Conference Paper

Buffer Allocation Using Exact Linear Programming Formulations and Sampling Approaches

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Abstract

Several sampling approaches have been proposed to analyze flow lines with stochastic processing times and finite buffer capacities. If the number of buffers between the stations is given, the system's performance can be evaluated via a linear programming formulation. This work presents several mixed integer programming approaches to optimize the buffer allocation in flow lines with stochastic processing times. The processing times are sampled according to different approaches. The objective is to minimize the overall number of buffer spaces obtaining at least a given goal production rate. Numerical experiments are carried out in order to evaluate different sampling approaches and model formulations. The sampling approaches are compared regarding the robustness of the allocation decision with respect to the sample sizes.

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... 1. Integrated optimization methods based on samples of random variables are suggested by several authors. BAPs with discrete-material flow are modeled as Linear Programs (LPs) and Mixed-Integer Linear Programs (MILPs, Matta, 2008;Helber et al., 2011;Alfieri and Matta, 2012;Stolletz and Weiss, 2013). In contrast, Kolb and Göttlich (2015) address a BAP with a continuous flow of material by a sequential quadratic programming formulation. ...
... Even correlations can be considered . However, the computation times have thus far restricted the applicability to relatively small systems (Stolletz and Weiss, 2013). 2. Instead of using sampling, analytical results are utilized to capture the randomness, e.g., by Soyster et al. (1979). ...
... If exact generative methods are combined with approximate evaluative methods, the generative part typically consists of complete enumeration. Exceptions are the integrated methods of Matta (2008), Matta et al. (2014), Soyster and Toof (1976), and Stolletz and Weiss (2013), as well as the search algorithms of and Weiss et al. (2018) and the DP approach of Yamashita and Altiok (1998). ...
... i) Integrated optimization methods based on samples of random variables are suggested by several authors. BAPs with discrete-material flow are modeled as Linear Programs (LPs) and Mixed Integer Linear Programs (MILPs) (Helber et al., 2011;Matta, 2008;Stolletz and Weiss, 2013;Alfieri and Matta, 2012). In contrast, Kolb and Göttlich (2015) address a BAP with a continuous flow of material by a sequential quadratic programming formulation. ...
... Even correlations can be considered . However, the computation times have thus far restricted the applicability to relatively small systems (Stolletz and Weiss, 2013). ...
... If exact generative methods are combined with approximate evaluative methods, the generative part typically consists of complete enumeration. Exceptions are the integrated methods of Matta (2008), Matta et al. (2014), Soyster and Toof (1976) and Stolletz and Weiss (2013), as well as the search algorithms of and Weiss et al. (2018) and the DP approach of Yamashita and Altiok (1998). ...
Article
Flow production lines with finite buffer capacities are used in practice for mass production, e.g., in the automotive and food industries. The decision regarding the allocation of buffer capacities to mitigate throughput losses from stochastic processing times and unreliable stations is known as the Buffer Allocation Problem (BAP). This paper classifies and reviews the literature on the BAP with respect to different versions of the optimization problem. It considers the detailed characteristics of the flow lines, the objective function, and the constraints. Moreover, a new classification scheme for solution methods is presented that differentiates between explicit solutions, integrated optimization methods, and iterative optimization methods. The characteristics of test instances derived from realistic cases and test instances used in multiple references are discussed. The review reveals gaps in the literature regarding the considered optimization problems and solution methods, especially with a view on realistic lines. In addition, a library, “FlowLineLib”, of realistic and already used test instances is provided.
... Integrated optimization methods formalize the BAP into a mathematical programming model, in which the performance evaluation based on samples or analytic results are included, and solve the model to the optimum. Some works used sample-based optimization and formulated BAP with mixed integer linear programming (MILP) (Matta, 2008;Stolletz and Weiss, 2013;Matta et al., 2014). ...
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The buffer allocation problem (BAP) in production flow lines is very relevant from a practical point of view and very challenging from a scientific perspective. For this reason, it has drawn great attention both in industry and in the academic community. However, despite the problem relevance, no exact method is available in the literature to solve it when long production lines have to be dealt with, i.e., in practical settings. This work proposes a new Mixed Integer Linear Programming (MILP) formulation for exact solution of sample-based BAP. Due to the huge number of variables and constraints in the model, an algorithm based on Benders decomposition is proposed to increase the computational efficiency. The algorithm iterates between a simulation module that generates the Benders cuts and an optimization module that involves the solution of an updated MILP model. Multiple Benders cuts after each simulation run are generated by exploiting the structural properties of reversibility and monotonicity of flow line throughput. The new MILP formulation is tighter than the state-of-the-art model from a theoretical point of view, and order of magnitude of computation time saving is also observed in the numerical results.
... For example, Soyster et al. (1979) use an analytical representation of the problem. Other examples build a MILP to find a sample-exact solution, e.g., Matta (2008), Helber et al. (2011), Alfieri and Matta (2012), Stolletz and Weiss (2013). ...
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Full-text available
The buffer allocation problem (BAP) for flow lines has been extensively addressed in the literature. In the framework of iterative approaches, algorithms alternate an evaluative method and a generative method. Since an accurate estimation of system performance typically requires high computational effort, an efficient generative method reducing the number of iterations is desirable, for searching for the optimal buffer configuration in a reasonable time. In this work, an iterative optimization algorithm is proposed in which a highly accurate simulation is used as the evaluative method and a surrogate-based optimization is used as the generative method. The surrogate model of the system performance is built to select promising solutions so that an expensive simulation budget is avoided. The performance of the surrogate model is improved with the help of fast but rough estimators obtained with approximated analytical methods. The algorithm is embedded in a problem decomposition framework: several problem portions are solved hierarchically to reduce the solution space and to ease the search of the optimum solution. Further, the paper investigates a jumping strategy for practical application of the approach so that the algorithm response time is reduced. Numerical results are based on balanced and unbalanced flow lines composed of single-machine stations.
... According to the classification in Weiss, Schwarz, and Stolletz (2018), it can be solved through explicit solutions (Basu, 1977;Martin, 1990;L. Li, Qian, Du, & Yang, 2016), integrated optimization methods (Alfieri & Matta, 2012;Stolletz & Weiss, 2013;Weiss & Stolletz, 2015;Kolb & Göttlich, 2015), and iterative optimization methods (C. Papadopoulos, O'Kelly, & Tsadiras, 2013;Kose & Kilincci, 2015;S. ...
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One of the main problems in production systems is the buffer sizing. Choosing the right buffer size, at each production stage, that allows to achieve some performance measure (usually throughput or waiting time) is known as Buffer Allocation Problem (BAP), and it has been widely studied in the literature. Due to its complexity, BAP is usually approached using decomposition methods, under very strict system assumptions, or using simulation-optimization techniques. In this paper, the approximated mathematical programming formulation of the BAP simulation-optimization based on the time buffer concept is used. Using this approximation, buffers are modeled as temporal lags (time buffers) and this allows to use Linear Programming (LP) instead of Mixed Integer Linear Programming (MILP) models. Although LP models are easier to solve than MILPs, the huge dimension and the complex solution space topology of the time buffer approximated BAP calls for ad hoc solution algorithms. To this purpose, a row-column generation algorithm is proposed, which exploits the theoretical properties of the time buffer approximation to reduce the solution time. The proposed algorithm has been compared with a standard LP solver (ILOG CPLEX) and with a state-of-the-art MILP solver and it proved to be better than the LP solver in most of the cases, and more robust than the MILP solver with respect to computation time. Moreover, the LP model (for flow lines) is able to solve the BAP also for assembly/disassembly lines.
... In general, most of the policies aiming at the control of work-in-progress (WIP) in tandem lines (Subramaniam et al., 2009) or general flow lines (Sepehri and Nahavandi 2007) by means of thresholds or levels Roof, 1998, Wang andPrabhu, 2006) can be more easily evaluated. Given that buffer capacity may be considered as a particular type of threshold, configuration decisions for long lines such as the buffer allocation problem (BAP) (Gershwin 2000;Papadopoulos et al. 2013), analysed with analytical techniques (Colledani and Tolio 2005;Shi and Gershwin (2014)), simulation (McNamara et al. 2013), genetic algorithms (Dolgui et al. 2002), programming formulations (Stolletz and Weiss 2013) or search algorithms (Nahas et al. 2009;Vergara and Kim 2009), may inspire the definition of methods to set the optimal values of the thresholds to guarantee the required performance measures. Indeed, particular features influencing the BAP can be taken into account, such as downtimes (Enginarlar et al. 2002) or cash-flow-oriented decisions (Helber 2001). ...
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... proposed a first generalisation for the cases provided in Matta (2012a, 2012b), . Other researchers have recently used integrated mathematical programming models to deal with various simulation-optimisation problems Stolletz 2011a, 2011b;Helber et al. 2011;Schwarz and Stolletz 2013;Stolletz and Weiss 2013;Tan 2015;Weiss, Matta, and Stolletz 2017;Stolletz 2013, 2015); however, no formal properties of the methodology have been investigated so far. ...
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... The optimal solution from this method is the global optimal based on one simulation sample path. Other examples of enhancement of the DEO approach can be found in Tan (2015), Stolletz and Weiss (2013). Pedrielli (2013); Pedrielli, Matta, and Alfieri (2015a); Pedrielli, Matta, and Alfieri (2016) proposed a more general framework, i.e. not tailored to a specific simulation optimization problem. ...
... The time buffer approximation has been tested with the buffer allocation problem (BAP) for open queueing networks and to size the number of customers in a closed queueing network (Alfieri and Matta 2012a;Alfieri, Matta, and Pedrielli in press;Schwarz and Stolletz 2013;Stolletz and Weiss 2013). For the BAP case, the characterisation of the approximate model asymptotic properties has also been developed (Pedrielli 2013). ...
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... The numerical study in Sect. 4.1 supports this claim in the case of the BAP [see also Stolletz and Weiss (2013)]. ...
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