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Forecasting the unit cost of every product type in a factory is an important task. However, it is not easy to deal with the uncertainty of the unit cost. Fuzzy collaborative forecasting is a very effective treatment of the uncertainty in the distributed environment. This paper presents some linear fuzzy collaborative forecasting models to predict the unit cost of a product. In these models, the experts’ forecasts differ and therefore need to be aggregated through collaboration. According to the experimental results, the effectiveness of forecasting the unit cost was considerably improved through collaboration.
Cost forecasting means different things at different stages of the product life cycle. In product design, the designer needs to know whether the product will be economically produced. After a product goes into mass production, forecasting the unit cost is the basis of financial and production planning activities. When a product enters the market, the followup customer service and maintenance costs must also be taken into account.
Accurately predicting the unit cost of each product type is a very important task in any factory. If the unit cost is less than expected, then the efforts and investment of cost reduction are not necessary. Conversely, if the unit cost is more than expected, then the profitability of the product will be overestimated, resulting in the wrong investment and production decisions. However, forecasting the unit cost is not an easy task because of the uncertainty of the unit cost, mainly due to the cost of human operations in the production of products, which is sometimes unstable. In addition, there is not much relevant literature in the unit cost forecasting. On the other hand, several recent studies (e.g. [
In the proposed methodology, the stakeholders are a group of domain experts, such as the product engineer, factory managers and accounting department staff. These experts apply fuzzy linear regression methods to predict the unit cost of a product. A fuzzy linear regression equation can be converted into a linear or nonlinear programming problem in a variety of ways. Furthermore, within the conversion process some parameters need a subject setting. As a result, forecasts obtained by the experts may be very different and therefore requires a collaborative mechanism to deal with the following issues:
(1) How to integrate these forecasts?
(2) How experts can refer to the forecasts of others to modify their own?
In response to this issue, the methods presented in this study are as follows:
(1) Some linear fuzzy regression models for the unit cost forecasting are proposed and compared.
(2) Development of dedicated software to pass the forecast of each expert to other experts for their reference. In the meantime, the software can integrate different forecasts using the hybrid fuzzy intersection and back propagation network approach.
(3) In reference to the forecasts of others, each expert subjectively modifies the parameters in the fuzzy linear regression method.
The objectives of this study are as follows:
(1) To enhance the accuracy of the unit cost forecast. In other words, the forecasts obtained must be very close to the actual values.
(2) To improve the precision of the unit cost forecasting. Namely, a very small range containing the actual value can be estimated.
(3) The application of an instance to compare the advantages and disadvantages of different linear fuzzy collaborative forecasting models.
The organization of this study is described as follow.
Carnes [
Although there have been some literature about fuzzy collaborative intelligence and systems, but very few directly related to fuzzy collaborative forecasting. Shai and Reich [
In short, the existing approaches have the following problems:
(1) The unit cost forecasted by the existing methods may be lower than the actual value, resulting in overestimated profits if the financial plan is based on the forecasts.
(2) For precision in the unit cost forecasting, the narrowest scope containing the actual value is required; however, this has rarely been discussed.
(3) The peak and average unit costs 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 peak value [
(4) The existing fuzzy linear regressionback propagation network methods selected particular fuzzy linear regression methods, but did not explain the reasons or compared with other fuzzy linear regression methods.
The parameters used in the proposed methodology are defined in advance.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
Prior to predict the unit cost of a product, we emphasize at the outset that the reduction in the unit cost follows a learning process, which is the assumption of this study.
According to Gruber [
The unit cost can be calculated as
A fuzzy linear regression equation can be fitted in various ways. For example, in Tanaka and Watada [
The second method for fitting a fuzzy linear regression equation is Peters’ method [
The third method for fitting a fuzzy linear regression equation is Donoso’s quadratic nonpossibilistic method [
The fourth method for fitting a fuzzy linear regression equation is Chen and Lin’s nonlinear programming method [
Model I
Model II
A mechanism is required to combine the fuzzy forecasts. The aggregation mechanism consists of two steps. In the first step, fuzzy intersection is applied to aggregate the fuzzy forecasts into a polygonshaped fuzzy number, in order to improve the precision of forecasting. Every fuzzy forecast contains the actual value. As a result, the intersection of the fuzzy forecasts also contains the actual value. Besides, the intersection has a narrower range than those of the original regions. Therefore, the forecasting precision measured in terms of the average range is indeed improved after intersection, which is one of the basic mechanisms of fuzzy collaborative forecasting. Fuzzy intersection combines
The fuzzy intersection of two triangular fuzzy numbers.
The result of this step is a polygonshaped fuzzy number that specifies the narrowest range of the fuzzy forecast. However, in practical applications a crisp forecast is usually required. Therefore, a crisp forecast has to be generated from the polygonshaped fuzzy number. For this purpose, a variety of defuzzification methods are applicable [
The configuration of the back propagation network used is established as follows:
(1) Inputs: 2
(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 chosen from 1~4
(4) Output: the crisp forecast.
(5) Network learning rule: Delta rule.
(6) Network learning algorithms: There are many advanced algorithms for training a back propagation network, e.g. the Fletcher–Reeves algorithm, the Broydon–Fletcher–Goldfarb–Shanno algorithm, the Levenberg–Marquardt algorithm, and the Bayesian regularization method [
(7) Number of epochs per replication: 10,000.
(8) Activation function: Logsigmoid function.
(9) Number of initial conditions/replications: Because the performance of a back propagation network 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.
The LevenbergMarquardt algorithm was designed for training with secondorder speed without having to compute the Hessian matrix. It uses approximation and updates the network parameters in a Newtonlike way. When training a back propagation network, the Hessian matrix can be approximated as:
Some performance measures of fuzzy collaborative forecasting are defined as follows.
(1) The average range (AR):
Mean absolute percentage error (MAPE):
(2) Root mean squared error (RMSE):
(1) Maximum percentage improvement (MPI):
(2) Average percentage improvement (API):
These functions can easily be extended to involve more than two objects.
An example is given in
An example.
t  c_{t} (US$) 

1  2.57 
2  1.61 
3  1.76 
4  1.28 
5  1.53 
6  1.19 
7  1.32 
8  1.32 
9  1.61 
10  1.32 
Some linear fuzzy collaboration forecasting models for used to predict the unit cost.
In this model, the two objects use the same fuzzy linear regression method (WT(
Therefore, the quality of collaboration with respect to the forecasting precision can be evaluated as
In order to evaluate the forecasting accuracy, the forecasts by the two objects are defuzzified using the center of gravity (COG) method, and then are compared with the actual values:
while in the fuzzy collaborative forecasting method, the forecasts by the two objects are aggregated using the fuzzy intersection and back propagation network approach to generate a single crisp value:
Therefore, the quality of collaboration with respect tothe forecasting accuracy can be evaluated as
In this model, both objects use Peters(
After collaboration, the forecasting precision and accuracy are both improved:
The quality of collaboration in the two aspects can be evaluated as
and
respectively.
In this model, both objects use Donoso(
(
(
Then their forecasting performances are
Comparatively, the forecasting performance of the fuzzy collaborative forecasting method is
Therefore, the quality of collaboration can be evaluated as
It is worth noting that the performance of this collaboration model is not as good as expected.
In this model, the two objects use the same method CL1(
(
(
The
Then, the forecasting performances are compared with those of the linear methods WT(0.3) and WT(0.6). The results are summarized in
Comparison of the performances of CL1(






WT(0.3)  0.56  0.16  0.1  0.19 
CL1(3, 0.3)  0.56  0.16  0.1  0.19 
WT(0.6)  1.14  0.24  0.15  0.31 
CL1(2, 0.6)  1.12  0.23  0.14  0.3 
Obviously, the use of a nonlinear objective function may change the optimal solution. The quality of collaboration is evaluated as follows:
Then, the quality of collaboration in FCF(CL1(3, 0.3), CL1(2, 0.6)) is compared with that in FCF(WT(0.3), WT(0.6)). The results are shown in
Comparison of FCF(WT(0.3), WT(0.6)) and FCF(CL1(3, 0.3), CL1(2, 0.6)).
FCF(WT(0.3), WT(0.6))  FCF(CL1(3, 0.3), CL1(2, 0.6))  



50% 


25% 

71%  71% 

64% 


67%  67% 

58% 


52% 


36% 

As can be seen from this table, the use of a nonlinear model CL1(
This model assumes that both of the two objects use CL2(
(
(
The performances of the two objects are evaluated as
Through the collaboration of the two objects, FCF(CL2(3, 0.3, 2), CL2(2, 0.5, 3)) achieves a better forecasting performance:
The quality of collaboration is assessed as follows:
In this model, one of the two objects uses CL1(
(
(
The forecasting performance of FCF(CL1(3, 0.3), CL2(2, 0.5, 3)) is evaluated as
The quality of collaboration is assessed as follows:
In this section, the performances of the fuzzy collaborative forecasting models are compared. First, the forecasting accuracy considering the average range of forecasts, the performances of different models are compared in
The forecasting accuracy of the fuzzy collaborative forecasting models.
Secondly, in order to compare the forecasting accuracy of the models, three indicators—MAE, MAPE, and RMSE are considered. The comparison results in the three indicators are shown in
The quality of collaboration of the fuzzy collaborative forecasting models.
The forecasting accuracy (MAE) of the fuzzy collaborative forecasting models.
The forecasting accuracy (MAPE) of the fuzzy collaborative forecasting models.
The forecasting accuracy (RMSE) of the fuzzy collaborative forecasting models.
The quality of collaboration of the fuzzy collaborative forecasting models.
Forecasting the unit cost of every product type in a factory is an important task. After the unit cost of every product type in a factory is accurately forecasted, several managerial goals (including pricing, cost down projecting, capacity planning, ordering decision support, and guiding subsequent operations) can be simultaneously achieved. However, it is not easy to deal with uncertainty in the unit cost. This paper presents some fuzzy collaborative forecasting models based on a few wellknown fuzzy linear regression methods to predict the unit cost of a product. An example is used to illustrate the applicability of the proposed methodology. According to the experimental results,
(1) The effectiveness of the unit cost forecasting was greatly improved through the collaboration of the experts, especially when using FCF(CL2(
(2) With respect to the quality of collaboration on the forecasting precision, only one performance measure is proposed and the proposed performance measure can effectively compare the differences among the models.
(3) With respect to the forecasting accuracy on the forecasting accuracy among the performance measures, the one that considers MAPE can effectively compare the differences among the models.
The contribution of this study includes the following:
(1) Six fuzzy collaborative forecasting models for the unit cost forecasting are investigated. From this, the most effective one can be identified.
(2) More performance measures on the quality of collaboration have been proposed.
This work is partially supported by National Science Council of Taiwan.