Principle Investigator: Prof. Semyon M. MeerkovInstitution: University of MichiganAward Number: DMI-0245377Program: Manufacturing Enterprise SystemsProject Title: GOALI: Quantitative Methods for Designing LeanBuffering in Production Systems: Theory andApplicationsLean Buffering in Serial Production Lines withNon-exponential MachinesEmre Enginarlara, Graduate StudentJingshan Lib, Senior Research EngineerSemyon M. Meerkova, ProfessoraDepartment of Electrical Engineering and Computer ScienceUniversity of Michigan, Ann Arbor, MI 48109-2122bManufacturing Systems Research LaboratoryGM Research and Development Center, Warren, MI 48090-9055ABSTRACT: In this paper, lean buffering (i.e., the smallest level of buffering necessary andsufficient to ensure the desired production rate of a manufacturing system) is analyzed for thecase of serial lines with machines having Weibull, gamma, and log-normal distributions of up- anddowntime. The results obtained show that: (1) the lean level of buffering is not very sensitive tothe type of up- and downtime distributions and depends mainly on their coefficients of variation,CVupand CVdown; (2) the lean level of buffering is more sensitive to CVdownthan to CVupbut thedifference in sensitivities is not too large (typically, within 20%). Based on these observations,an empirical law for calculating the lean level of buffering as a function of machine efficiency,line efficiency, the number of machines in the system, and CVupand CVdownis introduced. Thisempirical law leads to a reduction of lean buffering by a factor of up to 4, as compared with thatcalculated using the exponential assumption on up- and downtime distributions.12004 NSF Design, Service and Manufacturing Grantees and Research Conference/SMU - Dallas, Texas1 INTRODUCTION1.1 Goal of the StudyThe smallest buffer capacity, which is necessary and sufficient to achieve the desired throughput ofa production system, is referred to as lean buffering. In (Enginarlar et al. 2002, 2003), the problemof lean buffering was analyzed for the case of serial production lines with exponential machines, i.e.,the machines having up- and downtime distributed exponentially. The development was carriedout in terms of normalized buffer capacity and production system efficiency. The normalized buffercapacity was introduced ask =NTdown, (1)where N denoted the capacity of each buffer and Tdownthe average downtime of each machine inunits of cycle time (i.e., the time necessary to process one part by a machine). Parameter k wasreferred to as the Level of Buffering (LB). The production line efficiency was quantified asE =P RkP R∞, (2)where P Rkand P R∞represented the production rate of the line (i.e., the average number of partsproduced by the last machine per cycle time) with LB equal to k and infinity, respectively. Thesmallest k, which ensured the desired E, was denoted as kEand referred to as the Lean Level ofBuffering (LLB).Using parameterizations (1) and (2), Enginarlar et al. (2002, 2003) derived closed formulas forkEas a function of system characteristics. For instance, in the case of two-machines lines, it wasshown that (Enginarlar et al. 2002)kexpE=2e(E−e)1−E, if e < E,0, otherwise.(3)Here the superscript exp indicates that the machines have exponentially distributed up- and down-time, and e denotes machine efficiency in isolation, i.e.,e =TupTup+ Tdown, (4)2where Tupis the average uptime in units of cycle time. For the case of M > 2-machine serial lines,the following formula had been derived (Enginarlar et al. 2003):kE(M ≥ 3) =e(1−Q)(eQ+1−e)(eQ+2−2e)(2−Q)Q(2e−2eQ+eQ2+Q−2)×ln³E−eE+eEQ−1+e−2eQ+eQ2+Q(1−e−Q+eQ)(E−1)´, if e < E1M−1,0, otherwise,(5)whereQ = 1 − E12h1+(M−3M−1)M/4i+µE12h1+(M−3M−1)M/4i− EM−2M−1¶exp(−ÃE1M−1− e1 − E!). (6)This formula is exact for M = 3 and approximate for M > 3.Initial results on lean buffering for non-exponential machines have been reported in (Enginarlar etal. 2002). Two distributions of up- and downtime have been considered (Rayleigh and Erlang). Ithas been shown that LLB for these cases is smaller than that for the exponential case. However,(Enginarlar et al. 2002) did not provide a sufficiently complete characterization of lean bufferingin non-exponential production systems. In particular, it did not quantify how different types ofup- and downtime distributions affect LLB and did not investigate relative effects of uptime vs.downtime on LLB.The goal of this paper is to provide a method for selecting LLB in serial lines with non-exponentialmachines. We consider Weibull, gamma, and log-normal reliability models under various assump-tions on their parameters. This allows us to place their coefficients of variations at will and studyLLB as a function of up- and downtime variability. Moreover, since each of these distributions isdefined by two parameters, selecting them appropriately allows us to analyze the lean buffering for26 various shapes of density functions, ranging form almost delta-function to almost uniform. Thisanalysis leads to the quantification of both influences of distribution shapes on LLB and effectsof up- and downtime on LLB. Based of these results, we develop a method for selecting LLB inserial lines with Weibull, gamma, and log-normal reliability characteristics and conjecture that thesame method can be used for selecting LLB in serial lines with arbitrary unimodal distributionsof up- and downtime.1.2 Motivation for Considering Non-exponential MachinesThe case of non-exponential machines is important for at least two reasons:3First, in practice the machines often have up- and downtime distributed non-exponentially. As theempirical evidence (Inman 1999) indicates, the coefficients of variation, CVupand CVdownof theserandom variables are often less than 1; thus, the distributions cannot be exponential. Therefore,an analytical characterization of kEfor non-exponential machines is of theoretical importance.Second, such a characterization is of practical importance as well. Indeed, it can be expected thatkexpEis the upper bound of kEfor CV < 1 and, moreover, kEmight be substantially smaller thankexpE. This implies that a smaller buffer capacity is necessary to achieve the desired line efficiency Ewhen the machines are non-exponential. Thus, selecting LLB based on realistic, non-exponentialreliability characteristics would lead to increased leanness of production