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<ul><li><p>fy, Cam</p><p>In-process inventory</p><p>Nonlinear optimization</p><p>prese</p><p>for p</p><p>ith d</p><p>tion</p><p>is in</p><p>are p</p><p>ng lin</p><p>& 2009 Elsevier B.V. All rights reserved.</p><p>t of mstorageer forine, or</p><p>to the second machine, etc. It is desirable to nd ways to</p><p>der a, we</p><p>foroptimizing buffer space through a nonlinear programming</p><p>Contents lists available at ScienceDirect</p><p>journal homepage: www.e</p><p>Int. J. Productio</p><p>ARTICLE IN PRESS</p><p>Int. J. Production Economics 122 (2009) 7257400925-5273/$ - see front matter & 2009 Elsevier B.V. All rights reserved.</p><p>doi:10.1016/j.ijpe.2009.06.0401.2. Literature review</p><p>Substantial research has been conducted on produc-tion line evaluation and optimization (Dallery and</p><p> Corresponding author.E-mail addresses: chuanshi@mit.edu (C. Shi),</p><p>gershwin@mit.edu (S.B. Gershwin).optimize buffer space allocation to make factories mostefcient and most protable. Tools for rapid design of</p><p>approach. In our objective function, we consider bothbuffer space cost and average inventory cost.without buffers as a single machine.) As it is shown inFig. 1, material ows in the direction of the arrows, fromupstream inventory to the rst machine for an operation,to the rst buffer where it waits for the second machine,</p><p>inventory cost.In this paper, aiming at maximizing prots un</p><p>production rate constraint for production linespresent an accurate, fast, and reliable algorithmtransfer line, is organized with machines connected inseries and separated by buffers. Fig. 1, for instance, is a six-machine line. Squares represent machines or sequences ofmachines without buffers while circles represent buffers.(In the following, we treat a sequence of machines</p><p>capital investment, oor space of the line, and inventory.In this paper, we assume that the manufacturing processand machines have already been chosen. Therefore, thedecision variables are sizes of buffer spaces. Productionline cost comes from buffer space cost and average1. Introduction</p><p>1.1. Problem</p><p>A manufacturing system is a seportation elements, computers,other items that are used togeth(Gershwin, 1994). A production lachines, trans-buffers, and</p><p>manufacturingow line, or</p><p>production lines are especially important for productswith short life cycles.</p><p>Generally, designers of such production lines want toeither optimize the production rate, or the prot, ofthe line. However, material ow may be disrupted bymachine failures. The inclusion of buffers increasesthe average production rate of the line by limiting thepropagation of disruptions, but at the cost of additionalAn efcient buffer design algorithmprot maximization</p><p>Chuan Shi, Stanley B. Gershwin </p><p>Department of Mechanical Engineering, Massachusetts Institute of Technolog</p><p>a r t i c l e i n f o</p><p>Article history:</p><p>Received 20 October 2008</p><p>Accepted 25 June 2009Available online 16 July 2009</p><p>Keywords:</p><p>Flow line</p><p>Buffer allocation</p><p>Prot maximization</p><p>a b s t r a c t</p><p>In this paper, we</p><p>size optimization</p><p>inventory cost w</p><p>nonlinear produc</p><p>strained problem</p><p>Numerical results</p><p>both short and loor production line</p><p>bridge, MA 02139-4307, USA</p><p>nt an effective algorithm for maximizing prots through buffer</p><p>roduction lines. We consider both buffer space cost and average</p><p>istinct cost coefcients for different buffers, and we include a</p><p>rate constraint. To solve the problem, a corresponding uncon-</p><p>troduced and a nonlinear programming approach is adopted.</p><p>rovided to show the efciency and accuracy of our algorithm for</p><p>es.</p><p>lsevier.com/locate/ijpe</p><p>n Economics</p></li><li><p>ARTICLE IN PRESS</p><p>B1 B2</p><p>N2N1</p><p>M1 M2 M3</p><p>Inventory space</p><p>n2Average inventory n1</p><p>ction</p><p>C. Shi, S.B. Gershwin / Int. J. Production Economics 122 (2009) 725740726Gershwin, 1992). There are many studies focusing onmaximizing the production rate but few studies concen-trating on maximizing the prot. In production lineoptimization, there are two distinct approaches: thesimulation based approach and the numerical evaluationapproach. It is desirable to develop numerical methodssince they are much faster than simulation. For a briefdescription of simulation methods, see Gershwin andSchor (2000). Here we describe some literature on non-simulation methods.</p><p>As early as 1967, Buzacott derived the analyticformula for the production rate for two-machine, one-buffer lines in a deterministic processing time model(Buzacott, 1967). The invention of decomposition meth-ods with unreliable machines and nite buffers(Gershwin, 1987) and its corresponding DDX algorithm(Dallery et al., 1988) enabled the numerical evaluationof the production rate of lines having more than twomachines. Methods have been found for the exactnumerical analysis of some small lines with more thantwo machines.1 However, they are severely limited. Inthis paper, we use decomposition for the approximateanalysis and optimization of much larger systems. DeKoster (1987) introduced an aggregation method toestimate the production rate of long lines. Park (1993)developed a two-phase heuristic algorithm to solve thetotal buffer space minimization problem. But hismethod could not always nd the optimal solutionsand did not always converge. Schor (1995) and Gersh-win and Schor (2000) presented an efcient bufferallocation algorithm that applied a primaldualapproach to minimize the total buffer space under aproduction rate constraint. They also studied the protmaximization of a line through a nonlinear program-ming method that was fast and accurate, but they didnot consider the production rate constraint in the latterproblem.</p><p>More recently, Huang et al. (2002) considered a ow-shop-type production system and used a dynamicprogramming approach to maximize its production rateor minimize its work-in-process under a certain bufferallocation strategy. Diamantidis and Papadopoulos (2004)also presented a dynamic programming algorithm foroptimizing buffer allocation based on the aggregationmethod given by Lim et al. (1990). Although their dynamic</p><p>Fig. 1. A produprogramming methodology brought new approaches toproduction line design, they did not attempt to maximizethe prots of lines. Chan and Ng (2002) compared four</p><p>1 Gershwin and Schick (1983) derived an analytical solution for a</p><p>three-machine line with unreliable machines and small buffers. There</p><p>are also numerical methods for exact analysis of lines that are slightly</p><p>longer with small buffers (Tan, 2002).buffer allocation strategies and presented a modied onefor production rate maximization. Shi and Men (2003)introduced a hybrid algorithm based on hybrid nestedpartitions and a Tabu search method for production lineoptimization. However, they also focused on maximizingthe production rate of the line under a total buffer spaceconstraint, rather than the prot of the line. Smith andCruz (2005) solved the buffer allocation problem forgeneral nite buffer queueing networks in which theyminimize buffer space cost under the production rateconstraint, but they did not consider the average inven-tory cost. One paper that considers both buffer space costand average inventory cost is Dolgui et al. (2002). Theirbuffer allocation problem consists in determining buffercapacities considering the production rate of the line, thebuffer acquisition and installation cost, and the inventorycost. For that problem, they proposed a genetic algorithmwhere tentative solutions were evaluated with an approx-imate method based on the Markov-model aggregationapproach. They did not have the production rate con-straint in their problem. Some practical considerationsin optimization of ow production systems were reportedin Tempelmeier (2003). In addition, some metaheuristicmethods were adopted to deal with the scheduling andbalancing problems for production lines or assembly lines(Bautista and Pereira, 2007; Jin et al., 2006).</p><p>The optimization problem becomes much harder ifthe production rate constraint is considered in productionline design because the production rate is a nonlinearfunction of buffer sizes. Our problem includes theproduction rate constraint and aims at maximizingthe prot for the line. As introduced in Section 1.1, weconsider both buffer space cost and average inventorycost in our objective function, and assign distinct costcoefcients to different buffers. The average inventory ofthe line, and consequently the lines cost, are alsononlinear functions of buffer sizes. Hence, we havenonlinear elements in both our objective function andconstraints.</p><p>1.3. Outline of paper</p><p>The paper is organized as follows. The model we studyin our research and important features of its behavior aredescribed in Section 2. The proposed algorithm is derived</p><p>B3 B4 B5</p><p>N3 N4 N5</p><p>M4 M5 M6</p><p>n3 n4 n5</p><p>line example.in Section 3, following by numerical results showing itsaccuracy and efciency in Section 4. We summarize thepaper and propose future research in Section 5. Inaddition, we provide the continuous variable version ofthe solution of two-machine line evaluation in AppendixA, the proofs about our algorithm for a special case inAppendix B, and a brief introduction to the P% surfacesearch in Appendix C.</p></li><li><p>ARTICLE IN PRESS</p><p>C. Shi, S.B. Gershwin / Int. J. Production Economics 122 (2009) 725740 7272. Problem statement, assumptions, and notation</p><p>2.1. Model of the line</p><p>The model described here is the deterministic proces-sing time model of Gershwin (1987, 1994). We make allthe assumptions and approximations of that model,follow all his conventions, and use his notation. Weoutline the key features of the model below.</p><p>In our model, we denote machine i by Mi and buffer iby Bi. The line consisting of kmachines and k 1 buffers iscalled a k-machine, k 1-buffer line, or k-machine line forshort. Processing times of all machines are equal,deterministic, and constant. Time is scaled so thatoperations take one time unit. Transportation time isnegligible compared to the operation time.</p><p>In addition, Ni, the size of buffer Bi; 8i 1; . . . ; k 1, aredecision variables. Therefore, there are k 1 decisionvariables for a k-machine, k 1-buffer line. Machines areunreliable and are parameterized by probabilities offailure and repair. Specically, the parameters of machineMi are pi, the probability of a failure during a time unitwhile the machine is operating; and ri, the probability of arepair during a time unit while the machine is down. As aconsequence, the times to failure and to repair aregeometrically distributed. By convention, repairs andfailures occur at the beginnings of time units and changesin the buffer levels take place at the ends of time units.Machine parameters are xed. We let P be the productionrate of a line. Although the production rate P is a functionof machines and their reliability, we vary only buffer sizes,so we write P PN1; . . . ;Nk1, or PN for short, whereN is the vector N1; . . . ;Nk1. PN is a nonlinear functionof buffer sizes N, and is calculated numerically bydecomposition for lines having more than two machines.The numerical method is described in Gershwin (1994).</p><p>The prot of a k-machine, k 1-buffer line can beformulated as</p><p>Profit APN1; . . . ;Nk1 Xk1i1</p><p>biNi Xk1i1</p><p>cini K ,</p><p>where A40 is the prot coefcient associated with theproduction rate PN, bi and ci are cost coefcientsassociated with the buffer space and average inventoryfor the ith buffer, respectively, and K stands for all costsother than those due to buffer sizes, average inventory,and raw material. Since K is independent of N, we simplifythe formulation above and write our objective function as</p><p>JN1; . . . ;Nk1 APN1; . . . ;Nk1 Xk1i1</p><p>biNi Xk1i1</p><p>cini. (1)</p><p>In the following, we refer to J as the prot of the line. Tosimplify terminology, the rst item on the right side ofEq. (1) can be seen as the total revenue of the line; whilethe other two items together can be interpreted as thetotal cost of the line. Allowing different buffers to havedifferent cost coefcients is realistic as we know that, forexample, the cost of buffer space in a clean room is muchexpensive than elsewhere in a factory.2.2. Monotonicity and concavity of PN</p><p>A common intuition in the line design eld is theconcavity of PN, though there is no analytical result thatconclusively shows that the lines we study in this paperexhibit the concavity property. However, some research insimilar systems indicates that this is a reasonableassumption. Extensive numerical experiments supportthis assumption as well. Shanthikumar and Yao (1989a)pointed out monotonicity and concavity properties incyclic queueing networks with nite buffers. In theirresearch, service processes were exponential, with ratesthat are increasing functions of the number of customersin the queue. In Shanthikumar and Yao (1989b), theyestablished the concavity of the throughput in a multicellsystem. They considered two types of cells. The rst typeis an open queueing network with limited buffer capacityof the Jackson (1963) type, while the second type is anordered-entry system with heterogeneous servers andlimited buffer capacity. They showed that the productionrate of each cell of either type is an increasing and concavefunction of its buffer allocation. Anantharam and Tsoucas(1990) proved stochastic concavity of throughput in aseries of queues with single exponential servers, nitebuffers, and communication blocking (blocking beforeservice). Meester and Shanthikumar (1990) showed theconcavity of throughput in tandem queueing systems withnite buffers and exponential servers. Dallery et al. (1994)generalized their work by establishing the concavity ofthroughput as a function of buffer sizes (and initialconditions, for a closed network) for a greatly extendedclass of distributions of service times. Glasserman and Yao(1996) found that the throughput was a concave functionof buffer parameters in serial lines with general blockingand synchronized service. They studied similar systemsunder a wide variety of blocking mechanisms, anddemonstrated concavity for systems with exponentialprocessing times, and other qualitative properties forgeneral service time distributions.</p><p>In addition, some work based on the concavity of theproduction rate for the same or similar systems has beenpublished. Park (1993) assumed the concavity of theproduction rate over both a buffer and a vector of buffersin his study of buffer size optimization. Gershwin andSchor (2000) studied the same model and established aprimaldual algorithm for buffer space allocation inproduction lines basing on the assumption of concavePN. Levantesi et al. (2001) presented an algorithm forbuffer allocation in production lines with the sameassumption. Jeong and Kim (2000) applied that propertyto assembly systems with unreliable machines and nitebuffers after an empirical study through simulation toshow the concavity of the throughput for their systems. So(1997) also mentioned the concavity of the productionrate in his study on optimal buffer allocation strategy forunpaced production lines with machines having differentservice times and nite buffers.</p><p>It is also necessary to assume that PN is monotoni-cally increasing in N. Schor (1995) provided a relativelydetailed survey on this property. As an ex...</p></li></ul>