John G Wilson
email: jwilson@ivey.uwo.ca

Bayesian Modelling for Product Testing and Release

Keywords: Product Development, Prototype Testing, Product Release, Reliability Growth

Deciding when to release a new product requires a tradeoff between costs, potential profits and the underlying reliability of a product. Many new products go through a "Test Analyze and Fix" process. When a failure occurs, an immediate design "fix" may occur or the product might undergo a minimal fix with design changes being delayed until later when many changes can be introduced at the same time. We introduce a Bayesian model that allows for the introduction of managerial knowledge and experience. Unlike most approaches, we do not build in an assumption that the product always improves throughout the process.

Introduction

Background

The mean time between failures of a new product (MTBF) is an important characteristic that has a direct impact on revenue streams and costs. For products designed for the US military, historically MTBF has to reach a specified level before a product will be accepted. In the commercial world, there is a balance between MTBF and the timing of a new product release: release the product too early and the reliability problems will damage a company's reputation and involve many repair and replacement costs; release the product after extensive and time intensive development to raise MTBF and market advantage and share might be lost. Before formally incorporating costs and revenues into a decision model of how to manage the development phase of a new product and the timing of product release, it is essential to have a flexible statistical process for modelling the phases of test development.

A complex newly designed system generally undergoes several stages of development testing before it is put into operation. After each stage of testing, changes are made to the design with the hope that the new design leads to a longer period of performance. This procedure is referred to as reliability growth. It is the result of iterative Test Analyse and Fix processes which are conducted to discover deficiencies and to verify that corrective action will prevent recurrence in the further test phases. A "fix" refers to a design change that improves the reliability of the system After fixes have been implemented, the system reliability will jump to a higher value. Usually, the estimate of the reliability jump is not straightforward since the untested fixes require a specific prediction rule for them to "follow" in a growth model. In addition, the test data of a specific test prototype is only one set of data of all possible situations from the same design basis, e.g. design specifications and environmental specifications (Ireson et al. [START_REF] Ireson | Handbook of Reliability Engineering and Management[END_REF]).

Literature

The first reliability growth model appeared in the late 1950's. Duane [START_REF] Duane | Learning curve approach to reliability monitoring[END_REF] analysed the reliability data of five different complex systems and demonstrated that the cumulative failure rate versus the cumulative operation time fell close to a straight line when plotted on a log-log scale. Crow [START_REF] Crow | Reliability Analysis for Complex Repairable Systems[END_REF] at Army Materiel Systems Activity (AMSAA) suggested that Duane's postulate be stochastically represented as a Non-Homogeneous Poisson Process (NHPP) with Weibull intensity 𝜆(𝑡) = 𝛼𝛽𝑡 𝛽-1 where𝛼, 𝛽 > 0. This is called the AMSAA model. For 0 < 𝛽<1, 𝜆(𝑡) is decreasing implying reliability growth. Bayesian extensions for the AMSAA model were proposed by Higgins and Tsokos [START_REF] Higgins | A quasi-Bayes estimate of the failure intensity of a reliability-growth model[END_REF] and Guida et al. [START_REF] Guida | Bayes inference for a non-homogeneous Poisson process with power intensity law (reliability)[END_REF]. Yu et al. [START_REF] Yu | Predictive analyses for nonhomogeneous Poisson processes with power law using Bayesian approach[END_REF] consider predictions using this model using noninformative priors. A pseudo-Bayes approach has been discussed by Singpurwalla [START_REF] Singpurwalla | A Bayesian scheme for estimating reliability growth under exponential failure times[END_REF] under the stochastic ordering of prior and posterior distributions. Robinson and Dietrich [START_REF] Robinson | A new nonparametric growth model[END_REF][START_REF] Robinson | A nonparametric-Bayes reliability-growth model[END_REF] proposed two nonparametric growth models in which no specific functional form is assumed for the change of process failure rate. Crow's Engineering Judgement Model [START_REF] Crowe | Reliability Growth Assessments Utilizing Engineering Judgements[END_REF] (EJM) considered techniques for projection of future expected reliability based on a delayed fixes testing plan. In the EJM model it is assumed that the effectiveness factors (EF) of fixes for all distinct failure modes in the system are given and all failures observed in RGDT are not fixed until the end of the testing program.

Wayne and Modarres [START_REF] Wayne | A Bayesian model for complex system reliability growth under arbitrary corrective actions[END_REF] assume that the failure times between fixes are exponential and that a fix results in a known increase in mean time between failures. (MTBF's). Conjugate priors are used to model knowledge of the parameters. Wayne [START_REF] Wayne | Modeling Uncertainty in Reliability Growth Plans[END_REF] discusses using a beta prior distribution for the factor by which MTBF increases after a fix. Our model generalises this to assuming that the increase in MTBF is itself a random variable and that MTBF's follow a pattern similar to that of AMSAA NHP approach. For software projects, predicting the expected number of software defects is important and Bayesian methods are making inroads. Rana et al. [START_REF] Rana | Analyzing defect inflow distribution and applying Bayesian inference method for software defect prediction in large software projects[END_REF] discuss models including the Weibull model which is similar to the NHPP above and discuss finding prior distributions. Chen et al. [START_REF] Chen | Multi-stage Bayesian evaluation method of reliability growth for large complex system[END_REF] consider a multi-stage system with NHPP at each stage and models growth factors. Non-Bayesian approaches to modelling RDGT using power law growth models such as NHPP are still areas of active research--see, e.g., Xu et al. [START_REF] Xu | An approach for reliability demonstration test based on power-law growth model[END_REF].

Bayesian methods for estimating the reliability of systems are examined in Li et al. [START_REF] Li | Multiple priors integration for reliability estimation using the Bayesian melding method[END_REF] and Ruiz et al. [START_REF] Ruiz | Data analysis and resource allocation in Bayesian selective accelerated reliability growth[END_REF]. A comparison of classical versus Bayesian methods can be found in Kamranfar et al. [START_REF] Kamranfar | Statistical inference for a repairable system subject to shocks: classical vs. Bayesian[END_REF]. Pollo et al. [START_REF] Pollo | Objective Bayesian Inference for Repairable System Subject to Competing Risks[END_REF] introduce an "objective Bayesian prior" that produces a posterior that is a product of gamma functions. Wayne et al. [START_REF] Wayne | A Bayesian Model for Complex System Reliability Growth under Arbitrary Corrective Actions[END_REF] model a posterior distribution for the failure intensity Optimal product release times for a product consisting of subsystems connected in series are modeled in Li et al [START_REF] Li | A Joint Modeling Approach for Reliability Growth Planning Considering Product Life Cycle Cost Performance[END_REF].

In distinction to the above literature, we don't even require that the reliability grows…some fixes don't work! However, we do assume that on average fixes work and the expected reliability grows according to a power law. The parameters of this power law are unknown and we assume that we can obtain prior distributions for the parameters. However, we translate these parameters into quantities that are intuitive and will enable engineers to arrive at reasonable priors. This approach has the advantage of being transparent and intuitive. However, mathematically, arriving at conclusions will now require numerical methods.

A Bayesian Product Improvement Model

We will start with a description of the basic model and assumptions. For most of this paper, we consider the model where a fix is incorporated after each failure. Thus, as the machine fails and is repaired, its reliability grows. This is discussed in $2.1, where model assumptions are discussed in detail. The likelihood function for the data is formulated in $2.2. A prior density function 𝜋(𝛼, 𝛽) over the parameters 𝛼 and 𝛽 of the Weibull intensity 𝜆(𝑡) = 𝛼𝛽𝑡 𝛽-1 is required. Ideally, this should incorporate expert opinion and knowledge. We will provide the likelihood function for the observed data. Then we will describe how to come up with reasonable prior distributions for the underlying parameters. This is an important feature of the model since it allows for the incorporation of expert managerial judgement. This is particularly important in product development where, initially, not much data is available. Most product developers will have a feel for the possible values of the reliability growth parameter β. This is discussed in $2.3. Information about the parameter α is not so intuitive. Instead, in $2.4, we propose the product developer provide information about the expected mean time between failures. This can then be translated into prior information for α. In $2.6, we combine the material from $2.3 and $2.4 to provide a joint prior for the underlying parameter.

Model Development

After each failure and fix, the time to the next failure is assumed to be exponential but with a new parameter since the machine has been "improved". This is the assumption made in Wayne and Modarres [START_REF] Wayne | A Bayesian model for complex system reliability growth under arbitrary corrective actions[END_REF].

Suppose the system testing stops at some predetermined number of failures n. Let 𝑡 1 , 𝑡 2 , … , 𝑡 𝑛 be the cumulate test times of n successive failure modes observed up to 𝑡 𝑛 and let 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 be the times between failures i.e. 𝑥 1 = 𝑡 1 and 𝑥 𝑖 = 𝑡 𝑖 -𝑡 𝑖-1 , for i =2,…,n.

So the number of failures in each stage is limited to one. We note that the test plan of the most popular reliability growth model using NHP belongs to this class of problem.

During a particular stage, the distributions of the times to failure are assumed exponential. The exponential parameters are not assumed to be equal. In Wayne and Modarres [START_REF] Wayne | A Bayesian model for complex system reliability growth under arbitrary corrective actions[END_REF], the assumption is that each fix results in a known fractional increase in meant time to failure. We generalise this by assuming that exponential parameter for stage i after a fix is assumed to be the value of a gamma random variable with mean 𝛼𝛽𝑡 𝑖-1 𝛽-1 and variance 𝜎 2 . We will assume that the value 𝜎 2 is provided-although extension to a prior over this parameter is straightforward. This allows for the incorporation of the major feature of the AMSAA model. In addition, in distinction to other models, the actual failure rate does not necessarily decrease from stage to stage. This is particularly important when considering the development of software products: a fix is new computer code which may itself be faulty and decrease the reliability of the product. However, overall the reliability tends to improve as fixes are incorporated. The parameters 𝛼, 𝛽 and σ will depend on managerial specifications and knowledge and are key components. There are various ways to obtain information that can help set prior distributions. For instance, Wayne [START_REF] Wayne | A Bayesian model for complex system reliability growth under arbitrary corrective actions[END_REF] discusses developing prior distributions for the factor by which MTBF increases after a fix and suggests the use of historical information such as that found in Trapnell [START_REF] Trapnell | Update on Reliability Fix Effectiveness Factors[END_REF] and in Brown [START_REF] Brown | Development and validation of methodology for fix effectiveness during product development[END_REF] On finding prior distributions, we need to obtain the posterior distributions for the parameters and the posterior expected value of various quantities of interest such as the expected time to failure once all the test data and fixes have been incorporated. Since we are making as few assumptions as possible and trying to capture managerial intuition, these are algebraically messy and ultimately need to be evaluated using numerical methods.

The Likelihood Function

At the nth failure the data observed will be the times 𝑡 1 , 𝑡 2 , … , 𝑡 𝑛 . After simplification, the likelihood function -a constant times the probability of observing the data 𝑡 1 , 𝑡 2 , … , 𝑡 𝑛 , can be written as:

∏ 𝛼𝛽𝑡 𝑖-1 𝛽-1 𝑛 𝑖=2 {1 + (𝑡 𝑖 -𝑡 𝑖-1 )𝜎 2 𝛼𝛽𝑡 𝑖-1 𝛽-1 } -(1+(𝛼𝛽𝑡 𝑖-1 𝛽-1 ) 2 )
.

Sufficient statistics of fixed dimensionality do not exist for the above likelihood function so that no natural conjugate prior can be found (Schlaifer and Raiffa [START_REF] Schlaifer | Applied statistical decision theory[END_REF]). Thus, inference based on a continuous joint prior generally requires numerical procedures.

From a practical viewpoint, Bayes methods are attractive because they allow for incorporating experience or technical considerations in the estimation procedure especially when the test data and the test procedure is expensive. This is greatly facilitated when a clear physical meaning can be attached to the model parameters. Generally, for selecting a prior distribution, it must adequately represent the state of prior knowledge about the parameter and not be a computational burden. Based on a technical judgement, if we believe the variability between prototypes will be relatively small it may be reasonable to take σ as a common constant parameter of the model and we do that in the sequel.

Prior for 𝛃

We note that a shape parameter β less than 1 means a system whose expected failure rate is decreasing (small β means rapid reliability growth) whereas a shape parameter β greater than 1 means a system whose expected failure rate is increasing. In many practical situations, bounds for β can follow on the basis of prior knowledge of the underlying failure mechanism. For instance, guidance on practical ranges for β can be found in the following:

• The data collected by Duane [START_REF] Duane | Learning curve approach to reliability monitoring[END_REF] and Schafer et al. [START_REF] Schafer | Reliability growth study[END_REF] show that β is "about 0.5 and is rarely less than 0.3 regardless of the initial and final MTBF's"

• For equipment exhibiting a good fit, the Crow's growth parameter β ranged from 0.579 to 0.794 (Gates et al. [START_REF] Gates | Reliability Growth Projections[END_REF])

• The Duane model provides a reasonably good fit to most of the data sets and the estimated growth rates are within the generally accepted range or 0.3 to 0.9 (Gates et al [START_REF] Gates | Reliability Growth Projections[END_REF])

When no further information exists, it is reasonable to represent prior knowledge by using a uniform prior pdf over the interval (0.3,0.9), the widest range for β as indicated by Gates et al. [START_REF] Gates | Reliability Growth Projections[END_REF].

Prior for α

Since the scale parameter α in this model does not provide a clear physical meaning, we suggest an alternative approach to find a reasonable joint pdf for α and β as follows.

For the failure process considered here, the expected failure time at time t is 𝛼𝛽𝑡 𝛽-1 . We note that generally a development test program is never begun without some information regarding potential failure rates. So, the mean life time at some beginning time 𝑡 0 can be viewed as a process parameter 𝜇 0 . Without loss of generality, suppose that 𝑡 0 = 1. The physical meaning of 𝜇 0 should allow an engineer to formulate prior knowledge. However, if no prior information is available from a similar system then a commonly used estimate of 10% to 20% of the requirement mean time between failures can be uses as an initial reliability (Morris and MacDiarmid [START_REF] Morris | Reliability Growth Testing Effectiveness[END_REF]). Using this, the prior for 𝜇 0 .can be chosen as uniform over some interval (𝑐 1 , 𝑐 2 ):

𝜋 𝜇 0 (𝜇 0 ) ∝ 1 𝑐 2 -𝑐 1
for 𝑐 1 < 𝜇 0 < 𝑐 2 .

Joint prior for α and β

Assuming the random variables 𝜇 0 and 𝛽 are independent, the joint density function of 𝜇 0 and 𝛽 is given by

𝜋 𝜇 0 ,𝛽 (𝜇 0 , 𝛽) ∝ 1 𝑐 2 -𝑐 1 1 𝛽 2 -𝛽 1
for 𝑐 1 < 𝜇 0 < 𝑐 2 and 𝛽 1 < 𝛽 < 𝛽 2 . By making a change of variable transformation, this induces an equivalent prior over (𝛼, 𝛽) given by 𝜋(𝛼, 𝛽) ∝ 1 𝛽𝛼 2 over the region 0<𝛽 1 < 𝛽 < 𝛽 2 <1 and 0<𝑐 1 ≤

1 𝛼𝛽 ≤ 𝑐 2 .
that 𝜇 0 .and β are independent, the joint density function is then equal to

Putting it all together

The likelihood from 2.1 is now multiplied by the prior from 2.5 to form a quantity proportional to the posterior distribution. From this, various posterior quantities such as the posterior MTBF, posterior means of α and β may be obtained via integration. However, the resultant integrals have no closed form and the integrations, of necessity, must be numerical.

Applying the model

To illustrate the application of the Bayesian model, we use the 15 failure data points from Crow's [START_REF] Crow | A methodology for managing reliability growth during operational mission profile testing[END_REF] example. These data were generated by the computer simulation of a NHPP with α=0.42 and β = 0. It is important to note that we know a priori that the Bayes model does not fit this data since it was generated assuming the time between failures is non-exponential and there is no variability between prototypes. Thus, the simulated model is at best an approximation to the model which most authors suggest. As will be seen, even for this unfavourable data, the proposed Bayesian model performs well.

We assumed that β was uniform over (0.45, 0.55), 𝜇 0 was uniform over (4.5 and 5) and σ was small. The table below summarises the estimation results. Included are the results from applying the Bayesian model and the estimated using maximum likelihood estimation of α and β in Crow's model.

Conclusions

Planning for new product release requires a reasonable model for the modelling of reliability growth. An approach to product reliability growth modelling that allows for managerial input, variability between prototypes and incorporates historically relevant observations has been presents. In the case of limited data, a Bayesian approach combined with expert judgement, often provides superior results. A limitation of the Bayesian approach is arriving at the prior distributions. This depends on the expert knowledge of the product developers. For cases where development time is short, misspecifications can lean to costly errors. Continuing work involves expanding the model to more general situations and performing extensive numerical comparisons. Now that a feasible model for tracking product reliability growth has been established, a next step is to explicitly incorporate costs and revenues to optimise the testing and fix process.

1 (𝑐 2 - 1 𝛽𝛼 2

 1212 𝑐 1 )(𝛽 2 -𝛽). Changing variables the joint distribution of 𝛼 and β can be shown to be proportional to where 0 < 𝛽 1 ≤ 𝛽 ≤ 𝛽 2 < 1 and 0<𝑐 1

  5. These are (cumulative test times): 0.2, 11.2, 37.2, 39, 48.4, 53.4, 90.2, 91.6, 151.4, 159.4, 197.2, 240.2, 323.6, 361.2, 381.6.

Table 1 .

 1 Comparison of Model to Estimates from Crow's Model where underlying data is generated from a known distribution

	Model	True σ	True β
	True Model	0.42	0.5
	Bayesian Estimates	0.40463	0.522
	MLE for Crowe Model	0.4690	0.5829

Acknowledgment

This work was accomplished with the assistance of T. Chu and forms part of his thesis


Elena Tappia

Emilio Moretti
email: emilio.moretti@polimi.it

Marco Melacini

A Simulation Analysis of Part Feeding to Assembly Stations with Vertical Robotic Storage and Retrieval Systems

Keywords: Parts Feeding, Factory Logistics, Mobile Robots, Supermarket Warehouses, Discrete Event Simulation

The evolution of customer requirements has led companies to pursue mass customization and personalization strategies, increasing the complexity of part feeding operations in assembly systems. At the same time, emerging technologies, based on mobile robots, promise to support material handling activities in an efficient yet flexible manner. This paper studies part feeding to assembly stations with Vertical Robotic Storage and Retrieval Systems (VRSRS), a new technology that allows automating both the storage and the internal transportation activities with the same fleet of mobile robots. We develop a Discrete Event Simulation model of a system made of a Central Warehouse replenishing supermarkets which, in turn, supply mixed-model assembly stations. Through this model, we carry out experiments showing a clear trade-off between the number of robots, which affects the overall investment in the system, and the supermarkets size, which affects space occupation in the shop floor, where space is a critical resource. We also estimate the replenishment lead time and the downtime of assembly stations, showing that this system performs best when small production orders are issued and could therefore be a suitable solution to support part feeding activities in a mass customization context.

Introduction

The evolution of customer requirements has led companies to pursue mass customization and personalization strategies. Although beneficial from a sales perspective, the implementation of such strategies may increase the complexity of operations within assembly plants [START_REF] Emde | Just-in-time vehicle routing for in-house part feeding to assembly lines[END_REF]. In fact, it calls for a sharp rise in the number of part variants and it entails assembling goods in small batches, so as to offer a wide product range without overly increasing inventories [START_REF] Fathi | A modified particle swarm optimisation algorithm to solve the part feeding problem at assembly lines[END_REF]. In this context, companies are adopting mixed-model assembly lines, where a single line is able to assemble several product variants or models [START_REF] Faccio | The impact of production mix variations and models varieties on the parts-feeding policy selection in a JIT assembly system[END_REF]. To cope with these challenges, and since the available space at the border of the lines is normally scarce [START_REF] Faccio | Macro and micro-logistic aspects in defining the parts-feeding policy in mixed-model assembly systems[END_REF], part feeding systems are shifting towards frequent deliveries of small quantities of parts, like boxes or totes, to assembly stations [START_REF] Emde | Scheduling the replenishment of just-in-time supermarkets in assembly plants[END_REF][START_REF] Golz | Part feeding at high-variant mixed-model assembly lines[END_REF]. This part feeding policy is often supported by the use of supermarkets, i.e. decentralized warehouses located in the shop floor allowing to reduce delivery lead times [START_REF] Battini | Just-in-Time supermarkets for part supply in the automobile industry[END_REF][START_REF] Sali | Line feeding optimization for Just in Time assembly lines: An application to the automotive industry[END_REF]. The related academic literature is mainly focused on supermarkets design (e.g. [START_REF] Emde | Optimally locating in-house logistics areas to facilitate JIT-supply of mixed-model assembly lines[END_REF][START_REF] Nourmohammadi | Design of stochastic assembly lines considering line balancing and part feeding with supermarkets[END_REF]) and on the scheduling, loading, and routing of tugger trains for parts delivery to assembly stations (e.g. [START_REF] Diefenbach | Loading tow trains ergonomically for just-in-time part supply[END_REF][START_REF] Emde | Scheduling in-house transport vehicles to feed parts to automotive assembly lines[END_REF]). Some research gaps still need to be addressed. In particular, while emerging technologies, based on mobile robots, promise to support material handling activities in an efficient yet flexible manner, there is a gap about the application of such technologies in part feeding systems. Among these technologies, Vertical Robotic Storage and Retrieval Systems (VRSRS) are extremely interesting in this context because they allow automating both the storage and the internal transportation activities with the same fleet of mobile robots, thus seamlessly integrating the two activities. This is possible since the robots, carrying totes, can both roam the shop floor and navigate inside the storage racks, moving horizontally and vertically (Figure 1). This system has been studied in literature in relation with distribution centers only by [START_REF] Azadeh | Design, modeling, and analysis of vertical robotic storage and retrieval systems[END_REF]: they develop a queuing network-based model of VRSRS, but they neglect the possibility for robots to navigate outside the racks. Only one contribution has addressed the deployment of VRSRS in a factory environment [START_REF] Moretti | A performance model for mobile robotbased part feeding systems to supermarkets[END_REF], but it models the replenishment of supermarkets from the central warehouse, without considering the transportation of totes to assembly stations. This research aims to assess the performance of a part feeding system to assembly stations based on VRSRS. We study the system through Discrete Event Simulation (DES), and we measure its performances in terms of replenishment lead time and downtime of assembly stations due to lack of materials. The remainder of this paper is organised as follows. Section 2 gives an overview of the system under analysis, while Section 3 describes the developed simulation model. Section 4 reports the results of our analysis: we test alternative configurations of the part feeding system, differing among each other due to the supermarkets size, and we evaluate the impact of production order size (i.e. number of assembled products per order) on system performance. Finally, Section 5 includes conclusions and directions for future research.

System Description

Figure 2 shows the layout of the system considered in this paper. It consists of a central warehouse (CW) replenishing supermarkets which, in turn, supply mixed-model assembly stations. Each type of stations can make multiple product models. We assume that there are two stations of each type, facing away from each other (e.g. station "A" and station "B" in the figure), and they share the same stock. The product models made by one type of stations cannot be assembled by stations of a different type.

All warehouses are VRSRS made of one aisle (supermarkets) or multiple aisles (CW), with two single-deep storage racks per aisle. Each rack is divided into storage columns and storage levels. Totes are stored randomly within the warehouses, i.e. the probability of accessing any column or level is based on the uniform distribution. Each CW aisle has an input/output (i/o) point for robots, while each supermarket has an input point, facing the CW, and an output point, facing assembly stations. Robots handle mono-item totes, denoted as "full totes", and perform single-command cycles inside storage racks: they enter the storage aisle input point, travel horizontally to reach the storage column, climb the rack up to the storage level where they retrieve or store the tote, and then travel vertically and horizontally towards the output point.

To avoid deadlocks, one robot at a time can access a supermarket or a CW aisle.

Robots can also travel outside the racks, following the bi-directional paths shown in Figure 2, so as to replenish supermarkets and to feed assembly stations with full totes. Supermarket replenishment works according to the kanban system, as in previous studies (e.g. [START_REF] Emde | Scheduling the replenishment of just-in-time supermarkets in assembly plants[END_REF]): whenever a tote is retrieved from the supermarket, its replenishment from the CW is triggered. Similarly, part feeding to assembly stations is controlled through a two-bin inventory system [START_REF] Limère | Optimising part feeding in the automotive assembly industry: deciding between kitting and line stocking[END_REF]: two totes per each part type are stored between each couple of stations of the same type and, whenever a tote is empty, its replenishment from the supermarket is triggered. Each supermarket supplies four assembly stations.

Finally, we assume that the management of empty totes at the assembly stations is out of scope. For instance, empty totes could be stacked on pallets next to each assembly station and forklift trucks could be used to periodically collect them.

Simulation Model

The DES model of the system was implemented in MATLAB-Simulink, using the Si-mEvents library. As shown in Figure 3, it is made of four main parts. The first part, highlighted in green, models the generation of robots at the start of the simulation and their travel inside the CW. Robots are modelled as entities of the system: after they are generated, they queue up before the CW in a First In First Out (FIFO) robot queue, waiting to be matched with tote replenishment orders. An order can be issued by either an assembly station or a supermarket. In the former case, if the robot queue in front of the supermarket supplying the assembly station is empty, a robot travels from the CW robot queue to that supermarket, in order to replenish the station as explained further in this section. In the latter case, instead, the robot travels from the CW robot queue to the input point of the CW aisle where the tote required by the supermarket is stored. This travel is modelled as an infinite server, assuming that there are no interferences among robots outside the racks. The same assumption is made for all the other robot movements outside the racks. Thus, the robot travel time from the queue to the CW aisle only depends on the selected aisle and on the robot speed. As in [START_REF] Azadeh | Design, modeling, and analysis of vertical robotic storage and retrieval systems[END_REF], a fixed speed equal to 1 m/s is assumed. Then, to simulate the robot movement inside the racks, each of the 7 CW aisles is modelled as a single server, so as to allow its access to one entity at a time.

The CW maximum height is set equal to 10 m. To compute the number of storage levels and all the dimensions of the CW, the assumptions made by [START_REF] Azadeh | Design, modeling, and analysis of vertical robotic storage and retrieval systems[END_REF] about storage locations size are taken into account. The second part of the model, highlighted in yellow in Figure 3, simulates the travel of the robot from the i/o point of a CW aisle to the input point of a supermarket and inside the supermarket. The travel from the CW to the supermarket is modelled as an infinite server, whose service time depends on the robot speed and on the distance along the path shown in Figure 2 (120 m for the cross-aisle connecting the CW with the supermarket area, plus the distance from the CW aisle i/o point to this cross-aisle and from this cross-aisle to the supermarket input point). Three supermarkets, evenly distributed along the shop floor, are considered in the model. Supermarkets are modelled as single servers, whose service times depend on the same assumptions made for the CW aisles.

After storing the tote and exiting the supermarket from its output point, robots queue up in front of the supermarket, waiting for a replenishment order. When a replenishment order is received from an assembly station, as soon as the supermarket is idle and the required tote is available, the robot enters the supermarket, picks up the tote and exits the supermarket. A signal is sent to the CW to trigger the tote replenishment. The maximum supermarket height is set equal to 5 m, accounting for the real industrial applications. Based on the supermarket storage capacity, the number of storage columns is determined. We assume that the supermarket is full at the start of the simulation.

The model section highlighted in blue in Figure 3 simulates the travel of the robot between the supermarket output point and the assembly station, the assembly activity and the ensuing consumption of parts. The distance between a supermarket and a station is assumed to be fixed (15 m). After delivering the full tote to the assembly station, the robot either returns to the queue in front of the supermarket, if this queue is empty, or travels to the queue in front of the CW. After a tote is delivered to the assembly station, new entities are generated, corresponding to the parts stored in that tote, that queue up before the station. As regards the assembly activity, we assume that each type of assembly station can make four different product models and each model is made of two parts replenished through the robot-based feeding system: a first part specific of that model and a second one shared with another model assembled at the same station. The assembly of a product may start when three conditions have been verified through a gate: a signal has been received to start the assembly (as explained later in this section) and the two parts are available at the station. These parts are subtracted from the total stock available at the station through an entity terminator, so as to model parts consumption: when all the parts in a tote have been used up, the replenishment from the supermarket is triggered. The number of parts stored in a tote is assumed to be equal to 10, for part types specific of one product model, and to 20, for part types shared by two models. Each couple of stations of the same type is modelled as a dual server, meaning that one product at a time can be assembled at each station. Service times of the assembly stations are exponentially distributed, as in previous simulation analyses of production systems (e.g. [START_REF] Kundu | A study of a kanban based assembly line feeding system through integration of simulation and particle swarm optimization[END_REF]), with an average of 30 seconds.

The model part highlighted in pink in Figure 3 simulates the arrival of production orders (i.e. the demand). An entity generator creates orders with exponentially distributed inter-arrival times. An order is assigned to one of the product models through an entity output switch, according to a uniform distribution. Then, an entity replicator generates a number of entities corresponding to the production order size, i.e. the number of assembled products per order. The order size is fixed for all the orders during a simulation. Based on the product model, the entities of an order are assigned to an assembly station and wait in a FIFO queue before a single server. When the first entity enters this server, a signal is sent to the assembly station (model section highlighted in blue) to trigger assembly activities. After they exit the single server, entities are terminated.

Each run of the model simulates an 8-hours shift. The model was verified and then validated by comparing its output, in terms of assembled products per shift, with the theoretical output, i.e. the overall demand per shift. This comparison was made considering a number of robots high enough to avoid that robots were the bottleneck in the system. It was found that the difference between the simulation output and the theoretical one is always below 5%, thus mostly due to the stochasticity of the assembly time.

4

Results Discussion

Relationship between Supermarkets Size and Number of Robots

The main design variables of this system are the supermarkets size, S s , i.e. the number of totes that can be stored in each supermarket, and the number of robots, N R , needed to fulfill all the replenishment orders. Once fixed the number of supermarkets, S s determines space occupation in the shop floor, where space is a critical resource, while robots are the main component of the investment in this system [START_REF] Moretti | A performance model for mobile robotbased part feeding systems to supermarkets[END_REF]. Thus, the first experiments are aimed at studying the relationship between these two variables.

Figure 4 shows the results when the average demand is 960 orders per shift and the order size is 10 products per order. The colored areas inside the chart represent the difference between the system output, i.e. assembled products per shift, and the required output, i.e. overall demand. In particular, the dark blue area corresponds to the combinations of S s and N R for which the system is able to fulfill demand (difference below 5%, that can be ascribed to the stochasticity of demand). Results show a trade-off between S s and N R : by increasing N R , a lower S s is needed to fulfill demand because the throughput capacity of the system increases; vice versa, the higher S s the lower the needed N R , since more totes are stored in the shop floor and thus a lower speed of supermarket replenishment activities has a lesser impact on overall system performances.

Results also show that, given an adequate investment in the robot fleet, the system can fulfill demand with extremely small supermarkets (in the scenario of Figure 4, even less than 50 totes per supermarket, corresponding to less than 5 m 2 per supermarket).

Relationship between Order Size and System Performance

One of the effects of mass customization is a smaller size of production orders. Therefore, with the second group of experiments we study how system performances change when varying the number of assembled products per order and the number of robots. Performance measures encompass the average replenishment lead time (LT) and the average downtime of the assembly stations due to lack of parts (DT). LT is defined as the average time elapsed between a request for tote replenishment by an assembly station and the tote delivery to the station. DT is defined as the percentage of the 8-hours shift during which assembly stations are not working due to lack of materials.

In order to plot results and make them comparable, the overall demand in terms of assembled products per shift was fixed for all the scenarios and the number of orders per shift was varied accordingly. For instance, Figure 5 shows results for a demand equal to 9,600 products per shift and S S equal to 150 totes: LT and DT improve when decreasing the production order size, going from the red areas (where LT can be as high as 100 seconds and DT can reach 15% of the 8-hours shift) to the dark blue ones (LT close to 30 seconds and DT close to 0%). Therefore, results show that this system performs better with a lower order size. This is due to the nature of the system itself, which is designed to handle small quantities of several part types: big production orders would overly stress the system since they would require a high quantity of one or few part types, exceeding the quantity stored at the border of the stations and possibly even the quantity stored in the supermarkets.

Conclusion

New material handling technologies, based on mobile robots, promise to improve the efficiency of part feeding activities in the context of mass customization and personalization. This paper is the first to study a part feeding system to assembly stations based on VRSRS. We develop a DES model of a system made of a CW replenishing supermarkets that, in turn, supply mixed-model assembly stations.

Through the experiments we conduct, we show the trade-off between the number of robots and the supermarkets size, concluding that, if an adequate investment in the robot fleet is made, this system is able to effectively supply assembly stations even with extremely small supermarkets, thus saving space in the shop floor. Furthermore, we show that this system performs best, in terms of replenishment lead time and downtime of assembly stations, with small production orders, and therefore may be suitable to support part feeding activities in a mass-customized system.

The main limitations of this study are related to the assumptions made in the model development; future research could enlarge the experiments performing sensitivity analyses to understand how our results change when varying parameters. Moreover, future developments could include cost analyses or model extensions to test different layouts of the system, considering a different number of supermarkets or even direct deliveries of totes from the CW to assembly stations.

Fig. 1 .

 1 Fig. 1. Mobile robots inside (a) and outside (b) the storage racks (source: Exotec Solutions)

Fig. 2 .

 2 Fig. 2. Part feeding system layout

Fig. 3 .

 3 Fig. 3. DES model of the robot-based part feeding system, implemented in Simulink

Fig. 4 .

 4 Fig. 4. Relationship between supermarkets size and number of robots

Fig. 5 .

 5 Fig. 5. Relationship between order size and LT (on the left), DT (on the right)


Florian Defèr
email: florian.defer@fir.rwth-aachen.de

Günther Schuh

Volker Stich

Towards a unified reliability-centered information logistics model for production assets

Keywords: risk analysis system, asset management, criticality analysis, intelligent support system, risk management, information logistics model, Internet of production, digital shadow

Reliability-centered maintenance for production assets is a well-established concept for the most effective and efficient disposition of maintenance resources. Unfortunately, the approach takes a lot of effort and relies heavily on the knowledge of individuals. Reliability data in Computerized Maintenance Management System (CMMS) is scarce and almost never used well. An automated risk assessment system would have the potential to contribute to the dissemination and effective use of risk information and analysis. The individuality of production setting, however, prevents current systems from being practically relevant for most industries. The presented approach combines ontologies to store and link knowledge, an information logistics model displaying the various information streams, and the Internet of production to take the different user systems and infrastructure layers into account. The provided model of a reference digital shadow for risk information and a detailed information logistics model will help software companies to improve reliability software, standardize and enable assets owners to establish a customized digital shadow for their production networks.

Introduction

The percentage of productive maintenance work is stated by WIREMAN at 25 to 35 percent [START_REF] Wireman | Benchmarking best practices in maintenance, reliability and asset management: Updated for ISO 55000[END_REF], which corresponds to about 9 hours of productive maintenance work in a continuous shift system per day per full time equivalent (FTE). Reliability-centered approaches in maintenance provide a powerful tool to identify the right maintenance strategy for each asset, choosing the most effective improvement projects and giving comprehensive guidance for operational disposition. The approach was introduced in the 1960s aircraft industry and published by NOWLAN and HEAP in 1978 [START_REF] Nowlan | Reliability-Centred Maintenance[END_REF]. BLOOM states that only 40 percent of the introduced reliability concepts in production plants are successful. In addition, a high percentage of these implementation attempts are very superficial and therefore the real success rate is much lower [START_REF] Bloom | Reliability Centered Maintenance (RCM)[END_REF]. Within the following paragraphs, five main reasons are given why the implementation of a more than 40 year old concept is still a big challenge. First, however, the next paragraph provides a brief introduction to reliability and risk for production assets.

Reliability and risk in production and maintenance

There is currently no generally valid definition of risk in the literature, as the broad range of the term allows for a wide variety of perspectives [START_REF] Ni | Some extensions on risk matrix approach[END_REF]. In terms of asset reliability, risk is closely linked with the performance of the assets. For example, POLLANZ defines risk as "negative influence on the achievement of objectives" [START_REF] Pollanz | Konzeptionelle Überlegungen zur Einrichtung und Prüfung eines Risikomanagementsystems: droht eine Mega-Erwartungslücke?[END_REF]. Consequently, risk in maintenance management can be described as the product of the probability of occurrence and the impact on the effectiveness of the plant or asset [START_REF] Leidinger | Wertorientierte Instandhaltung_ Kosten senken, Verfügbarkeit erhalten[END_REF].

Risk = probability of occurrence × impact on the effectiveness

The task of maintenance is to identify the reasons for these deviations and to take countermeasures [START_REF] Al-Najjar | A Model to Diagnose the Deviation in the Maintenance Performance Measures[END_REF].

Holdbacks for the implementation of reliability concepts

The holdbacks for the implementation of reliability-based approaches can be clustered into five groups: Increasing system complexity, complex failure modes, subjective risk identification, complex risk aggregation, and insufficient data. In the following sections, these clusters are briefly explained.

Increasing system complexity:

The concept of the value chain defined by Porter in 1985 [START_REF] Porter | Competitive advantage: Creating and sustaining superior performance ; with a new introduction[END_REF] no longer holds true for today's companies and is transforming more and more into complex value-added networks [START_REF] Allee | Reconfiguring the value network[END_REF]. This means that companies can have many different value-added chains within the same production plant, which, in addition to generating various potentials, leads to different, interconnected and more complex risks [START_REF]Asset management: Part 1: Specification for the optimized management of physical assets 03[END_REF]. The increasingly close networking of different systems and production areas resulting from Industry 4.0 also adds new and complex risks for manufacturing companies [START_REF] Tupa | Aspects of Risk Management Implementation for Industry 4.0[END_REF].

Complex failure modes: Reliability concepts deal with the individual error causes of each component and can therefore define the ideal action to be taken instead of just having to replace the component regularly [START_REF] Nowlan | Reliability-Centred Maintenance[END_REF]. Limitations exist if the error mechanism or the overall system is not properly understood [START_REF] Nowlan | Reliability-Centred Maintenance[END_REF], which is more and more difficult given the increasing complexity of assets. These patterns cannot be predicted for a single asset, but can only be predicted by looking at reference plants [START_REF] Geldermann | The reference installation approach for the estimation of industrial assets at risk[END_REF].

Subjective and dynamic risk identification:

In many cases, a risk determination is only possible post ex, since the multitude of risks cannot be estimated and data for easily comparable investments do not exist. This not only means that in most cases only a risk diagnosis is possible, as risks cannot be predicted. But also that risks that have already occurred are heavily over or underestimated (recall bias, illusory correlation etc.) [START_REF] Kahneman | Prospect theory: An analysis of decision under risk[END_REF]. Since vulnerability is not static, it is even more difficult to measure risk [START_REF] Green | Using risk tolerance criteria to determine safety integrity levels for safety instrumented functions[END_REF]. For example, age and quality of maintenance have an enormous influence on how susceptible a production asset is to failure [START_REF] Geldermann | The reference installation approach for the estimation of industrial assets at risk[END_REF].

Complex risk aggregation and interdependencies:

The difficulty lies in the coupling of effects, since individual events can represent an acceptable risk, but are extremely intensified by a coupling effect [15]. It is not possible to estimate overall risks, thus procedures for risk aggregation are necessary [START_REF] Gleißner | Quantifizierung komplexer Risiken -Fallbeispiel Projektrisiken[END_REF].

Insufficient data: However, many risks cannot be predicted for the individual asset and can only be predicted by looking at reference plants [START_REF] Geldermann | The reference installation approach for the estimation of industrial assets at risk[END_REF]. This makes applying failure data from one system to another difficult even for identical standard systems [START_REF] Eruguz | A survey of maintenance and service logistics management: Classification and research agenda from a maritime sector perspective[END_REF]. Hence, it is necessary to have a huge amount of data and find an adequate way to take variations in the environment, process and so on into account. The assessment of risks is usually expressed in language, which makes it even harder to analyze [START_REF] Murè | Fuzzy Application Procedure (FAP) for the risk assessment of occupational accidents[END_REF].

Derivation of research needs

Production risks are mostly determined in a manual, workshop-like process, which takes place a few times in the life cycle of the assets. The gathered information of these workshops are subjective, scattered across information silos and difficult to compare. Modern information systems offer the possibility to make generalizations based on large data sets and to derive recommendations for action. The possibility of learning from data represents a great potential not only within the own organization, but also beyond. This potential is not used in current organizations and information systems.

The various internal and external data silos must be linked and in order to do so, use cases have to be derived from actual business needs. Risk assessment is one of the most crucial concepts for maintenance management. An automated risk modelling could further dissemination and enable a more efficient and effective maintenance organization. For the establishment of a comprehensive risk modeling, there needs to be a data logistics model to fuse the different data types and needs together.

Risk modelling and knowledge preservation

ROMEIKE identifies two groups of methods for risk identification, analysis and evaluation. The search methods (e.g. checklists, loss databases, SWOT analysis) are particularly suitable for already known or trivial risks. The search methods are suitable for complex and unknown risks and are further subdivided into analytical methods and creativity methods. In the context of this work, the analytical methods will be considered in the following, as they are suitable for "identifying future and previously unknown risk potentials". [START_REF] Romeike | Risikomanagement. Studienwissen kompakt[END_REF] The creativity methods are therefore not suitable for this task, as the aim of this paper is to lay out a foundation for an automated determination of risks. Current information systems are not suitable for a creative performance, but are very capable of finding patterns in huge sets of data [20].

The most commonly used and broadly accepted as the tool of choice for risk assessment is the Failure Mode and Effect Analysis (FMEA), which considers a wide range of factors and delivers compelling results. FMEA considers the severity (S), occurrence (O) and detection (D) ratings to determine the risk priority number (RPN) [START_REF]Risk assessment: A practical guide to assessing operational risks[END_REF].

Risk Priority Number (RPN) = Severity (S) × Occurrence (O) × Detection (D) (2)

KÖNIG adapted the method, which originally came from quality management, in order to be able to effectively prioritize risks. With the Risk Mode and Effect Analysis (RMEA) approach, the risks are combined into a risk cube, which consists of different risk classes according to the probability of occurrence and detection. [START_REF] König | Management betrieblicher Risiken bei produzierenden Unternehmen[END_REF] There are many other approaches to risk measurement and aggregation like Failure Mode, Effect and Cause Analysis (FMECA) [START_REF] Carpitella | A combined multi-criteria approach to support FMECA analyses: A real-world case[END_REF], Socio Technical Reliability Analysis (STRA) [START_REF] Filho | Management Tool for Reliability Analysis in Socio-Technical Systems -A Case Study[END_REF], Fuzzy Fault Tree Analysis (FFTA) [START_REF] Kuzu | Application of Fuzzy Fault Tree Analysis (FFTA) to maritime industry: A risk analysing of ship mooring operation[END_REF], Monte Carlo Simulation (MCS) [START_REF] Resteanu | Monte Carlo Simulation for Reliability Centered Maintenance Management[END_REF] and evaluation laboratory (DEMATEL) [START_REF] Si | DEMATEL Technique: A Systematic Review of the State-of-the-Art Literature on Methodologies and Applications[END_REF].

The weak point of all those models still prevails. They all rely on the estimation of the failure rates for the assets [START_REF] Garg | Maintenance management: literature review and directions[END_REF] or simulate random numbers (e.g. for MCS). In addition, they presume that all possible failure modes are known to the asset owner. It is therefore necessary to develop a data model for risk identification and comparison between companies and production sites. Current ontology-based approaches are limited in the level of detail they provide and do not include a sufficient digital reference shadow for asset risk information.

Information logistics model for risks of production assets

The current state of the ontology-based risk model is presented in this section, which starts off with a brief description of risk in ontology. Subsequently, the transfer of one example into the information logistics model notation gives an outlook on the results.

Risk management ontology for production assets

The application of an ontology-based risk management is not yet sufficiently extended to production assets because of a lacking depth of detail for a reference architecture and comparison of different assets. For example, the models do not take environmental conditions into adequate account, although they are an important factor influencing the wear and tear and the functional efficiency of a machine [START_REF] Karl | Bedarfsermittlung und Planung von Rekonfigurationen an Betriebsmitteln[END_REF].

For the definition of the risk management ontology for production assets (Fig. 1), the procedure following STUCKENSCHMIDT was chosen [START_REF] Stuckenschmidt | Ontologien: Konzepte, Technologien und Anwendungen[END_REF]. The functions of an information system are implemented according to KRCMAR: Collect, structure, process, provide, transform, communicate and use data, information and knowledge [31]. The ontology notation of MILVICH is used for Fig. 1 [32]. Accordingly, classes are displayed as dark grey ellipses. Related terms are represented by lighter grey ellipses and are connected by a straight, continuous grey arrow. In order to keep the ontology readable, the classes have been divided into sub-concepts only to the extent necessary for understanding the approach. Accordingly, not every class has assigned terms. Furthermore, in Fig. 1, information containers are displayed with a rectangular grey frame. These are connected to a class by a dotted grey arrow and describe an identification property (displayed on the connection) and its type (string, value, array, and so on). The name and type can be filled with the corresponding values of the application when using the ontology.

Exemplary transfer into an information logistics model

For the implementation and future usage, the ontology needs to be transformed into an information logistics model. Fig. 2 displays an excerpt of the model for condition monitoring, which has an impact on the occurrence (O) as well as detection (D) of a risk.

The information logistic model notation of NIENKE was used for Fig. 2 [START_REF] Nienke | Ontologie für Energieinformationssysteme produzierender Unternehmen[END_REF]. It consists of roles that are characterized by the generation of information (source) and the use of or need for information (sink). A vertical line visually separates the two parts of a role. Each role has both elements. A special role is the decision role or target role, which has a certain need for information in order to decide. An oval shape characterizes the role with a dark edge. In addition, the decision questions of the target role are noted above the role. Information objects are exchanged between roles and systems through information flows. The information objects are described under a role or a system on the source side. To improve clarity and structuring, the data records are grouped together into information objects and listed accordingly. Fig. 2 displays the data transfer from the asset to a data base. The data base is then fed into an analysis system which derives a decision template to present to the specific role. With each step more data is considered, aggregated and analyzed. Since not all data is numerical, available and easy to process, the focus must lie on the possibility to fill the gaps by reference assets. Even if it results in less accuracy for the analysis.

Conclusion

Automated risk assessment systems have the potential to contribute to the dissemination and effective use of risk information and analysis. Especially in complex production environments, it provides an effective and efficient way to optimize maintenance and investment spending, as well as the effectiveness of the whole plant. The provided reference digital shadow for risk information linked to a detailed information logistics model will help software companies to improve reliability software, standardize and enable assets owners to establish a customized digital shadow for their production networks. The future goal should be to extend the model with a variety of functions and information. Therefore, further research and case studies need to be conducted. For the moment, the model's functions focus on the process industry, and other industries can be implemented in the future, as they have similar business requirements.

Fig. 1 .

 1 Fig. 1. Risk management ontology for production assets (top level excerpt)

Fig. 2 .

 2 Fig. 2. Information logistics model for condition monitoring (excerpt)


Sara Linderson
email: linderso@kth.se

Monica Bellgran

Seyoum Eshetu Birkie

Insights from a Top-down Lean Subprogram Deployment in a Production Group: A Tactical Perspective

Keywords: Sustainable Implementation, Global Operations Management, Strategy Management

Global production companies spend a noticeable amount of resources on developing lean subprograms deployed within the corporate group. Despite the objective of achieving overall improvements, the expected performance might default due to inefficient deployment processes. Unforeseen lag or resistance to centralized programs has in previous studies shown to be influenced by contextual characteristics of the recipient, such as absorptive capacity, national culture, and factory size. How should deployment plans be designed while considering the characteristics within the corporate group? The research purpose here, is to explore the intentions and reasoning behind the designed deployment plan within a corporate group. Based on insights from a real case, this paper presents three various tactical deployment approaches of possible eight theoretical configurations, here called 'horizontal,' 'parallel,' and 'delegated' deployment. They differ principally by cascading approach, degree of freedom, and centralization of targets. The concept of 'deployment tactics' is introduced as a way to understand how the deployment plans are arranged given the contextual characteristics, and shows that the context dependency in lean subprogram deployment can be managed in various ways within a company.

Introduction

Many corporate production groups develop a tailored lean program as a way to coordinate and reach consensus about the company's dialect on lean [START_REF] Netland | Exploring the phenomenon of company-specific production systems: One-best-way or own-best-way?[END_REF]. Ideally, the benefits from a shared program come with making an impact and improving business performance in all sites within the corporate group. However, developing and deploying such a corporate lean program is a challenging task when it comes to creating a balance between global conformity and local adaptation [START_REF] Netland | Multi-plant improvement programmes: a literature review and research agenda[END_REF]. Can we assure that centralized attempts to implement lean in corporate groups do not become a waste in itself?

The existing lean transfer literature emphasizes mainly major lean implementation programs [START_REF] Bellgran | A Corporate Perspective on Global Management and Development of Lean Production Systems: A case study[END_REF][START_REF] Netland | Company-specific production systems and competitive advantage A resource-based view on the Volvo production system[END_REF][START_REF] Scherrer-Rathje | Lean, take two! Reflections from the second attempt at lean implementation[END_REF]. The research presented in this is paper investigates implementation on the next level, i.e. how strategic lean subprograms are deployed within a corporate group as part of their overall lean program. This view is somewhat highlighted in [START_REF] Danese | The transfer process of lean practices in multiplant companies[END_REF], who recognize different types of transfer approaches in a global lean setting [START_REF] Danese | The transfer process of lean practices in multiplant companies[END_REF]. Lean subprograms support the lean journey in different ways, e.g. by campaigns to reinforce a shared mindset, or by deploying assessment frameworks to track and steer the development progress. It also implies that the corporate production group could have several lean subprograms deployed at the same time, all to achieve overall business improvements. The number of failed lean subprogram cascading in the organization is somewhat unknown, but the success rate of lean subprograms that can be inferred from previous research is not higher than 30 % [START_REF] Jadhav | Exploring barriers in lean implementation[END_REF]. With the widespread impact lean has on global production companies, it is necessary to gain further understanding and knowledge to improve the efficiency of the deployment programs. The exploratory research presented here aims to attain further insight on intentions and reasoning behind top-down lean deployment approaches by investigating different deployment arrangements within a corporate production group. We suggest that the managerial perspective of sustainable production in corporate groups includes how to utilize deployment tactics to improve the efficiency of such top-down deployments.

2

Theoretical Framework

Lean Management in Global Corporations

As already mentioned, many global corporations develop a company-specific production system (XPS) conceptualized in structures considerably inspired by the Toyota Production System [START_REF] Netland | Exploring the phenomenon of company-specific production systems: One-best-way or own-best-way?[END_REF]. Despite the plethora of definitions, this study views lean production as a holistic system combining tools, methods, principles, and strategies [START_REF] Shah | Lean manufacturing: Context, practice bundles, and performance[END_REF]. In contrast to the well-known challenges of lean implementation [START_REF] Netland | Coordinating Production Improvement in International Production Networks: What's New[END_REF], the management of lean in global settings derives additional challenges. A network composed of multiple production sites allocated worldwide leads to rather complex structures to manage since the sites are not isolated from each other [START_REF] Scherrer | Linking QFD and the manufacturing network strategy: Integrating the site and network perspectives[END_REF]. Previous research often investigates the challenges with integration and organization among the production sites, as well as the balance in decision-making between headquarters and sites, thus, takes primarily a coordination perspective [START_REF] Netland | Multi-plant improvement programmes: a literature review and research agenda[END_REF][START_REF] Scherrer | Linking QFD and the manufacturing network strategy: Integrating the site and network perspectives[END_REF]. Balancing the decision-making vertically is dominantly deciding the degree of autonomy of the site in either centralized, integrated, or decentralized structures [START_REF] Scherrer | Linking QFD and the manufacturing network strategy: Integrating the site and network perspectives[END_REF]. However, inspired by [START_REF] Danese | The transfer process of lean practices in multiplant companies[END_REF], this study takes primarily a configurational view to learn about the joint influence of contextual characteristics in lean deployment [START_REF] Danese | The transfer process of lean practices in multiplant companies[END_REF]. Structured processes for implementing or reinforcing lean through corporate programs can achieve positive global outcomes [START_REF] Netland | Effects of a production improvement programme on global quality performance: The case of the Volvo Production System[END_REF]. Such tailored XPS programs aims for a reduced heterogeneity in mindset and increased global conformity of the corporation's lean journey [START_REF] Netland | Multi-plant improvement programmes: a literature review and research agenda[END_REF]. However, a single program is seldom sufficient [START_REF] Scherrer-Rathje | Lean, take two! Reflections from the second attempt at lean implementation[END_REF], and a lean journey builds on the implementation of several subprograms to continuously improve the business in a strategic way [START_REF] Demeter | Transferring lean knowledge within multinational networks[END_REF]. Since a deployment process of subprograms relies on its recipients, a business' competitive advantage relies on how well institutionalized they are by the production sites within the corporate group [START_REF] Netland | Multi-plant improvement programmes: a literature review and research agenda[END_REF]. Thus, the deployment process becomes crucial to realize the strategic purpose of the program.

A lean subprogram is for this paper defined as a multi-site improvement program to improve the business performance by reinforcing the global structure for XPS or harmonization of lean mindset within the network. Examples of lean subprograms are maturity assessments, capability assessments, and innovative implementation processes.

The stream of lean transfer literature has presented several variables influencing the transfer of lean practices within a corporate group [START_REF] Danese | The transfer process of lean practices in multiplant companies[END_REF][START_REF] Demeter | Transferring lean knowledge within multinational networks[END_REF]. Coordination aspects are dominantly in focus by investigating the challenges with a resource-based view [START_REF] Netland | Coordinating Production Improvement in International Production Networks: What's New[END_REF][START_REF] Maritan | Heterogeneity and transferring practices: Implementing flow manufacturing in multiple plants[END_REF]. In contrast, this study focusses on joint effects of the absorptive capacity of the recipient [START_REF] Maritan | Heterogeneity and transferring practices: Implementing flow manufacturing in multiple plants[END_REF], factory size [START_REF] Shah | Lean manufacturing: Context, practice bundles, and performance[END_REF][START_REF] Netland | Critical success factors for implementing lean production: The effect of contingencies[END_REF], and national culture [START_REF] Danese | The transfer process of lean practices in multiplant companies[END_REF][START_REF] Netland | Critical success factors for implementing lean production: The effect of contingencies[END_REF]. For instance, a collectivistic culture (e.g., in China) is characterized by precedence to follow orders that could somewhat facilitate the deployment of a program initially [START_REF] Danese | The transfer process of lean practices in multiplant companies[END_REF][START_REF] Netland | Critical success factors for implementing lean production: The effect of contingencies[END_REF][START_REF] Boscari | Implementation of lean production in multinational corporations: A case study of the transfer process from headquarters to subsidiaries[END_REF]. Similarly, previous studies showed initial resistance to deployed programs in individualistic cultures (e.g., in the US) [ibid.].

Methodology

A single case study, aligned with views of Yin (2009), has been followed approximately for one year [START_REF] Yin | Case study research: Design and methods[END_REF]. The case was bounded to the deployment of a centralized lean subprogram within the production branch of the corporate group. The globally initiated lean subprogram had the purpose of assessing and improving a particular set of lean capabilities in the entire organization, which implies both operational and service functions. The lean capability assessment framework (LCF) was cascaded from the global lean office via lean managers at a division's level to a large number of sites in 17 countries. Delimitations of the scope resulted in inclusion of the three divisions clustered geographically by the market (Asia, America, and Europe). The collection of data was gathered with a mixed-method approach. Five semi-structured interviews with global decision-makers in lean promoting functions were conducted. Interviews were executed face-to-face or virtually (approx. 1-1.5 h). Intentions behind the decisions of the designed deployment plan, and the expected outcomes of the arrangement was the main focus during interviews. Furthermore, questions gathering insight regarding any issues or ad-hoc solutions was asked. The data collection consisted of a transcript of the recorded interviews, documents, and communication material used internally. Triangulation of data assured verification of contradictory statements and was after that, deductively analysed with a focus on the decisions and intentions behind the plan for deploying the LCF.

Results

Global Deployment of a Lean Subprogram

The case company had more than a decade of lean experience, although a more centralized corporate lean program was initiated about 2016. The lean journey consisted of several deployed lean subprograms before the LCF studied here, such as lean maturity assessments and campaigns. A shift from a rather tool-based framework resulted in the categorization of lean practices into bundles called basic requirements, business process management, Six Sigma, and additional elements. Inadequate lean capabilities, recognized as a general point to improve in the corporate group, triggered the strategic decision to develop a lean capability framework (LCF). The LCF had the purpose of assessing and elevating lean knowledge based on a certification system where the employees' theoretical knowledge, and for some, a demonstration in practice, was tested.

The assessment framework was highly standardized, with no room for local adaptation. The framework included sets of criteria clustered in 10 different categories, which together constituted the bundle of 'basic requirements.' Depending on a set of intraorganizational roles, various assessment requirements were formulated in the criteria.

Together with the launch in early 2019, the decision that 80 % of all targeted employees should complete the basic requirements until the end of 2020 got communicated. A global directive was that leaders should go first to acquire the sufficient prerequisites needed to coach the employees in their teams. However, the execution details for the deployment of the LCF to a local level (i.e., the sites) was delegated to the division's lean managers to decide. The assessment progress was regularly followed-up globally. Further description of the deployment plans is found in the following section.

Deployment plans

The reasoning behind the plan for the deployment is described for the three divisions Asia, America, and Europe separately. Some of the contextual characteristics are similar between the division, such as the dominant individualistic culture in America and Europe (see Table 1). Table 1 also presents the decided deployment variables, which includes cascading approach, degree of freedom, and centralization of target. * Based on the same categorization as Netland (2016) [START_REF] Netland | Critical success factors for implementing lean production: The effect of contingencies[END_REF]. **Small (<49), Medium (<249), Large (<1000), and Huge (>1000).

Asia:

The deployment plan was to rigorously cascade the LCF, starting with the management team at the division level, which included all factory managers. The management team on site level was then responsible for choosing three categories that the whole site should mutually focus on for the first year. The option to choose was to enable adjustments for lean maturity at the site level, local site needs, and ensure engagement by enhancing the ownership of the program. The intention was not to deploy the LCF further vertically in the organization until a completed horizontal deployment.

The horizontal deployment was a deliberate approach and explained to be essential to ensure sufficient capability among the leaders. Two of the most lean mature sites were located within the Asian division and there where pilots prior to the deployment. America: The deployment plan focused mainly on the time when the LCF should be introduced to the various roles. Factory managers were prioritized to start ahead by getting introduced to the LCF first. After some months, the LCF was introduced to support functions in parallel, and at the same time, individuals on the shop floor in

The freedom to choose among the categories was deliberately to adapt to 'the entrepreneurial spirit of the population.' The population within the production network in the American region was generally lean immature due to many new acquisitions (60 % of the people within the last three years), meaning that many sites were entirely new to the corporate production program. The segregated population was considered as a significant factor in the planning of the deployment approach within the region. The plan to deploy the LCF to various roles in parallel to each other was believed to avoid slowing down the sites that were further in their lean transformation, and therefore, would want to progress quicker in the LCF deployment. Insights from pilots within the division were expressed to be important and had influenced the deployment plan.

Europe: The deployment plan of deliberately letting the sites be responsible for their own deployment plan meant that no steering took place at the division level. The idea of deliberately delegating the responsibility for the deployment to the sites was decided with consensus among the sites in the European division. The reasoning behind the decision was to integrate the deployment of the LCF to existing plans at the site level. An rigorous work with local plans for elevating lean maturity were developed prior the LCF without the steering of the regional lean team, and thus, the reasoning was not to interfere with those plans through top-level steering. Also, another underlying purpose of not steering the deployment from division level, was to prevent 'deployment-inertia,' meaning that the sites had experienced several previous deployments of centralized programs and experienced frustration of performing additional activities that interfered with existing plans. The lean maturity within the region is roughly classified as medium. There were no pilots prior the deployment of LCF, unlike the other division.

Analysis and Discussion

Despite a centralized global deployment, a partially delegated decision-making for the cascading approach at a divisional level yielded in three various deployment plans. These plans differed prominently in the cascading approach, degree of freedom, and centralization of targets. Interestingly, the deployment plans were arranged to fulfill the strategic purpose of the centralized lean subprogram with the intention to prevent predicted deployment inefficiencies caused by the contextual characteristics within the division. For instance, the American division predicted that 'a very heavy-handed centralized approach' would not work for their employees. Therefore they had a higher degree of freedom when choosing among the various categories (classified as low in Figure 1.). This reasoning aligns with the findings by [START_REF] Danese | The transfer process of lean practices in multiplant companies[END_REF] regarding the dominance of the individualistic culture within the division [START_REF] Danese | The transfer process of lean practices in multiplant companies[END_REF].

Taking contextual variables into account when planning the arrangement of the deployment could be described as a tactical deployment approach (see Figure 1). To exemplify this further, the Asian division planned a rigorous top-down cascading deployment with a tactical stage-gate principle. With a focus on the sequence when deploying top-down along the organization hierarchy, the stage-gate principle promoted a completed horizontal deployment before the program was cascaded vertically (classified as a strong cascade approach in Figure 1). The importance of leaders' capability to support employees were highly motivated and aligned with Liker's principle 'grow leaders that thoroughly understand the work, live the philosophy, and teach it to others' [17]. However, the top-down cascading was presumably enabled by the significant large proportion of the collectivistic culture within the Asia division, perhaps avoiding issues with stickiness in the initiation phase [START_REF] Danese | The transfer process of lean practices in multiplant companies[END_REF]. Also, the 'horizontal' deployment tactic could work oppositely, to hold back, as a tactic to prevent workers to routinely implement practices without involvement by managers. Another enabler that supported the 'horizontal' tactical approach was the initial focus on a rather low amount of categories, considering the time needed to adjust for the impact of the change that a new program made. The freedom of choice regarding the option to choose categories was rather narrow, however, not absent.

The American and European tactical approach differed from the Asian by having a weaker cascading approach. The similarity could be related to the similar dominance of individualistic national culture since both the American and European predicted a possible resistance to a centralized global subprogram. However, the European 'delegated' tactical approach could probably be related to the large/huge factory sizes in this division. The American 'parallel' tactical approach differs in that sense with a high degree of centralized common targets in the division. Moreover, since large production sites have a more complicated structure with administrative tasks, it may affect the implementation of lean practices negatively [START_REF] Shah | Lean manufacturing: Context, practice bundles, and performance[END_REF]. It could, therefore, imply requirements of a higher degree of coordination to deploy a lean subprogram to large or huge production sites successfully. Thus, the reasoning behind the delegated deployment tactic in Europe could be valid, considering the risk of superficial implementation, and thus, the absence of expected performance improvement, as discussed in [START_REF] Netland | Multi-plant improvement programmes: a literature review and research agenda[END_REF].

Conclusion

Many corporations that aim for centralized efforts to deploy lean within the production group, experience challenges with resistance and unexpected inefficiencies while doing so [START_REF] Boscari | Implementation of lean production in multinational corporations: A case study of the transfer process from headquarters to subsidiaries[END_REF]. Sustainable implementation implies that centralized lean subprograms cascaded within the corporate group should not result in superficial implementations, nor cause inertia for global initiatives [START_REF] Netland | Multi-plant improvement programmes: a literature review and research agenda[END_REF]. The tactical view presented here helps to cope with the challenges of polishing deployment plans to fulfill the strategic purpose of the program by minimizing deployment inefficiencies. The three various tactical deployment approaches, of eight theoretically possible configurations, highlights variables seen as important in the planning phase of the deployment. However, further research would provide more comprehensive generalizations and validate the relevance of the remaining five theoretical configurations. Beyond the need to strengthen the external validity, investigating which of the various deployment tactics are most successful and under what circumstances, deeper discussions are required. Although some early findings indicate following recommendations to managers; (1) It could be valuable for the corporate management to closely consider the vitality of a global completion target, since undesirable effects, such as 'certification-hunting', might counteract the strategic purpose and lead to superficial implementation, (2) Site level management could benefit from being aware of the global approach to understand local requirements, for instance, decentralized goal setting, and (3) The global lean office could consider how to mitigate possible risks with resistance to programs with a tailored deployment approach, i.e. allow different intra-organizational arrangements. However, at the same time consider support for various requirements of particular cascading approaches, for example, with deployment principles (e.g. stage-gate principle) to enable flexibility in the arrangements, rather than 'one size fits all'. In other words, one approach is not necessarily the most suitable for all. The tactical perspective allows for more thoughtful arrangements where the influential contextual characteristics of each site within a corporate group could be taken into consideration. One must remember, the ultimate purpose of excelling in efficient and sustainable deployment is to avoid the paradoxical situation of lean deployment become a waste in itself.

Fig. 1 .

 1 Fig. 1. Framework for the arrangement of deployment plans for three tactical deployment approaches.

Table 1 .

 1 Contextual characteristics and deployment variables at the various divisions.

	Context characteristics	ASIA	AMERICA	EUROPE
	National Culture*	Majority collectivistic	Majority individualistic	Majority individualistic
	Factory Size**	Small/ Medium/	Small/Medium/	Small/Medium/
		Large	Large	Huge
	Deployment Variables
	Cascading Approach	Top-down cascad-ing with a focus on the deployment se-quence among hier-archical levels	Cascading with a focus on the tim-ing for initiation of the program among different roles.	Delegation of the responsibility to the sites, to plan the deployment. A focus on enhanc-ing integration.
	Degree of freedom	Low	High	High
		Steering on the divi-sion level. Common categories on site level. Recommen-dation provided. Lower levels could not choose.	Steering on the di-vision level. Op-tion to choose on individual level. Recommendations provided.	No steering on the division level. Option to choose on site level. No recommendations provided.