Pooling heterogeneous products for manufacturing environments

Abstract

In a stochastic environment pooling naturally leads to economies of scale, but heterogeneity can also create variability. In the article, we investigate this trade-off in the case of a manufacturing environment. Pooling for queueing systems has been widely investigated in the literature on the design of service systems; however, much less attention has been given to manufacturing systems where jobs are given a due date upon arrival. In such systems it is not the elapsed time until the actual completion of the job that counts, but rather the due date lead time that can be promised to the customer in order to guarantee a high service level. The purpose of the article is to get a deeper understanding about how pooling strategies and lead-time decisions can be implemented to attain a high due-date performance. To this end, we develop a simulation and analytical study to determine the benefits of pooling manufacturing systems with heterogeneous demand streams. Our work allows managers to identify the characteristics of production systems such that a pooling strategy would be beneficial. Our results demonstrate that when a due-date setting and scheduling mechanism is implemented, heterogeneity does generally not lead to deterioration of performance, as previously observed in service environments. Our studies also reveal that the benefits of pooling in terms of the expected sojourn time obtained by a simple analytical treatment can serve as a good prediction of the benefits of pooling on the average due date lead time in a wide range of situations.

Appendix

For the comparison of the production configurations with different number of order types, we add an additional superscript n to the appropriate notation. In order to show that $\varepsilon (E\left[ S_g^{n}\right] )\le \varepsilon (E\left[ S_g^{n+1}\right] )$, it is sufficient that $E\left[ S_g^{d,n}\right] \le E\left[ S_g^{d,n+1}\right]$ and $E\left[ S_g^{p,n}\right] \ge E\left[ S_g^{p,n+1}\right]$.

The inequality $E\left[ S_g^{d,n}\right] \le E\left[ S_g^{d,n+1}\right]$ can be written as follows:

$$\begin{aligned} \dfrac{n}{n+1}\le \dfrac{\sum _{i=1}^n k^{\tfrac{i-1}{n-1}}}{\sum _{i=1}^{n+1} k^{\tfrac{i-1}{n}}}. \end{aligned}$$

(18)

The right side of inequality (18) is an increasing function of k and it achieves the minimum (i.e. $\dfrac{n}{n+1}$) when $k=1$. Then, the inequality is satisfied for any value of $k\ge 1$.

Based on Eqs. (11) and (12), the global expected sojourn time for system p with n order types can be expressed as:

$$\begin{aligned} E\left[ S_g^{p,n}\right] =\rho \tau _1 k \sum ^n_{i=1} \left[ k^{\frac{i-1}{n-1}}(n-i\rho )(n-(i-1)\rho )\right] ^{-1}+\tau _1 k\left[ \sum ^n_{i=1} k^{\frac{i-1}{n-1}}\right] ^{-1}. \end{aligned}$$

(19)

Equation (19) shows that increasing the number of order types between the order types with the minimum and maximum expected processing times will decrease the variance of the processing times of the pooled facility and will consequently lead to a decrease in the global expected sojourn time. Therefore, $E\left[ S_g^{p,n}\right]$ is decreasing in n.