Determination of dynamic safety stocks for cyclic production schedules

Abstract

Safety stocks are necessary to accommodate supply or demand uncertainties. Typically, safety stocks are held constant throughout the management horizon. In this paper, we study the determination of dynamic safety stock levels under cyclic production schedules. Currently, companies often operate according to contract orders with fixed and cyclic delivery dates and varying quantities. In this context, companies have attempted to implement cyclic production modes for their cyclic delivery requirements. Indeed, cyclic production scheduling has many benefits, such as respecting just-in-time principles by synchronising production with demand and improving shop floor control. The objectives of this paper are twofold: (1) to propose a new dynamic approach for determining safety stock levels that is adapted to cyclic production and (2) to demonstrate the limitations of traditional approaches under nonstationary demands. Our dynamic approach is based on modelling a repetitive production sequence comprising many manufacturing orders. Our model estimates the dynamic safety stock levels necessary to accommodate cyclic manufacturing orders with uncertain quantities and start dates using a Monte Carlo simulation approach. For the case in which all demands follow normal probability distributions, we validated the results of our simulation model by comparing them to numerically approximated theoretical results. We then compared the proposed dynamic simulation approach with the traditional approach. The results demonstrate that the dynamic approach is more efficient in terms of simultaneously minimizing the required safety stocks and improving the service level by decreasing the probability of stock outs. Finally, we applied our simulation approach to a case in which all demands follow uniform probability distributions.

Appendix

1.1 Appendix 1: In this appendix, we determine the theoretical probability distribution of the gap Gi

X_ui is the latest start date that covers delivery numbers 1 to u covered by MO_i.

$$ {X}_{ui}={\lambda}_{ui}-{\sum}_{v=1}^u\frac{y_{vi}}{P{R}_i} $$

λ _ui :

Delivery date of delivery number u in the set of deliveries covered by MO_i,

y_vi:

Quantity of delivery number v in the set of deliveries covered by MO_i, and

PR _i :

Production rate of MO_i.

LSD_i represents the latest start date of an MO_i if we want to produce all the items on time.

$$ LS{D}_i=\underset{u=\mathrm{1..}{U}_1}{\mathit{\operatorname{Min}}}\;{X}_{ui} $$

It is more complex to determine the probability distribution of LSD_i because it is the minimum of a set of normal distributions (Hill 2011). Let us determine the probability that LSD_i is less than or equal to a certain value a. If we assume that all X_ui are independent, we can formulate the following relations.

$$ {\displaystyle \begin{array}{l}\Pr \left( LS{D}_i\le a\right)=\Pr \left(\underset{u=\mathrm{1..}\ {U}_i}{\min }{X}_{ui}\le a\right)\\ {}\kern3.599998em =1-\Pr \left({X}_{1i}>a\ and\ {X}_{2i}>a\ and\dots and\ {X}_{U_ii}>a\ \right)\\ {}\kern3.599998em =1-\prod \limits_{u=1}^{U_i}\Pr \left({X}_{ui}>a\right)\\ {}\kern3.599998em =1-\prod \limits_{u=1}^{U_i}\frac{1}{\sigma_{Xui}\sqrt{2\pi }}{\int}_a^{\infty }{e}^{-\frac{1}{2}{\left(\frac{x-{\mu}_{Xui}}{\sigma_{Xui}}\right)}^2} dx\end{array}} $$

Thus, the CDF of the gap G_i can be determined as follows.

$$ {\displaystyle \begin{array}{l}\Pr \left({G}_i\le a\right)=\Pr \left({t}_i-\underset{u=\mathrm{1..}\ {U}_i}{\min }{X}_{ui}<\frac{a}{P{r}_i}\right)\\ {}\kern.3em =\Pr \left({t}_i<\frac{a}{P{r}_i}+\underset{u=\mathrm{1..}\ {U}_i}{\min }{X}_{ui}\right)={\int}_{-\infty}^{\infty}\Pr \left({t}_i=s\right)\Pr \left(\underset{u=\mathrm{1..}\ {U}_i}{\min }{X}_{ui}>s-\frac{a}{P{r}_i}\right) ds\\ {}\Pr \left({G}_i\le a\right)={\int}_{-\infty}^{\infty}\Pr \Big({t}_i\\ {}=s\Big)\Pr \left({X}_{1i}>s-\frac{a}{P{r}_i} and\ {X}_{2i}>s-\frac{a}{P{r}_i} and\dots and\ {X}_{U_ii}>s-\frac{a}{P{r}_i}\right) ds\\ {}\Pr \left({G}_i\le a\right)={\int}_{-\infty}^{\infty}\Pr \left({t}_i=s\right)\Pr \left({X}_{1i}>s-\frac{a}{P{r}_i}\right)\Pr \left({X}_{2i}>s-\frac{a}{P{r}_i}\right)\dots \Pr\;\left({X}_{U_ii}>s-\frac{a}{\Pr_i}\right)\; ds\\ {}\begin{array}{l}\;\\ {}\Rightarrow \Pr \left({G}_i\le a\right)={\int}_{-\infty}^{\infty}\frac{1}{\sigma_{t_i}\sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{s-{\mu}_{t_i}}{\sigma_{t_i}}\right)}^2}\left({\prod}_{u=1}^{U_i}\frac{1}{\sigma_{Xui}\sqrt{2\pi }}{\int}_{s-\frac{a}{P{r}_i}}^{\infty }{e}^{-\frac{1}{2}{\left(\frac{x-{\mu}_{Xui}}{\sigma_{Xui}}\right)}^2} dx\right) ds\end{array}\end{array}} $$