# A Bi-objective Based Measure for the Scalability of Reconfigurable Manufacturing Systems

Audrey Cerqueus, Xavier Delorme

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Audrey Cerqueus, Xavier Delorme. A Bi-objective Based Measure for the Scalability of Reconfigurable Manufacturing Systems. IFIP International Conference on Advances in Production Management Systems (APMS), Sep 2021, Nantes, France. pp 544-552, 10.1007/978-3-030-85902-2_58. emse-03360785

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# A bi-objective based measure for the scalability of reconfigurable manufacturing systems

Audrey Cerqueus¹[0000-0003-0430-5971] and Xavier Delorme¹[0000-0002-9465-1255]

Mines Saint-Etienne, Univ Clermont Auvergne, CNRS, UMR 6158 LIMOS, Institut Henri Fayol, 42023 Saint-Etienne, France
{audrey.cerqueus, delorme}@emse.fr

**Abstract.** The reconfigurable manufacturing systems aim to efficiently respond to demand changes. One of the key characteristics of these systems is the scalability, i.e. the ability to modify the volume of the throughput in order to fit to the demand variability. The design of the RMS has a high impact on its scalability. In the literature, there are only few indicators to evaluate the scalability of a system and most of them are a posteriori measures. In this article, we propose a new measure to assess the scalability since the design phase of the RMS. We present experimental results on state-of-the-art instances to validate our approach. They show that the proposed measure evaluates accurately the scalability.

**Keywords:** Scalability · Reconfigurable manufacturing systems · Multi-objective indicator.

## 1 Introduction

In a context of high volatility of market conditions and increased customization leading to smaller batches, manufacturing companies have to react quickly and efficiently to changes in order to remain competitive. Reconfigurable Manufacturing Systems (RMS) have been introduced in [10] to answer to this need of adaptability. Basically the main purpose of RMS is to manage shorter product lifecycles while keeping longer production system lifecycles.

RMS are production systems composed of serial stages with identical parallel resources, which can be automated (for example Computer Numerical Control (CNC) or Reconfigurable Manufacturing Tools (RMT)), or other resources such as workers or cobots. A gantry and a conveyor are generally used to move the products in this grid of stations [9]. A RMS layout is schematized in Figure 1.

The efficiency of RMS relies on six key features: modularity (ability to reuse machines and tools), integrability (ability to rapidly and efficiently connect new modules), diagnosability (ability to identify automatically a problem in the production system), convertibility (ability to change the system for new products), customization (ability to produce different parts in a family) and scalability (ability to adapt the volume of production). The four first ones are
<figure><img src="image_3.png" /><figcaption>Fig. 1. RMS layout as seen by [9]</figcaption></figure>

mainly related to technological issues while the two last ones are related to
organizational issues. [11] states that scalability might be the most important
feature to deal with the uncertain demand or ramp-up phase. This adaptation
can be achieved by two levers, either (a) adding or removing parallel resources
on the stations, or (b) processing a full reconfiguration of the system by changing
the tools used in RMT or CNC machines[15]. A scalable system of good quality
must be able to adapt quickly, incrementally (in small steps) and cost-effectively
in order to provide at any time the exact capacity needed. In this paper, we focus
on the first lever, which is the one allowing for the shorter and less expensive
reconfiguration times, and we propose a new indicator to assess the scalability of
an RMS, integrating these three aspects, at a strategic level and independantly
of the initial state of the system.

The remaining of the paper is organized as follows: Section 2 presents the
related literature, the proposed scalability measure is explained in Section 3 and
experimental results are analyzed in Section 4. Section 5 concludes this paper.

## 2 Related literature

The articles dealing with scalability of RMS can be separated in two categories,
according to the decision level : operational or strategical.

The first category contains works dealing with the planning of the configu-
rations. They focus on how to use the scalability of RMS to adjust production
throughput over time to face the evolution of demand. [3, 4] present a dynamic
method to assess different reconfiguration policies on various scenarios of de-
mand evolution. This method deals with each reconfiguration independently.
[19] presents a heuristic method to minimize the number of machines required
for successive reconfigurations. An extension of this method has been presented
in [11], integrating buffers in RMS. The scalability level of different configura-
tions of RMS are evaluated a posteriori with the production throughput gain
when adding a given number of machines and rebalancing the line. Since the
method is a heuristic, it is difficult to evaluate if the throughput comes from
the scalability level of configurations or from the performance the optimization
method. Based on the experimental results obtained, the authors stated that
a lower number of stages leads to higher throughput and gain, mainly because
it leads to a more reliable system. [5] has also used simulation to study the
advantages of using RMS when dealing with unreliable production systems. [6] proposed a new approach for production planning to realize capacity scalability and functionality changes in planning processes. A prototypical application is developed to prove the applicability of their method. [8] presents a simulation-based method to optimize the production planning of RMS taking into account the variations of demand. [7] uses Petri nets to model an auto-adaptive RMS based on multi-agents to adjust production capacity.

The second category of studies considers the scalability of the RMS at the design phase. [10] and [18] studied the impact of different system configurations on throughput and scalability. [17] questioned the link between the balancing of the production systems and its productivity and scalability. They highlighted that unbalanced RMS can generate smaller steps of capacity changes. More recently, [13] proposed a classification of the main root causes leading to convertibility and scalability. Also, whereas RMS have been initially introduced for discrete manufacturing, [1] proposed to extend the definition of scalability to integrate process manufacturing and to calculate the average wasted capacity for a given curve of demand. In [19], a metric based on the smallest possible incremental capacity change is presented to evaluate a priori scalability on the first reconfiguration. However this metric is dependent on the current state of the system and does not allow to take into account the subsequent reconfigurations. Indeed, [15] states there is a need for new performance measures of scalability.

Finally, some works try to integrate the two decision level. In [12], the problem of design multi-product and scalable RMS for multiple production periods is addressed, minimizing design and reconfiguration costs while fulfilling demand. The authors presented two approaches: a up- and downgrading method based on approximate demand in each period; and RMT selections and reconfigurations based on long-term demand estimations. A similar problem of design and reconfiguration planning was considered in [2] where the authors use the scalability of a mono-product RMS to minimize the energy cost.

## 3 Hypervolume based indicator for scalability

In this study, we consider reconfigurations based on the same assignment of tasks to the stages. A configuration is thus defined by these two pieces of information: the balancing and the number of resources in each stage, and a reconfiguration consists in varying the number of resources on the stages, e.g. by switching on/off some resources. The cycle time of a stage is the workload (i.e., the sum of the processing time of tasks assigned to it) divided by the number of resources assigned to the stage. The takt time of a configuration is defined by the stage with the highest cycle time. Obviously, the only interesting configurations are those with the highest productivity for the same number of resources. A set of configurations can be derived for each feasible balancing, i.e. such that each tasks is assigned to one and only one stage, respecting the precedence constraints. For a given balancing, these configurations can be obtained by an iterative method:
starting with one resource on each stage and gradually adding a resource in the bottleneck stages (i.e. the stages with the highest cycle time).

Knowing the whole set of available configurations, we can sort them by increasing number of resources and calculate the smallest capacity increment starting from each configuration by looking at the resources gap with the next configuration. We can thus obtain the average value of this stability measure among all the configurations, however it would not take into account some important features such as the available range of productivity or the efficiency of configurations. Actually, a scalable balancing should ideally provide a set of highly efficient configurations (i.e., with few idle times) covering a large range of possible market demands with a small increment between them. These characteristics are very similar to the ones sought from a set of trade-offs in the fields of multi-objective optimization and we can thus use the same metrics. Here, we will focus on one of the most used multi-objective metric which is the hypervolume [20]. For an optimization problem with multiple objectives to minimize, the hypervolume is the area above the set of points to evaluate and below a reference point (point such that it is not possible to have highest value along any objective). It is denoted $\mathcal{H}$ in the following.

Two objectives are considered in our scalability measure: the takt time (to assess the productivity of the configurations) and the number of resources (to evaluate their cost). These two criteria are to be minimized. A reference point for the hypervolume computation can easily be determined since a configuration cannot have a takt time strictly greater than the sum of the processing times and the number of resources in bounded (otherwise there would be an infinity of configurations derived from a balancing).

To evaluate the quality of a set of configurations, we compare the objective values obtained for the specific set associated with a balancing (denoted by $F_a$), with the best possible values, i.e. with the whole set of feasible balancings and all derived configurations ($F_r$, it is a Pareto front [14]):

$$HV = 1 - \frac{\mathcal{H}(F_a)}{\mathcal{H}(F_r)}.$$

$HV$ is in $[0,1]$ and a low value indicates that the configurations derived from the considered balancing are close from the best possible values ($F_r$) and thus indicates that this set of configurations is of good quality.

## 4 Experimental results

We conducted experiments to evaluate the proposed indicator and to compare it with, on one hand, some classical line balancing indicators (takt time, number of stages, idle time and smoothness index computed on the configuration with one resource per station) and on the other hand the average scalability value from [19] on all configurations. This scalability indicator for a configuration is either the smallest incremental capacity in percentage (i.e. the number of bottleneck stages over the total number of resources in the system) if it is possible to add resources
in all bottleneck stations, or the number of stages otherwise (corresponding to the creation of a whole new line).

For an instance, this comparison is done for all feasible balancings, obtained by a total enumeration under the following assumptions:

- There cannot be more than three resources per stage.

- The total number of resources for a configuration cannot exceed 50% of the number of tasks.

- At most 50% of the tasks can be assigned to the same stage.

The 10 smallest instances from [16] are used as a benchmark. We limited the time for the enumeration to 5 minutes, on a computer with an Intel Core i7, with a 2.60 GHz processor with 16 GB of RAM. Under these conditions, only the 5 smallest instances completed the process under the time limit. For the other, we gradually reduced the number of operations, following a random order, until the resulting instance finished the process under the time limit. Table 1 summarizes the initial number of operations of the considered instances and the size of the instances solved by our process.

<table><thead><tr><td>Instance name</td><td>Size</td></tr></thead><tbody><tr><td>Mertens</td><td>7</td></tr><tr><td>Bowman8</td><td>8</td></tr><tr><td>Jaeschke</td><td>9</td></tr><tr><td>Jackson</td><td>11</td></tr><tr><td>Mansoor</td><td>11</td></tr></tbody></table>

<table><caption>**Table 1.** Description of the benchmark instances size (number of tasks). The table on the left are the instances for which the execution finished within the time limit and the one on the right contains the instances that needed to be reduced for the experiments.</caption><thead><tr><td>Instance name</td><td>Original size</td><td>Reduced size</td></tr></thead><tbody><tr><td>Mitchell</td><td>21</td><td>14</td></tr><tr><td>Roszieg</td><td>25</td><td>13</td></tr><tr><td>Heskia</td><td>28</td><td>11</td></tr><tr><td>Buxey</td><td>29</td><td>12</td></tr><tr><td>Sawyer30</td><td>30</td><td>11</td></tr></tbody></table>

Table 2 shows the correlation between the classical line balancing indicators and both the average scalability indicator from [19] and the proposed hypervolume metric *HV*. Since all indicators are to be minimized, the high score indicates a strong correlation. The table shows that neither the average scalability nor the *HV* indicator are correlated with the classical line balancing measures, except with the number of stages. The strong correlation with the number of stages can be explained by the fact that a low number of stages most likely allows to derive more configurations within the limit of number of resources per stage or in the system. The *HV* indicator is highly impacted by the number of configurations in the set. When computing the average scalability indicator, there is always a configuration for which the indicator corresponds to the creation of a new line (thus with a particularly high value) which has less impact if the set of configurations is large. However, the number of stages alone does not give any guaranty on the productivity of the configurations. It is also interesting to note that the average scalability is almost unrelated to the smoothness index which is the most usual indicator used to assess if a solution is well-balanced. Thus
this table shows that the classical line balancing indicators cannot be used to evaluate the scalability of a RMS.

<table><caption>**Table 2.** Correlation of the classical SALBP indicators with the scalability indicators</caption><thead><tr><th></th><th>takt time</th><th>nb of stages</th><th>idle time</th><th>smoothness</th></tr></thead><tbody><tr><td>avgScalability</td><td>-0.372</td><td>0.951</td><td>0.237</td><td>0.093</td></tr><tr><td>HV(F<sup>a</sup>)</td><td>-0.220</td><td>0.981</td><td>0.432</td><td>0.292</td></tr></tbody></table>

<table><caption>**Table 3.** Correlation of the hypervolume with the average scalability from [19]</caption><thead><tr><th>Instance</th><th>avg Scalability</th><th>Instance</th><th>avg Scalability</th></tr></thead><tbody><tr><td>Mertens</td><td>0.920</td><td>Mitchell</td><td>0.895</td></tr><tr><td>Bowman8</td><td>0.959</td><td>Roszieg</td><td>0.922</td></tr><tr><td>Jaeschke</td><td>0.953</td><td>Heskia</td><td>0.946</td></tr><tr><td>Jackson</td><td>0.927</td><td>Buxey</td><td>0.922</td></tr><tr><td>Mansoor</td><td>0.954</td><td>Sawyer30</td><td>0.942</td></tr></tbody></table>

Table 3 shows a strong correlation between the average scalability indicator from [19] and the proposed hypervolume metric *HV*, for each instance of the benchmark, with low variability. This means that on most of the balancings, the two measures agree of the assessment of the scalability, but not on all of them. Indeed, for a balancing with a high number of configurations derived, the average scalability indicator will be high, but the *HV* indicator can be quite low if the takt time of these configurations is high (when the stages are unbalanced, the productivity is then low).

<table><caption>**Table 4.** Minimum and average value of the hypervolume metric among all balancings</caption><thead><tr><th>Instance</th><th>min HV</th><th>avgHV</th><th>Instance</th><th>min HV</th><th>avgHV</th></tr></thead><tbody><tr><td>Mertens</td><td>0.0%</td><td>0.3%</td><td>Mitchell</td><td>4.4%</td><td>72.3%</td></tr><tr><td>Bowman8</td><td>3.1%</td><td>59.9%</td><td>Roszieg</td><td>4.8%</td><td>72.4%</td></tr><tr><td>Jaeschke</td><td>3.6%</td><td>48.3%</td><td>Heskia</td><td>5.8%</td><td>70.6%</td></tr><tr><td>Jackson</td><td>6.6%</td><td>61.7%</td><td>Buxey</td><td>8.4%</td><td>72.0%</td></tr><tr><td>Mansoor</td><td>7.4%</td><td>61.9%</td><td>Sawyer30</td><td>5.8%</td><td>71.1%</td></tr><tr><td colspan="4">Average on all instances</td><td>4.99%</td><td>59.05%</td></tr></tbody></table>

Finally, Table 4 shows the minimum and average values of the hypervolume indicator *HV* among all the balancings for each instance. Since the indicator *HV* is based on the ratio of the hypervolume on the set of configurations derived from the same balancing and the set of best possible values, the minimum *HV* evaluates the quality of the optimal balancing (according to the *HV* metric) with respect to a case where every reconfiguration would be possible. Here we have an average gap of 5% which corresponds to the cost of the assumption to only consider sets of configurations based on the same balancing. This cost
seems really low by comparison with the additional costs of reconfiguration times associated with a change of tools. The average HV evaluates the quality of a random balancing which is significantly higher (nearly 60%). This highlights the potential gains of optimizing this indicator during the design phase.

# 5 Conclusion

Scalability is a one of the main characteristics of RMS but its evaluation at the design step has not received a lot of attention. Only two works have considered this issue: (a) [19] has defined a measure based on the smallest increment required to increase the capacity starting from for a given configuration, and (b) [13] has tried to identify the core characteristics leading to the scalability. In this paper, we propose a new measure based on a classical multi-objective metric to assess the scalability level of a balancing by taking into account all the configurations which can be achieved.

The preliminary results show that this approach can be viewed as an extension of the smallest increment measure which is not dependent on a specific initial state. It also reveals that the usual line balancing criteria are mainly unrelated with the scalability, enhancing the need for dedicated metrics which could be used to optimize the scalability at the design step. This conclusion fits with the statement of [17] on the possibility to have a good scalability level in some unbalanced RMS. It is also interesting to note that, similarly to the conclusions reported in [11], in our setting a lower number of stages is strongly and positively correlated with the scalability even without taking the reliability into account.

A first perspective of this work would be to use the proposed measure to characterize what makes a system scalable. In addition, this work focuses on only one of the RMS characteristics. Even if a priori the different characteristics seem to be independent, it would be interesting to study their interaction. Finally, the development of methods to design RMSs optimizing their scalability together with classical performance measures would be needed.

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