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# Metamodeling of Deteriorating Reusable Articles in a Closed Loop Supply Chain

Eoin Glennane, John Geraghty

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Eoin Glennane, John Geraghty. Metamodeling of Deteriorating Reusable Articles in a Closed Loop Supply Chain. IFIP International Conference on Advances in Production Management Systems (APMS), Sep 2021, Nantes, France. pp.198-207, 10.1007/978-3-030-85874-2_21. hal-04030380

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# Metamodeling of Deteriorating Reusable Articles in a Closed Loop Supply Chain

Eoin Glennane¹ and John Geraghty²

Advanced Processing Technology Research Center (APT), School of Mechanical and Manufacturing Engineering, Dublin City University, Dublin 9, Ireland

¹eoin.glennane2@mail.dcu.ie

²john.geraghty@dcu.ie

**Abstract.** In this paper a closed loop supply chain for a reusable, deteriorating tool is presented. The tool is used in a manufacturing process on an item in a linear supply chain. A model is created for the linear item supply chain and the tools closed loop supply chain to analyse the interactions between them and various input parameters so that output responses of the system can be modelled. Three approaches are taken to model the system, a brute force factorial design, a modified version of a Latin hypercube space filling design, and a fast flexible space filling design. It is found that all three methods can describe responses that require only a few inputs well but cannot accurately predict more complex responses without all the relevant factors. Space filling designs should be used if more factors are needed as they minimise the total amount of simulations needed to produce an accurate model.

**Keywords:** Reusable Articles, Deteriorating Article, Closed Loop Supply Chain, ExtendSim, Space Filling Design, Metamodeling.

## 1 Introduction

Improving efficiency of supply chains, implementing reverse supply chains and remanufacturing are all topics of research that are becoming more important as the world's resources become scarcer. The desire for companies to embrace this movement has been driven by environmental, social, and financial motivations as more research proves the efficacy of implementing such systems (Mastos, 2019).

This paper presents supply chain that produces two items, one high and one low quality, using the same deteriorating tool. The company running the supply chain wants to know how well it is running. To do this a simulated version of the item processing supply chain as well as the tool supply chain is created. The company currently orders new tools to replenish the supply of tools for each line as they are needed but random ordering times can be costly, and less dependable when trying to meet production targets. Instead, an ordering policy for purchasing new tools according to a schedule should be created that aims to meet production targets at a high percentage of the time. To create an ordering policy that will meet its targets and can be adjusted according to multiple input factors is the ultimate goal for the company.
ExtendSim, a program for modelling discrete event, continuous, agent based, discrete
rate and mixed mode processes, was used to create and run the simulations. All simu-
lations were conducted on a desktop computer with 32gb of RAM, an ASUS Strix GTX
970 Graphics Card, and an Intel Core i5-9600K CPU @ 3.7GHz. Design-Expert, a sta-
tistical software package was used to design the experimental scenarios. Microsoft Ex-
cel, JMP (another statistical software analysis package) and Design Expert were used
to analyse the results of the simulations. In the final iteration of the model, each simu-
lation took 5-7 seconds to complete and over eight thousand simulations were con-
ducted for a total computational time of roughly 13.3 hours. On a higher specification
CPU this time may be reduced.

## 2 Objectives

The main objective of the paper is to analyse and compare multiple metamodeling
methods applied to a Closed Loop Supply Chain (CLSC) that included a reusable arti-
cle. A sufficiently accurate metamodel of a CLSC could aid in the decision-making
process and allow a company to predict the outcome of choosing certain production
parameters or the effect of setting certain ordering policies without having to simulate
each individual scenario which could number in the 1000's or 10'000s and could take
hours or days to run for more complex supply chains.

The objective of the metamodeling methods is to minimise the number of simulations needed to produce an accurate metamodel. As a system becomes more complex, the computational time needed to complete a single run increases exponentially hence the need to minimise the total number of simulations. Within each metamodel, the goal is to minimise and maximise certain responses such as the time an item spent queueing or the time between new tool orders, respectively.

## 3 Literature Review

Closed Loop Supply Chains are a key component of this study. While the items being processed by the model are not in a closed loop, the tools used to process the items can be classed as a reusable product (Carrasco-Gallego et al., 2012) and also a deteriorating product (Moubed et al., 2021) inside a closed loop supply chain. Singh & Saxena, 2013, explored a mathematical approach to a very similar problem in which remanufacturing of a pair of deteriorating items of two different qualities was integrated into a closed loop supply chain. This paper takes a simulation-based approach.

Metamodeling has become an important tool for operational efficiency in many differ-
ent types of industries, from design of vegetative filter strips (Lauvernet & Helbert,
2020) to satellite visibility prediction (Wang et al., 2019) and more. With the goal of
simulating how a complex system reacts to inputs accurately and quickly, metamodel-
ing, if conducted correctly, can output a simple model of a system and remove the need
to conduct further simulations. The Latin Hypercube design has been shown to work with this type of model, one that includes random demand while minimising system outages (Chen et al., 2019). Factorial designs are useful and quick at examining models with multiple independent variables (Haerling (Adamson) & Prion, 2020) however the total amount of simulations can increase significantly as a model becomes more complex.

# 4 Model Development

With the goal of examining how a reusable article interacts with a supply chain, model development began in ExtendSim with the linear, forward supply chains for the items. The model changed many times throughout the course of model development as more about the interactions between certain aspects of the model were understood. The mechanisms by which the tool returned to the supply chain for use changed multiple times over the course of the model development but always kept to a similar pattern.

The tool would be kept in a queue until an item entered the queue. The tool and item would leave the queue together, the item would be processed while the tool waited in another queue for the item to finish processing. Once the item was finished being processed it was again, released from the second queue with the tool, at which point the item would exit the supply chain, having been processed, while the tool was redirected to be artificially deteriorated. Depending on the numerical value for the tool's quality post degradation, the tool would then be sent back to its original queue to wait for another item or be sent to an identical queue to be used in a lower quality production line or discarded entirely.

The model went through three major iterations, being finalised once all the factors chosen could be input from a database into the system correctly and the responses of the system were calculated and input into a database correctly. The final HQ Item supply chain is shown in Figure 1.

<figure><img src="image_4.png" /><figcaption>Fig. 1. - Finalised Item Supply Chain</figcaption></figure>

Item creation was a factor that seemed to have the most effect on the responses in the system and was one of the hardest areas to balance so that all factors and responses could be measured adequately. Process Time for the Machine was set to a lognormal distribution with a mean of 1 and a standard deviation of 0.1 while the distribution for
the “Create” Block was an exponential distribution that was varied over the simulations between a mean of 1.1 and 1.25. The Lognormal distribution has been shown to model activity time(Trietsch et al., 2012) in real scenarios while the Exponential distribution has been shown to model interarrival times in processes such as the one in this model(Helbing et al., 2006) in industry settings. One of these responses is measured by the “Information Block” directly after the “Queue Matching” block labelled “HQ Tool Item” in Figure 1.

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** - Tool Creation and Return</figcaption></figure>

The number of tools currently in the system was tracked using a "sensor" block as seen in Figure 2. With the sensor block located just after tool creation, it can track how many items have entered the supply chain, while the "hq exit" variable tracks the number of tools that have left this specific supply chain. When the number tracked by the sensor incremented below the limit set by the user or by the scenario criteria, a signal would be sent to purchase new tools for the system.

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** – (a.) Tool Degradation and (b.) New Tool Creation</figcaption></figure>

When each tool is detached from a processed item, it goes through the degradation and return process as shown in Figure 3 (a.). The “Get” and “Set” blocks, accompanied by math and equation blocks, simulate the degradation of the tool by incrementing the condition variable of each individual tool that passes through by a certain amount. Once a tool is degraded, its new condition variable is checked, and the tool is either returned to its original pool of tools or sent to the low quality production line. For example, if a slightly used tool entered the “Get” block with a value of 0.62 and was degraded by 0.05 to a value of 0.57, and the cut-off point for high quality tools was 0.6, then the tool would be sent on the downward path to be used in the lower quality production chain.
One of the criteria that was chosen as an important variable was the cut-off point at which a tool was demoted to another supply line or discarded altogether. Creating new tools for the system is the area that took the greatest number of iterations to complete fully before it behaved in a way that could be analysed correctly and easily, the final iteration of which is seen in Figure 3 (b.). Whenever the sensor block's reading for "number of items currently in the area" goes below a threshold as set by the user/scenario, a signal is sent to send new tools into the supply chain to be used. Using another sensor block, an information block and an "Unbatching" block, the system, when it senses this signal, opens the gate for a single tool to be allowed through. These single tools passed through the information block which measures the average amount of time between each of the orders, then passes through the "Unbatching" block, splitting the tool into multiple tools according to a variable set by the user/scenario. This factor of the number of tools the individual tool is split into is referred to as the Batch Size.

The number of factors in the final design was six. The names and limits of each of the factors were decided based on human observation of the model, considering the response of the system over multiple iterations of model design. Initial Tools (4,15), Low Inventory Limit (0,4), HQ Tool Cut-off (0.3,0.6), LQ Tool Cut-off (0,0.2), Batch Size (1,6), and Item Mean (1.1,1.25) were the factors and limits chosen. The number of responses for the system was four and included Time between Orders (for tools) and Item Queue Time for each of the two supply lines.

# 5 Metamodeling

With the limits for the factors chosen and the responses ready to calculate, the metamodeling could begin. Two iterations of metamodeling occurred, with the result of the first iteration being a reduction in the number of factors from eight to the final six. The reduction of the number of factors included in the model reduced the number of simulations required for the Design Expert metamodel significantly. As the design expert model is a factorial one, the number of simulations required by the model is equal to two to the power of the number of factors. With eight factors, 256 simulations must be run to create that model. However, to reduce any outlier and randomness in the model, it was decided that ten repetitions of each scenario had to be run and have the results averaged which increased this to 2560 simulations. The reduction of the number of factors from eight to six reduced the total number of simulations needed from 2560 to just 640, a reduction of 75%. This adjustment of the number of factors also influenced both JMP space filling metamodel designs. Both space filling designs had the total number of simulations required lowered from 800 to 600, a reduction of 25%.

The first metamodel created in JMP was a Latin hypercube-based design. This type of design has been shown to produce accurate metamodels and assist in the decision making process when multiple variables and responses are taken into account (Ben Ali et al., 2018). One problem that was encountered in this path was that the Latin Hypercube generator in JMP only allowed continuous values for each of its factors. While testing
this data in ExtendSim, many problems occurred with duplication of items and tools when a factor such as batch size had a non-integer value attached, so changes to the Latin Hypercube Model had to be made. Each of the factors that required only integer values had its generated results rounded to the nearest integer, simulating that the data was categorical. Three Categorical factors (Initial Tools, Low Inventory Limit and Batch Size) and three Continuous factors (HQ Tool Cut-off, LQ Tool Cut-off and Item Mean) were included in the experimental design.

While this may have compromised the integrity of the metamodeling method, the other option was to not use the method altogether. This modified approach did not contain any of the duplication errors and the results produced were error free in initial testing. This method is mentioned from here on as the Rounded Latin Hypercube (RLHC) method. The Fast Flexible Filling Design (FFFD) in JMP allowed for categorical data alongside continuous data so no cleaning of the input factors for the simulations was required. This was chosen alongside RLHC for comparison.

# 6 Simulation Setup

Databases were created in ExtendSim so that any number or combination of runs could be setup to simulate back-to-back. The amount of runs that ExtendSim conducted was always equal to the number of rows in the factors database so once the run button was pressed, ExtendSim ran all the simulations required. The data recorded in each run was written into another database for the responses of the system at the end of each run. This data was exported to excel where each of the ten duplicate runs were averaged. The averaged results could then be exported back into Design Expert and JMP as needed and analysis of the results was conducted.

# 7 Results

The aim of using the metamodels is to create an accurate model of the system while minimising the amount of computation needed to make it accurate. As each of the three methods needed roughly the same amount of simulation time (600-640 runs each) to complete their models in the final iteration, they all perform the same in this aspect. The way in which they must be compared is then by the accuracy of their models for each of the responses. Comparison of the $R^2$ values for each of the models for each of the responses was conducted and tabulated in Table 1. An $R^2$ value above 0.90 is deemed to be accurate and can be classified as successfully emulating the simulation, provided it is not disqualified in the validation stage.

The Design Expert metamodel, would be expected to achieve very accurate results when predicting all four values in the simulations, whereas the FFFD or RLHC would not be expected to predict LQ Time between Orders or LQ Item Queue Time very well in comparison to the other two criteria. This is the case as can be seen in Table 1,
however, the DE model, while producing a high R² value for two of the criteria, does not predict an accurate value for the scenario (highlighted in red).

<table><caption>**Table 1.** Predicted vs Simulated Results</caption><thead><tr><th rowspan="2"></th><th>Sim Results</th><th colspan="2">FFFD Predicted</th><th colspan="2">RLHC Predicted</th><th colspan="2">DE Predicted</th></tr><tr><th>Results</th><th>R<sup>2</sup></th><th>Sim</th><th>R<sup>2</sup></th><th>Sim</th><th>R<sup>2</sup></th><th>Sim</th></tr></thead><tbody><tr><th>HQ TBO</th><td>28.84</td><td>0.921</td><td>29.09</td><td>0.934</td><td>28.86</td><td>0.995</td><td>29.7</td></tr><tr><th>LQ TBO</th><td>201.7</td><td>0.453</td><td>116.99</td><td>0.580</td><td>294.97</td><td>0.962</td><td>43.5</td></tr><tr><th>HQ IQT</th><td>0.95</td><td>0.921</td><td>1.18</td><td>0.934</td><td>1.18</td><td>0.908</td><td>1.71</td></tr><tr><th>LQ IQT</th><td>0.41</td><td>0.453</td><td>0.52</td><td>0.580</td><td>0.48</td><td>0.887</td><td>0.43</td></tr></tbody></table>

Validation of the models was conducted by taking a random model scenario and running it ten times to get an average result and comparing each of the model's predictions to the averaged result of the simulation. The ten simulations were run with the following factors: Initial Tools = 10, Low inventory Limit = 2, HQ Tool Cut-off = 0.45, LQ Tool Cut-off = 0, Batch Size = 4, Item Mean = 1.175. In Design Expert, the confirmation tab allows the user to input numbers for each factor and will output the expected responses. In JMP, a profiler was constructed using a partial least squared regression method for both JMP models that allowed the user to input values for the factors and receive estimated responses.

## 8 Discussion

Modelling the HQ time between orders seemed to be the one area that all three models successfully completed, all yielding R² values greater than 0.9 as well as all having accurate (<5% error) estimations in comparison to the averaged simulation results. The other three responses were not as close in estimation except for the Design Expert model in predicting the low quality queue time response. This estimation only yielding an error of 3.6% in comparison to the simulations while the RLHC and FFFD responded with errors of 15.4% and 25.6% respectively.

Estimating the low quality time between orders was the worst predicted response for all models with the best model (FFFD) yielding a value 42% lower than the simulated value while the RLHC and Design Expert Model estimated +46% and -78% of the simulated value. Even though the Design Expert model had an R² value that indicates that is has produced an accurate model, the predicted value is very far off the simulated result. The same phenomenon occurred with the design expert model for predicting the high quality item queue time with an R² value of 0.9081 yet the estimate is 79.7% greater than the simulated value. The RLHC and FFFD both yielded values ~24% greater than the measured value for high quality item queue time.
The models producing good results for “HQ time between orders” in comparison to the other three responses may be due to the number of variables that influence the results. For example, the high quality item line only takes item inputs from one area and new tool inputs from one area, whereas the low quality line takes item inputs from one area but takes new tool inputs from two areas, one being controlled by an ordering policy while the other is semi-random inputs of tools from the high quality line. Figure 4 (a.) illustrates this type of occurrence in the lower quality line as a group of tools all get demoted to the lower line in a small time frame, resulting in a huge number of LQ tools and a large time frame for the LQ tools to diminish again. This type of influx into the low quality line is something that none of the models consider. If another factor were added to the models that represented this behaviour in the system, there could be increase accuracy in the three models’ predictions.

<figure><img src="image_7.png" /><figcaption>Fig. 4. – (a.) Tool Tracking Mid-Simulation and (b.) Linear rate of demotion of HQ Tools</figcaption></figure>

The RLHC model for the low quality responses was poor. With an $R^2$ value of 0.5797 for both LQ responses, some important factor is clearly missing. While some factors such as initial tools, low inventory limit and batch size had little to no effect on each response, factors such as mean items and HQ/LQ tool cut-off had a significant impact.

There are a few areas of the model that could be looked at to address this issue, the first being the rate at which tools leave the high quality area and enter the low quality area. The linear rate of addition of tools to the LQ tool supply, as seen in Figure 4 (b.), could be an example of a factor that could be incorporated into the model specifically for the low-quality responses. The same theory could be applied to the FFFD model.

Randomness in the ExtendSim simulation may also be a contributing factor to the lack of accuracy of the metamodels. Item generation, Item process time, and the rate of tool degradation were all influenced by random numbers while also being three areas of the model with a huge influence on results. While only the mean item generation was included as a factor in each of the models, adding these three areas of randomness as factors to the models could produce more accurate results in the metamodeling process.

Equations to calculate each of the responses for each of the metamodeling methods based on the input factors provided were created. Table 2 contains the “HQ Time between Orders” equations for each of the metamodel types.
<table><caption>**Table 2.** - HQ Time Between Orders Equations</caption><thead><tr><th rowspan="2">DE</th><th colspan="2">Multiplier</th><th rowspan="2">Factor</th></tr><tr><th>RLHC</th><th>FFFD</th></tr></thead><tbody><tr><td>-0.082</td><td>-8.13</td><td>-13.69</td><td>Intercept</td></tr><tr><td>0.03</td><td>22.34</td><td>24.86</td><td>*Mean Items +</td></tr><tr><td>4.54</td><td>7.16</td><td>7.41</td><td>*Batch Size +</td></tr><tr><td>0.089</td><td>-38.79</td><td>-37.46</td><td>*HQ Tool Cut-off +</td></tr><tr><td>0</td><td>-0.036</td><td>0.051</td><td>*Initial Tools +</td></tr><tr><td>0</td><td>-0.045</td><td>0.14</td><td>*Low Inventory Limit</td></tr><tr><td>-10.17</td><td>0</td><td>0</td><td>*HQ Tool Cut-off *Batch Size +</td></tr><tr><td>6.35</td><td>0</td><td>0</td><td>*Batch Size *Mean Items</td></tr></tbody></table>

## 9 Conclusions

All three models, Design Expert, RLHC, and FFFD, accurately predict the response with the least amount of input: High Quality Time between Orders. However, the more complex responses of the system are not accurately captured in any of the three models. One of the reasons for this may be the lack of relevant factors analysed by the simulation. Another reason could be that there is too much unaccounted randomness in the system for the models to accurately predict the more complex performance indicators. The use of space filling designs such as RLHC and FFFD perform well with high numbers of factors and should be explored further. The Design Expert model performs well but the number of simulations required increases exponentially with the number of factors present.

## 10 Further Work

Only two types of space filling designs for metamodel creation were explored in this paper, a modified version of the Latin Hypercube and JMPs Fast Flexible Filing Design. Both methods are one-shot, non-sequential designs so sequential methods (Crombecq et al., 2011) may be able to create accurate models with even lower numbers of simulations required.

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