New fuzzy method for improving the precision of productivity predictions for a factory

Abstract

Predicting productivity is crucial in managing a factory. However, optimizing the precision of productivity prediction is not easy because two concerns must be addressed simultaneously: the hit rate and the average range. To enhance the precision of predictions of the productivity of a factory, a new fuzzy method is proposed in this paper. First, the hit rate and average range are merged into the cost for inclusion (CFI) to evaluate the precision of productivity prediction more reasonably. Subsequently, the CFIs of a traditional probabilistic method and four existing fuzzy methods are compared. According to the results of the comparison, the 100 (1 – α) % inclusion interval is used to reduce the CFI at the expense of a slight decrease in the hit rate. Subsequently, to improve the hit rate and CFI for testing and unlearned data, a selectively widened inclusion interval is adopted. Some theoretical properties of the proposed methodology are proven. In addition, data from two real cases, a dynamic random access memory (DRAM) factory and a wafer foundry, were used to illustrate the applicability of the proposed methodology. According to the experimental results, the wafer foundry exhibited a higher maturity level of productivity learning than the DRAM factory. In addition, the proposed methodology outperformed the existing probabilistic and fuzzy methods in minimizing the CFI (i.e., maximizing the precision). Further, the selectively widened inclusion interval was proven to be an effective means for trading the training performance for the testing performance.

Appendix

1.1 LP Model I

$${\text{Min}}\,Z_{1} = \sum\limits_{t = 1}^{T} {\left( {p_{3} - \frac{{b_{1} }}{t} - p_{1} + \frac{{b_{3} }}{t}} \right)}$$

(A1)

subject to

$$\ln (P_{t} ) \ge p_{1} - \frac{{b_{3} }}{t} + s\left( {p_{2} - \frac{{b_{2} }}{t} - p_{1} + \frac{{b_{3} }}{t}} \right)$$

(A2)

$$\ln (P_{t} ) \le p_{3} - \frac{{b_{1} }}{t} + s\left( {p_{2} - \frac{{b_{2} }}{t} - p_{3} + \frac{{p_{1} }}{t}} \right)$$

(A3)

$$0 \le b_{1} \le b_{2} \le b_{3}$$

(A4)

$$p_{1} \le p_{2} \le p_{3}$$

(A5)

1.2 QP Model I

$${\text{Max}}\,Z_{2} = \bar{s}$$

(A6)

subject to

$$\sum\limits_{t = 1}^{T} {\left( {p_{3} - \frac{{b_{1} }}{t} - p_{1} + \frac{{b_{3} }}{t}} \right)} \le Td$$

(A7)

$${ \ln }(P_{t} ) \ge p_{1} - \frac{{b_{3} }}{t} + s_{t} (p_{2} - \frac{{b_{2} }}{t} - p_{1} + \frac{{b_{3} }}{t})$$

(A8)

$$\ln (P_{t} ) \le p_{3} - \frac{{b_{1} }}{t} + s_{t} \left( {p_{2} - \frac{{b_{2} }}{t} - p_{3} + \frac{{p_{1} }}{t}} \right)$$

(A9)

$$\bar{s} = \frac{{\sum\nolimits_{t = 1}^{T} {s_{t} } }}{T}$$

(A10)

$$0 \le b_{1} \le b_{2} \le b_{3}$$

(A11)

$$p_{1} \le p_{2} \le p_{3}$$

(A12)

$$0 \le s_{t} \le 1$$

(A13)

1.3 QP model II

$$\begin{aligned} {\text{Min}}\,Z_{3} &= w_{1} \sum\limits_{t = 1}^{T} {(\ln P_{t} - p_{2} + \frac{{b_{2} }}{t})^{2} } \\ &\quad +\, w_{2} \sum\limits_{t = 1}^{T} {\left( {\left( {\ln P_{t} - p_{3} + \frac{{b_{1} }}{t}} \right)^{2} + \left( {\ln P_{t} - p_{1} + \frac{{b_{3} }}{t}} \right)^{2} } \right)} \end{aligned}$$

(A14)

subject to (A2)–(A5).