A Credibility-Based MPS/MRP Integrated Programming Model Under Complex Uncertainty

Abstract

Master production schedules (MPS) and Material requirements planning (MRP) are two important planning modules in the production planning system. In this paper, a new fuzzy credibility-based double-sided chance-constrained programming (FCDCP) model is proposed for the capacitated MPS/MRP integrated programming problem under complex uncertainty. The proposed model supports simultaneous lot sizing and scheduling decisions and can deal with uncertainty presented as fuzziness in both right-hand and left-hand sides of constraints. To tackle the FCDCP model, the corresponding defuzzification methods based on credibility measure are proposed for transforming the fuzzy MPS/MRP integrated programming model into a crisp equivalent mixed-integer programming model. When decision-makers decide on MPS/MRP integrated programming problem, the optimal lot sizing and scheduling solutions from the FCDCP model can not only provide an effective method for production planners to deal with the uncertainty in the production system, making the production plan more reasonable and accurate, but it can also help decision-makers obtain the optimal production plans and determine the best allocation of production resources, effectively reducing the total cost of the production system. Finally, an industrial example from a transmission manufacturer is provided to show the applicability and flexibility of the model.

Appendices

Appendix A: Proof of Theorem 1

Proof

(1): Because $\alpha \le 1/2$ and $Cr\left\{ {\hat{\xi } \le r} \right\} \ge \alpha$, from Eq. (17), it is easy to know that $a_{1} \le r \le a_{2}$, and then we have $\frac{{r - a_{1} }}{{2(a_{2} - a_{1} )}} \ge \alpha$, therefore conclusion(1) can be obtained.

(2) Similar to (1), if $\alpha > 1/2$ and $Cr\left\{ {\hat{\xi } \le r} \right\} \ge \alpha$, we have $a_{2} \le r \le a_{3}$, according to Eq. (17), it is easy to prove the conclusion(2).

The graphical explanation of Theorem (2) is shown in Fig. 10.

Appendix B: Proof of Theorem 2

Proof

For a given fuzzy number $\hat{\xi }$ = ($a_{1} ,a_{2} ,a_{3}$) is denoted by the membership function $\mu$. It is easy to know that for any $\lambda \in (0,1)$, there exist two values on the x-axis for the inverse function of $\mu (x)$, they are $(1 - \lambda )a_{1} + \lambda a_{2}$ and $(1 - \lambda )a_{3} + \lambda a_{2}$, respectively, and $(1 - \lambda )a_{1} + \lambda a_{2}$ < $(1 - \lambda )a_{3} + \lambda a_{2}$, which can be viewed as two points of $\user2{Ax}$ in real line R, as shown in Fig. 10. To ensure that $Cr\left\{ {{\text{f}} (\user2{x}, {\varvec{\hat{\xi }}}) \le 0} \right\} \ge 1/2$, the smaller value, i.e.,$(1 - \lambda )a_{1} + \lambda a_{2}$ is valid according to Eq. (18).

Furthermore, since $\user2{A,x} \in \user2{R}^{n}$, therefore $\user2{Ax} \in \user2{R}$, and $Cr\left\{ {{\text{f}} (\user2{x}, {\varvec{\hat{\xi }}}) \le 0} \right\} = Cr\left\{ {\user2{Ax} \le \hat{\xi }} \right\} \ge \alpha$. According to Theorem (2), we have $2(1 - \alpha )a_{2} + (1 - 2\alpha )a_{1} - {\user2{Ax}} \ge 0$. As can be seen from the geometric relationship shown in Fig. 10, the following formula can be deduced:

$$ \lambda = \frac{{{\user2{Ax}} - a_{1} }}{{a_{2} - a_{1} }} \le \frac{{2(1 - \alpha )a_{2} + (1 - 2\alpha )a_{1} - a_{1} }}{{a_{2} - a_{1} }} = 2(1 - \alpha ) $$

Therefore, the proof of Theorem (3) is complete.

Appendix C:. Proof of Theorem 3.

Proof

Since both $h_{k}^{ + } ({\mathbf{x}})$ and $h_{k}^{ - } ({\mathbf{x}})$ are non-negative, and $h_{k} ({\mathbf{x}}) = h_{k}^{ + } ({\mathbf{x}}) - h_{k}^{ - } ({\mathbf{x}})$, and we have

$g({\mathbf{x}},\hat{\varvec{\xi }}, {\varvec{\hat{\eta }}}) = g({\mathbf{x}},\hat{\varvec{\xi }}) - \hat{\eta }$, $Cr\left\{ {g({\mathbf{x}},\hat{\varvec{\xi }}, {\varvec{\hat{\eta }}}) \le 0} \right\}{ = }Cr\left\{ {g({\mathbf{x}},\hat{\varvec{\xi }}) \le \hat{\eta }} \right\}$.

According to Theorem (1), $g({\mathbf{x}},\hat{\varvec{\xi }})$ is also a triangular fuzzy variable, and can be denoted as the following form:

$$ g({\mathbf{x}},\hat{\varvec{\xi }}) = \left[ {\begin{array}{*{20}c} {\sum\limits_{k = 1}^{n} {\left[ {h_{k}^{ + } (x)\xi_{k1} - h_{k}^{ - } ({\mathbf{x}})\xi_{k3} } \right]} + h_{0} ({\mathbf{x}})} \\ {\sum\limits_{k = 1}^{n} {\left[ {h_{k}^{ + } (x)\xi_{k2} - h_{k}^{ - } ({\mathbf{x}})\xi_{k2} } \right]} + h_{0} ({\mathbf{x}})} \\ {\sum\limits_{k = 1}^{n} {\left[ {h_{k}^{ + } (x)\xi_{k3} - h_{k}^{ - } ({\mathbf{x}})\xi_{k1} } \right]} + h_{0} ({\mathbf{x}})} \\ \end{array} } \right] $$

And also, $\hat{\eta }{ = (}b_{1} {,}b_{2} ,b_{3} {)}$, according to Eq. (22), it is easy to prove that Eqs. (29) and (30) are true.