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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). ...

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). ...

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). ...

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. ...

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). ...

An analytical model for evaluating the performance of two-machine continuous flow systems with finite buffer capacity, multiple up and down states and generalised thresholds is presented in this paper. The idea is that the machines can behave differently above or below certain buffer levels named thresholds. In addition, when the buffer level crosses one of the thresholds, certain changes of state of the machines can happen. With the method presented in the paper it is possible to consider all the different types of two machine lines which can be modelled with multiple up and down states, like for example cases with machines having phase-type failure and repair time distributions, serial/parallel machines and quality control machines. The proposed approach, allows to include system control by means of thresholds in the system model and provides a way to analyse the performance of a wide range of two-machine systems. Some of these cases are proposed in the numerical examples at the end of the paper. Moreover, the proposed two-machine line can be used as a building block for the analysis of larger systems, including systems with loops.

... 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. ...

Simulation optimisation has gained a great attention due to its success in the design of complex manufacturing systems. In this paper, we look at manufacturing as a special class of queueing systems and propose the Discrete Event Optimisation (DEO) methodology, which provides a formal way to develop integrated mathematical models for the simultaneous simulation and optimisation. In the case, the obtained model is a mixed integer linear programming model; the methodology provides a formal way to generate approximations of them. The analytical properties of DEO models are analysed for the first time in the framework of sample path optimisation and mathematical programming. The methodology represents a reference for the use of mathematical programming as a way to model simulation optimisation for queueing systems. The applicability of the DEO methodology to complex problems is showed using the task and buffer allocation problem in a production line.

... 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). ...

Pull policies are considered to be among the most efficient control strategy. Setting the correct parameters to maximise their efficiency is, however, not a trivial task. Simulation–optimisation techniques have received particular attention as a means to solve this problem. Nevertheless, they require the iterative solution of an optimisation model to generate the parameter values and a discrete event simulator to evaluate the resulting system performance. In the framework of simulation-optimisation, this paper proposes a combined solution of the optimisation and simulation problems for the optimal operation of pull control systems under several control strategies. Numerical experiments were performed to evaluate the performance of the proposed technique.

... The numerical study in Sect. 4.1 supports this claim in the case of the BAP [see also Stolletz and Weiss (2013)]. ...

The allocation of buffer space in flow lines with stochastic processing times is an important decision, as buffer capacities influence the performance of these lines. The objective of this problem is to minimize the overall number of buffer spaces achieving at least one given goal production rate. We optimally solve this problem with a mixed-integer programming approach by sampling the effective processing times. To obtain robust results, large sample sizes are required. These incur large models and long computation times using standard solvers. This paper presents a Benders Decomposition approach in combination with initial bounds and different feasibility cuts for theBufferAllocation Problem,which provides exact solutionswhile reducing the computation times substantially. Numerical experiments are carried out to demonstrate the performance and the flexibility of the proposed approaches. The numerical study reveals that the algorithm is capable to solve long lines with reliable and unreliable machines, including arbitrary distributions as well as correlations of processing times.

Solving the buffer allocation problem (BAP) in long production lines is difficult because it is an NP-hard problem. In this paper, we propose a new approach to solve the BAP for long series-parallel unbalanced production lines with the objective of minimising the average work-in-process subject to a throughput rate constraint. Instead of directly optimising a long line, this method decomposes the original long line into several small decoupled subsystems and adds relation condition variables between the subsystems. After several iterations between subsystem optimisation and condition updating, this method can obtain the optimal or near-optimal solution of the original system with a significantly improved computational efficiency. Extensive numerical experiments demonstrate the accuracy and efficiency of this approach. Finally, several examples and a multi-factorial experimental analysis are provided to show the influence of the decomposition strategy, initial values, and parameters of the target systems on the performance of the proposed method.

The buffer allocation problem consists of a dynamical description of the underlying production process combined with stochastic processing times. The aim is to find optimal buffer sizes averaged over several samples. Starting from a time-discrete recursion we derive a time-continuous model supplemented with a stochastic process. The new model is used for simulation and optimization purposes as well. Numerical experiments show the efficiency of our approach compared to other optimization techniques.

Buffer capacity allocation problems for flow-line manufacturing systems with unreliable machines are studied. These problems arise in a wide range of manufacturing systems and concern determining buffer capacities with respect to a given optimality criterion which can depend on the average production rate of the line, buffer cost, inventory cost, etc. Here, this problem is proven to be NP-hard for a tandem production line and oracle representation of the revenue and cost functions, and NP-hard for a series-parallel line and stepwise revenue function.

In this paper, we consider the problem of buffer space allocation for a tandem production line with unreliable machines. This
problem has various formulations all aiming to answer the question: how much buffer storage to allocate between the processing
stations? Many authors use the knapsack-type formulation of this problem. We investigate the problem with a broader statement.
The criterion depends on the average steady-state production rate of the line and the buffer equipment acquisition cost. We
evaluate black-box complexity of this problem and propose a hybrid optimization algorithm (HBBA), combining the genetic and
branch-and-bound approaches. HBBA is excellent in computational time. HBBA uses a Markov model aggregation technique for goal
function evaluation. Nevertheless, HBBA is more general and can be used with other production rate evaluation techniques.

Optimization-via-simulation consists in applying iteratively two detached models until an optimality condition is reached: a simulation model for predicting the system performance, and a model for generating potential optimal solutions. Mathematical programming representation has been recently used to describe the behavior of discrete event systems as well as their formal properties. This paper proposes explicit mathematical programming representations for jointly sim- ulating and optimizing discrete event systems. The main advantage of such models is the rapidity of searching for the optimal solution, given to the explicit knowledge of objective function and constraints. Three types of for- mulations are proposed for solving the buffer allocation problem in flow lines with finite buffer capacities: an exact mixed integer linear model, an approximate LP model and a stochastic programming model. Numerical analysis shows that the computational time required to solve resource allo- cation problems can be significantly reduced by using the proposed formulations.

The most important models and results of the manufacturing flow line literature are described. These include the major classes of models (asynchronous, synchronous, and continuous); the major features (blocking, processing times, failures and repairs); the major properties (conservation of flow, flow rate-idle time, reversibility, and others); and the relationships among different models. Exact and approximate methods for obtaining quantitative measures of performance are also reviewed. The exact methods are appropriate for small systems. The approximate methods, which are the only means available for large systems, are generally based on decomposition, and make use of the exact methods for small systems. Extensions are briefly discussed. Directions for future research are suggested.

This paper presents a linear programming approach to analyze and optimize flow lines with limited buffer capacities and stochastic processing times. The basic idea is to solve a huge but simple linear program that models an entire simulation run of a multi-stage production process in discrete time, to determine a production rate estimate. As our methodology is purely numerical, it offers the full modeling flexibility of stochastic simulation with respect to the probability distribution of processing times. However, unlike discrete-event simulation models, it also offers the optimization power of linear programming and hence allows to solve buffer allocation problems. We show under which conditions our method works well by comparing its results to exact values for two-machine models and approximate simulation results for longer lines.

Analytical models for the dynamics of some discrete event systems are introduced where the system trajectories are solutions to linear and mixed-integer programs

Although simple random sampling is the standard sampling procedure in Monte Carlo simulation, such practice is questioned in this paper. In any Monte Carlo application, sampled distributions are assumed to be known. Using simple random sampling, sample histograms or, equivalently, sample moments will vary at random, thus producing an imprecise description of the known input distribution, and consequently increasing the variance of simulation estimates. This problem can be avoided with descriptive sampling, here proposed as a more appropriate approach in Monte Carlo simulation than simple random sampling. Descriptive sampling is based on a deterministic and purposive selection of the sample values—in order to conform as closely as possible to the sampled distribution—and the random permutation of these values. As such, it represents a fundamental conceptual change in Monte Carlo sampling, departing from the 'principle' that sample values must be randomly generated in order to describe random behaviour. The basis of this new idea, examples of its use and empirical results are presented.

This paper describes an empirical study of the variability of simulation estimates, which produced some interesting new results. Simulation estimates are determined by the input samples. Any input sample can be divided into two basic features: the set of input values and their sequence. Based on this idea, the individual contribution of each feature to the estimates' variance can be empirically studied. This is done following a two-way factorial experiment. Using the standard random sampling approach, the set of values was found to affect most simulation estimates in a common way, and to play a relevant role in their variability. The sequence effect, however, was found to be problem-dependent. Apart from providing a better understanding of the estimates' variability, this study contributes to the proposal of a new sampling approach in simulation: descriptive sampling.

Simulation is a numerical technique accepted to be time consuming in the development phase of the model, in the execution of experiments and in the output analysis. Nevertheless, simulation is widely used to analyze the detailed performance of production systems with the main purpose of determining the optimal size and proper management policies. This paper uses analytical techniques for the analysis of closed production flow lines with limited buffer capacities and random processing times instead of the traditional simulation. In particular, the paper addresses the problem of deriving structural properties by representing the production system with a linear programming model in which objective function and constraints analytically describe the system behavior.

This paper describes efficient algorithms for determining how buffer space should be allocated in a flow line. We analyze two problems: a primal problem, which minimizes total buffer space subject to a production rate constraint; and a dual problem, which maximizes production rate subject to a total buffer space constraint. The dual problem is solved by means of a gradient method, and the primal problem is solved using the dual solution. Numerical results are presented. Profit optimization problems are natural generalizations of the primal and dual problems, and we show how they can be solved using essentially the same algorithms.