Peter Nielsen

Zbigniew Michna
email: zbigniew.michna@ue.wroc.pl

Ngoc Dung

Ngoc Anh

Dung Do

An Empirical Investigation of Lead Time Distributions

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Supply chain management has long been one of the leading topics in both management and academia. The focus is the management of activities a cross a chain of companies and thus partitioning the supply chain into a number of echelons. As with all fields there are number of assumptions build in to the way supply chain management literature addresses the planning and control activities [START_REF] Otto | Does supply chain management really pay? Six perspectives to measure the performance of managing a supply chain[END_REF]. This paper investigates lead times and the associated assumptions found in Supply chain literature.

In practice most research into supply chain management assumes that lead times follow one of two forms. Either lead times are assumed to be constant (see e.g. Chen et al. 2000) or they are assumed to be i.i.d. [START_REF] Kim | Quantifying the bullwhip effect in a supply chain with stochastic lead time[END_REF][START_REF] Michna | The bullwhip effect in supply chains with stochastic lead times[END_REF]. Rather than attempt to model the impact of a given type of lead time behavior on a supply chain, this paper investigates the actual lead time behavior in a manufacturing company. The aim is to identify the actual lead time behavior and subsequently in further research model this behavior and its impact on supply chain performance.

The remainder of the paper is structured as follows. First, a brief literature review of the current-state of supply chain management with regards to modeling lead time behavior is presented. Second, a method for analyzing the distributions of lead times is presented before it is applied to data from a company. Finally implications from the case investigation and future avenues of research are presented. [START_REF] Otto | Does supply chain management really pay? Six perspectives to measure the performance of managing a supply chain[END_REF] define six specific approaches to address supply chain management with varying focus. This paper focuses on the perspective [START_REF] Otto | Does supply chain management really pay? Six perspectives to measure the performance of managing a supply chain[END_REF] would term Systems Dynamics and papers related to this perspective. Systems Dynamics is characterized by focusing on distortion of demand patterns for various reasons (demand forecasting, non-zero lead time, supply shortage, order batching, and price fluctuation [START_REF] Duc | A measure of the bullwhip effect in supply chains with stochastic lead time[END_REF] and typically this distortion is quantified by the bullwhip effect (variance of downstream orders / variance of upstream demand e.g.

State-of-the-art

Chen et al. (2000)).

This research is narrowly focused on lead times and the assumptions build into analytical or simulation based models for determining the bullwhip effect in a given supply chain. It is also based on the notation that IT can be used to estimate and register lead times in real supply chains (Arshinder and Deshmukh, 2008). Chen et al. ( 2000) is one of the main contributions to quantifying the bullwhip effect in supply chains. However, in the work of Chen et al. (2000) lead times are actually assumed to be constant. The same goes for the control theory approach used in [START_REF] Dejonckheere | Measuring and avoiding the bullwhip effect: A control theoretic approach[END_REF]. This is not the case in the more recent work of [START_REF] Duc | A measure of the bullwhip effect in supply chains with stochastic lead time[END_REF]. Here the bullwhip effect is quantified for a system with stochastic lead times. The lead times are assumed to be stationary and i.i.d.. The same assumption is found in the work by [START_REF] Kim | Quantifying the bullwhip effect in a supply chain with stochastic lead time[END_REF], who in their work use a similar analytical approach to Chen et al. (2000) and [START_REF] Duc | A measure of the bullwhip effect in supply chains with stochastic lead time[END_REF]. This is very significant as Chattfield et al. ( 2004) note that stochastic lead times are major source of bullwhip effect. [START_REF] Kim | Quantifying the bullwhip effect in a supply chain with stochastic lead time[END_REF] choose another approach by assuming that both demand and lead times are stochastic, but rather than predict demands and assume stochastic lead times they choose to predict lead time demand. Another interesting aspect addressed by [START_REF] Chaharsooghi | LT variance or LT mean reduction in supply chain management: Which one has a higher impact on SC performance[END_REF] is determining for a specific supply chain what has the largest effect; reducing the average lead time or reducing the variance of lead time? Chaharsooghi and Heydari (2010) use simulating and multivariate models to conclude that lead time variance is in fact a major cause of bullwhip effect. This underlines the importance of determining actual lead time behavior to reduce the bullwhip effect in supply chains. If lead times are not in fact i.i.d. the bullwhip effect will in all likelihood be higher than expected, and also higher than standard models will be able to explain. Nielsen et al. (2010a) offers an approach to improve supply chain planning through the use of RFID technology to track and thus be able to provide higher quality information, while Sitek and Wikarek (2013) offers approaches for improving planning through optimization. The main issue in applying optimization methods for supply chain management is the quality of the information available for the methods.

It is interesting to note that there is very limited research that actually investigates lead times, regardless the fact that quite some research has been conducted assuming a given lead time behavior. This research addresses this gap and suggests a method for determining whether lead times are in fact i.i.d..

3

Method of analysis

The method for analysis is composed so that it addresses both the independence of observations (in this case only investigated as L i is independent of L j , where L j is a lead time observations lagged arbitrary to L i ) and whether any set of observations stem from the same distribution as any other set.

The data used is from a manufacturing company and represents the sales for the ten most frequently sold (in terms of number of order lines) make-to-stock products over a two year period. The most frequently sold product has 6,967 orders in the period and the tenth most sold has 2,158 orders. So no testing is done on less than 2158 observations. Data is only cleaned in so far that any lead time that is more than six standard deviations from the mean are removed as the tests methods tend to over fit to the few extreme distributions. The outliers are removed in one step.

The analysis methodology addresses both the aspect of identical distributions and independently distributions. From literature two combined assumptions have been identified and thus the analysis methodology must address both these aspects. This means that there are two hypotheses that must be tested:

1) Lead times are identically distributed, i.e. for any given practical purpose it is possible to assume that for a given planning horizon the lead time distribution is the same for the whole period.

2) Lead times are independently distributed. This may have different implications, but is typically taken to mean that any lead time observation does not depend on any previous observed lead time value. It may however also be taken to mean that lead times is not dependent on any other variable. In this research only the first aspect is investigated.

To investigate both these aspects the analysis methodology shown in Figure 1 is used. Kolmogorov-Smirnov's (KS) test is used to determine whether or not the lead times stem from identical distributions. The KS test is a widely used robust estimator for identical distributions [START_REF] Conover | Practical Nonparametric Statistics[END_REF] and does not suffer from some of the weakness of other tests such as Chi-squared. The method (as seen in Figure 1 and 2) relies on comparing samples of lead times and using KS test to determine whether not these pairwise samples are identical. In this research a 0.05 significance level is used and the ratio of pairwise comparisons that pass this significance test is the output from the analysis. Different sample sizes are used to determine if the lead times can be assumed to be similar distributed in smaller time periods, and thus if it is fair to sample previous lead time observations to estimate lead time distributions for planning purposes. For the (in)dependence tests Box-Jenkins autocorrelation function is used [START_REF] Box | Time Series Analysis: Forecasting and Control[END_REF]. In this case the test of independence is run on the mean lead time achieved on any given actual delivery date. The mean value is used as it makes no Sample 1

SL 1 = {t | t ∈ [1, Z]} Vs. Sample 2 SL 2 = {t | t ∈ [Z+1, 2Z]} Sample 3 SL 3 = {t | t ∈ [2, Z+1]} Vs. Sample 4 SL 4 = {t | t ∈ [Z+2, 2Z+1]} Sample 2j-1 SL 2j-1 = {t | t ∈ [j, Z+j-1]} Vs. Sample 2j SL 2j = {t | t ∈ [Z+j, 2Z+j-1]} Sample 2J-1 SL 2J-1 = {t | t ∈ [J, Z+J-1]} Vs. Sample 2J SL 2j = {t | t ∈ [Z+J, 2Z+J-1]} … … … …
SL: a sample of lead time Z: sample size of a sample of lead time J: number of pairs for pairwise comparison sense to chronologically order lead times for individual orders in smaller time intervals than one day. It is for all practical purposes not reasonable in most manufacturing environments to discuss manufacturing lead time in smaller intervals than whole days, as the data is not sampled with that level of detail. Only the first lag is considered as no systematic behavior is expected beyond piecewise increasing/decreasing lead times in the particular case study. In other contexts seasonality or customer order cycles could be incorporated and other lags should be included in the analysis.

Case application of method

The following section explores the results of the case study, with the results from applying the methodology to the case data shown in Figure 3.

Fig. 3. An overview of first order autocorrelation

As seen in Figure 3 the first order autocorrelation values for the ten products range from 0 to 0.4. For practical purposes it would reasonable to assume that lead time set that have first order autocorrelation values below 0.2 is in fact independently distributed. Out of the ten products, six products have first order autocorrelations less than 0.2, while the remaining ten products have first order autocorrelations in the interval 0.25-0.42 indicating some form of dependence in the mean values of the lead times. It is also interesting to note that all the first order autocorrelation values are positive, indicating that to some extend large average lead times are follow by large average lead times. This clearly underlines that the average lead times may be linked to capacity issues, despite the products being purely make-to-stock. If capacity constraints are periodically present, one would expect to find that the average lead times are in fact dependently distributed, with positive autocorrelation values for a number of lags. In this aspect it is also interesting to note that the four products that are not independently distributed are in fact the third, fourth, seventh and tenth most sold products. So there is no indication that the number of observations is a relevant criterion when determining whether a product is in fact independently distributed. Furthermore it should be noted that the products are all produced on the same production line, with very similar components and process times. This could indicate that the company in periods of constrained capacity / material shortages chooses to deliver specific products, e.g. caused by prioritization of certain customers.

As can be seen from Figure 3 there is a clear tendency that the larger the sample size used for the pairwise comparison with the KS test for identical distributions, the less of the samples are significantly identically distributed. However, to determine whether the distributions can reasonably be considered to be identical one needs a benchmark. For this reason a benchmark study is conducted. In the benchmark study 10,000 pairs of distributions in sizes 50, 100, 150 and 200 (the same samples sizes as shown in Figure 3) are generated and compared. Three distributions are chosen for the experiment; normal, exponential and uniform distributions, and the sample values are rounded to nearest integer (as lead times are integers). The benchmark value is then how many of these samples are actually for a given distribution and sample size found to be similar when they are known to be sampled from the same distribution. The benchmark values can be seen in Table 1. The indication from the values in Table 1 is that even for small samples (50 observations) more than 95% of the comparisons should be significantly the same using a KS test. None of the sampled lead times achieve this high level of confidence. It is noteworthy that the KS test performs equally well for all three benchmark distributions, so no bias can be expected in the test due to the shape of the lead time distribution. It would be fair to conclude that most of the products exhibit a behavior that indicates that for reasonable sample sizes the lead times actually stem from the same distribution. The data covers around 500 work days of observations, meaning that even the product with the highest order frequency only has approximately 15 orders/day, assuming a lead time of 7 days, that would mean that on average 105 orders can be observed during a normal lead time and thus in c. 80% of the cases the lead time distribution for the next 7 days would be the same as for the previous average lead time period. This means that for most of the products it is actually fair to assume that the lead time distribution for the next lead time period can be estimated from the lead times observed during the last lead time period. It however also means that lead time distributions should be updated with relatively high frequency, which can lead to nervous planning systems. It is also interesting to note that the products with high first order autocorrelation values exhibit the poorest performance on the test for identical distributions as well. This is not unexpected as the KS test for similar distributions also tests the mean of the samples. If there is a tendency that lead times increase /decrease systematically, it is also fair to assume that the distributions change. Here it is critical to note that the shape of the lead time distribution may actually be the same, an obvious subject for further research. Taking together with the knowledge that demand distributions may exhibit similar time dependent distributions even within a planning period (Nielsen et al., 2010b) the planning problem is very much more complex that the assumptions allow for.

Exponential

Implications & further research

There are a number of implications that can be inferred from the presented analysis for both academia and practitioners. First, it is obvious that lead times are, in the presented case, not constant as assumed in most supply chain models (see e.g. Chen et al. ( 2000)). Second, some of the products exhibit lead times that neither stem from identically nor independently distributions, while a number (six out of ten) can in fact reasonably be assumed to be i.i.d. For research purposes this indicates that even assuming that lead times are i.i.d. may be an oversimplification compared to real-life conditions. It also underlines the folly of assuming that lead times are constant, as this is a gross oversimplification. For practitioners there is also a significant implication. Specifically that lead times are in all likelihood not constant and thus that this uncertainty should be included in the planning and control approaches used. Another interesting implication is that the more information used to estimate the lead time distribution, the worse the estimate of the distribution in fact becomes. This is due to the pairwise comparison of identical distributions indicating that large sample sizes seldom lead to the same lead time distributions. In practice this means that companies should update their lead time information frequently and disregard old observations.

Fig. 1 .

 1 Fig. 1. Analysis methodology for investigating whether lead times are i.i.d.

Fig. 2 .

 2 Fig. 2. Sampling for pair-wise comparison of similar lead time distributions.

Table 1 .

 1 Benchmark overview of three different distributions, with four different samples of in each case 10,000 pairwise comparisons.

			Normal	Uniform
	Sample of 50	0.962	0.964	0.960
	Sample of 100	0.961	0.964	0.961
	Sample of 150	0.959	0.962	0.961
	Sample of 200	0.960	0.962	0.961

This is completely contrary with the conclusions in e.g. Chen et al. (2000) and Michna et al. (2013) when stationary behavior is assumed on demand. Future research will focus on investigating lead time distributions dependence on other parameters, specifically whether lead times depend on order sizes. The presented research is partly supported by the Polish National Centre of Science under the grant UMO-2012/07/B/HS4/00702.


Mathieu Bettwy

Karine Deschinkel

Samuel Gomes

An optimization tool for process planning and scheduling

Keywords: Process planning, Process scheduling, Genetic algorithm

Process planning and scheduling are one of the most important functions to support flexible planning in a manufacture. The planning and scheduling should be solved simultaneously and not sequentially for productivity improvements in manufacturing. In this paper, we propose an optimization tool based on genetic algorithm (GA) approach to help person in charge of process planning and scheduling to find the most promising sequence of operations considering a choice of machines on which to perform the operations. Minimizing makespan is the evaluation criteria.

Introduction

Many industries are trying to best optimize the whole system to deal with a global manufacturing industry more competitive. This will require to reconsider the supply chain. As a result, companies must migrate from separated planning processes toward the integrated planning process to provide competitive products while reducing costs and / or production time.

We define the problem as an integrated problem of process planning and scheduling (IPPS). A lot of work has been done on this subject and various approaches have been used to solve the problem.

Tan (2000) [6] presents a review of the research in the process planning and scheduling area and discusses the extent of applicability of various approaches. They show that the efficient planning considering the alternative machines results in reduced lead-time and in improved overall machine utilization. Moona and Seo (2005) [4] develop an evolutionary algorithm (EA)based to solve some flexibility problems on shop floor.

Yuan and Xu (2013) [7] show an heuristic algorithm to figure out large-scale shop floor problem.

Li, Shao, Gao and Qian (2010) [3] develop a hybrid algorithm (HA) based-approach has been developed to facilitate the integration and optimization of process planning and scheduling in same time.

Kim and Choi (2014) [2] propose to solve the backward on-line job change scheduling problem, a production system-based simulation methodology.

However, the main weakness of the models introduced from this two decades is that they consider alternative machines for each operation for a fixed assembly sequence. The originality of this paper is to consider a set of alternative assembly sequences and to provide the best one optimizing multiple criteria. These criteria values are obtained through the resolution of the IPPS based on genetic algorithm method. In this paper, we consider a IPPS problem for a plant composed of a set of alternatives machines, multiple product flows and various assembly sequences. The alternative machines have different capabilities and require unequal processing time for an operation. Section 2 is dedicated to the problem definition. Section 3 gives details of our approach. The paper ends with a conclusion where our contribution is summarized and planned future work is discussed.

Problem definition

In many manufactures, operations have a different possible way to be done with a set of alternative process plans. Fig. 1 shows an example of different flow to produce in a shop floor.

Shoop floors include several machines, which have different functions, processing time, and capabilities. Therefore, actual optimization should be done by considering the available machines and their ability. Fig. 2 shows the different steps of our proposed methodology for determining process planning and scheduling.

Our study is based on previous work done by ROBERT [5] concerning the tool "Orasse Product". This tool aims to guide the users during the definition of the assembly sequence of the generic product of a family. It goes from the concept with its list of components and the modular product architecture (from the adaptation of the FAST diagram) to the assembly sequence defined by the assembly planner thanks to Orasse Product features. From an assembly and kinematic scheme (first quarter of Fig. 2), "Orasse Product" helps assembly planners to build promising assembly sequences. In this tool, physical contacts and precedence constraints between components are modeled by a directed graph and algorithms based on partitioning matrix and graph theory are developed in order to guide the assembly planner during the constitution of the best assembly sequence. Based on the same methodology, we intent to develop "Orasse Process" (Fig. 3). In case of "Orasse Process", we consider an operations graph deduced from an assembly sequence generated by "Orasse Product". Next step consists in determining a process planning and a process scheduling associated to this operations graph regarding to specific criteria. For the moment we have limited our approach to minimizing a single criterion : the makespan.

Algorithms to solve IPPS

Genetic algorithm (GA) is well suited to deal with optimization problem with a large searching space as in the IPPS (several assembly sequences, alternatives machines, mutli criteria). Moreover GA produces a set of near optimal solutions compared to other optimization methods which give only one solution. This can be useful for a person in charge of the shop floor production to select a solution among a reduced number of acceptable solutions taking into account particular criteria. A theoritical foundation of GA and their convergence to an optimal solution can be found in [START_REF] Goldberg | Genetic Algorithms in Search, Optimization and Machine Learning[END_REF].

In this work, each chromosome is represented by a string of priorities: one distinct priority value for each operation. This string of priorities induces a specific order to derive a feasible schedule. The Makespan algorithm (AF) explains in details how the schedule is constructed and how the assignment of machines on operations is determined. The value of the resulting makespan is used as the fitness of the chromosome in the Genetic Algorithm.

Makespan algorithm

The Makespan algorithm (AF) computes the fitness function by considering the priorities P i assigned to each operation i. Input data is a directed graph of operations denoted by G = (O, E) where O is the set of nodes and E is the set of arcs. Each node corresponds to an operation and a precedence constraint between two operations is modeled by an arc. The set of machines and their processing time for each operation are assumed to be known (filled or modified by the user) and data are saved in a matrix T where T (i, j) is the time to operation i on machine j and T (i, j) = 0 if operation i is not feasible on machine j. The Makespan algorithm (AF) processes the operations in the order given by the priorities and assign a machines with the minimum time among available machines for the corresponding operation. An unique sequence with a given makespan σ is generated.

Genetic Algorithm

Genetic Algorithms (GAs) are stochastic, population-based search algorithms to deal with multiobjective optimization problems. GAs start by initializing a set (population) containing a selection of chromosomes (individuals). A fitness associated to each individual allows to distinguish between better and worse individuals. A GA iteratively tries to improve the average fitness of a population by construction of new populations. A new population consists of individuals (children) built from the old population (parents) by the use of re-combination operators (crossover and mutation operators). Better individuals have higher probability to be selected for re-combination than other individuals. After some criterion is met, the algorithm returns the best individuals of the population. In our case, each chromosome is represented by a string of priorities (an integer between 1 and the number of operations). Figure 4 gives an example of chromosome. The initial population contains individuals randomly generated. The crossover and mutation operators are represented in figure 5. The fitness (here the makespan) of each chromosome of the population is evaluated through the execution of algorithm (AF). After a given number of iterations without improvement of the average fitness of the population, the algorithm stops and provides the best individuals. As we consider a set of alternatives assembly sequences and consequently a set of alternatives operations sequences, we have to repeat the optimization process for different populations. At the end, our optimization tool will provide a set of promising schedules to the person dealing with the shop floor production.

Conclusion

In this paper, a methodology based on genetic algorithm is proposed to solve IPPS problem by considering various assembly sequences for a same product and a set of machines for a same operation. We intent to enrich our model by introducing many other criteria as transportation times for all pairs of machines, set-up times between operations, ergonomy and so on. Subsequently we will incorporate specific knowledge of the IPPS in the GA, so which generally improves its efficiency. A software "Orasse Process", proposing users the whole method, is under development.

Fig. 1 :

 1 Fig. 1: Production flows in a shop floor

Fig. 3 :

 3 Fig. 3: Aims of Orasse process

Fig. 4 :

 4 Fig. 4: Chromosome representation

Fig. 5 :

 5 Fig. 5: crossover and mutation

  Starting time of operation iFi: Ending time of operation i Dj: Date of availability of machine j Assignment A(i, j) = 1 if operation i is assigned to machine j else A(i, j) = 0

	Algorithm 1 AF Algorithm
	Input:
	G: Operations Graph (with O : set of nodes, and E set of arcs)
	M: Set of Machines
	T: Operating time
		Machine 1 Machine 2 ... Machine j ... Machine M
	Operation 1 T(1,1)	T(1,2) ... T(1,j) ... T(1,m)
	Operation 2 T(2,1)	T(2,2) ... T(2,j) ... T(2,m)
		...	...	...	...	...	...	...
	Operation i	T(i,1)	T(i,2) ... T(i,j) ... T(i,m)
		...	...	...	...	...	...	...
	Operation N T(n,1)	T(n,2) ... T(n,j) ... T(n,m)
	P: set of priorities; Pi = priority associated with operation i
	Output:
	Process scheduling
	Si: 4:	Dj = 0
	5:	for all operation i ∈ O do
	6:	A(i, j) = 0
	7:	end for
	8: end for
	9: Process scheduling :
	10: while O = ∅ do
	11:	if all nodes ∈ O have predecessor then
	12:	Impossible assignment
	13:	else
	14:	Select i ∈ O with no predecessor and with the highest priority (minimum
		value of Pi)
	15:	Select machine j with smallest T (i, j)
	16:	A(i, j) = 1
	17:	Si = Dj
	18:	Fi = Si + T (i, j)
	19:	Dj = Fi
	20:	O ← O \ i
	21:	E ← E \ (i, v), v ∈ O
	22:	end if
	23: end while
	24: σ=max

σ : Makespan 1: Initialization : 2: G (O , E ) ← G(O, E) 3: for all machine j ∈ M do j∈M Dj


Professor Ilkyeong Moon
email: ikmoon@snu.ac.kr

Sanghoon Shin

Dongwook Kim

Integrated Assembly Line Balancing with Skilled and Unskilled Workers

Keywords: assembly line balancing, genetic algorithm, skilled/unskilled workers

In this paper, we extend the general assembly line balancing problem by designing an integrated assembly line and addressing the number of workstations and simultaneous assignments of skilled and unskilled workers. We develop a mixed integer program that minimizes the sum of total annual workstation costs and annual salaries of skilled and unskilled workers within a predetermined cycle time. Because this problem is NP-hard, we also develop a genetic algorithm to obtain efficient solutions for large problems. Numerical experiments demonstrate the efficiency of the random key-encoded genetic algorithm.

Introduction

Most manufacturing companies employ workers based on their experience or skill set while also trying to reduce production costs. Therefore, assembly line problems have been widely researched during the latest period of industrial development. In his graduate thesis, [START_REF] Bryton | Balancing of a continuous production line[END_REF] introduced the idea of line balancing, and many studies have undertaken this subject over the subsequent 60 years. Various new assembly line balancing problem types such as two-sided, parallel, mixed-models and others, have emerged as have effective solution algorithms. Hoffmann [START_REF] Hoffman | Assembly line balancing with precedence matrix[END_REF] presented optimal line balances by operation on a matrix of zeros and ones called a precedence matrix.

Because the installation of an assembly line requires large investments, assembly line balancing problems can be distinguished based on objectives addressed through such as cost-or profit-oriented models. One of the cost-oriented models involves selecting equipment specific to the tasks assignments to workstations. The manufacturing of a product depends on resources such as equipment and manpower as well as processes. The term assembly line design problem is frequently used where these decisions are related. Pinto et al. [START_REF] Pinto | Assembly line balancing with processing alternatives: an application[END_REF] considered a model that combines the balancing problem with process alternatives that reduce task time. Bukchin and Tzur [START_REF] Bukchin | Design of flexible assembly line to minimize equipment cost[END_REF] suggested a model of equipment alternatives that minimizes the total equipment costs for a given cycle time. The cost-oriented model can be extended by considering.

Since Salveson [START_REF] Salveson | The assembly line balancing problem[END_REF] formulated the assembly line balancing problem (ALBP) mathematically, numerous heuristic methods and procedures have been introduced. Corominas et al. [START_REF] Corominas | Balancing assembly line with skilled and unskilled workers[END_REF] considered the process of rebalancing the line at a motorcycle assembly plant. They suggested a mathematical model based on a binary linear program to minimize the number of unskilled workers required. Chong et al. [START_REF] Chong | Solving Assembly Line Balancing Problem using Genetic Algorithm with Heuristics-Treated Initial Population[END_REF] compared a heuristicstreated initial population and a randomly generated initial population of genetic algorithm for solving ALBP. Moon et al. [START_REF] Moon | Integrated assembly line balancing with resource restrictions[END_REF] introduced the integrated assembly line balancing problem with resource restrictions. They considered multi-skilled workers as resources and formulated a mixed integer linear program.

In this paper, we deal with the problem of designing an integrated assembly line with simultaneous assignment of skilled and unskilled workers. Although Corominas et al. [START_REF] Corominas | Balancing assembly line with skilled and unskilled workers[END_REF] considered labor, they did not consider integrated assembly line balancing at the same time. Their objective was to minimize the number of unskilled workers for each workstation. Moon et al. [START_REF] Moon | Integrated assembly line balancing with resource restrictions[END_REF] studied an integrated assembly line balancing problem, but they did not consider reducible task times through cooperation, a problem that arises when limited resources affect the operation time for every task. The selection of both skilled workers, whose salaries depend on their competencies, and unskilled workers, who can reduce operation time to perform tasks, is addressed in this study. In addition, the assignment of tasks to workstations with precedence restrictions is also considered as the resource issues are evaluated.

Mathematical Model

For the integrated assembly line balancing with skilled/unskilled workers, the following assumptions were made:

(1) The skilled workers are multi-skilled with different salaries.

(2) The skilled workers must be assigned to one workstation.

(3) The skilled workers can be assigned to tasks depending on their skills. (4) The unskilled workers cannot be assigned alone to any workstations. [START_REF] Hoffman | Assembly line balancing with precedence matrix[END_REF] The task time can be reduced by assigning an unskilled worker to a task. (6) An unskilled worker can be assigned to only one task. (7) A skilled worker can be assigned to a single workstation. [START_REF] Salveson | The assembly line balancing problem[END_REF] The precedence constraints determine the sequence in which the tasks can be processed. (9) There is a limitation for assigning tasks at a station. Using these assumptions, we present a mixed integer linear program for an integrated assembly line balancing problem with skilled and unskilled workers. The following notation is used to explain the model:

Notation , i j indices of tasks (i,j = 1,
i w s Z      Objective function 1 1 1 1 1 W S I S W w sw isw w s i s w Min FC F LC Y LA Z = = = = =   ⋅ + + ⋅     ∑ ∑ ∑∑∑ (1)
Subject to

1 1 1 S W isw s w X i = = = ∀ ∑∑ (2)
, ,

isw isw Z X i s w ≤ ∀ (3) 1 1 S sw s Y w = ≤ ∀ ∑ (4) ( ) 1 1 ( ) 0 ( , ) S W isw jsw ij s w s X s X i j P = = ⋅ -⋅ ≤ ∀ ∈ ∑∑ (5) 1 1
( )

I W i isw i isw i w o X r Z C s = = ⋅ -⋅ ≤ ∀ ∑∑ (6) 1 , I isw sw i X M Y s w = ≤ ⋅ ∀ ∑ (7) 1, 1 0 w I S isw i i A s X w = ≠ = = ∀ ∑ ∑ (8) 1 1
( )

I W isw isw i w X Z n s = = + ≤ ∀ ∑∑ (9) 1 , S isw s s X F i w = ⋅ ≤ ∀ ∑ (10) , isw X , sw Y {0,1} isw Z ∈ (11)
The objective function ( 1) is to minimize the sum of the total annual workstation costs and the annual salaries of the skilled and unskilled workers. Constraints (2) ensure that every task is performed by one skilled worker at exactly one workstation. Constraints (3) restrict that an unskilled worker might be assigned to task i when a skilled worker is assigned to task i. Constraints (4) restrict that a skilled worker is assigned to exactly one workstation. Constraints (5) ensure the implementation of the precedence relationships between the precedence task sets identified by 𝑃𝑃 (𝑖𝑖,𝑗𝑗) ; for example, if task i is the immediate predecessor of task j, it must be assigned a higher operating index than task j. Constraints [START_REF] Moon | Integrated assembly line balancing with resource restrictions[END_REF] represent that the total maximum operating time of each workstation within a given cycle time. Constraints [START_REF] Pinto | Assembly line balancing with processing alternatives: an application[END_REF] define the task assignment at the workstation with a worker. 𝑋𝑋 (𝑖𝑖𝑖𝑖𝑤𝑤) can be greater than zero when skilled worker w is assigned to a workstation. Constraints [START_REF] Salveson | The assembly line balancing problem[END_REF] guarantee that a skilled worker cannot be assigned to a workstation with a task that the worker cannot perform according to the skilled worker's available task set 𝐴𝐴 𝑤𝑤 . Constraints (9) restrict the number of skilled and unskilled workers to the predetermined number for the workspace. Constraints (10) determine the total number of workstations to be used. Constraints (11) express the binary nature of the variables.

Genetic Algorithm

We developed a random key-encoded genetic algorithm with a procedure that slightly differs from the general one. The suggested genetic algorithm is summarized in following sentences.

Step 1. Generate an initial population of randomly constructed chromosomes with random keys for task and unskilled worker assignments that represent solutions to the problem.

Step 2. For the task arrangement chromosome, the offspring is generated using the convex hull crossover. Step 3. Subject the offspring to mutation based-on probability to create slight changes in offspring structure. This mutation procedure can avoid premature convergence to a local optimum. Step 4. Reorder the offspring to obtain fitness values, which are used for measuring the quality of each chromosome in comparison with others.

Step 5. Apply a selection procedure. The chromosome with the best fitness value joins the population and the one with the poorest fitness value is removed

Step 6. Terminate the algorithm when predetermined generations were reached or when the best individual does not improve more than 0.01% for predetermined generations.

Representation chromosomes

The proper representation of a solution plays a key role in the development of a genetic algorithm. A string that consists of real integers is a solution (a chromosome).

Traditionally, chromosomes are simple binary strings; however, this simple representation is not well suited for reproduction and other procedures. In this study, we use a chromosome that consists of two parts to represent task precedence and unskilled worker assignments. One of these offers solutions for assigning the tasks to the workstations through a random key, while the other offers solutions for unskilled worker assignments through randomly generated binaries as shown in Figure 1.

We used a random key to the task assignment for maintaining precedence relationships. If tasks do not have any predecessors, then their random keys are compared for selection and assigned to a workstation. For example, because Tasks 1 and 2 do not have any predecessors, as shown in Figure 1, they are candidates for selection. However, Task 2 is selected for a workstation assignment because the random key of Task 2 is larger than that of Task 1. After selection, the value of the random key is assigned a negative value to prevent it from being selected again.

For the skilled worker assignment, we used a simple heuristic that sorts on the basis of task precedence restrictions and the skilled workers' available task sets. The steps for the simple heuristic are as follows:

Step 1. Calculate operating time of cumulative tasks by observing the chromosome and divide workstations using the predetermined cycle time.

Step 2. Arrange the skilled worker candidates for each workstation.

Step 3. Select the workstation with the fewest candidates. If a tie occurs, select a workstation arbitrarily.

Step 4. Select the worker with the lowest salary.

Step 5. If the index of skilled worker candidates is empty for any workstations, find the appropriate combination of skilled workers.

Objective and fitness functions

The evaluation function in a genetic algorithm plays a role similar to that of the environment in natural evaluation. Each individual in the population represents a potential solution to the problem. A fitness function is computed for each string in the population and the objective is to find the string with the minimum fitness function value. For a given string, one can locate the minimum relevant cost. Equation ( 1) is used as a fitness function in the proposed genetic algorithm.

Reproduction, crossover, and mutation

The genetic operators such as crossover and mutation produce a second generation population of solutions. The method of the genetic operators -reproduction, crossover, and mutation-can change the set of the next population.

Reproduction is the biological process of producing new offspring from parents. In this study, parents are chosen by a rank mechanism. Unlike other reproduction approaches, a ranking approach offers a smoother selection probability curve. This prevents organisms from being dominated at early points in evolution. The crossover operation is a diversification mechanism that enables the genetic algorithm to examine unvisited regions and generate a solution. In this paper, we show the convex hull crossover operator for the task assignment and the one-cut point crossover operator for the unskilled worker assignment. We select two relative chromosomes for the crossover. Figure 2 shows an example to illustrate it. For the unskilled worker assignment, the position of the cut point is randomly generated from the range [1, (length of the chromosome -1)]. We generate the offspring simply by exchanging the appropriate parts of their parents.

Fig. 2. Example of the convex hull crossover procedure

In a genetic algorithm, mutation is a background operator that produces a spontaneous random change in various chromosomes. Mutation plays the crucial role of replacing genes lost from the population during evolution so that they can maintain diverse populations, thereby preventing the population from being too rigid to adapt to a dynamic environment. In this paper, we used a swap mutation operator for the task assignment and a random point mutation operator for the unskilled worker assignment.

The random point mutation selects a random point from the range [1, (length of the chromosome -1)] where the gene is subsequently changed.

Termination

The terminating condition was determined when 50,000 generations are produced or when individual fitness was not bettered by more than 0.1% over 2,000 generations. For the parameters of the proposed genetic algorithm, the crossover and mutation operators were employed with the parameters listed in Table1. A pilot test was conducted to find appropriate parameter values.

Computational Experiments

The mixed integer linear program was implemented and solved using LINGO version 10.0. The proposed random key-encoded genetic algorithm was developed using C# with .Net framework version 4.0. Both numerical experiments were conducted on a Pentium IV 3.2GHz processor with 2GB RAM on the Microsoft Windows XP operating system. The mathematical model requires significant time to find an optimal solution. Moreover, this model was not able to solve problems as large as those with 32 tasks. To validate results, the mathematical model was compared with the proposed genetic algorithm for small problems.

The experiment was conducted for large problems: 61 tasks and 29 workers. The predetermined cycle time is 350 minutes and the annual operating cost for each workstation is $180,000/year. This assembly line is composed of 12 workstations, 23 skilled workers, and 23 unskilled workers. The total operating cost, which combines expenses of workstations as well as those of skilled and unskilled workers, is $3,188,000. The average computational time of 10 evaluations is 1,533 seconds. In addition, we solved 19 different examples. The result of the mathematical model and proposed genetic algorithm is shown in Table 2.

Conclusions

Moon et al. [START_REF] Moon | Integrated assembly line balancing with resource restrictions[END_REF] proposed the concept of integrated assembly line balancing problem, and this study extends the idea by considering skilled and unskilled worker assignments. The skilled workers have multiple competencies and commensurate salaries, and unskilled workers, with lower salaries, are assigned to help them. We developed a mathematical model as an MILP for the integrated assembly line balancing problem with skilled and unskilled workers based on a task precedence relationship. Furthermore, we developed an efficient genetic algorithm based on random key-encoded chromosomes.

The proposed genetic algorithm overcame the computational burden of the MILP. Therefore, the genetic algorithm has been shown to be a more helpful alternative than an MILP for designing a cost-effective assembly line with suitable worker assignments.

Fig. 1 .

 1 Fig. 1. Representation and initialization for a sample problem

Table 1 .

 1 Parameters of genetic operators

	Parameters	Value
	Population size	100
	Convex hull crossover probability	0.7
	One-cut point crossover probability	0.6
	Swap mutation probability	0.4
	One point mutation probability	0.4

Table 2 .

 2 Comparison of mathematical model and proposed genetic algorithm

			Mathematical Model	Genetic Algorithm
	No	Examples
			Objective	Time(sec)	Objective	Time(sec)
	1	9 Tasks 8 Workers (1)	390,000	105	390,000	30
	2	9 Tasks 8 Workers (2)	406,000	110	406,000	33
	3	9 Tasks 8 Workers (3)	358,000	201	358,000	25
	4	11 Tasks 8 Workers (1)	523,000	190	523,000	21
	5	11 Tasks 8 Workers (2)	473,000	85	473,000	63
	6	11 Tasks 8 Workers (3)	476,000	187	476,000	52
	7	12 Tasks 8 Workers (1)	578,000	153	578,000	68
	8	12 Tasks 8 Workers (2)	509,000	221	509,000	96
	9	12 Tasks 8 Workers (3)	572,000	427	572,000	111
	13	21 Tasks 15 Workers (1)	746,000	2,807	746,000	155
	14	21 Tasks 15 Workers (2)	746,000	3,848	746,000	109
	15	21 Tasks 15 Workers (3)	-	-	756,000	91
	16	32 Tasks 15 Workers (1)	-	-	1,191,000	434
	17	32 Tasks 15 Workers (2)	-	-	1,118,000	598
	18	32 Tasks 15 Workers (3)	-	-	1,148,000	572
	19	61 Tasks 29 Workers (1)	-	-	3,188,000	1,533

adfa, p. 1, 2014. © Springer-Verlag Berlin Heidelberg 2014

Acknowledgments

This work was supported by the BK21 Plus Program (Center for Sustainable and Innovative Industrial Systems) funded by the Ministry of Education, Korea.


Marco Macchi
email: marco.macchi@polimi.it

Alessandro Pozzetti
email: alessandro.pozzetti@polimi.it

Luca Fumagalli
email: luca1.fumagalli@polimi.it

Industrial implementation of models for joint production and maintenance planning

Keywords: Production Planning, Maintenance planning, Economic Production Quantity (EPQ), Manufacturing Industry

The paper aims at investigating the industrial applicability of models for joint production and maintenance planning. Scientific community has paid attention to this issue for decades and more recently many researchers proposed models to optimize such kind of planning. Nevertheless each scientific work is based on its peculiarities and hypothesis, which might prevent the applicability within different industrial contexts. This is also due to the very theoretical roots of many models, often tested only within numeric or simulation scenarios. This paper aims at presenting a case study analysis where the industrial test-bench is analysed to understand how these types of model are adaptable and extendable to a real context. To support the case study analysis, the main literature on the topic is reviewed, with the purpose to provide the background for the model deployment and test within the selected industrial context.

Introduction

Production and maintenance management are responsibilities of different functions often corresponding to different departments of a manufacturing company. This could lead to an un-optimized management that does not seek for the overall maximization of operations performance, but only of production and maintenance. Conversely, a joint vision would allow to get an optimized management, in coherence with the single objectives of the two departments, but also aligned with the overall goal of the company. In this regard, all practices, rules and, in general, models that allow a joint and coordinated management are herein considered as "models" for joint production and maintenance planning in different time horizons; further on, within this research, a specific concern is given to mathematical models for optimization.

Considering these models, it is worth pointing out that in literature many works are dedicated to the optimization of production and the optimization of maintenance, but a practical point of view on a joint approach is not so diffused. Recently, models have been distinguished based on the time horizon they refer to in their planning scope. In particular, according to [START_REF] Budai | Maintenance and Production: A Review of Planning Models[END_REF], models are used for the (i) long term strategic and maintenance concept, (ii) medium term planning (i.e. tactical level) and (iii) short term scheduling; besides, another category of scientific works is specifically focused on control and performance indicators.

This paper deals with models supporting decisions at tactical level and it is based on a case study. The case allows studying the industrial applicability of such kind of models within a real context. This is aligned with the long-term objective of our research: contributing to fulfill the gap between the scientific theories and industrial practices with regard to the joint production and maintenance management. After a brief literature review (section 2), the model developed for industrial applicability is presented with related testing cases (section 3) and then used within a manufacturing industry case study (section 4); conclusions are eventually provided (section 5).

Literature Review

Brief state of the art analysis

Literature background is mostly based on two papers which revise the state of the art [START_REF] Budai | Maintenance and Production: A Review of Planning Models[END_REF][START_REF] Ben-Daya | Integrated production, quality & maintenance models: an overview[END_REF], presenting a classification of models for aggregate planning and scheduling.

According to [START_REF] Budai | Maintenance and Production: A Review of Planning Models[END_REF], a first group of models -strategic decision models -is used during the system design phase, in order to identify at a high level the type of maintenance policy that should be applied in view of the production requirements of the designed system. The second group of models -tactical decision models -is focused on decisions over a mid-term (i.e. between one month and one year as time horizon). Such models support decisions for important maintenance interventions, deciding whether and when they should be performed based on their impact on the production plan for what concern the delivery of the demand of the planned period; this kind of models often considers the finite capacity of the maintenance crew in terms of hours that they make available for the activities of the maintenance plan. The last group of models -short term scheduling -considers a short time horizon, in order to schedule the interventions on lines or on singles machine based on an elaborated plan; this type of models takes care of deciding when a planned intervention must be scheduled within (often) one day timeframe. [START_REF] Ben-Daya | Integrated production, quality & maintenance models: an overview[END_REF] propose to analyze the scientific works by means of three areas of interest: Quality, Maintenance, Production. The authors then divide the works based on how these areas of interest are considered, by a joint approach, within the mathematical models provided by the papers surveyed from literature. Focusing on the relationships between two areas, Production and Quality are worth to be firstly discussed. They are intrinsically linked, considering that a production process is imperfect. This implies that scraps and WIPs are generated in an un-optimized manner, thus economically impacting on the performances of the production process. A quality factor is then included in models for the Economic Production Quantity (EPQ). These models are a variant of models for Economic Order Quantity (EOQ), defined when a production process is considered (instead of acquisition process). The imperfection of the process is generally modelled by its deterioration as a function of time with, for instance, a linear or exponential model. Quality problems might be then detected by inspections, often considered by the models, including their cost. This approach allows to optimize the EPQ and the inspection interval to minimize the overall costs. Instead, for what concern the joint approach on Production and Maintenance areas, the main highlight concerns the relationship between production and maintenance in planning, often seen by separated approaches, pursuing an optimal maintenance plan, given the production plan or vice-versa. Also in this case, the EPQ models are used, considering the tradeoff between preventive maintenance cost and the cost associated to the risk of corrective maintenance during the production period (in some cases also including the risk of defective products, that is the lack of quality). Further on, still within the relationship between Production and Maintenance areas, it is worth considering that the negative effect of stoppages might be also mitigated by the material kept on stock, which can be used to supply a given demand when production is not possible: in this regard, a good number of models considering the production systems with buffer capacity is proposed in literature. Eventually, the Maintenance and Quality areas are also sometime jointly addressed while neglecting the Production area. This is done basing on typical approaches as the following ones: a study of how maintenance, as an imperfect activity, impact on quality; an orientation towards using maintenance to reduce the quality gap with respect to a target value.

Selection of reference models from literature

Starting from the literature surveys of [1,2,3], their references have been considered for more in depth analysis (respectively, [START_REF] Budai | Maintenance and Production: A Review of Planning Models[END_REF] cited 88 papers, [START_REF] Ben-Daya | Integrated production, quality & maintenance models: an overview[END_REF] 116 papers, [START_REF] Schimdt | Scheduling with limited machine availability[END_REF] 45 papers). The references, integrated with other papers published during the more recent years, allowed us to select a sample of 90 papers as background for our research.

Firstly, the analysis of this sample allows to assert that not all the scientific works provide a validation or verification of the proposed model. Moreover, only few of the sampled papers verify the models by means of an application in real world (6 papers, out of 90 papers of the sample), while the large amount of works considers validation only through numeric example (59 out of 90) or simulation (15 out of 90). This result further justifies our intention to concentrate on industrial applicability. Moreover, the objective of this analysis was to identify some reference models to be used for the definition of a model candidate to be tested in the industrial case study. Thus, as the 90 papers present optimization models for joint maintenance and production planning and scheduling, models were initially classified according to 7 clusters. The clusters are primarily inspired by the classification of [START_REF] Budai | Maintenance and Production: A Review of Planning Models[END_REF]. Amongst them, 3 clusters are grouping the tactical models, as main concern of our research: EPQ models with quality issues; EPQ models with failure issues, Models considering systems with buffer capacity. EPQ with failure issues is the cluster of models this research has been concentrating on: more than a quarter of papers in our sample presents EPQ models with failure issues; based on this simple statistic, these models can be reasonably considered as the most addressed by the scientific community. Thus, focusing only on the EPQ models with failure issues, we identified some decision drivers in order to fine-tune the selection of models. The linearity and easy understandability of the models have been considered as relevant properties for the industrial application, assuming that they would enable a more transparent approach to the industrial user. Moreover, production and maintenance have been focused as main target, not neglecting necessarily models that consider also quality as third area covered by the model. Eventually, the type of production system assumed by the model was also taken into account, in order to select the joint planning models that apply on properly detailed systems, thus at least comparable to real industrial systems.

3 Deployment of the proposed model

Overview of the proposed model

Literature analysis led to select some of the most recent models: [START_REF] Sitompul | An integrated hierarchical production and maintenanceplanning model[END_REF][START_REF] Najid | An integrated production and maintenance planning model with time windows and shortage cost[END_REF]. [START_REF] Sitompul | An integrated hierarchical production and maintenanceplanning model[END_REF] propose a model oriented to follow a given production optimization. The maintenance plan is subordinated to a strict constraint: the realization of the entire demand of each type of product. Backlog is not considered and demand must be strictly respected. The model considers a series of machines (a production line), introducing flexibility to represent both the system as a whole and the single phases and stations. Finally, each machine is subject to a Preventive Maintenance (PM) only once within the production planning bucket. [START_REF] Sitompul | An integrated hierarchical production and maintenanceplanning model[END_REF] has been selected as good reference of a model that properly takes care of production priorities. [START_REF] Najid | An integrated production and maintenance planning model with time windows and shortage cost[END_REF] propose a model oriented to follow a given maintenance plan. Optimal PM frequency is then fixed and must be respected. Based on a certain maintenance behavior of the system, represented through failures' models, the model optimizes the production plan of orders, provided within a given time window (i.e. defined by the earliest and latest time for production). Backlog is considered through the related backorder costs; also overtime, if needed, can be exploited. Finally, the system is represented as a whole, even if failures' models provides a good lever for detailed modelling of the system. [START_REF] Najid | An integrated production and maintenance planning model with time windows and shortage cost[END_REF] was selected as good reference of a model that give adequate priority to maintenance plan. The two models are, thus, representative of two different attitudes, that is priority to production and priority to maintenance. Both models then reveal pros and contra. [START_REF] Sitompul | An integrated hierarchical production and maintenanceplanning model[END_REF] model is good for a well deployment of the production system model by considering the machines / subsystems, while it lacks in a detailed modelling for overtime work and flexibility in the number of PM within a production time (only one PM is allowed). [START_REF] Najid | An integrated production and maintenance planning model with time windows and shortage cost[END_REF] model, instead, is good for its modelling capability of overtime work, but does not allow decomposition of the system and represents PM as fixed in its scheduling. Hence, this research aimed at the preparation of a new model as a fusion of the features provided by the two models.

Hence, based on their pros and contra, requirements for a new model were firstly elicited. Requirements come from the need to support a decision maker within a manufacturing industry with a good flexibility to cope with both production and maintenance planning issues. A company producing white-goods offered a test-bench to this end. The company operates a mass-production. The production system of the test-bench is a manufacturing line; therefore, the system reliability is in accordance to a series logic: a stoppage of one of the machines in the line implies the stoppage of the entire line. This strongly requires an accurate maintenance planning, coordinated with production, to optimize availability of the line with the purpose to guarantee the production targets. Further details of the application of the model in the case study are proposed in section 4. Then, the following are the requirements identified for the deployment of a new model: i) frequency of PM is not fixed a-priori; ii) one PM always occur at the first period in order to restore the system at an "As-Good-As-New" state; iii) PM can occur more than once (generally twice) within the production time horizon; iv) the production system is considered divided in sub-systems, like machines, for quantifying the maintenance performances; v) the production system is considered as a whole for the production performance; vi) backlog is allowed; vii) overtime work is also exploitable. Overall, according to the requirements, the [START_REF] Sitompul | An integrated hierarchical production and maintenanceplanning model[END_REF]'s model has been extended by the introduction of two decision variables -overtime work and backlog -and some new specific constraints (derived from [START_REF] Najid | An integrated production and maintenance planning model with time windows and shortage cost[END_REF]).

Testing scenarios

An experimental plan was defined for a numerical validation of the developed model. From the literature, considering the typical variables involved in the analysis, the followings have been considered for the plan: production mix variability, demand predictability, aggregate planning strategy. Each variable can be considered with two levels: high (H) or low (L) value for variability and predictability, Level or Chase for the planning strategy. Thus, with the two levels and three variables, 2 3 scenarios are possible. Nevertheless, some key scenarios have been considered, out of all possible scenarios, envisioning their meaningfulness in some real industry. This led to identify three experimental conditions: i) Level strategy, high demand predictability, low mix variability; ii) Chase strategy, high demand predictability, low mix variability; iii) Chase strategy, low demand predictability, high mix variability. These scenarios were then created with numerical examples and applied to the proposed model, after representing the uncertainty (subsequent to demand predictability) based on Monte Carlo simulation. The numerical examples helped validating the new model, which has achieved similar or even better performances than the original models of [START_REF] Sitompul | An integrated hierarchical production and maintenanceplanning model[END_REF][START_REF] Najid | An integrated production and maintenance planning model with time windows and shortage cost[END_REF]. Indeed, the case study corresponds to one of the scenarios used for numerical testing, namely the first one featuring a Level strategy, high demand predictability and low variability of the production mix (like working under conditions of a "quasiconstant" mix, with a low uncertainty for what concern the demand prediction). In this scenario, it has been seen that stocking material is a solution to prefer than requiring overtime work. To this end, the model proposed by [START_REF] Najid | An integrated production and maintenance planning model with time windows and shortage cost[END_REF] is not able to flexibly manage the preventive maintenance scheduling and result in a schedule that cut considerably available working time, forcing than to consider overtime working (even in a scenario where nominal capacity can satisfy a stable demand). The new model, instead, manage the risk of possible stoppages, balancing such risk, creating a scheduling that more reasonably consider the exploitation of the production capacity of the system. The new model was, thus, considered ready for an application in industry and judged as good reference in order to further work on a test in a real case.

Manufacturing industry case study: analysis and results

By applying the model to the case the results theoretically achieved are equal to an overall saving of 57% of maintenance costs, compared with the AS-IS situation. This seemed a very interesting result, considering that in literature models for optimization of maintenance and production scheduling are claimed to be able to provide a saving of around 30% [START_REF] Cassady | Integrating Preventive Maintenance Planning and Production Scheduling for a Single Machine[END_REF]. Starting from this potential, the analysis of the case was further directed to critically understand the reasons, identifying the hypothesis laying behind this behavior. Indeed, what is worth to be herein discussed is the sensitivity analysis, performed on the model with the purpose to check how it behaves. The methodology for case study analysis, that has been thus applied, followed some steps related with sensitivity analysis: i) the results given by the model have been compared with the actual practices of the company, ii) great differences between what suggested by the model and what presently ongoing as practices in the company were carefully checked, iii) the weaknesses of the model, namely some restrictive hypothesis not originally identified, were highlighted as a final outcome.

Analysis on production capacity variation

First variable checked by sensitivity is production capacity. Production capacity has been intended by the company as a reduced time available for operations (shortly, in the remainder, operating time), resulting from either a reduction of shifts or closure days. This was compliant with the hypothesis of use of the model. As a general consideration on the model behavior, it has been observed that the production capacity results correlated with production system utilization: a decrease of production capacity corresponds to an almost linear increase of utilization. Further on, the proposed model considers the PMs scheduled within operating time; their further impact should be then measured: subsequently, when decreasing production capacity, it has been noted that the model suddenly proposes to use overtime work in order to guarantee the demand; on the other hand, the variation of production capacity does not show any correlation with the plan of PMs suggested by the model, that remains unchanged. This can be explained basically considering the ratios of costs adopted for experimentation: the overtime work is costing less than backlogs and PMs; then, the model shows a capability to adapt its planning strategy based on cost minimization. In other words, in order to counterbalance the decreasing capacity, the overtime work is the resource used because is the less costing.

In comparison with the actual practices of the company, what suggested by the model was instead judged as not properly aligned. In fact, the practice of the company is to try to avoid the overtime work when the nominal production capacity is reduced and, instead, to neglect some PM interventions in comparison with the original plan, in order to save some operating time for production. This has been proved, based on experience over the years, not impacting on availability of the line. After discussing such a behavior, the root cause of misalignment of the model with the real conditions was identified: the reliability of the machines is in fact considered by the model as independent from the production capacity, hence the operating time of the line. More precisely, our model followed what normally proposed in literature: to consider the reliability only as a function of the calendar time. But reliability is also a function of the type of use, hence the operating time and other minor variables, especially true with electromechanical systems, as the machines of the line, that should be normally assumed behaving under so called Operating Dependent Failure (ODF). Thus, the case study highlighted how the type of use and the operating time should not be neglected, but instead should be taken into account by the model in order to represent a reliability behavior that is influenced by the capacity that the system deploys for achieving the production targets.

Analysis on maintenance parameters variation

Maintenance parameters, time and costs, are the second type of variables checked by sensitivity analysis, to verify their effect on the comprehensive cost of the plan, and the number of maintenance interventions. In this regard, it is worth underlining that the failure distribution model adopted for the case is a Weibull function, according to well-known reliability theory, while the parameters of the Weibull have been derived based on an analysis of the historical data recorded in the company's maintenance information system at machine level. Considering an increase of cost of Corrective Maintenance (CM) interventions or of time needed to fix problems (equivalent to say that failures become more critical events), the model pushes to carrying on more PMs. More specifically, this trend is observed for the machines of the production system whose occurrence of failure is modelled by a Weibull with a form factor larger than 1, representing the behavior in a wear-out phase of the bathtub curve, well-known in the reliability theory. Conversely, for other machines, where the form factor is equal to 1 (i.e. the failure occurs based on a memory-less phenomenon, modelled by an exponential distribution), the model does not suggest PM, that practically would be not necessary, being the failures not depending on the elapsing of time. Eventually, when the form factor of the Weibull is minor than 1, the model leads to reduce as much as possible the number of PMs, that has not positive impact on the reduction of failures. All in all, the behavior observed with different Weibull models, changing based on different form factors, is aligned with expectations from reliability theory, in particular, with literature on maintenance cost optimization models.

Nevertheless, a misalignment with actual practices of the company was detected, in the cases of Weibull with form factor minor than 1. Further investigation then aimed at checking whether practices of the company might be considered wrong, or if some data introduced in the model were not aligned with the actual hypothesis. Indeed, the machines whose failures are characterized by a Weibull distribution with form factor minor than 1 are currently maintained with much more PMs than what suggested by the model. The mathematical formulation behind our optimization model, instead, considers that for these machines it is not worth carrying out PMs. In particular, according to how the model behaves based on its laying assumptions, the machines are considered "As-Good-As-New" after each PM and, in this case, they would start to suffer of more failures due to the case of infant mortality (i.e. the wear-in phase of the bathtub curve, correspondent to a form factor of the Weibull function minor than 1). According to this behavior, the model thus suggests to carry out less PMs than the ones presently carried out in the company. Nevertheless, this is a blind application of the mathematical model that risks to create a bias and even errors in maintenance planning. The most reasonable root cause of the error is using the Weibull analysis at a machine level. This hides the different failure modes that the machine can suffer from, and how the PMs contrast such failure modes. In effect, in reality, not all the PMs are an overhaul intervention on the machine; instead, they are of different types (e.g. minor regulations, cleaning, etc…) and address only few failure modes, thus being far from the hypothesis of returning the machines to an "As-Good-As-New" condition. Therefore, the model certainly shows one of its drawback when the overall behavior of the machine is modelled with Weibull with factor form < 1, while some PMs are carried out just to contrast few failure modes that, at a detailed level, could result modelled by a Weibull with factor form > 1. Making a detailed modelling of behavior of the machine is well discussed in reliability analysis, and, from our result, it seems worth of consideration within models for joint production and maintenance planning. Indeed, by the herein presented case study, it has been verified the strong impact it can have on the industrial usability of such kind of optimization model.

Conclusions

The paper introduced the topic of joint production and maintenance planning and scheduling by summarizing three main surveys in the research area. A new model based on existing scientific works has been presented by means of a brief explanation, neglecting the mathematical formulation, out of the scope of the present dissertation while focusing on its use in an industrial application. Indeed, the industrial application revealed some quite relevant differences between the recommendations resulting from the model and the actual practices carried on by the company. This has allowed to highlight some weaknesses of such kind of models, that we deem quite interesting in terms of possible future research.

On the whole, the analysis revealed how the impact of variance or uncertainty of inputs related with maintenance issues can be quite serious, possibly more relevant than the uncertainties due to production inputs. Unfortunately, the maintenance data are the ones that are often less considered, hence less modelled, in the companies. The case study then arises a warning on a proper management of maintenance data and, in particular, on maintenance performance, which can be really affecting overall plant performance and, more specifically, a coordinated production and maintenance plan.

Acknowledgements. The authors would like to thank Alessandra Del Centina for her support in the research through the deployment of her master of science thesis.


Marco Macchi
email: marco.macchi@polimi.it

Alessandro Pozzetti
email: alessandro.pozzetti@polimi.it

Luca Fumagalli
email: luca1.fumagalli@polimi.it

Exploring the Integration of Maintenance with Production Management in SMEs

Keywords: Maintenance management, production management, integration, SME, manufacturing. Production Management Interaction

The paper presents the results of an exploratory research based on 10 SMEs used as case studies with the purpose to observe the state of practices with regard to the integration of maintenance with production management. The research intends to provide an evaluation of the quality of integration by means of a maturity assessment method. The resulting evidences allow an initial concern on strengths and weaknesses of maintenance management and its relationship with production management in SMEs.

Introduction

The importance of maintenance in manufacturing has been increasing in the recent years [START_REF] Cholasuke | The status of maintenance management in UK manufacturing organisations: results from a pilot survey[END_REF] considering its effect on the long-term improvement of equipment availability, product quality and production costs [START_REF] Swanson | Linking maintenance strategies to performance[END_REF] and the growing concern that profit and productivity can be improved when maintenance potential is exploited [START_REF] Pinjala | An empirical investigation on the relationship between business and maintenance strategies[END_REF]. Nonetheless, maintenance is not yet developed in many industries: according to [START_REF] Alsyouf | Maintenance practices in Swedish industries: Survey results[END_REF], reporting a survey on 118 companies in Sweden, 55 % from the mechanical industry, relevant weaknesses emerge; we achieved similar evidences in Italy through a survey on 128 manufacturing companies [START_REF] Tesem | Offrire valore gestendo la manutenzione come un'impresa: utopia o sfida reale?, yearly report[END_REF]. Nonetheless, it is worth remarking a positive issue with regard to the use of TPM (Total Productive Maintenance), as a good share of companies in the mechanical industry declares TPM as standard practice. Since TPM is an opportunity space for better integration of maintenance with production [START_REF] Cooke | Implementing TPM in plant maintenance: some organisational barriers[END_REF], the aim of this paper is to study the relationship between production and maintenance function within mechanical plants. This is discussed in a wide number of publications; even so, literature is still lacking attention on the integration of maintenance and production within SMEs. Thus the paper aims at providing the results of an exploratory research carried out in 10 SMEs in the wire drawing industry, a sector of the mechanical industry. This industry operates downstream from the steel-making industry and, according to the Comité Européen de la Tréfilerie, in Europe it is composed of nearly 500 active companies, with about 50,000 employees.

The study starts with the design of the exploratory research (section 2), its implementation is then described (section 3) and the results are discussed (section 4) before concluding with limits and next steps for future work (section 5).

2

Design of the exploratory research

Scope and framework

Figure 1 summarizes research scope and conceptual framework used to drive the next step of research implementation.

Fig. 1. Reseach scope and conceptual framework

The framework is based on maintenance management with special regard to TPM practices [START_REF] Cooke | Implementing TPM in plant maintenance: some organisational barriers[END_REF][START_REF] Ahuja | Total productive maintenance: literature review and directions[END_REF]; further on, due to the scarcity of SMEs-related literature, interviews to 2 SMEs as pilot cases have been carried on, to identify the most important matters perceived in regard to the integration between production and maintenance. As first result of the pilots, the framework includes the information system as key resource to support information and procedures required for the maintenance tasks and the interaction with the production function. Further on, it is worth remarking the logic for continuous improvement as a perceived need in SMEs: in this regard, the machines installed in the company are the entities driving the interaction between production and maintenance function; the human resources are the key resources to contribute to its successful implementation. Other relevant activities were identified in the pilots: i) failure reporting and analysis, enabling the integration by capturing information and knowledge from maintenance and production through simple "tools" (e.g. Ishikawa diagram as suggested by TPM); ii) opportunistic maintenance, whenever the failure creates a repair "opportunity" leading to consider the possibility to anticipate some preventive repairs; iii) maintenance planning and scheduling, that should be driven by the machines' criticality; iii) execution of maintenance activities, requiring training of operators, even production ones for autonomous maintenance; iii) performance measurement, to assess the achievement of production targets through Key Performance Indicators (KPIs).

2.2

Target process area

The conceptual framework was re-organized by defining 6 maintenance process areas as reference model for the implementation step (table 1).

Tab. 1. Process areas and matters of interest for integration within each area

Process area

Matters of interest for integration

Maintenance Planning and Scheduling

Production and maintenance function should share common objectives and coordinate accordingly plans and schedules of machines' usage, considering all machines and a prioritization approach to plan and schedule maintenance activities. Furthermore, it is opportune to consider the availability of production operators, with the purpose to develop autonomous maintenance plans.

Coordination for Maintenance Execution and Reporting

Production operators need the necessary knowledge to manage a failure in a standardized way, to solve problems by means of an autonomous maintenance execution, or to identify the right person to restore the machines to the normal condition. Reporting is also required for the necessary recordings of activities, to suit further needs of failure analysis; reporting starts with the request issued by production operators.

Failure Analysis and Opportunistic Maintenance

When a failure happens, it is possible to allocate unscheduled preventive maintenance as repair "opportunity"; this requires a short-term coordination between production and maintenance function.

Failure Analysis and Continuous Improvement

Failure analysis could bring to new procedures that, inside maintenance and production function, promote continuous improvement.

Human Resources Management

Human resources are a lever for effective integration: production operators should be trained on basic functions of the machines they work in, and their failure modes to autonomously make some activities, and to further collaborate with maintenance by sharing ideas for continuous improvement.

Performance Measurement

Performance measurement is needed to check the effectiveness of improvements: KPIs are an important "tool", shared between production and maintenance function, to control the adequacy of improvements.

Implementation of the exploratory research

The questionnaire

The process areas were used to guide the generation of a questionnaire for the case study analysis: the questionnaire was structured with sets of questions, one for each area, while the pilots were helpful for transforming knowledge available from literature in questions understandable in practice. The questionnaire was then used to lead a discussion with target employees in each case study, allowing to gather open answers.

Table 2 provides a sample of questions for the Failure Analysis and Continuous Im-provement process area. Accordingly to the related tasks expected within an organization, the questions are classified by indicating the responsibilities, either only within the maintenance function (M) or within the interaction with production function (M-P). The questions are also classified considering the capabilities of the maintenance function that the practices contribute to: the definition of capabilitiesmanagerial, organizational and technological (MAN, ORG and TEC)is based on previous works of the authors, as better explained in the next section 3.2.

Tab. 2. Sample of questions taken out from the questionnaire and their classification When you decide some improvement, who is consulted to finally assess the decision?

 
Who is in charge to check performances to monitor the improvements after implementing new procedures?  

Maturity assessment method

The quality of a business process can be evaluated through the concept of maturity. Indeed, maturity has been proposed to assess how business processes are carried on; correspondingly, maturity assessment methods have been developed with the purpose to assess the state of practices of such processes [START_REF] Macchi | A maintenance maturity assessment method for the manufacturing industry[END_REF]. Maturity models lead to the provision of a normative description of the good / best practices [START_REF] Volker | Developing a maturity model for infrastructural asset management systems[END_REF], which helps to develop a rank of practices in the target processes. In particular, making a rank of practices is possible through the identification of a given maturity or capability level [START_REF]CMMI Product Team: CMMI -SVC, Version 1.3[END_REF]: a company, subject to a quality audit, is assigned a maturity / capability level based on the observation of its practices in the target processes. Using terms originally developed for maturity assessment methods, a Maturity Level (ML) can be assigned to a process, or to a set of processes, so called Process Area (PA); several models have been used for the definition of maturity, many of which are based on the Capability Maturity Model (CMM) and its later integrations [START_REF]CMMI Product Team: CMMI -SVC, Version 1.3[END_REF]. As such, adopting the MLs can be considered a way to assess tangiblyby using a scorethe state of practices in a business process / process area. Some authors of this paper have already experienced the use of maturity assessment: a concept of maturity assessment for maintenance management was presented in [START_REF] Garetti | Investigating the organizational business models of maintenance when adopting self diagnosing and self healing ICT systems in multi site contexts[END_REF] and tested in case studies from different industries [START_REF] Fumagalli | Evaluating the role of maintenance maturity in the adoption of new ICT in the process industry[END_REF][START_REF] Gomez Fernandez | A scorecard approach to investigate the IT in the Maintenance Business Models[END_REF][START_REF] Macchi | Towards e-Maintenance: maturity assessment of maintenance services for new ICT introduction[END_REF]. More recently, the method is presented in its theory [START_REF] Macchi | A maintenance maturity assessment method for the manufacturing industry[END_REF], inherited in the present paper in the main assumptions. Summarizing them, PAs are assessed in terms of their managerial, organizational and technological capabilities: when making a quality audit of business processes, one can analyze, at the most aggregated level, the MLs reached by a company in terms of such capabilities; afterwards, the MLs can be split to detect the criticalities through the maturity profiles, firstly, of the PAs and, after further decomposition, of their component processes. The difference of this paper, with respect to the original concept of [START_REF] Macchi | A maintenance maturity assessment method for the manufacturing industry[END_REF], is due to the extended scope of analysis: the method is herein adopted with the purpose to study the interaction between maintenance and production management instead of only focusing on maintenance. Another difference is related to the intended use of the method: it is to support a rough analysis aimed at the identification of two MLs, high and low. This rank does not achieve a precise granularity, as it would be possible when more established approaches are adoptede.g., those recommended by CMMI, with 5 to 6 levels. The limited use proposed for the present paper was sufficient for the purpose of the exploratory research: the maturity assessment helps making a rough rank of practices in the target processes; more precise granularity is required if the aim is to assist companies improving their practices.

The following procedure was then adopted for each case study: i) the case is initially analyzed through a narrative description of the interviews carried on (step 1: "written report of the interview"); the narrative description is the starting point to synthesize the state of practices in the target PAs and the emergent capabilities (step 2: "summary of the observed practices in the target PAs and subsequent capabilities"; the PAs are defined accordingly with previous Table 1); comparing all the cases, and considering the behavior resulting from the practices currently observed, the PAs are classified according to two MLs, i.e. high and low maturity; this enables to make evident their contribution to capabilities and, more in general, the quality of interaction between production and maintenance (step 3: "maturity assessment of target PAs and subsequent capabilities"). Table 3 provides the definition of managerial, organizational and technological capabilities as used in this research; Table 4 reports a sample of result for a case study, showing the outcome of step 2 and 3.

Tab. 3. Definition of managerial, organisational and technological capabilities

Capability Definition

Managerial Decision making capabilities within the planning and control cycle, considering also the responsibilities, either only within the maintenance function or involving the interaction with the production function.

Organizational Definition of organisational roles and duties, organisational relationships between production and maintenance function, and mechanisms to support knowledge management, motivation and growth of human resources.

Technological Support provided by ICT tools and techniques/methods for data analysis, with particular concern on their effective use in the company's practices (i.e. it is interesting how tools/techniques/methods are effectively adopted, not solely their availability).

Tab. 4. Sample of a case study result

Process area Sample of evidences emergent for the company's capabilities ML

Maintenance Planning and Scheduling (MAN): the maintenance plan is updated every year but there are weaknesses in the planning process: the decisions do not result from any collaboration with production function, while the plan changes are based only on special, big events worthy of remark.

L Coordination for Maintenance Execution and Reporting

(ORG): the company adopts a standardized reporting process, with the purpose to gather all inputs from the work request issue to the work order execution; organizational roles are identified with their task responsibilities along the process.

H

Failure Analysis and Continuous Improvement (TEC): time series of past failures are not stored in electronic means; their trends and statistical properties cannot be analyzed.

L

Results of the exploratory research

The outcome of case study analysis is now presented focusing on main features to characterize each capability (Table 5): the features are synthesizing the most meaningful observation of practices emerged through the maturity assessment in different PAs; to comply with privacy issues, the SMEs are named by an anonymous number.

Tab. A good maturity is evident for the organizational capability, especially in: i) the cross-functional communication, facilitated by the simple organizational structure; ii) the standardization of the reporting process, with task responsibilities of different functions along the process; this enables gathering all inputs required for work order management, from work request issue to order execution. Conversely, serious weaknesses regard the involvement of production operators in the execution of autonomous maintenance: in almost all the companies, production operators are forbidden to execute any maintenance activity on machines; further on, training only aims at guaranteeing skills to correctly use the machines and to identify particular symptoms that the failure has occurred or is close to occur; all in all, there is a clear separation between production and maintenance functions at the execution level.

A poor maturity is evident in most of the sampled companies for what concern the technological capability, due to a low maturity in: i) the scheduling of opportunistic maintenance, not relying on a tool that, based on the information of machines' expected workload and maintenance downtimes, could support the planner in more in-formed decisions; ii) the storage of failure reports, kept in paper forms in case of nonmature companies; this causes hard consultation and analysis of past events. On the other hand, the IT support to condition monitoring of machines is exploited in half of the sampled companies: this is a promising issue for mature practices.

Managerial capability did not result mature, except for maintenance scheduling. Maintenance scheduling is eased in all companies by cyclic programs in case of regular stoppages for frequent activities (every Saturday) or longer durations (once or twice a year): the maintenance function can manage the programs autonomously, and there is no real need for coordination with the production function, thanks to a decoupling mechanism defined through simple rules (e.g. Saturday is only dedicated to maintenance, and it is scheduled once every week, or every more weeks). On the other hand, many weaknesses arise, such as the followings: i) maintenance planning is carried on once a year, even if decisions concerning preventive maintenance frequencies are not aligned with the actual requirements, being not driven neither by an analysis of failure trends nor by the production needs arising from the demand forecast; ii) scheduling of opportunistic maintenance is not engineered in its decision criteria, because the analysis of maintenance downtimes is usually missing and, further on, scheduling is not based on a priority list of activities, which could guide opportunistic decisions; iii) performance management is not currently observed; in particular, management driven by performances such as OEE is missing; as positive remarks, it is however opportune to point out that some KPIsi.e. production yield and productivity -, which can be related to the OEE componentsi.e. quality, availability and performance -, are used in the most mature cases.

Conclusions

The exploratory research of this paper has provided first evidences on practices used in manufacturing SMEs to improve maintenance thanks to its integration with production: a wide number of weaknesses, concerning managerial and technological capabilities, are revealed, while organizational capabilities seem more robust. Future works will continue the research focusing in other districts and industries. The purpose will be to enlarge the sample and make a cross-analysis of different SMEs context. It will be relevant to identify the common weaknesses, then the correspondent gaps, to establish a further research phase aimed at technology transfer helpful to SMEs.

  failure analysis is done in your tasks, do you analyze the trends and statistical properties of past failures?  

EXECUTION OF MAINTENANCE ACTIVITIES PLANNING AND SCHEDULING OF MAINTENANCE ACTIVITIES HUMAN RESOURCES FAILURE ANALYSIS FAILURE REPORTING PERFORMANCE MEASUREMENT CONTINUOUS IMPROVEMENT OPPORTUNISTIC MAINTENANCE MACHINES KEY RESOURCE Legend KEY ACTIVITY

	Precedence
	Contribution to
	successful
	implementation

  5. Cross-analysis of main features emerging in the case studies

	Features / ORG	1	2	3	4	5	6	7	8	9	10
	Cross-functional communication	H	H	H	H	H	H	H	H	H	H
	Task standardization along the	H	H	H	L	H	H	H	H	H	H
	reporting process
	Production operator involvement in	L	L	L	H	L	L	L	L	H	L
	maintenance
	Features / TEC	1	2	3	4	5	6	7	8	9	10
	IT support to condition monitoring	L	H	L	L	H	L	H	H	L	H
	IT support to opportunistic mainte-	L	L	L	L	L	L	H	H	L	L
	nance
	IT support to failure analysis	L	L	L	L	L	L	H	L	L	H
	Features / MAN	1	2	3	4	5	6	7	8	9	10
	Alignment of maintenance planning	L	L	L	L	L	L	H	H	L	H
	with requirements
	Decoupling of production and	H	H	H	H	H	H	H	H	H	H
	maintenance programs
	Decision criteria for opportunistic	L	L	L	L	L	L	H	H	L	L
	maintenance
	Standard KPIs driving performance	L	L	L	L	L	L	H	H	H	H
	management

INFORMATION SYSTEM

Maintenance Management