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Multilevel hybrid method for optimal buffer sizing and inspection stations positioning
 Fatima Zahra Mhada^{1},
 Mohamed Ouzineb^{2},
 Robert Pellerin^{3} and
 Issmail El Hallaoui^{4}Email author
 Received: 4 June 2016
 Accepted: 26 November 2016
 Published: 30 November 2016
Abstract
Designing competitive manufacturing systems with high levels of productivity and quality at a reasonable cost is a complex task. Decision makers must face numerous decision variables which involve multiple and iterative analysis of the estimated cost, quality and productivity of each design alternative. This paper adresses this issue by providing a fast algorithm for solving the buffer sizing and inspection positioning problem of large production lines by combining heuristic and exact algorithms. We develop a multilevel hybrid search method combining a genetic algorithm and tabu search to identify promising locations for the inspection stations and an exact method that optimizes rapidly (in polynomial time) the buffers’ sizes for each location. Our method gives valuable insights into the problem, and its solution time is a small fraction of that required by the exact method on production lines with 10–30 machines.
Keywords
 Production
 Inspection
 Quality
 Combinatorial optimization
 Metaheuristics
Background
The system we consider in this paper consists of n machines, n fixedsize buffers, and \(m+1\) inspection stations (one inspection station is at the end of the line and m are internal) in series. We explore the impact of quality constraints on the system performance and (numerical) complexity. The objective is to simultaneously minimize the combined storage and shortage costs, determine the optimal buffers’ sizes, and specify the optimal number and positions of the m internal inspection stations. The resulting mathematical model is an intractable combinatorial nonlinear optimization model; it is difficult to find an exact solution in a reasonable time, especially when the production line is large. This paper develops an efficient evolutionary heuristic for this complex model. While the number of machines considered in the literature does not exceed 10 (to the best of our knowledge), this paper aims at solving larger production lines and a variable number of inspection stations.
In the recent past, several authors have investigated improving productivity with production control policies. Most studies assume that all the parts produced are conforming, which is unrealistic. Standard quality analysis models usually separate the problem of quality preservation in production lines (via the positioning of inspection stations, for example) and the optimization of production (via Kanban, CONWIP etc.). However, these two problems are interdependent (see Kim and Gershwin 2005, 2008; Colledani and Tolio 2005, 2006, 2009, 2011). We illustrate this by considering the Kanban strategies introduced by Toyota in the 1960s that have since become the paradigm of “lean manufacturing”. These strategies essentially advocate areas of limited storage between successive machines in a production line. The storage areas allow a degree of decoupling between machines to increase the productivity of the line. The idea is to limit the impact of shortages on downstream machines and to avoid blocking machines that are upstream from one that is broken down. These areas should be limited because they are associated with frozen capital: they introduce storage costs and extended transit times for parts in the workshop.
In an ideal “just in time” scenario, there would be no intermediate storage and finished parts would be pulled from the system as they are produced. In reality, machine breakdowns and shortages of raw material or operators make it impossible to ensure continuous workshop output. The sizes of the storage areas should depend on the likelihood of such events and the estimated costs of the associated service interruptions. If we decide the amount of storage without considering the quality dimension, we risk creating storage areas that contain important quantities of defective parts. These defective parts correspond to misused productionline time, and they undermine the effectiveness of the intermediate storage in terms of increasing productivity. Moreover, the costs associated with storage impact the budget available for improvements in quality and vice versa. In a similar manner, Inman et al. (2013) show that production design impacts quality and vice versa. Also, the location of an inspection station affects both the expected production cost per item and the production rate of the line. The quality control is obviously critical to any production system and has significant managerial impact. As a result, it is crucial to develop strategies to allocate inspection or quality control stations into a manufacturing processes that can help preventing all wastes resulting from unidentified defective items being processed (see Shetwan et al. 2011; Raviv 2013).
In a real manufacturing context, simulation can be used to determine the optimal buffers’ sizes and the optimal positions of the inspection stations by considering all possible scenarios. However, this is not practical when the systems are complex: the number of scenarios is \((^n_m) \times 100^n\) for a system with n machines and m inspection stations where each buffer of equal size is discretized to 100 levels. Note that the system is required to have an inspection station at the end of the line. Since the position of this station is known, it is not included in the optimization. An alternative to simulation is to develop a realistic model and a fast optimization technique to solve it. The real model is however very complex because many factors impact quality and production. So, we need elaborate some simplifying assumptions in such a way that the resulting model is still realistic and can be solved optimally.
Researchers have investigated the integration of quality aspects and production policies for a single unreliable machine producing a single product type. The system starts in an “incontrol” state producing conforming items and then switches to an “outofcontrol” state and starts producing nonconforming items. Integrated models can be classified as: (1) integrated production and quality management (e.g., Hajji et al. 2010; Kutzner and Kiesmüller 2013; Mhada et al. 2014b; Naebulharam and Zhang 2014; Matta and Simone 2016; 2) integrated production and maintenance management (e.g., BenDaya 2002; Dhouib et al. 2012; RiveraGómez et al. 2013; Jafari and Makis 2015; Nourelfath et al. 2016); and (3) integrated production, quality, and maintenance management (e.g., Radhoui et al. 2009, 2010; Njike et al. 2012; Zequeira et al. 2008; RiveraGmez et al. 2016; Bouslah et al. 2016). Some researchers (see Mandroli et al. 2006) focus on determining the optimal inspectionstation position in n serial production lines with or without (1) scrapping, (2) reworking, and (3) offline repairs, without considering the buffer sizing.
A large part of published articles related to the inspection suppose the inspection reliable and without errors. They assumed that the produced items are subject to 100% inspection and there are no inspection errors. Some researchers have incorporated quality inspection errors caused, among others, by human failure in their models (see Hsu and Hsu 2013; Duffuaa and Khan 2002; Khan et al. 2014).
In the context of serial production lines with production and inspection machines that follow Bernoulli reliability and quality assumptions, Meerkov and Zhang (2010, 2011) provide important insight into the nature of production and quality bottlenecks. Such systems are encountered in automotive assembly and painting operations where the downtime is relatively short and the defects are a result of uncorrelated random events (Ju et al. 2013).
Mhada et al. (2014a) consider a situation where every machine has to satisfy a demand specified as good parts per time unit. Moreover, the inventory size takes into account that the stock is a mixture of good and defective parts, and defective parts are generally eliminated at inspection stations. Mhada et al. (2014a) improve the continuous models of Kim and Gershwin (2005, 2008). The former model is more fluid in the sense that quality is treated as a continuous flow, whereas in the Kim and Gershwin models quality is considered to be discrete. They considered a line with 4 machines.
Mhada et al. (2014a) develop decomposition methods to reduce the analysis of the line to a series on an equivalent machine that can be isolated and sequentially analyzed. They consider just one inspection station. Ouzineb et al. (2013, 2014) generalize the model proposed in Mhada et al. (2014a) by optimizing the number of internal inspection stations (\(m \ge 1\)) and their positioning, and develop an exact search method to solve it. The exact search method (ESM) is exhaustive: the search is guaranteed to generate all possible locations of inspection stations. For each location, the problem is reformulated as a network flow optimization problem that can be efficiently solved by a fast polynomial algorithm. This method is efficient for small instances, but it may need days for reasonably large instances. The solution time increases exponentially with the size of the problem, i.e. with n and m. For larger problems, we expect that the solution time may span over several months. We can have an idea on that from Table 4.
The present paper efficiently adapts a space partitioning technique that decomposes the search space into multiple levels. At each level, we use two wellknown metaheuristics to explore some potential regions. A genetic algorithm (GA) is used to identify a population of promising locations of inspection stations. Tabu search (TS) then searches intensively around by exploring locations that are neighbors to some newborns in the population. The population evolves dynamically in a sense that worst locations are discarded and good ones are added during the search. We use a quick shortest path procedure (from ESM) to determine the optimal buffers’ sizes and evaluate the objective function (the fitness) for each considered location. From time to time, GA replenishes the population. GA facilitates thus the exploration by guiding the search to unvisited regions with good potential, and this gives a certain diversification in terms of the regions to explore. The new approach is hence a multilevel hybrid heuristic (MHH) that provides certain balance between diversification and intensification. This hybrid approach is effective when the number of solutions is huge: it finds optimal or nearoptimal solutions in a fraction of the time required by ESM. We also discuss some important properties of the production line and its sensitivity to system parameters.
Similar ideas have been used to solve some highdimensional combinatorial optimization problems in other contexts. In Ouzineb et al. (2011), the authors apply a GA to solve an optimal design model for assembly/disassembly manufacturing networks. The objective is to maximize the production rate subject to a total cost constraint, the machines are chosen from a list, and the buffers’ sizes are within a predetermined range. Nahas et al. (2014) applies a GA–TS algorithm to optimize the nonhomogeneous redundancy of multistate seriesparallel systems. In Ouzineb et al. (2010), the authors use space partitioning to solve two design optimization problems: the first is the expansion scheduling of multistate seriesparallel systems, and the second is the redundancy allocation of binarystate seriesparallel systems. We adapt space partitioning and GA–TS techniques to efficiently solve the design problem addressed in this paper.
The remainder of this paper is organized as follows. “Problem formulation and discussion” section presents the problem statement and discusses some theoretical properties, and “Methods” section describes our approach. “Numerical results” section presents numerical results for ESM (Ouzineb et al. 2013, 2014) and our method. “Conclusions and future work” section provides concluding remarks.
Problem formulation and discussion
Let d be the demand rate for conforming parts, \(x_i\) the inventory level for buffer i, \(\tilde{d}_i\) the longterm average number of parts pulled per unit of time from stock \(x_i\), \(c_p\) the storage cost per time unit and per part, \(c_I\) the inspection cost per pulled part, and \(a^n_{des}\) the required availability rate of conforming finished parts.

\(a_i, i=1 \ldots n1\) is the total workinprogress availability coefficient at buffer \(x_i\); this coefficient has a lower bound of \(max\left[ \left( \frac{r}{(r+p)}\right) ^{i1},\frac{(r+p)\,\tilde{d}_i }{r \, k}\right]\) and an upper bound of \(min\left( \left[ \frac{r+p}{r}\right] ^{(ni)} a_{n}^{des} , 1\right)\). These bounds are proved in Ouzineb et al. (2014) (see Propositions 3.3 and 3.5 for more details). Note that \(a_n=a_n^{des}\).

\(\tilde{d}_i, i=1 \ldots n\) is the longterm average number of parts pulled per unit time from stock \(x_i\). It depends on the positions of the inspection stations. Let \(e_j, j=1 \ldots m\) be the positions of the inspection stations, i.e.,Then we have$$\begin{aligned} \lambda _i = \left\{ \begin{array}{l l} 1 &{}\quad \text {if} \; i \in \{e_j,\, j=1 \ldots m\} \\ 0 & \quad{} \text {otherwise} \end{array}\right\} ,\quad i = 1 \ldots n1. \end{aligned}$$(3)$$\begin{aligned} \tilde{d}_i= \left\{ \begin{array}{l l} d(1+\beta )^n &{}\quad \text {if} \quad 1 \le i \le e_1 \\ d(1+\beta )^{ne_1} &{} \quad \text {if} \quad e_1< i \le e_2 \\ .&{} \quad .\\ .&{} \quad .\\ d(1+\beta )^{ne_{m1}} &{} \quad \text {if} \quad e_{m1}< i \le e_m \\ d(1+\beta )^{ne_{m}} &{} \quad \text {if} \quad e_{m} < i \le n \end{array}\right\} , \end{aligned}$$(4)

and finally$$\begin{aligned}&T^{(i)}(a_i,a_{i1},\lambda )= c_p\,\left( \frac{k\, \frac{(r(1a_{i1})+p)}{a_{i1}}}{\sigma _i (k\frac{\tilde{d}_i}{a_i})\,\frac{(r+p)}{a_{i1}}}\frac{k\ (1a_i)}{\sigma _i (k\frac{\tilde{d}_i}{a_i})} \left[ \frac{1}{\sigma _i}\frac{(1a_i)\,\frac{(r+p)}{a_{i1}}}{\sigma _i^2(k\frac{\tilde{d}_i}{a_i})} \right] \right. \\&\quad \left. ln\left[ \frac{\frac{(r(1a_{i1})+p)}{a_{i1}}\,\frac{\tilde{d}_i}{a_i}}{r \, (k\frac{\tilde{d}_i}{a_i}) }\frac{\sigma _i\, \frac{(r(1a_{i1})+p)}{a_{i1}}\,\frac{\tilde{d}_i}{a_i}}{\frac{(r+p)}{a_{i1}}\, r\,(1a_i) }\right] \right) ,\quad i=1, \ldots , n1, \end{aligned}$$(5)with \(\sigma _i=\frac{ ({\frac{(r+p)}{a_{i1}}})\, \frac{\tilde{d}_i}{a_i}k\,r }{(k\frac{\tilde{d}_i}{a_i})\,\frac{\tilde{d}_i}{a_i}}\), \(\rho _n=\frac{r (k\frac{\tilde{d}_n}{a_{n}^{des}})}{\frac{(r(1a_{n1})+p)}{a_{n1}}\,\frac{\tilde{d}_n}{a_{n}^{des}}}\), \(\mu _n=\frac{(r(1a_{n1})+p)}{a_{n1} (k\frac{\tilde{d}_n}{a_{n}^{des}})}\), and \(z_n(a_{n}^{des}) = \frac{\ln \left[ \frac{1}{\rho _n }\left( 1\frac{(1 \rho _n)}{(1a_{n}^{des})(\frac{(r+p)}{(r(1a_{n1})+p)})}\right) \right] }{\mu _n(1 \rho _n)}.\)$$\begin{aligned}&T_F(a^n_{des}, a_{n1},\lambda ) = \\&\quad \frac{ \rho _n \, c_p\,\left( \frac{k\,(1 \exp (\mu _n(1 \rho _n)\,z_n(a_{n}^{des}))}{1 \rho _n}\frac{(r+p)}{a_{n1}} z_n(a_{n}^{des})\exp (\mu _n(1 \rho _n)\,z_n(a_{n}^{des})\right) }{\frac{(r+p)}{a_{n1}} (1 \rho _n \,\exp (\mu _n(1 \rho _n)\,z_n(a_{n}^{des})))} \end{aligned}$$(6)
Proposition 1
The total cost as a function of \(c_I\) is a piecewise linear function. The slope of the line depends on the number of inspection stations m and their positions \(\lambda\).
Proof
The cost function J is likely to be convex as conjectured by Conjecture 1 and confirmed by the numerical results in “Characteristics of the production line” section. The convexity of the cost function is used to reduce the number of iterations. Actually, the local minimum of a convex function is also a global minimum, and there are many specialized methods and mathematical tools for optimizing convex functions.
Conjecture 1

The minimal total cost J is a convex function of \(\lambda\).

The minimal total cost J is a convex function of the number of inspection stations m.

If we divide the production line into two parts with the same number of machines and we fix the number of stations in the first part, the minimal total cost J is a convex function of the number of internal stations in the second part.
Methods
In Ouzineb et al. (2013, 2014), authors present an exact search methd (ESM) for the problem studied in this paper. For a given location of inspection stations, they show that the optimizing of the buffers’ sizes with ESM can be quickly done in polynomial time. Actually, the bottleneck of ESM is the number of possible locations. This number could be reduced by using the property of convexity (Conjecture 1) but nevertheless, it would stay very large. The innovation in this paper is to smartly select potential locations using an evolutionary algorithm combined with tabu search. We summarize ESM in “Exact search method (ESM)” section and then present our new approach in “Multilevel hybrid heuristic (MHH)” section.
Exact search method (ESM)
For a fixed \(\lambda\), Ouzineb et al. (2013, 2014) shows that the term \(T^{(i)}(a_i, a_{i1},\lambda )\), and thus the objective function itself, is separable by the variables \(a_i\). ESM reformulates then the problem defined by (7)–(11), for a given \(\lambda\), as a shortest path problem defined on a network (described below). It uses a standard shortest path algorithm to find the optimal buffers’ sizes. For this, we discretize the continuous variables \(a_i\) to take discrete values 1, 2, ..., 100%. For a fixed \(\lambda\), the value of m is determined straightforwardly. So, the constraints (8), (10), and (11) are satisfied. They do not depend on a and consequently could be omitted.

\(N_0\) is connected to the nodes \(N^j_1\), \(j = 1,\dots ,100\) representing machine 1. The weight of each arc \((N_0,N^j_1)\) is null. \(N_n\) is connected to the nodes \(N^j_{n1}\), \(j = 1,\dots ,100\) representing machine \(n1\). The weight of each arc \((N^j_{n1},N_n)\) is \(c^{j}_{n1,n}=T_{F} (a^{des}_n, j, \lambda )+c_I \times \tilde{d}_n\), \(1\le j\le 100\).

Each node \(N^j_i\) representing machine i is connected to each node \(N^k_{i+1}\) representing machine \(i+1\) by an arc \((N^j_i,N^k_{i+1})\) that is weighted \(c^{jk}_{i,i+1}=T^{(i)} (k,j,\lambda )+ c_I\times \lambda _i\times \tilde{d}_i\), \(i = 1,\dots ,n2\) and \(1\le j,k \le 100\).
Nodes, and all incident arcs, corresponding to values that are not in the intervals imposed by constraints (9) are removed from the network. Algorithm 1 presents ESM. It finds optimal solutions but takes hours for instances with 20 machines. It is unable to solve some larger instances with 30 machines in one month!
Multilevel hybrid heuristic (MHH)
Our approach combines ideas from ESM and metaheuristic techniques. The choice of a search space and a neighborhood structure is a critical step in metaheuristic design (see Gendreau 2002). Our search space is the space of all possible line structures, i.e. the vectors (a, \(\lambda\), m) that are feasible solutions to the system (8)–(11). As discussed above, the most determinant element of these vectors is \(\lambda\) because if we know \(\lambda\), we can compute m in a straightforward manner and a in a polynomial time. Consequently, our search space S in this section is composed of all possible \(\lambda\) values. We reduce it by: (1) partitioning this space into a set of disjoint subspaces as explained in “Partitioning the search space” section; (2) applying a hybrid heuristic to selected subspaces in order to find potential solutions; we use Algorithm 2 above to evaluate the fitness of each selected \(\lambda\) (in a subspace) and to find the optimal buffers’ sizes a. See “Hybrid heuristic” section for more details. The multilevel hybrid heuristic pseudocode is then given in “MHH pseudocode” section. This approach finds, as reported in the numerical results, nearoptimal solutions in a fraction of the time required by ESM.
Partitioning the search space
We refer hereunder to \(\lambda\) as a solution vector or simply solution. Each solution vector is assigned an address, also called a level, as explained below.
Definition 1
We divide the solution vector \(\lambda\) into two equal parts and create its address r as follows: \(r = Address(\lambda )= \sum _{l=1}^{\lfloor n/2 \rfloor }\lambda _l\). A search subspace of address r, denoted \(S_r\), is defined to be the set of solutions that have the same address, i.e. equal to r.
The lower bound of r is 0 and its upper bound is m. It is clear that \((S_r)_{1\le r\le m}\) is a partition of the space S, i.e., \(S_{r_1} \bigcap S_{r_2} =\emptyset \;\forall \, r_1\ne r_2\) and \(S=\bigcup _{r=0}^{m} S_r\).
Example
Hybrid heuristic
The goal here is to locate promising regions in a given subspace \(S_r\). Our hybrid optimization is based on a combination of a GA, which provides diversification, and TS, which provides intensification. The hybrid heuristic pseudocode is presented in Algorithm 3 and a flowchart is given in Fig. 3. \(N_{rep}\) and \(N_c\) are GA tuned by experimentation. The main steps are explained in the paragraphs that follow.
 1.
Copying the string elements belonging to the first part from \({\varvec{\lambda }}_1\);
 2.
Copying the rest of the string elements from \({\varvec{\lambda }}_2\).
Example of 1point crossover operator
Parent string \({\varvec{\lambda }}_1\)  0  0  0  1  0  0  1  0  0  0  0  1  0  0  0  0  0  1  0  1 
Parent string \({\varvec{\lambda }}_2\)  0  0  0  0  1  0  1  0  0  0  0  1  1  0  0  0  0  0  0  1 
Child string  0  0  0  1  0  0  1  0  0  0  0  1  1  0  0  0  0  0  0  1 
Observe that the child has the same address as its parents.
In Step 4, we use TS, in which we iteratively move from the current solution \(\lambda\) to a new solution in its neighborhood Neighborhood(\({\varvec{\lambda }}\)) until some stopping criterion has been satisfied. For a given \(\lambda\), we define Neighborhood(\({\varvec{\lambda }}\)) = {solutions (inspection stations configurations) obtained by applying a single move to \({\varvec{\lambda }}\)}. Moving to a new solution involves changing the positions of two inspection stations; the number of stations does not change. Note that we stay in the same level when we move to a neighbor solution. We use Algorithm 2 to evaluate the fitness of each new solution, and we use theses fitness values to compare different solutions.
TS uses a memory structure: a potential solution is declared tabu so that the algorithm does not visit it again; this prevents cycling. At each iteration, we select the best solution \({{\varvec{\lambda '}}}\) in a subspace V(\({\varvec{\lambda }}\)) \(\subset\) Neighborhood(\({\varvec{\lambda }}\)) as a tabu solution. The subspace V(\({\varvec{\lambda }}\)), called the effective neighborhood, is obtained by eliminating the tabu solutions from Neighborhood(\({\varvec{\lambda }}\)). The tabu solutions are stored in a shortterm memory, called a tabu list, which contains the solutions that have been visited in the recent past. We use a variablesize tabu list (tabu tenure) because this is generally more efficient (Gendreau 2002; Ouzineb et al. 2008). Finally, our stopping criterion is defined by the maximum number of local iterations (mnli), i.e. the number of visited neighbors that do not improve the bestknown solution. The value of mnli is determined by the experimentation.
MHH pseudocode
In MHH, at each iteration we move from the current level to a new level. We divide the search space into a set of disjoint subspaces (levels) using the partitioning technique of “Partitioning the search space” section and then apply the heuristic presented in “Hybrid heuristic” section to each selected subspace. Algorithm 4 presents the proposed pseudo code and Fig. 4 the corresponding flowchart.
Numerical results
Parameters
n  p  r  k  \(\beta\)  d  \(c_p\)  \(c_I\)  \(a_n^{des}\)  \(max_t (s)\)  \(N_{c}\)  \(N_{rep}\)  mnli  

10 machines  10  0.2  0.9  4  0.1  1  1  2  0.95  10  2  3  5 
20 machines  20  0.2  0.9  9  0.1  1  0.1  0.2  0.95  600  2  5  5 
30 machines  30  0.2  0.9  22  0.1  1  0.1  0.2  0.95  3600  2  5  5 
40 machines  40  0.2  0.9  9  0.05  1  0.1  0.2  0.95  3600  2  2  5 
Comparing MHH to ESM

Remove Step 4.

Replace Step 5 by: If the new solution is better than the worst solution currently in the population, it replaces that solution; otherwise, it is discarded.
Results for I20
m  MHH cost  GA cost  Running time (s)  

ESM  MHH/Best  MHH  RPCPU  GA  
1  8.3125  9.23149  15.3  34.3  177.6  −91.39  457.0 
2  6.2645  6.90041  105.7  77.4  440.5  −76.01  983.0 
3  5.9802  6.28064  421.7  422.0  1011.9  −58.33  1524.4 
4  6.0663  6.32419  1601.2  591.7  1191.7  25.58  1364.0 
5  6.2772  6.51936  4743.3  571.2  879.4  81.46  1402.9 
6  6.5230  6.76087  11,108.7  631.4  992.0  91.07  1649.9 
7  6.7959  7.02709  20,613.0  635.2  1108.9  94.62  1710.7 
8  7.0965  7.37334  29,790.2  960.7  1560.7  94.76  1255.8 
9  7.4298  7.67294  35,764.9  476.5  1076.5  96.99  1176.9 
10  7.7947  8.05445  35,767.6  109.0  709.0  98.02  977.7 
Results for I30
m  MHH cost  Running time (s)  

ESM  MHH/Best  MHH  RPCPU  
1  40.7297  31.2  109.6  829.3  −96.24 
2  22.9847  389.4  1186.7  1554.6  −74.95 
3  20.0887  3524.2  3779.7  3926.9  −10.25 
4  19.0494  18,769.2  4470.1  5671.9  69.78 
5  18.9133  115,432.0  6584.2  8065.0  93.01 
6  19.1256  421,630.0  10,034.2  13,634.2  96.77 
7  19.4257  –  10,713.9  14,313.9  – 
8  19.7315  –  12,300.6  15,900.6  – 
9  20.1073  –  10,549.2  14,149.2  – 
10  20.3278  –  11,128.0  14,728.0  – 
For I30, MHH finds “optimal” solutions in a reasonable time. ESM cannot solve the instances with \(m \ge 7\). For example, for \(m=7\), ESM did not converge after a month of running time. We believe that the MHH solution is optimal or nearoptimal for \(m = 7\) and beyond, but we cannot confirm this. Large problems require fast heuristics such as MHH that provide good results in an acceptable time.
Results for I40
m  ESM cost  MHH cost  Running time (s)  

ESM  MHH/Best  MHH  
1  25.2938  25.2938  109,147  194.46  387.83 
2  16.0119  16.1078  1105,72  1008.23  1013.94 
3  13.3200  13.3208  13,200,38  2336.34  2623.08 
4  12.5685  3932.31  5827.07  
5  12.3450  6092.55  6920.45  
6  12.5043  5266.51  11,511.62  
7  12.6973  10,604.86  16,425.48  
8  12.8781  18,929.32  23,346.94  
9  13.1145  12,666.15  25,374.23  
10  13.3623  15,501.52  31,699.22 
MHH and ESM results for I20 with computational time limited to one hour
m  MHH cost  ESM cost 

1  8.3125  8.3125 
2  6.2645  6.2645 
3  5.9802  5.9802 
4  6.0663  6.0663 
5  6.2772  6.4972 
6  6.5230  7.0839 
7  6.7959  7.4549 
8  7.0965  7.8745 
9  7.4298  8.4677 
10  7.7947  8.8909 
Characteristics of the production line
Sensitivity analysis
Parameters
n  p  r  k  d  \(c_p\)  \(a_n^{des}\)  

20 machines  20  0.2  0.9  9  1  0.1  0.95 
Inspection cost \(c_I\)
Optimal location of inspection stations
\(c_I\)  0.2  0.3  0.5  0.6  0.8  1.2  1.3  2  2.7  3.3  3.9  4.5  5.2  5.7 

Optimal m  3  3  2  2  2  2  1  1  1  1  1  1  1  1 
Optimal positions  2; 7; 18  2; 7; 19  3; 11  3; 13  3; 14  4; 19  5  6  7  8  9  10  11  11 
Fraction \(\beta\) of nonconforming to conforming parts
Optimal station locations
\(\beta\)  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  0.11 

Optimal m  1  1  1  1  2  2  3  3  4 
Optimal positions  18  18  18  16  8; 18  6; 18  4; 10; 19  2; 7; 18  2; 4; 8; 18 
Conclusions and future work
We proposed an efficient hybrid approach, based on ideas from an exact method and metaheuristics, for the buffer sizing problem in unreliable homogeneous production lines with several inspection stations. This is a difficult mixed integer nonlinear program. Our approach combines a genetic algorithm and tabu search to identify profitable configurations (locations of inspection stations). For these locations, we use an exact approach to decide the optimal buffers’ sizes. Our final goal is to find an optimal or nearoptimal design as rapidly as possible. This hybrid approach provides a balance between diversification and intensification and works well on homogeneous production lines with up to 40 machines. MHH is significantly faster than ESM and produces solutions that are equally good.
Future research should focus on developing an optimization method for more realistic nonhomogeneous production lines and solving it by combining simulation and optimization. The model developed here could be used as an approximation model for more complex real life design problems. Fast optimization techniques, like the one proposed in this paper, would help in rapidly selecting potential scenarios/configurations for a more realistic simulation, and hence reducing the solution time without losing solution applicability in real life context.
Declarations
Authors' contributions
FM developed the theoretical study, designed the numerical study and participated in the writing of the paper. MO coded the algorithms, carried out the numerical testing and participated in the writing of the paper. RP participated in the paper coordination and helped to draft the manuscript. IEH participated in the design of the paper, coordinated the development works and finalized the writing. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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