-297.
	• Yoshimura, M., Fusimi, Y., Izui, K., Nishiwaki, S., Decision-making support system for human
	resource allocation in product development projects, International journal of Production Research,
	2006, 44 (5), 831-848.

• Saaty, T.L., The Analytic Hierarchy Process,

N.Y., McGraw Hill, New York, 1980.

• Shen, M., Tzeng, G.-H., Liu, D.-R., Multi-criteria task assignment in workflow management systems, Page 22 of 27 http://mc.manuscriptcentral.com/tprs Email: ijpr@lboro.ac.uk

Table 4 .

 4 Relative importance of skill k for project i

	skill	1	2	3		4	5
	project
	1	0.25	0.4	0	0.35	0
	2	0	0.3	0.15	0.5	0.05
	3	0.2	0	0.1	0.45	0.25
	4	0.1	0.35	0.15	0.15	0.25
	5	0.25	0	0.3	0.2	0.25
		Table 5. Resource skill level c jk
	skill
	resource	1	2	3		4	5
	1	0.3	0.5	0.4	0.077	0.8
	2	0.4	0.7	0.077	0.8	0.3
	3	0.8	0.077	0.5	0.5	0.4
	4	0.4	0.5	0.7	0.6	0.077
	5	0.077	0.8	0.3	0.6	0.3
	6	0.4	0.3	0.6	0.077	0.8
	7	0.4	0.077	0.6	0.3	0.7
	8	0.8	0.3	0.077	0.5	0.4
		Table 6. Learning curve parameter β k
	skill	1	2	3	4	5
		0.7	0.75	0.55	0.5	0.65

Table 7 .

 7 Social relationships matrix α jl

	resource
	resource	1	2	3	4	5	6	7	8
	1		0.2	0.8	0.077	0.6	0.2	0.077	0.6
	2			0.8	0.6	0.077	0.2	0.4	0
	3				0.6	0.2	0.8	0.4	0.2
	4					0.8	0.4	0.077	0.6
	5						0.2	0.6	0.2
	6							0.077	0.6
	7								0.077
	8

Table 8 .

 8 Pareto solutions to submit to ELECTRE III Table10. Pair wise comparisons between objectives with respect to the company satisfaction

	Solution	Method used	F 1	F 2	F 3
	1	l.g.p.	0.156	3.383	2.000
	2	l.g.p.	0.139	3.383	11.800
	3	l.g.p.	0.151	3.228	16.000
	4	l.g.p.	0.132	3.321	16.000
	5	l.g.p.	0.181	3.198	1.000
	6	ε	0.146	3.349	15.200
	7	w	0.143	3.377	13.800
	8	w	0.154	3.374	10.400
	9	ε	0.150	3.349	14.400
	10	w	0.162	3.310	13.200
	11	w	0.137	3.357	15.600
	12	w	0.138	3.364	15.200
	13	ε	0.176	3.265	4.400
	14	ε	0.175	3.214	9.600
	15	w	0.179	3.242	4.400
		Table 9. Saaty Scale
	Numerical values	Verbal scale			Explanation
	1	Equal importance of both elements			Two elements contribute equally
	3	Moderate importance of one element over another	Experience and judgment favour one element over another
	5	Strong importance of one element over another		An element is strongly favoured
	7	Very strong importance of one element over another		An element is very strongly dominant
	9	Extreme importance of one element over another	An element is favoured by at least an order of magnitude
	2,4,6,8	Intermediate values			Used to compromise between two judgments

Table 11 .

 11 Pair wise comparisons between objectives with respect to Ob 1

	Ob 1	Ob 2	Ob 3
	Ob 2	1	2
	Ob 3	1/2	1

Table 12 .

 12 Pair wise comparisons between objectives with respect to Ob 2

	Ob 2	Ob 1	Ob 3
	Ob 1	1	2
	Ob 3	1/2	1

Table 13 .

 13 Pair wise comparisons between objectives with respect to Ob 3

			Ob 3	Ob 1	Ob 2
			Ob 1	1	3
			Ob 2	1/3	1
			Table 14. Unweighthed supermatrix
			goal			project objectives
	Cluster node labels	company satisfaction project quality	increase of the resources skill	satisfaction of the work team
	goal	company satisfaction	0		0	0	0
		project quality	0.701		0	0.667	0.250
	project objectives	increase of the resources skill	0.193		0.667	0	0.750
		satisfaction of the work team	0.106		0.333	0.333	0
			Table 15. Objective functions weights
		Ob 1		Ob 2		Ob 3
		0.413		0.337		0.250

Table 16 .

 16 Indifference threshold q

	Page 27 of 27
	solution	F 1		F 2		F 3
	1	0.030		0.541		0.260
	2	0.026		0.541		1.534
	3	0.029		0.516		2.080
	4	0.025		0.531		2.080
	5	0.034		0.512		0.130
	6	0.028		0.536		1.976
	7	0.027		0.540		1.794
	8	0.029		0.540		1.352
	9	0.029		0.536		1.872
	10	0.031		0.530		1.716
	11	0.026		0.537		2.028
	12	0.026		0.538		1.976
	13	0.033		0.522		0.572
	14	0.033		0.514		1.248
	15	0.034		0.519		0.572
	Table 17. Resources assignment for the solution 9
	skill	1	2	3	4	5
	project
	1	6;8	4;8		4;5
	2		4;5	1;5	2;4	1;8
	3	2;3		1;3	2;3	1;3
	4	3;6	4;5	4;5	4;5	3;6
	5	3;6		1;3	2;3	1;6
	Table 18 Resources assignment for the solution 10
	skill	1	2	3	4	5
	project
	1	3;8	4;5		4;5
	2		1;5	1;5	2	1;5
	3	3;8		1;3	2	1;6
	4	6;8	6;8	4;5	4;5	6
	5	3		4;6	2	1;6

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