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Advances in Mechanical Engineering

Volume 2013 (2013), Article ID 234571, 13 pages

http://dx.doi.org/10.1155/2013/234571

## A Fuzzy Collaborative Forecasting Approach for Forecasting the Productivity of a Factory

Department of Industrial Engineering and Systems Management, Feng Chia University, 100 Wenhwa Road, Seatwen, Taichung 408, Taiwan

Received 6 February 2013; Accepted 3 July 2013

Academic Editor: Jerry Fuh

Copyright © 2013 Yi-Chi Wang and Toly Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Productivity is always considered as one of the most basic and important factors to the competitiveness of a factory. For this reason, all factories have sought to enhance productivity. To achieve this goal, we first need to estimate the productivity. However, there is considerable degree of uncertainty in productivity. For this reason, a fuzzy collaborative forecasting approach is proposed in this study to forecast the productivity of a factory. First, a learning model is established to estimate the future productivity. Subsequently, the learning model is converted into three equivalent nonlinear programming problems to be solved from various viewpoints. The fuzzy productivity forecasts by different experts may not be equal and should therefore be aggregated. To this end, the fuzzy intersection and back propagation network approach is applied. The practical example of a dynamic random access memory (DRAM) factory is used to evaluate the effectiveness of the proposed methodology.

#### 1. Introduction

Seeing a problem from various perspectives ensures that no parts are ignored when solving the problem. To this end, the concepts of collaborative computing intelligence and collaborative fuzzy modeling were proposed, and the so-called fuzzy collaborative systems are being established [1, 2]. Many existing fuzzy collaboration systems have been used for clustering (e.g., [1, 3]), that is, the so-called fuzzy collaborative clustering systems. The applications of fuzzy collaborative intelligence in forecasting remain to be investigated. A fuzzy collaborative forecasting approach is proposed in this study to forecast the productivity of a factory, which is important because of the following reasons [4].(1)The productivity and financial performance of a factory are closely related.(2)Bailey [5] contended that employees often perform below their maximum potential, and therefore measuring the productivity is important.(3)Productivity is extremely important for the sustainable development of an enterprise [6].

Traditionally, productivity is calculated by dividing a fixed output by the required input. The units of input and output may be different. In addition, both input and output can be made up of a number of different items that need to be integrated somehow, that is, the composite productivity. Nevertheless, measurement of productivity is straightforward in most factories because the most manufactured items exhibit a high degree of uniformity. Godard and Prevost [7] introduced a model for integrated measurement and analysis of productivity (SIMAP), in which the global productivity ratio is the multiplicative product of three constant factors: the price structure, the mix of output quantities, and variations in productivity. Han and Halpin [8] divided the existing methods of productivity estimation into two categories: deterministic methods and simulation methods. Huang [9] evaluated the productivity of Taiwan semiconductor manufacturers using the Malmquist productivity index. For the same purpose, M. Y. Huang and S. Y. Huang [10] proposed a three-stage Malmquist data envelopment analysis (DEA) method. Inputs considered in their study include R&D expenditures, fixed assets, labor costs, and operating costs. The most widely used measure of productivity is labor productivity [11]. With well-trained employees, factory productivity usually increases over time, but there is a limit. For these reasons, any improvement in productivity can be seen as a learning process. Wright [12] studied the impact of learning on productivity in the aircraft manufacturing industry. The productivity of a factory should be compared with past values of productivity for the same factory and with those of competitors. Black and Lynch [13] found that higher productivity is associated with how that work practice is actually implemented within the establishment. In recent years, the development of data warehouses has enabled researchers to look more directly at productivity [14]. Bandyopadhyay et al. [15] designed an integrated system to help provide information used in human resource decision-making such as administration, payroll, recruiting, training, and performance analysis.

Brandt [11] proposed a formula to forecast the long-term productivity, but that formula requires input and output estimates, and therefore does not simplify this problem. In particular, the long-term outputs are affected by price fluctuations and are very difficult to estimate. To solve this problem, this study adopts the following method. Productivity basically follows a learning process, so a learning model can be established for it and can be used to predict the future productivity. However, productivity is affected by many factors including standardizing, quality differences, scrap rates, new workers, layoffs, and incentive plans [4], and therefore there is a considerable degree of uncertainty in the long-term productivity. To consider this uncertainty, this study uses a fuzzy value to represent each forecast of productivity. A fuzzy value has a range and therefore is able to appropriately express uncertainty. Subsequently, a group of experts in this field is formed. Each expert applies a fuzzy linear regression equation to forecast the productivity of a factory. Some parameters in the fuzzy linear regression equation need a subject setting. As a result, the forecasts obtained by the experts may be very different and therefore require a collaboration mechanism to aggregate them. In the proposed methodology, the fuzzy intersection and back propagation network approach is applied to aggregate the forecasting results.

The novel features of the proposed methodology include the following.(1)The proposed methodology was the first attempt we have had at considering the uncertainty in the long-term productivity.(2)To forecast the productivity of a factory from various perspectives ensures that no parts are ignored when solving the problem.(3)A fuzzy linear regression equation is fitted with three nonlinear programming models with six adjustable parameters. That offers the experts a great deal of flexibility in expressing their opinions (see Table 1).

The differences between the proposed methodology and the previous fuzzy collaborative forecasting methods are summarized in Table 2.

The aim of this study is(1)to improve the accuracy of productivity forecasting. In other words, the forecasted value should be as close as possible to the actual value;(2)to improve the precision of productivity forecasting. In other words, a narrow interval containing the actual value is to be established; (3)the use of a practical example to compare the performances of various methods when forecasting the productivity of a factory.

The rest of this paper is organized as follows. Section 2 reviews the existing methods in fuzzy collaborative forecasting. The problems faced by the existing methods are also discussed. Section 3 describes the details of the proposed methodology for forecasting the productivity of a factory. A practical example is used to evaluate the effectiveness of the proposed methodology. Finally, the concluding remarks and a view on the future are given in Section 4.

#### 2. Related Work

In the limited literature about fuzzy collaborative intelligence and systems, Shai and Reich [20, 21] defined the concept of infused design as an approach for establishing effective collaboration between designers from different engineering fields. Büyüközkan and Vardaloǧlu [22] applied the fuzzy cognitive map method to the collaborative planning, forecasting, and replenishment (CPFR) of a supply chain. The initial values of the concepts and the connection weights of the fuzzy cognitive map are dependent on the subjective belief of the expert and can be modified after collaboration. According to Poler et al. [23], for collaborative forecasting, the comparison of collaboration methods and the proposing of software tools are still very lacking. Pedrycz and Rai [24] discussed the problem of collaborative data analysis by a group of agents having access to different parts of data and exchanging findings through their collaboration. A two-phase optimization procedure was established, so that the results of communication can be embedded into the local optimization results. In recent years, Chen [25] used a hybrid fuzzy linear regression-back propagation network approach to predict the efficient cost per die of a semiconductor product. This method first gathered a group of experts in the field. Then, each expert uses a fuzzy linear regression equation to predict the future unit cost. The result is a fuzzy value and can be regarded as a nonsymmetric interval forecast. A crisp forecast is rarely equal to the actual value, while a fuzzy forecast contains the actual value. The fuzzy forecasts obtained by different experts are aggregated using a fuzzy intersection, resulting in a polygon-shaped fuzzy number, which can be defuzzified using a back propagation network. Chen [26] applied the same method to predict the foreign exchange rate. Chen [27] considered the case in which each expert has only partial access to the data and is not willing to share the raw data he/she owns. The forecasting results by an expert are conveyed to other experts to modify their settings, so that the actual values will be contained in the fuzzy forecasts after collaboration. All fuzzy collaborative intelligence methods seek the consensus of the results. In this field, Ostrosi et al. [3] defined the concept of consensus as the overlapping of design clusters of different perspectives. Similarly, Chen [6] defined the concept of partial consensus as the intersection of the views. Cheikhrouhou et al. [28] thought that collaboration is necessary because of the unexpected events that may occur in the future demand. Chen and Wang [29] proposed an agent-based fuzzy collaborative intelligence approach, in which software agents rather than domain experts are used to improve the efficiency of collaboration. Most fuzzy collaborative forecasting methods established nonlinear programming models to consider the opinions of experts and generate fuzzy forecasts. Such a practice cannot distinguish between the different expert opinions and is not easy to find the global optimal solution. In order to solve some problems and to improve the performance of semiconductor yield forecasting, Chen and Wang [30] proposed a quadratic-programming based fuzzy collaborative intelligence approach.

In short, to forecast the productivity using the existing approaches has the following problems.(1)The productivity forecasted by the existing methods may be lower than the actual value, resulting in overestimated outputs if the revenue plan is based on the forecasts.(2)For precision in productivity forecasting, the narrowest scope containing the actual value is required; however, this has rarely been discussed.(3)The upper bound and average value are forecasted separately, which is problematic because it is possible that the forecast becomes invalid in the sense that the average value may be higher than the upper bound [19].

#### 3. Methodology

##### 3.1. Productivity Forecasting Using Fuzzy Linear Regression

The productivity of a factory at time can be evaluated as follows: where and are input and output items, respectively; and . Their costs and prices at time are denoted by and , respectively. The learning model of productivity is where is the productivity at period , is the asymptotic/final productivity, is the learning constant, and is homoscedastic, serially noncorrelated error term.

As can be seen from (2), after converting to logarithms, the productivity learning model becomes a linear regression problem. Fitting historical data to obtain the parameters in the model is the conventional approach, under the assumption of where “” denotes that it is an estimate. We can obtain where

The shortage of this model is that the uncertainty and variation inherent in the learning process are not easy to consider [32]. To solve this problem, all parameters can be given in triangular fuzzy numbers instead [33–35], and the model becomes a fuzzy one. At first, assume , , and is the fuzzy productivity forecast which can be defuzzified if necessary. After log normalization, which is a fuzzy linear regression problem. The parameters can be obtained by solving the following nonlinear programming model [19], which is a modification from Tanaka and Watada’s linear programming model [16].

*Model 1. *Consider
where reflects the sensitivity of expert to the uncertainty of the fuzzy productivity forecast; . ranges from 0 (not sensitive) to (extremely sensitive); indicates the satisfaction level required by expert ; . In this model, the sum of ranges of fuzzy forecasts is minimized by fitting the historical data at a given level of satisfaction .

The second method for fitting a fuzzy linear regression equation is derived from Peters’ method [17], in which the following nonlinear programming problem is solved, aimed at the maximization of the average satisfaction level with the generalized average formula.

*Model 2. *Take
where is the desired range of every fuzzy forecast; ; represents the relative importance of the outliers in fitting the fuzzy linear regression equation; . When , the relative importance of the outliers is the highest and is equal to that of the nonoutliers.

The third method for fitting a fuzzy linear regression equation is Donoso’s quadratic nonpossibilistic method [18], in which the quadratic error for both the central tendency and each one of the spreads is minimized.

*Model 3. *Consider
where and belong to and add up to 1. If an expert’s concern is about precision, he/she should choose a large value and a small value, and vice versa.

##### 3.2. Aggregating the Forecasts Using Fuzzy Intersection and Back Propagation Network

The aggregation mechanism consists of two steps. In the first step, fuzzy intersection is applied to aggregate the fuzzy productivity forecasts into a polygon-shaped fuzzy number, in order to improve the forecasting precision. Fuzzy intersection combines fuzzy productivity forecasts in the following manner: where indicates the result of obtaining the fuzzy intersection of the fuzzy productivity forecasts by experts. If these fuzzy productivity forecasts are approximated with triangular fuzzy numbers, then the fuzzy intersection is a polygon-shaped fuzzy number (see Figure 1).

The result of this step is a polygon-shaped fuzzy number that specifies the narrowest range of the fuzzy productivity forecast that still contains the actual value. In this way, the forecasting accuracy is maximized. Subsequently, the fuzzy forecasts are defuzzified using a back propagation network with the following configuration, in order to optimize the forecasting accuracy measured in terms of mean squared error (MSE).(1)Inputs: parameters corresponding to the corners of the fuzzy forecasts of the experts. All input parameters have to be normalized before they are fed into the network. Extracting independent variables from the original inputs and using them as new inputs to the back propagation network may be a way to further enhance the accuracy of the network.(2)Single hidden layer: generally one or two hidden layers are more beneficial for the convergence property of the back propagation network.(3)The number of neurons in the hidden layer is , considering both effectiveness (forecasting accuracy) and efficiency (execution time).(4)Output: the crisp productivity forecast.(5)Network learning rule: delta rule.(6)Network learning algorithm: the Levenberg-Marquardt algorithm. (7)Activation function: the activation function for the hidden layer is the hyperbolic tangent sigmoid activation function, while for the other layers it is the linear activation function.(8)Number of initial conditions/replications: because the performance of a back propagation is sensitive to the initial condition, the training process will be repeated many times with different initial conditions that are randomly generated. Among the results, the best one is chosen for the subsequent analyses.(9)Convergence conditions: MSE < 10^{−6} or 10000 epochs have been run.

The Levenberg-Marquardt algorithm, which is much faster than the gradient descent algorithm for small-sized problems, is used for training the back propagation network. The Levenberg-Marquardt algorithm was designed for training with second-order speed without having to compute the Hessian matrix. When training a back propagation network, the Hessian matrix can be approximated as and the gradient can be computed as where is the Jacobian matrix containing the first derivatives of the network errors with respect to the weights and biases; is the vector of the network errors. The Levenberg-Marquardt algorithm uses this approximation and updates the network parameters in a Newton-like way where is the epoch number. However, overfitting may occur using the Levenberg-Marquardt algorithm, and the forecasting accuracy may be lower than expected for the untrained data.

#### 4. A Practical Example

A practical example from a dynamic random access memory (DRAM) factory is used to evaluate the effectiveness of the fuzzy collaborative forecasting approach (Table 3). DRAM is the most common kind of random access memory (RAM) for personal computers and workstations. The global DRAM market in 2013 is forecasted to reach 30 billion USD.

In the practical example, the multi-item productivity was calculated after the monetary values of inputs and outputs had been considered. The productivity of each period was evaluated in Table 4. After being converted to logarithms, the relationship between and was fitted with a linear regression. The coefficient of determination is 0.72, which confirms the learning property of productivity.

However, there is much uncertainty in . To deal with this problem, a fuzzy linear regression equation was fitted, according to (9).

Three experts from production control and industrial engineering departments were asked to submit their opinions about the fuzzy productivity forecasts, which were summarized in Table 5. Obviously, if an expert’s concern is about precision, he/she will choose a large value of and a small value of (Figure 2).

There were nine nonlinear programming problems to be solved. From the optimization result of each nonlinear programming problem, a corresponding fuzzy linear regression equation was constructed. The program coded using LINGO is shown in Algorithm 1. All nine fuzzy linear regression equations were applied to forecast the productivity of the DRAM factory. The results were summarized in Figure 3.

Subsequently, the fuzzy intersection of the forecasting results was obtained (see Table 6), which determined the narrowest range of the fuzzy productivity forecast. The forecasting precision, measured in terms of the average width of the fuzzy forecasts, has been significantly improved (see Figure 4).

Subsequently, the back propagation network was applied to aggregate the fuzzy forecasts into a single representative value, that is, the crisp productivity forecast. The program coded using MATLAB is shown in Algorithm 2. The back propagation network has 27 inputs and the same number of nodes in the hidden layer. The Levenberg-Marquardt algorithm was applied to train the back propagation network. The performance of a back propagation network is sensitive to the initial conditions; therefore, the training was repeated 10 times with different initial conditions (see Figure 5).

The gradient descent algorithm was also applied to train the back propagation network for a comparison. It can be seen from Figure 6 that the forecasting accuracy of the Levenberg-Marquardt algorithm (indicated by LM) is better than that of the gradient descent algorithm (indicated by GD).

The forecasting accuracy of the proposed methodology, in terms of mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean squared error (RMSE), was summarized in Table 7. The forecasting accuracy of each expert without collaboration is also shown in this table for a comparison. In all models, the center-of-gravity method was applied to defuzzify the productivity forecast

According to the experimental results, the following hold.(1)Obviously, after collaboration, the forecasting accuracy was significantly improved. This advantage is most obvious if MAPE is considered. On average, MAPE could be reduced by 99%.(2)On the other hand, the fuzzy collaborative forecasting approach also improved the forecasting precision, as shown in Figure 5. In other words, it is possible to precisely grasp the range of the future productivity, and the related planning and decision-making will not be misled.(3)Among the three nonlinear programming models, the forecasting accuracy of Model 3 was the best. However, the result can vary by case. On the other hand, the forecasting accuracy of Model 1 was quite close to that of Model 3, and they were both better than Model 2 in this aspect.(4)It is quite a natural approach to measure the quality of collaboration with the improvement in the forecasting performance. In Figure 7, we considered the case in which an expert was not involved in the collaboration. Obviously, the forecasting precision was not affected without the involvement of expert 3.(5)In Figure 8, we analyzed the impact of the number of experts on the forecasting accuracy. Obviously, with more experts, the forecasting accuracy could be improved. In addition, with the same number of experts, it did not matter which experts were involved.

#### 5. Conclusions and Directions for Future Research

Productivity is always considered as one of the most basic and important factors to the competitiveness of a factory. For this reason, factories around the world strive to enhance productivity. To this end, an accurate estimate of productivity is the prerequisite. Compared with other performance measures, productivity is easier to forecast in the long term because of its learning property. For this reason, a systematic procedure has been proposed to forecast the productivity of a factory. However, there is considerable degree of uncertainty in productivity that is not easy to cope with. For this reason, a fuzzy collaborative forecasting approach is proposed in this study to forecast the long-term productivity of a factory. First, a fuzzy productivity learning model is established to forecast the future productivity. Subsequently, the fuzzy productivity learning model is converted into three equivalent nonlinear programming problems to be solved from various viewpoints. The fuzzy productivity forecasts by different experts may not be equal and should therefore be aggregated. To aggregate these productivity forecasts, the fuzzy intersection and back propagation network approach is applied.

After applying the fuzzy collaborative forecasting approach to forecast the productivity of a DRAM factory in Taiwan, the following experimental results were obtained.(1)Compared with the existing methods, the fuzzy collaborative forecasting approach significantly improved the forecasting accuracy measured in terms of MAE, MAPE, and RMSE.(2)On the other hand, the fuzzy collaborative forecasting approach provided a range of possible productivity, so that the actual value fell within the range. Compared with the results without collaboration, the forecasting result was substantially more precise.(3)We also found that the number of experts involved in the collaboration process was influential to the forecasting accuracy and precision.(4)With the same number of experts, it did not matter which experts were involved.

The main contribution to the body of the knowledge is the following.(1)A systematic procedure is established in this study to forecast the long-term productivity of a factory through productivity learning modeling.(2)A novel fuzzy collaborative forecasting approach is proposed to consider the uncertainty in the productivity learning process and the viewpoints of multiple experts.

However, compared with some existing approaches, more data are required with the proposed methodology due to the need to incorporate expert opinions. In addition, it takes more time to develop the program codes for the proposed methodology than for the other approaches.

In future studies, the effects of the environmental factors on productivity can be considered. On the other hand, with more experts, the forecasting precision and accuracy can be improved. However, that may lead to more conflicts in the forecasts, and a larger network should be constructed, which means more computational costs for the proposed method. An alternative is to use software agents in future studies.

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