Process Flexibility for Multi-Period Production
Systems
April 21, 2017
Introduction.
In today’s market, firms are often required to offer their cus-
tomers more diverse product portfolios in order to keep ahead of the market
competition. The increase in product offerings, however, would invariably drive
up the variabilities of product demands. To deal with this challenge, it has been
observed that firms often adopt an operational strategy known as process flexibility (see Simchi-Levi (2010) and Cachon and Terwiesch (2009)). Also known
as capacity pooling, process flexibility allows the firm to quickly change the
production of different types of products from one plant to another with little
penalty in time and cost, thereby adapting itself to reduce the operational costs
under demand fluctuations.
Firms can choose to add different degrees of process flexibilities to their
production system. For example, a production system has full flexibility if
any of its plants is capable of producing any of the product (see Figure 1).
While a fully flexible system has important benefits such as capacity pooling,
it is often impractical due to its enormous implementation cost. As a result,
many manufacturers have chosen to add a sparse amount of process flexibility,
where each plant is only capable of producing a small subset of the products.
Surprisingly, researchers have observed that this practice is often enough to
guarantee excellent or even near-optimal performance. In the seminal paper of
Jordan and Graves (1995), the authors observed that by adding a little bit of
flexibility to the plants to create a chaining structure (also referred to as the
long chain, illustrated in Figure 1), one often obtains almost the same benefit
as a fully flexible system.
Much of the theoretical analysis on this topic focuses on a single-period
model (see, e.g., Chou et al. (2010, 2011), Simchi-Levi and Wei (2012), Wang
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Dedicated
Long Chain
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Figure 1: Examples of Process Flexibility Structures.
and Zhang (2015), Chen et al. (2015), Désir et al. (2016)). However, in practice,
firms often operate in a multi-period setting, where the unsatisfied demand is
backlogged into future time periods. The multi-period setting gives rise to
a complex multi-stage closed-loop dynamic optimization problem, where the
analysis of the single-period model does not carry through. This naturally leads
to the following research question: In a multi-period make-to-order environment,
can a sparse flexibility structure achieve a performance that is close to that of
the full flexibility structure?
Our Contributions.
In this paper, we provide an affirmative answer to the
question above. The first major contribution of our paper proves that under
realistic modeling assumptions, for any system with m plants and n products,
there exists a sparse structure with only m + n flexibility arcs, such that the
sparse structure achieves approximately the same performance as full flexibility
structure (consisted of mn arcs) as the plants becomes heavily utilized. In
addition, the proof of our result is constructive, thereby providing an efficient
algorithm for designing near-optimal, sparse flexibility structures.
Another major contribution of the paper is that we establish the necessity of
having at least m+n arcs for a flexibility structure to achieve a performance that
is close to that of full flexibility. More specifically, we construct example systems
in which any flexibility structure with at most m + n − 1 arcs has a performance
that is at least a constant factor away from the performance of full flexibility. As
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a result, our analysis not only echoes “a little bit of flexibility goes a long way”,
a recurrent theme in the process flexibility literature, but also quantifies exactly
how much flexibility is needed in the multi-period make-to-order environment.
Moreover, an important step in characterizing the performance of these systems
uses a lower bound on the performance of any flexibility structure, which we
believe is of independent interest.
References
Cachon, G., C. Terwiesch. 2009. Matching supply with demand . 2nd ed. McGraw-Hill.
Chen, Xi, Jiawei Zhang, Yuan Zhou. 2015. Optimal sparse designs for process flexibility
via probabilistic expanders. Operations Research 63(5) 1159–1176.
Chou, M. C., G. A. Chua, C.-P. Teo, H. Zheng. 2010. Design for process flexibility:
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43–58.
Chou, M. C., G. A. Chua, C.-P. Teo, H. Zheng. 2011. Process flexibility revisited: the
graph expander and its applications. Operations Research 59(5) 1090–1105.
Désir, Antoine, Vineet Goyal, Yehua Wei, Jiawei Zhang. 2016. Sparse process flexibility
designs: Is the long chain really optimal? Operations Research 64(2) 416–431.
Jordan, W., S. Graves. 1995. Principles on the benefits of manufacturing process
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Simchi-Levi, D. 2010. Operations Rules: Delivering Customer Value through Flexible
Operations. MIT Press.
Simchi-Levi, D., Y. Wei. 2012. Understanding the performance of the long chain and
sparse designs in process flexibility. Operations Research 60(5) 1125–1141.
Wang, X., J. Zhang. 2015. Process flexibility: A distribution-free bound on the performance of k-chain. Operations Research 63(3) 555–571.
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