Using linear programming to analyze and optimize stochastic flow lines

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. 1 Modeling flow lines with limited buffer capacities and random processing times Stochastic processing times at the stations of a flow line with limited buffer capacities can lead to blocking or starvation of the line’s bottleneck. In this case the throughput of the line falls below the production rate of the bottleneck operating in isolation (Gershwin, 1994, p. 117). In the design process for a flow line, one needs to quantify this impact of processing time variability on the line’s production rate and inventory level to efficiently allocate machines and buffers. In practice, discrete-event simulation (DES) is usually used to analyze the performance of a planned flow line. Several software packages with graphical user interfaces allow the planner to easily model a system at an arbitrary level of detail (Swain, 2007). DES offers a great degree of modeling flexibility with respect to probability distributions and other details of the line’s mode of operation. However, while modeling a flow line via DES is easy, a systematic optimization of the flow line design is not. A simple and relevant question is how to allocate a given total number of identical buffer spaces in a flow line so that the production rate is maximized. This question can usually not be answered efficiently using DES because of the long computation times of the simulation runs and the combinatorial nature of the decision problem (Gershwin und Schor, 2000). It is also possible to use analytic queueing models to derive exact or approximate closedform solutions or decomposition algorithms for flow lines with (un-)limited buffer capacities. The numerical effort for these methods is often negligible so that a systematic optimization of the line is possible (Gershwin und Schor, 2000). However, the mathematical assumptions required for these analytic models often restrict their use in practice. In addition, even a slight ? The authors thank the anonymous referees for their helpful comments and suggestions. 2 Helber, Stefan et al. modification of such an analytic model may easily exceed the capabilities of a practitioner who may therefore resort to DES. As a result, one rarely finds flow lines with limited buffer capacity that have been systematically optimized, as analytic queueing models are rarely understood and optimization based on DES is often too time-consuming. For the special problem of analyzing and optimizing flow lines with limited buffer capacity we propose a methodology that is about as simple as DES, but uses the optimization potential of mixed-integer linear programming. The key idea is to work with a discrete-time dynamic production-inventory model with continuous production quantities. This model approximates the behavior of a discrete-material production system operating in continuous time. Among the parameters of this model is the production capacity of a production stage during a (discrete) time period. It stems from a hypothetical simulation run in continuous time. In other words, the realizations of the stochastic processing times of the different jobs at a given production stage are transformed via sampling into corresponding realizations of production capacities for that production stage and the corresponding time period. If the number of these periods in the model is sufficiently large and some other conditions (to be explored in this paper) hold, the discrete-time model leads to a surprisingly accurate prediction of the production rate of the original flow line that operates in continuous time. To determine the production rate estimate within the context of our multi-stage discrete-time production-inventory model, we use the simplex algorithm of linear programming (LP). Our linear model can easily incorporate buffer allocation and/or machine selection decisions. In this case, additional integer decision variables for the buffer sizes are introduced. This leads to a mixed-integer problem that can be solved via branch&bound or branch&cut algorithms. Our approach therefore combines the flexibility of DES with respect to probability distributions of stochastic processing times with the optimization power of (mixed-integer) linear programming. The contribution of this paper is to describe how the method works and under which conditions it can be expected to yield precise production rate estimates. The literature on DES of production system is huge. Law und Kelton (1991) and Kelton et al. (2006), among others, give introductions to the methodology and describe simulation models of flow lines. Ho et al. (1979) introduced the concept of infinitesimal perturbation analysis were a single sample path (in continuous time) of a DES is used to determine a gradient of a performance measure for optimization purposes. A survey of the literature on analytic queueing models of flow lines with limited buffer capacity is given by Dallery und Gershwin (1992). The recent development in the field is presented in the book edited by Liberopoulos et al. (2006). Several monographs treat the analysis of manufacturing systems via queueing models (Buzacott und Shanthikumar, 1993; Gershwin, 1994; Tijms, 1994; Altiok, 1996). The literature on linear-programming based simulation of flow lines using a sample path of processing times is more limited. Abdul-Kader (2006) presents a linear programming (evaluation) model of an unreliable flow line in continuous time that is based on an earlier model by Johri (1987). In this model, the buffer capacity cannot be made a decision variable and only a fixed and given buffer allocation can be treated. The situation is similar for a model by Matta und Chefson (2005) which is based on an earlier model by Schruben (2000). In the model of a closed flow line by Matta und Chefson, the buffer capacity determines the number of constraints of the continuous time LP. We are not aware of continuous time LP models of stochastic flow lines in which the buffer size is a decision variable. However, in order to optimize the design of a flow line, the buffer size must be allowed to be a decision variable. For this reason we developed our discrete-time model which is presented in this paper. From a methodological point of view, our approach is very similar to the one presented by Helber und Henken (2007) for shift scheduling in contact centers. The remainder of this paper is structured as follows: In Section 2 we present and compare LP-based simulation approaches for stochastic flow lines. Section 3 presents results of a systematic numerical study to assess the accuracy of the proposed method. We summarize our results and give directions for further research in Section 4. Analysis and optimization of flow lines 3 2 Continuous vs. discrete time linear programming models of stochastic flow lines 2.1 Continuous time evaluation model As stated above, several modeling approaches have been proposed for continuous time LP models of flow lines. The key idea is to use for a sample part of processing times a set of real-valued decision variables to model the time at which processing of a workpiece w at a station k starts and/or ends. An example of such a model can be formulated as follows using the notation in Table 1, see also Matta und Chefson (2005):