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Contents lists available at ScienceDirect
Int. J. Production Economics
Int. J. Production Economics 122 (2009) 725–740
0925-52
doi:10.1
� Cor
E-m
gershw
journal homepage: www.elsevier.com/locate/ijpe
An efficient buffer design algorithm for production lineprofit maximization
Chuan Shi, Stanley B. Gershwin �
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
a r t i c l e i n f o
Article history:
Received 20 October 2008
Accepted 25 June 2009Available online 16 July 2009
Keywords:
Flow line
Buffer allocation
Profit maximization
In-process inventory
Nonlinear optimization
73/$ - see front matter & 2009 Elsevier B.V. A
016/j.ijpe.2009.06.040
responding author.
ail addresses: [email protected] (C. Shi),
[email protected] (S.B. Gershwin).
a b s t r a c t
In this paper, we present an effective algorithm for maximizing profits through buffer
size optimization for production lines. We consider both buffer space cost and average
inventory cost with distinct cost coefficients for different buffers, and we include a
nonlinear production rate constraint. To solve the problem, a corresponding uncon-
strained problem is introduced and a nonlinear programming approach is adopted.
Numerical results are provided to show the efficiency and accuracy of our algorithm for
both short and long lines.
& 2009 Elsevier B.V. All rights reserved.
1. Introduction
1.1. Problem
A manufacturing system is a set of machines, trans-portation elements, computers, storage buffers, andother items that are used together for manufacturing(Gershwin, 1994). A production line, or flow line, ortransfer 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 machineswithout buffers as a single machine.) As it is shown inFig. 1, material flows in the direction of the arrows, fromupstream inventory to the first machine for an operation,to the first buffer where it waits for the second machine,to the second machine, etc. It is desirable to find ways tooptimize buffer space allocation to make factories mostefficient and most profitable. Tools for rapid design of
ll rights reserved.
production lines are especially important for productswith short life cycles.
Generally, designers of such production lines want toeither optimize the production rate, or the profit, ofthe line. However, material flow 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 additionalcapital investment, floor 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 averageinventory cost.
In this paper, aiming at maximizing profits under aproduction rate constraint for production lines, wepresent an accurate, fast, and reliable algorithm foroptimizing buffer space through a nonlinear programmingapproach. In our objective function, we consider bothbuffer space cost and average inventory cost.
1.2. Literature review
Substantial research has been conducted on produc-tion line evaluation and optimization (Dallery and
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B1 B2 B3 B4 B5
N2 N3 N4 N5N1
M1 M2 M4 M5 M6M3
Inventory space
n2 n3 n4 n5Average inventory n1
Fig. 1. A production line example.
C. Shi, S.B. Gershwin / Int. J. Production Economics 122 (2009) 725–740726
Gershwin, 1992). There are many studies focusing onmaximizing the production rate but few studies concen-trating on maximizing the profit. 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.
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 finite 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 find the optimal solutionsand did not always converge. Schor (1995) and Gersh-win and Schor (2000) presented an efficient bufferallocation algorithm that applied a primal–dualapproach to minimize the total buffer space under aproduction rate constraint. They also studied the profitmaximization 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.
More recently, Huang et al. (2002) considered a flow-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 dynamicprogramming methodology brought new approaches toproduction line design, they did not attempt to maximizethe profits of lines. Chan and Ng (2002) compared four
1 Gershwin and Schick (1983) derived an analytical solution for a
three-machine line with unreliable machines and small buffers. There
are also numerical methods for exact analysis of lines that are slightly
longer with small buffers (Tan, 2002).
buffer allocation strategies and presented a modified 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 profit of the line. Smith andCruz (2005) solved the buffer allocation problem forgeneral finite 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 flow 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).
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 profit for the line. As introduced in Section 1.1, weconsider both buffer space cost and average inventorycost in our objective function, and assign distinct costcoefficients to different buffers. The average inventory ofthe line, and consequently the line’s cost, are alsononlinear functions of buffer sizes. Hence, we havenonlinear elements in both our objective function andconstraints.
1.3. Outline of paper
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 derivedin Section 3, following by numerical results showing itsaccuracy and efficiency 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.
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2. Problem statement, assumptions, and notation
2.1. Model of the line
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.
In our model, we denote machine i by Mi and buffer i
by Bi. The line consisting of k machines 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.
In addition, Ni, the size of buffer Bi; 8i ¼ 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. Specifically, 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 fixed. 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 ¼ PðN1; . . . ;Nk�1Þ, or PðNÞ for short, whereN is the vector ðN1; . . . ;Nk�1Þ. PðNÞ 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).
The profit of a k-machine, k� 1-buffer line can beformulated as
Profit ¼ APðN1; . . . ;Nk�1Þ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini � K ,
where A40 is the profit coefficient associated with theproduction rate PðNÞ, bi and ci are cost coefficientsassociated 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
JðN1; . . . ;Nk�1Þ ¼ APðN1; . . . ;Nk�1Þ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini. (1)
In the following, we refer to J as the profit of the line. Tosimplify terminology, the first 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 coefficients 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 PðNÞ
A common intuition in the line design field is theconcavity of PðNÞ, 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 finite 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 first 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, finitebuffers, and communication blocking (blocking beforeservice). Meester and Shanthikumar (1990) showed theconcavity of throughput in tandem queueing systems withfinite 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.
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 aprimal–dual algorithm for buffer space allocation inproduction lines basing on the assumption of concavePðNÞ. 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 finitebuffers 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 finite buffers.
It is also necessary to assume that PðNÞ is monotoni-cally increasing in N. Schor (1995) provided a relativelydetailed survey on this property. As an example, Fig. 2exhibits the monotonicity and concavity of PðNÞ for a three
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0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90
100
0.76 0.78 0.8
0.82 0.84 0.86 0.88 0.9
P(N
1,N
2)
N1
N2
Fig. 2. Monotonicity and concavity of PðNÞ.
2 We include the continuous variable version of the analytic solution
of the two-machine line in Appendix A.
C. Shi, S.B. Gershwin / Int. J. Production Economics 122 (2009) 725–740728
machine line with r1 ¼ 0:1, p1 ¼ 0:01, r2 ¼ 0:1, p2 ¼ 0:01,r3 ¼ 0:1, and p3 ¼ 0:01. Monotonicity and concavity ofPðNÞ are key assumption behind our algorithm.
2.3. Problem formulations
In this section, we introduce mathematical models.Our prime goal is a constrained problem, in which weaim at maximizing profits of production lines subjectto a production rate constraint. In order to solve theconstrained problem, we present a corresponding uncon-strained problem, in which we drop the production rateconstraint. We introduce the two problems here and leavethe reason for introducing the unconstrained problem tothe next section.
2.3.1. The constrained problem
The constrained problem is formulated as follows:
max JðN1; . . . ;Nk�1Þ
¼ APðN1; . . . ;Nk�1Þ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini
s.t. PðN1; . . . ;Nk�1Þ � P%,
Ni � 0; 8i ¼ 1; . . . ; k� 1, (2)
where P% is the required production rate. The firstconstraint is the production rate constraint. Note that itis nonlinear. The second constraint is called the buffersize constraint. It comes from the natural property ofbuffer sizes since it makes no sense to have negativebuffer sizes.
However, it is necessary for us to further limit thebuffer sizes. This is because we use the decomposition toevaluate the production rate of the line, and the decom-position is based on an analytical solution of thetwo-machine line (Gershwin, 1994). This analyticaltwo-machine-line evaluation requires buffer sizes to begreater than 2. For a line having buffer sizes equal to orless than 2, there are different ways to measure itsperformance and we do not discuss them in this paper.Therefore, we only focus on lines whose buffer sizes are allgreater than 2. In the following, let Nmin denote theminimum of the buffer size and we re-write the
constrained problem as
max JðN1; . . . ;Nk�1Þ
¼ APðN1; . . . ;Nk�1Þ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini
s.t. PðN1; . . . ;Nk�1Þ � P%,
Ni � Nmin; 8i ¼ 1; . . . ; k� 1. (3)
2.3.2. The unconstrained problem
In the unconstrained problem, we ignore the produc-tion rate constraint. Thus, the unconstrained problem is
max JðN1; . . . ;Nk�1Þ
¼ APðN1; . . . ;Nk�1Þ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini
s.t. Ni � Nmin; 8i ¼ 1; . . . ; k� 1. (4)
This is a convenient, although not quite accurate, namesince we still have the buffer size constraint. As we showin Section 3, the unconstrained problem can be solvedeasily by a gradient method. We will further illustrate therelationship between the two problems and reveal howwe take advantage of the unconstrained problem to solvethe constrained one.
3. Solution technique
In this section, we present our algorithm for solving theconstrained problem. Instead of solving it directly, weadopt a two-step strategy in which we introduce a newvariable A0. We replace A by A0 in the unconstrainedproblem (4) and solve it iteratively for different valuesof A0.
It is easy to solve the unconstrained problem through agradient method as we take advantage of the analyticalform of the two-machine-line evaluation, which enablesus to treat Ni as continuous variables. This is because theformulas for the production rate and average inventory ofthe two-machine line do not require Ni to be integers.2
This enables us to evaluate a two-machine line whosebuffer size is not an integer. As a consequent, though ourmodel is a deterministic processing time discrete statemodel, we can still treat Ni as continuous variables. Inaddition, since we evaluate PðNÞ by the decomposition andthe continuous variable version of the analytic two-machine-line evaluation (introduced in Appendix A), andtreat Ni as continuous variables, we are able to treat PðNÞand JðNÞ as continuously differentiable functions. Thisallows us to adopt a gradient method to solve theunconstrained problem, since @PðNÞ=@Ni; 8i is meaningfuland we can compute it numerically. However, we need toindicate that due to the lack of an analytical expression ofthe profit of a line having more than two machines, wecompute gradients according to a forward differenceformula.
It is important to point out that gradient methods areappropriate when the space being searched has a single
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maximum. This requires our objective function of theunconstrained problem, JðN1; . . . ;Nk�1Þ, to have only onemaximum point for the proposed optimization method towork correctly. Schor (1995) encountered the same issueand showed that JðN1; . . . ;Nk�1Þ has single maximumthrough numerical evidence and an intuitive argument.He introduced a two-step gradient method to solve ourunconstrained problem. His results were demonstratedto be correct when compared with both simulationand another optimization method (Seong et al., 1994).Gershwin and Schor (2000) confirmed this.
3.1. Algorithm derivation
Solving the unconstrained problem yields two cases:
1.
The unconstrained problem has a solutionðN%1 ; . . . ;N%
k�1Þ in which PðN%
1 ; . . . ;N%
k�1Þ � P%. In thiscase, the solution of the constrained problem isðN%
1 ; . . . ;N%
k�1Þ.
2.3 A feasible vector x is regular if the equality constraint gradients
rhiðxÞ; i ¼ 1; . . . ;m, and the active inequality constraint gradients
rgjðxÞ; j 2 MðxÞ, are linearly independent. Also x is regular in the
exceptional case where there are no equality constraints and all the
inequality constraints are inactive at x. MðxÞ is the set of active inequality
constraints, i.e., MðxÞ ¼ fjjgjðxÞ ¼ 0g (Bertsekas, 1999).
The solution of the unconstrained problem satisfiesPoP%. This is not the solution of the constrainedproblem. In this case, we consider the followingunconstrained problem:
max JðN1; . . . ;Nk�1Þ
¼ A0PðN1; . . . ;Nk�1Þ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini
s.t. Ni � Nmin; 8i ¼ 1; . . . ; k� 1 (5)
in which A of (4) is replaced by A0. Let ðN01; . . . ;N0k�1Þ be
the solution to this problem and P0 ¼ PðN01; . . . ;N0k�1Þ.
Then, we claim the following.
Assertion. The constrained problem
max JðN1; . . . ;Nk�1Þ
¼ A0PðN1; . . . ;Nk�1Þ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini
s.t. PðN1; . . . ;Nk�1Þ � P%
Ni � Nmin; 8i ¼ 1; . . . ; k� 1 (6)
has the same solution for all A0 in which the solution of theunconstrained problem (5) has P0oP%.
This is because the solution of problem (6) will satisfyPðN1; . . . ;Nk�1Þ ¼ P% so the objective function is equiva-lent to A0P%
�Pk�1
i¼1 biNi �Pk�1
i¼1 cini. Since the first term isindependent of all of the Ni, it has no effect on the solutionof the problem.
We claim in the assertion that if the optimal solutionof the unconstrained problem (4) is not the solutionof the constrained problem (3), then the solution ofthe constrained problem (3), ðN%
1 ; . . . ;N%
k�1Þ, satisfiesPðN%
1 ; . . . ;N%
k�1Þ ¼ P%. Therefore, to solve the constrainedproblem (3), we replace A by A0 in both problems (3) and(4) and solve problem (5) for different A0’s. We need tofind the value of A0 such that the solution to problem (5)satisfies PðN1; . . . ;Nk�1Þ ¼ P%. Then, this solution is thesame as that of the original constrained problem (3).
We prove this assertion by the Karush–Kuhn–Tucker(KKT) conditions of nonlinear programming. We firstconvert the constrained problem (3) into the minimiza-tion form:
min � JðN1; . . . ;Nk�1Þ
¼ �APðN1; . . . ;Nk�1Þ þXk�1
i¼1
biNi þXk�1
i¼1
cini
s.t. P%� PðN1; . . . ;Nk�1Þ � 0,
Nmin � Ni � 0; 8i ¼ 1; . . . ; k� 1. (7)
We have argued that we treat Ni as continuousvariables, and PðNÞ and JðNÞ as continuously differentiablefunctions. Let us consider the KKT conditions. A statementof the KKT conditions (Bertsekas, 1999) is that: Let x% be alocal minimum of the problem
min f ðxÞ
s.t. h1ðxÞ ¼ 0; . . . ;hmðxÞ ¼ 0,
g1ðxÞ � 0; . . . ; grðxÞ � 0, (8)
where f, hi, and gj are continuously differentiable func-tions from Rn to R. Assume that x% is regular.3 Then thereexists unique Lagrange multipliers l%
1 ; . . . ; l%
m andm%
1 ; . . . ;m%
r , satisfying the following conditions:
rxLðx%;l%;m%Þ ¼ 0,
m%
j � 0; j ¼ 1; . . . ; r,
m%
j gjðx%Þ ¼ 0; j ¼ 1; . . . ; r, (9)
where Lðx; l;mÞ ¼ f ðxÞ þPm
i¼1lihiðxÞ þPr
j¼1mjgjðxÞ is calledthe Lagrangian function.
Before we apply the KKT conditions to our problem, weneed to point out a necessary condition that guaranteesthe existence of Lagrange multipliers. One appropriate forour problem is the Slater constraint qualification forconvex inequalities (Bertsekas, 1999), which is: Let x% be alocal minimum of the problem (8), where f and gj arecontinuously differentiable functions from Rn to R, andthe functions hi are linear. Assume that the functions gj
are convex and that there exists a feasible vector x
satisfying gjðxÞo0; 8j 2 Mðx%Þ: Then x% satisfies the KKTconditions.
Let us now consider our constrained problem. Thereare no equality constraints in the problem, but there are k
inequality constraints:
g0ðNÞ ¼ P%� PðN1; . . . ;Nk�1Þ � 0,
giðNÞ ¼ Nmin � Ni � 0; 8i ¼ 1; . . . ; k� 1. (10)
Due to the concavity of PðNÞ, g0ðNÞ is a convex function. Allother giðNÞ are linear so that they are also convex. It is nothard to find a feasible vector to make our problem satisfythe Slater constraint qualification. Since the requiredproduction rate, P%, has to be feasible for the line, thereexists sufficiently large N such that PðN1; . . . ; Nk�1Þ4P% so
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g0ðN1; . . . ; Nk�1Þo0. In addition, giðN1; . . . ; Nk�1Þo0; 8i ¼1; . . . ; k� 1 because Nmin � Nio0; 8i ¼ 1; . . . ; k� 1. Hence,our constrained problem satisfies the Slater constraintqualification, and there exists unique Lagrange multipliersm%
i ; i ¼ 0; . . . ; k� 1, for our problem to satisfy the KKTconditions:
�rJðN%Þ þ m%
0rðP%� PðN%
ÞÞ þXk�1
i¼1
m%
i rðNmin � N%
i Þ ¼ 0
(11)
or
�
@JðN%Þ
@N1
@JðN%Þ
@N2
..
.
@JðN%Þ
@Nk�1
0BBBBBBBBBBBB@
1CCCCCCCCCCCCA
� m%
0
@PðN%Þ
@N1
@PðN%Þ
@N2
..
.
@PðN%Þ
@Nk�1
0BBBBBBBBBBBB@
1CCCCCCCCCCCCA
� m%
1
1
0
..
.
0
0BBBBB@
1CCCCCA� m%
2
0
1
..
.
0
0BBBBB@
1CCCCCA
� � � � � m%
k�1
0
0
..
.
1
0BBBBB@
1CCCCCA¼
0
0
..
.
0
0BBBBB@
1CCCCCA
, (12)
and
m%
i � 0; 8i ¼ 0; . . . ; k� 1, (13)
m%
0 ðP%� PðN%
ÞÞ ¼ 0, (14)
m%
i ðNmin � N%
i Þ ¼ 0; 8i ¼ 1; . . . ; k� 1, (15)
where N% is the optimal solution of our constrainedproblem.
Next, we show that finding the Lagrange multipliersm%
i ; i ¼ 0; . . . ; k� 1, and the optimal solution N% to satisfythe KKT conditions (12)–(15) is equivalent to solving theconstrained problem (3) by our algorithm. Suppose thatN% is an interior solution.4 In all our experiments, theoptimal solutions have this feature. In this case, bycondition (15), we know that m%
i ¼ 0; 8i ¼ 1; . . . ; k� 1.Hence, we can simplify the KKT conditions (12)–(15) to
�
@JðN%Þ
@N1
@JðN%Þ
@N2
..
.
@JðN%Þ
@Nk�1
0BBBBBBBBBBBB@
1CCCCCCCCCCCCA
� m%
0
@PðN%Þ
@N1
@PðN%Þ
@N2
..
.
@PðN%Þ
@Nk�1
0BBBBBBBBBBBB@
1CCCCCCCCCCCCA
¼
0
0
..
.
0
0BBBBB@
1CCCCCA
, (16)
m%
0 ðP%� PðN%
ÞÞ ¼ 0, (17)
where m%
0 � 0. We know, since N% is not the optimalsolution of the unconstrained problem, that rJðN%
Þa0.rJðN%
Þa0 means that not all @JðN%Þ=@Ni are equal to 0.
Thus, m%
0 a0 since otherwise condition (16) would be
4 An interior solution means that all N%
i in N% are greater than Nmin.
We discuss the case in which some N%
i are on the boundary, i.e.,
N%
i ¼ Nmin, in Appendix B.
violated. By condition (17), the optimal solution N%
satisfies PðN%Þ ¼ P%. Since g0 is the only active inequality
constraint, N% is regular. In addition, conditions (16) and(17) reveal how we could find m%
0 and N%. For every m%
0 ,condition (16) determines N% since there are k� 1equations and k� 1 unknowns. Therefore, we can thinkof N%
¼ N%ðm%
0 Þ. We search for a value of m%
0 such thatPðN%
ðm%
0 ÞÞ ¼ P%. As we indicate in the following, this isexactly what our algorithm does.
Replacing m%
0 by m040 in constraint (16) gives
�
@JðNÞ
@N1
@JðNÞ
@N2
..
.
@JðNÞ
@Nk�1
0BBBBBBBBBBB@
1CCCCCCCCCCCA
� m0
@PðNÞ
@N1
@PðNÞ
@N2
..
.
@PðNÞ
@Nk�1
0BBBBBBBBBBB@
1CCCCCCCCCCCA
¼
0
0
..
.
0
0BBBB@
1CCCCA, (18)
where N is the unique solution of (18). Note that N is thesolution of the following optimization problem:
min � JðNÞ ¼ �JðNÞ þ m0ðP%� PðNÞÞ
s.t. Nmin � Ni � 0; 8i ¼ 1; . . . ; k� 1 (19)
which is equivalent to
max JðNÞ ¼ JðNÞ � m0ðP%� PðNÞÞ
s.t. Nmin � Ni � 0; 8i ¼ 1; . . . ; k� 1 (20)
or
max JðNÞ ¼ APðNÞ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini � m0ðP%� PðNÞÞ
s.t. Nmin � Ni � 0; 8i ¼ 1; . . . ; k� 1 (21)
or
max JðNÞ ¼ ðAþ m0ÞPðNÞ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini
s.t. Ni � Nmin; 8i ¼ 1; . . . ; k� 1 (22)
or, finally,
max JðNÞ ¼ A0PðNÞ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini
s.t. Ni � Nmin; 8i ¼ 1; . . . ; k� 1, (23)
where A0 ¼ Aþ m0. This is exactly the unconstrainedproblem (5), and N is its optimal solution. Note thatm040 indicates that A04A. In addition, the KKT condition(17) indicates that the optimal solution of the constrainedproblem N% satisfies PðN%
Þ ¼ P%. This means that, forevery A04A (or m040), we can find the correspondingoptimal solution N satisfying condition (18) by solvingproblem (5), and, we need to find the A0 such that thesolution to problem (5), denoted as NA0 , satisfiesPðNA0
Þ ¼ P%. Then, m0 ¼ A0 � A and NA0 satisfy conditions(16) and (17). Hence, m0 ¼ A0 � A is exactly the Lagrangemultiplier satisfying the KKT conditions of our constrainedproblem, and N%
¼ NA0 is the optimal solution of ourconstrained problem. Consequently, solving the con-strained problem (3) through our algorithm is essentially
ARTICLE IN PRESS
N2
P > .88
A’=1000
A’=2000 A’=3000A’=4000
A’=5000A’=6000
Optimum
locus of cost optima 30 40 50 60 70 80 90
100
C. Shi, S.B. Gershwin / Int. J. Production Economics 122 (2009) 725–740 731
finding the unique Lagrange multipliers and optimalsolution of the problem. We have proven our assertion.
Therefore, in our algorithm, we conduct an one-dimensional search on A04A and stop after we find theA0 such that the solution to the unconstrained problemsatisfies PðN1; . . . ;Nk�1Þ ¼ P%. We conclude that it is alsothe desired solution of the constrained problem. Wesummarize our algorithm in Section 3.2.
3.2. Algorithm statement
10 20
1.
20 30 40 50 60 70 80 90 100Check the feasibility of the problem. We describe thisin Section 3.5.
N1
2.Fig. 3. An example of the algorithm.
Solve the unconstrained problem (4). If the solutionðN%
1 ; . . . ;N%
k�1Þ satisfies PðN%
1 ; . . . ;N%
k�1Þ � P%, stop. Thesolution of the constrained problem is ðN%
1 ; . . . ;N%
k�1Þ.This step is also the necessity check of the algorithm aswe point out in Section 3.5.
3.
Do a one-dimensional search on A04A to find A0 suchthat the solution of the unconstrained problemsatisfies PðN%1 ; . . . ;N%
k�1Þ ¼ P%. Stop. The desired solu-tion is ðN%
1 ; . . . ;N%
k�1Þ.
3.3. An example of the algorithm
We provide an example to show the algorithm. It is athree-machine, two-buffer line with parameters r1 ¼ 0:12,p1 ¼ 0:01, r2 ¼ 0:09, p2 ¼ 0:01, r3 ¼ 0:11, p3 ¼ 0:01, andthe profit function JðN1;N2Þ ¼ 1000PðN1;N2Þ � 0:5N1 � N2.Suppose first that the required production rate, P%, is 0.85.Solving the unconstrained problem and letting ðN1; N2Þ bethe optimal solution yield PðN1; N2Þ ¼ 0:8576. SincePðN1; N2Þ4P%
¼ 0:85, the solution ðN1; N2Þ is equivalentto the solution of the constrained problem and no furthersearch on A04A is needed.
Next suppose that the required production rate, P%, is0.88. In this case, the optimal solution of the uncon-strained problem does not satisfy PðN1; N2Þ � P%. Thus, weneed to conduct the one-dimensional search on A04A tofind the optimal solution of the constrained problem. So,we solve the unconstrained problem for A0 ¼ 1000;2000;. . . ;6000, and the optimal solutions for distinct A0 areshown in Fig. 3. Fig. 3 also tells that the requiredproduction rate P%
¼ 0:88. The locus of unconstrainedoptima of the cost function is sketched. Let ðNA0
1 ;NA0
2 Þ be theoptimal solution of the unconstrained problem for acertain A0. Note that for A0 ¼ 1000;2000, and 3000, theunconstrained optima have PðNA0
1 ;NA0
2 Þo0:88 while forA0 ¼ 4000;5000, and 6000, the unconstrained optimahave PðNA0
1 ;NA0
2 Þ40:88. Therefore, if the problem to besolved is to maximize 1000P � 0:5N1 � N2 subject toP � 0:88, then the solution is the intersection of P ¼ 0:88and the locus of unconstrained optima.
3.4. Detailed description of the algorithm
We describe in detail the algorithm we propose tosolve the constrained problem here. The algorithmincludes solving the unconstrained problem and theone-dimensional search on A04A, if necessary. We have
explained that we can treat the decision variables, Ni, ascontinuous variables, and JðN1; . . . ;Nk�1Þ as a continuouslydifferentiable function. Therefore, we adopt a gradientmethod that is based on the decomposition (Gershwin,1987) and the DDX algorithm (Dallery et al., 1988) to solvethe unconstrained problem.
3.4.1. Gradient method for the unconstrained problem
We solve the unconstrained problem with a gradientmethod. An initial guess ðN0
1; . . . ;N0k�1Þ is selected first. For
example, N0i can be chosen as the minimal value to satisfy
Pð1; . . . ;N0i ; . . . ;1Þ � P%. If this inequality is satisfied with
equality in all i, then the initial guess satisfiesPðN0
1; . . . ;N0k�1Þ � P%. This is how we choose the initial
guess in our implementation of the algorithm. However, itis helpful to indicate that our algorithm does not requirethat PðN0
1; . . . ;N0k�1Þ � P%. We have verified this with
experiments and the initial guesses ðN01; . . . ;N
0k�1Þ such
that PðN01; . . . ;N
0k�1Þ4P% lead to the same solution.
Then, we calculate the gradient direction to movein ðN1; . . . ;Nk�1Þ space. A line search is then conducted inthat direction until a maximum is encountered. Thisbecomes the next guess. A new direction is chosen and theprocess continues until no further improvement can beachieved. There is no analytical expression to computeprofits of lines having more than two machines. Conse-quently, to determine the search direction, we computethe gradient, g, according to a forward difference formula,which is
gi ¼JðN1; . . . ;Ni þ dNi; . . . ;Nk�1Þ � JðN1; . . . ;Ni; . . . ;Nk�1Þ
dNi,
(24)
where gi is the gradient component of buffer Bi, J is theprofit of the line and can be obtained by Eq. (1), and dNi isthe increment of buffer Bi. Since we treat Ni as continuousvariables, in the gradient calculation above, we choosedNi ¼ 0:01, which has proved to be a good choice in allexperiments we have conducted.
Apart from acquiring the gradient direction, we stillneed to determine the step size, a. We conduct a bisectionsearch to find a such that JðN1 þ ag1; . . . ;Nk�1 þ agk�1Þ, orJðNþ agÞ, is maximized. Then, we calculate the nextgradient and repeat. This process ends when there is no
ARTICLE IN PRESS
C. Shi, S.B. Gershwin / Int. J. Production Economics 122 (2009) 725–740732
improvement in profit or when all components of thegradient are sufficiently small. A block diagram of thisgradient method appears in Fig. 4.
3.4.2. One-dimensional search on A04A
To find A0 such that the solution of the unconstrainedproblem satisfies PðN1; . . . ;Nk�1Þ ¼ P%, we use the NewtonChord method (Isaacson and Keller, 1994), which is anefficient way to find t0 such that f ðt0Þ ¼ 0 for a givenfunction f ð�Þ. Thus, in our algorithm, for any particularvalue of A0, we define f ðA0Þ as
f ðA0Þ ¼ PðNoptA0 Þ � P%, (25)
where NoptA0 is the optimal buffer allocation associatedwith that A0 and P% is the required production rate. TheNewton Chord method in our algorithm consists of thefollowing three steps:
1.
Guess A00 and A01. Calculate the approximate slopes ¼f ðA01Þ � f ðA00Þ
A01 � A00. (26)
2.
Choose A02 so thatf ðA00Þ þ ðA0
2 � A00Þs ¼ 0 (27)
or
A02 ¼ �f ðA00Þ
sþ A00. (28)
3.
Repeat with A00 ¼ A01 and A01 ¼ A02 until jf ðA00Þj is smallenough.In our implementation, the termination criterion isjf ðA00Þj � 10�4. The first two values of A0 are A00 ¼ A andA01 ¼ Aþ 1000.
N is the solution.Stop.
Is J(Nnext) > J(N)?
Find a such that J(N+ag) is maximized.
Define Nnext = N + ag.
Calculate gradient g.
Specify initial guess.N = (N1,...,Nk-1).
Ye
No
Fig. 4. Block diagram of th
3.5. Implementation issues
Implementation issues about the algorithm include thefeasibility and necessity of the algorithm, the initializationof the DDX algorithm of the decomposition, and theconversion from continuous solutions to integers.
Before running the algorithm, we should ensure thatthe required production rate, P%, is feasible for the line tobe optimized. This means that P% should satisfy
P%omini
ri
ri þ pi
, (29)
where ri=ðri þ piÞ is the isolated production rate ofmachine Mi. If this fails, no set of buffers can satisfy theproduction rate constraint.
We also have to make sure that there is a need toconduct the one-dimensional search on A04A of ouralgorithm. This actually can be decided after we firstsolve the unconstrained problem. We have indicated thisin Section 3.1 and restate it as the second step of thealgorithm in Section 3.2. If the unconstrained problem (4)has a solution ðN1; . . . ;Nk�1Þ in which PðN1; . . . ;Nk�1Þ � P%,then we do not need to implement the one-dimensionalsearch on A04A and the solution is equivalent to that ofthe constrained problem.
Furthermore, since we apply the DDX algorithm for thedecomposition and it is an iterative algorithm, we mustinitialize it whenever it is called in our algorithm. We usethe most recent value of ðN1; . . . ;Nk�1Þ from the lastevaluation, instead of a standard initialization, to reducethe numbers of iterations and the two-machine-lineevaluations.
Finally, it is necessary to point out that, to use theresult of the algorithm in practical production line designfor factories, we have to convert it back to integers.The easiest way is to round each continuous solutionNi to the nearest integer. One can also check all 2k�1
Set N = Nnext.s
e gradient method.
ARTICLE IN PRESS
0.84
0.86
C. Shi, S.B. Gershwin / Int. J. Production Economics 122 (2009) 725–740 733
combinations of all N%
i and N%
i , i ¼ 1; . . . ; k� 1, where N%
i
is the minimal integer larger than N%
i , while N%
i is themaximal integer smaller than N%
i .
0.72
0.74
0.76
0.78
0.8
0.82
0 500 1000 1500 2000
P(N
*)
Value of A’
N2* becomes non-minimalN3* becomes non-minimal
N1* becomes non-minimal
Fig. 5. PðN%Þ vs. A0 .
4. Numerical results and analysis
Numerical results are provided to show not only theefficiency of our algorithm but also its implementationprocess. Hence, taking a four-machine, three-buffer line asan example, we first explain how PðN%
Þ changes with A0.Then, we apply the algorithm to both short and long linesto illuminate its efficiency. Computation speed is dis-cussed at the end of this section. In our implementationof the algorithm, we let Nmin be 2þ �, where � ¼ 10�6.In addition, the algorithm is written with Matlab and runfor all experiments on a computer with a 2.4 GHz IntelCore 2 Duo CPU.
Table 2Machine parameters of the five-machine line experiment.
Machine M1 M2 M3 M4 M5
r 0.11 0.12 0.10 0.09 0.10
p 0.008 0.01 0.01 0.01 0.01
4.1. System behavior
Let us consider a four-machine, three-buffer line,whose parameters are summarized in Table 1. All costcoefficients are set to be 1 so that profit is calculatedas
JðN1; . . . ;N3Þ ¼ A0PðN1; . . . ;N3Þ �X3
i¼1
Ni �X3
i¼1
ni. (30)
To study the system behavior of the algorithm, we run itfor this line to generate the curve of PðN%
Þ versus A0. Wevary the A0 from 0 to 2000 with a step size of 1 and thedesired curve is shown in Fig. 5. There are four segmentsin the curve. The flat segment stands for the case in whichthe optimal sizes of all buffers are Nmin. For A0 rangingfrom 0 to 249, the optimal buffer sizes for the line stay onthe boundary of the feasible region so they areðNmin;Nmin;NminÞ. The production rates associated withthose A0 are identical, forming the horizontal segment inthe curve. After A0 passes 249, one of the optimal buffersizes ðN%
2 Þ becomes greater than Nmin, i.e., it turns to anon-minimal value from Nmin. From that time, theproduction rate begins to monotonically increase withA0. Each time a new buffer becomes non-minimal, thederivative of the production rate curve changes, but thecurve remains continuous. Fig. 5 suggests that PðN%
Þ forwhich not all components in N% are 2þ � increases withA0 monotonically. This system behavior further verifies ouralgorithm; as A0 increases, PðN%
Þ increases monotonicallyso we can eventually find the A0 such that PðN%
Þ � P%
(for P% feasible).
Table 1Data for the system behavior experiment.
Machine M1 M2 M3 M4
r 0.11 0.12 0.10 0.09
p 0.008 0.01 0.01 0.01
4.2. Experiments on short lines
Our algorithm optimizes short lines very quickly,which can be shown by the following two experiments.We optimize a five-machine line and a six-machine line,and compare the optimal solutions of our algorithm withsolutions gained from searching the P ¼ P%
ðN1; . . . ;Nk�1Þ
surface in ðN1; . . . ;Nk�1Þ space.5
4.2.1. Experiment on a five-machine, four-buffer line
The machine parameters are listed in Table 2. Therequired production rate P% is 0.88. It is easy to check thatthe isolated production rate of the bottleneck of the line,machine M4, is greater than P%, so the problem is feasible.The profit is calculated as
J ¼ 2500PðNÞ �X4
i¼1
Ni �X4
i¼1
ni.
To verify the optimal solution, we conduct a search onthe P% surface in ðN1; . . . ;Nk�1Þ space. We search on thesurface around the optimal solution of our algorithm andthe step size for all buffers in our search is 0.01.Experimental results including optimal solutions bothfrom our algorithm and P% surface search, errors, and therounded integer optimal solution are generated in Table 3.We see that the maximal error is 0.23% and appears in n2.Computer time for this experiment is 2.69 s. The numberof the two-machine-line evaluations is 87666. (Thereason we provide this number is that it is not affected
5 For a brief introduction about the P% surface search, see
Appendix C.
ARTICLE IN PRESS
Table 3Results of the five-machine line experiment.
P%
surface
search
The
algorithm
Error (%) Rounded N%
Production
rate
0.8800 0.8800 0.00 0.8800
N%
1 28.85 28.8570 0.02 29
N%
2 58.46 58.5694 0.19 59
N%
3 92.98 92.9068 0.08 93
N%
4 87.39 87.4415 0.06 87
n1 19.0682 19.0726 0.02 19.1791
n2 34.3084 34.3835 0.23 34.7289
n3 48.7200 48.6981 0.04 48.9123
n4 31.9894 32.0063 0.05 31.9485
Profit ($) 1798.2 1798.1 0.006 1797.4
Table 4Results of the revised five-machine line experiment.
P%
surface
search
The
algorithm
Error (%) Rounded N%
Production
rate
0.8800 0.87995 0.00 0.87996
N%
1 30.90 31.1110 0.68 31
N%
2 65.02 64.7286 0.45 65
N%
3 79.02 78.9987 0.03 79
N%
4 96.59 96.1601 0.45 96
n1 20.6757 20.8461 0.82 20.7578
n2 39.9641 39.8214 0.36 39.9949
n3 41.7716 41.7820 0.03 41.8100
n4 33.7392 33.6399 0.29 33.6141
Profit ($) 1713.3 1713.8 0.03 1713.7
20
40
60
80
100
120
140
160
0 2 4 6 8 10 12 14
The
optim
al s
izes
of b
uffe
rs
Buffer size cost coefficient of buffer B3
The optimal size of buffer B1The optimal size of buffer B2The optimal size of buffer B3The optimal size of buffer B4
Fig. 6. Optimal buffer spaces vs. cost coefficient of B3.
Table 6Results of the six-machine line experiment.
P%
surface
search
The
algorithm
Error (%) Rounded N%
Production
rate
0.8800 0.8799 0.00 0.8799
N%
1 33.20 33.4002 0.60 33
N%
2 46.25 46.3292 0.17 46
N%
3 103.01 102.6612 0.34 103
N%
4 111.51 110.9841 0.47 111
N%
5 56.11 56.0119 0.17 56
n1 22.4904 22.6863 0.87 22.3744
n2 26.4580 26.5914 0.50 26.3037
n3 51.4389 51.4330 0.11 51.3898
n4 42.7788 42.7599 0.04 42.6693
n5 17.4193 17.4689 0.28 17.4469
Profit ($) 2096.7 2096.4 0.01 2097.8
Table 5Machine parameters of the six-machine line experiment.
Machine M1 M2 M3 M4 M5 M6
r 0.11 0.12 0.10 0.09 0.10 0.11
p 0.008 0.01 0.01 0.01 0.01 0.009
Buffer B1 B2 B3 B4 B5
bi 1 2 0.5 0.8 1
ci 1 1 2 1 1.5
C. Shi, S.B. Gershwin / Int. J. Production Economics 122 (2009) 725–740734
by the capability and performance of the computer thatruns the algorithm.) Hence, our algorithm offers accurateresults very quickly.
The optimal solution reveals that, since machine M4 isthe bottleneck, the optimal size of buffer B3 is greater thanthe optimal sizes of other three buffers. So, next wechange b3, the cost coefficient associated with buffersize of B3, to 2. This means that the line pays more for thebuffer size of B3. Thus, we expect the optimal solution forthe new line to have a smaller size for B3 while greatersizes for the other three buffers so as to satisfy theproduction rate constraint. Experimental results confirmour expectation (see Table 4). We see that the optimalbuffer size of B3 is reduced from 92.9068 to 78.9987, andoptimal buffer sizes of other three buffers increase. Themaximal error is 0.82% and appears in n1. Computertime for the revised experiment is 3.12 s. The number ofthe two-machine-line evaluations is 98 106. To furtherstudy this phenomenon, we conduct more experimentsfor this line by varying the cost coefficient of B3 from 0 to14 with a step size of 0.2, and report results in Fig. 6. Fig. 6indicates that as the cost coefficient of B3 becomes largerand larger, the optimal value of B3 becomes smaller tolimit the cost spent on it. Meanwhile the optimal values ofthe other three buffers get larger so the line maintains therequired production rate.
4.2.2. Experiment on a six-machine, five-buffer line
The machine parameters and cost coefficients areprovided in Table 5. The required production rate P% isstill 0.88, and it is easy to check that it is feasible for theline. The profit is calculated as
J ¼ 3000PðNÞ �X5
i¼1
biNi �X5
i¼1
cini.
Experimental results are presented in Table 6. Themaximal error between the solution of our algorithmand that of surface search is 0.87% and appears in n1.
ARTICLE IN PRESS
Table 7Machine parameters of the 12-machine line experiment.
Machine M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12
r 0.11 0.12 0.10 0.09 0.10 0.11 0.10 0.11 0.12 0.10 0.12 0.09
p 0.008 0.01 0.01 0.01 0.01 0.01 0.009 0.01 0.009 0.008 0.01 0.009
Table 8Results of the 12-machine line experiment.
P%
surface
search
The
algorithm
Error (%) Rounded N%
Production
rate
0.8800 0.8800 0.00 0.8799
N%
1 29.10 29.1769 0.26 29
N%
2 59.20 59.2830 0.14 59
N%
3 97.80 97.7980 0.002 98
N%
4 107.50 107.4176 0.08 107
N%
5 84.50 84.4804 0.02 84
N%
6 70.80 70.6892 0.17 71
N%
7 63.10 63.1893 0.14 63
N%
8 53.10 52.9274 0.33 53
N%
9 47.20 47.2232 0.05 47
N%
10 47.90 47.7967 0.22 48
N%
11 48.80 48.7716 0.06 49
n1 19.2388 19.2986 0.31 19.1979
n2 34.9561 35.0423 0.25 34.8194
n3 52.5423 52.6032 0.12 52.6833
n4 45.1528 45.1840 0.07 45.0835
n5 34.4289 34.4770 0.14 34.2790
n6 30.7073 30.7048 0.01 30.8229
n7 28.0446 28.1299 0.30 28.0902
n8 21.5666 21.5438 0.11 21.5932
n9 21.5059 21.5442 0.18 21.4299
n10 22.6756 22.6496 0.11 22.7303
n11 20.8692 20.8615 0.04 20.9613
Profit ($) 4239.3 4239.2 0.002 4239.5
6 http://www.mathworks.com/products/curvefitting/.
C. Shi, S.B. Gershwin / Int. J. Production Economics 122 (2009) 725–740 735
Computer time for this experiment is 7.72 s. The numberof the two-machine-line evaluations is 286 368.
4.3. Experiments on long lines
Next we apply our algorithm to lines with more than10 machines. In particular, we run an experiment for a 12-machine, 11-buffer line. Again, to verify our optimalsolution, we conduct a search on the P% surface. Wesearch on the surface around the optimal solution gainedby our algorithm but this time we set the step size in oursearch to 0.1 for all buffers to reduce search time. Machineparameters are shown in Table 7. We still use 0.88 as afeasible required production rate for this line. In addition,we set all cost coefficients to 1 and the profit is calculatedas
J ¼ 6000PðNÞ �X11
i¼1
Ni �X11
i¼1
ni.
Experimental results are generated in Table 8. We seethat the maximal error is 0.33% and appears in N%
8 . Itwould be further reduced if we decreased the step size ofthe P% surface search. Therefore, our algorithm providesaccurate optimal solutions for long lines as well. Compu-ter time for this experiment is 91.47 s. The number of thetwo-machine-line evaluations is 3 236 500.
4.4. Computation speed
Finally, we discuss the computation speed about ouralgorithm. Although we have shown that our algorithmoffers the optimal solution for a 12-machine, 11-bufferline within 2 min, it is important to observe the algorithmbehavior for longer lines. Thus, we run our algorithm for aseries of experiments for lines having identical machines.We vary the length of the line from 4 machines up to30 machines. Machine parameters are p ¼ 0:01 andr ¼ 0:1. In all cases, we choose a feasible production rateP%¼ 0:88, and the objective function is
JðNÞ ¼ APðNÞ �Xk�1
i¼1
Ni �Xk�1
i¼1
ni, (31)
where A ¼ 500k for the line of length k. Furthermore, weinitialize A00 ¼ A and A01 ¼ Aþ 1000. The numbers of thetwo-machine-line evaluations for lines with differentlengths are generated in Table 9 and Fig. 7, respectively.We fit a curve to those points in order to reveal therelation between the length of the production line and thecomputation effort of our algorithm. From those points inFig. 7, we guess that we can find an exponential curve to
fit them. Using the Curve Fitting Toolbox of Matlab,6 wefind
y ¼ 8:536� 105e0:1664k, (32)
where y denotes the number of the two-machine-lineevaluations and k denotes the length of the line. The curveis shown in Fig. 7 as well. It can be seen, from Table 9, thatit takes our algorithm about 3 min to find the optimalbuffer allocation for the 15-machine line, and about10 min to find the optimal buffer allocation for the20-machine line. However, the exponential curve impliesthat our algorithm needs much more time for lines withmore than 30 machines. As we have pointed out in Section1, in practice, it is possible that several machines arelocated adjacently in series and followed by a buffer. Sowhile it is not rare for a transfer line to have more than 30machines, the number of buffers is often much smaller.However, the exponential curve spurs us to furtherimprove our algorithm to make it more efficient for longerlines. We finally present the specific optimal solution forthe
ARTICLE IN PRESS
Table 9Numbers of the two-machine-line evaluations for lines with different
lengths.
Line length Production rate # of two-machine
evaluations
Computer
time (s)
4 0.8800 19 616 0.725
5 0.8799 47 136 1.428
6 0.8799 85 536 2.419
7 0.8800 168 010 4.500
8 0.8801 381516 9.974
9 0.8801 592 522 15.246
10 0.8799 1027 808 25.793
11 0.8800 1575 414 39.471
12 0.8800 2 329 460 61.903
13 0.8799 3 612 774 95.386
14 0.8799 5 309 640 138.166
15 0.8800 7 380 412 188.281
20 0.8800 27 540 720 686.022
25 0.8800 64 081036 1594.281
30 0.8799 122188 024 3061.463
0
2e+007
4e+007
6e+007
8e+007
1e+008
1.2e+008
1.4e+008
0 5 10 15 20 25 30 0
500
1000
1500
2000
2500
3000
3500
The
num
ber o
f the
two-
mac
hine
-line
eva
luat
ion
Com
pute
r tim
e, s
econ
ds
Length of production lines
fitting curve
Fig. 7. The number of the two-machine-line evaluations vs. the length of
production lines and its fitting curve.
20
30
40
50
60
70
80
90
100
0 5 10 15 20 25 30
The
optim
al s
izes
of b
uffe
rs
Index of buffers
Buffer sizesAverage inventories
Optimal buffer allocation
Rat
io o
f ave
rage
inve
ntor
y to
buf
fer s
ize
Fig. 8. Results of the 30-machine line. (a) Op
C. Shi, S.B. Gershwin / Int. J. Production Economics 122 (2009) 725–740736
30-machine line. The optimal sizes of buffers are shown inFig. 8(a). The curve of ni=Ni is illustrated in Fig. 8(b).
5. Conclusion
5.1. Summary
In this paper, we present an accurate, fast, and reliablealgorithm for maximizing profits through buffer spaceoptimization for production lines. In the cost function, weconsider both buffer space cost and average inventory costand assign different cost coefficients to different buffers.In addition, we include a production rate constraint in ourproblem. A nonlinear programming approach is adoptedto solve the problem. Experimental results are alsoprovided to show the accuracy and efficiency of ouralgorithm. Our algorithm is based on the assumption ofthe concavity of PðNÞ. Thus, one future research focus is toprove this assumption rigorously.
5.2. Future work
There are several research directions to which we canextend our algorithm in the future.
1.
0.3 0
0.4 0.4 0.4 0.4 0
0.5 0.5 0.5 0.5 0
tima
Assembly/Disassembly (A/D) systems.A/D systems are extensions of lines in which assemblyand disassembly take place. The first extension of ouralgorithm is to acyclic A/D systems.
2.
Systems with loops.Many production lines contain pallets or fixtures thattravel in closed loops. Many production lines arecontrolled by constant work-in-progress (CONWIP) torestrict inventory. Such systems have tokens that travelin closed loops. Thus, there is a need for machine linedesigners to invite algorithms to enhance the perfor-mance of such lines. Werner (2001) and Gershwin andWerner (2007) presented a decomposition algorithm8.42468.52468.6
0 5 10 15 20 25 30Index of buffers
bar n/N
ni / Ni curve¯
l buffer allocation, (b) ni=Ni curve.
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that enabled us to evaluate the production rate forlines with single loops. Zhang (2006) extended it tolines with multiple loops. So a good extension is thatwe take advantage of the decomposition algorithms tomaximize profits for lines with loops.
3.
Systems with machine selection.In the design of a production line, each operation hasvarious machine choices. Each machine has its owndistinct parameters and cost. Different machine com-binations lead to diverse production line performance.Nahas et al. (2009) developed a method to selectmachines and buffers in unreliable series–parallelproduction lines to maximize the production ratesubject to a total cost constraint. We want to decidewhich machine to choose for each operation as weselect buffers to maximize the profit for the line.4.
Systems with set-up cost for buffers.This means whenever we decide to establish a bufferbetween two machines (or machine sets), we intro-duce a fixed buffer set-up cost. After the buffer isestablished, the buffer space cost will be proportionalto its size. So, in this case, buffer space cost will be 0 ifNi ¼ 0 or ai þ biNi if Ni � Nmin for buffer Bi.5.
Systems with quality control.By taking account of quality control, we assume thatmachines generate both good parts and bad parts.Unfortunately, buffers delay the inspection of badparts. Consider a two-machine line. Suppose parts areinspected only after the second machine. So, when thefirst machine begins to generate bad parts, we will notknow it immediately if there are still some good partsin the buffer. We will only know that the first machineis generating bad parts after all good parts in the bufferare processed by the second machine and it begins toprocess the first bad part. During that delay, the firstmachine could have generated more bad parts. Thus, inthis case, buffers bring potential delay to inspection,which reduces the production rate and the profit ofthe line. Kim and Gershwin (2005) pointed out that inthe case of our example above, an increase of buffersize could either increase or decrease the productionrate for different lines.6.
Systems with additional buffer space constraints.In addition to the constraints in our constrainedproblem, we may encounter additional buffer spaceconstraints of the formPigijNi � hj. For instance,
consider a production line, in which two buffers, saybuffers B5 and B6, are in the clean room. Due to thehigh cost of buffer space in the clean room, there is amaximum total space constraint for B5 and B6 asN5 þ N6 � C, where C is a constant. This bringsadditional constraints to our problem, so we need tomake necessary modification to our algorithm so thatit can apply to the new problem.
These extensions are key complements to make ourmodel more practical for actual manufacturing systems infactories. A good understanding of our algorithm behavioron those extensions would be valuable in future produc-tion line design.
Acknowledgments
The authors are grateful for support from the MIT-Portugal Program and from the Singapore-MIT Alliance.
Appendix A
The continuous variable version of the solution of thedeterministic two-machine line is presented in thissection. Suppose we have a two-machine line withparameters r1, p1, r2, and p2, and the buffer size is N. Inthe two-machine case, the state of the system is s ¼
ðn;a1;a2Þ when n is the buffer level ð0 � n � NÞ and ai isthe repair state of machine i ði ¼ 1;2; ai ¼ 0;1Þ. pðn;a1;a2Þ
stands for the steady-state probability of that state.Gershwin (1994) showed that the steady-state probabilitydistribution is
pð0;0;0Þ ¼ 0, (A.1)
pð0;0;1Þ ¼ CXr1 þ r2 � r1r2 � r1p2
r1p2, (A.2)
pð0;1;0Þ ¼ 0, (A.3)
pð0;1;1Þ ¼ 0, (A.4)
pð1;0;0Þ ¼ CX, (A.5)
pð1;0;1Þ ¼ CXY2, (A.6)
pð1;1;0Þ ¼ 0, (A.7)
pð1;1;1Þ ¼CX
p2
r1 þ r2 � r1r2 � r1p2
p1 þ p2 � p1p2 � r1p2, (A.8)
pðN � 1;0;0Þ ¼ CXN�1, (A.9)
pðN � 1;0;1Þ ¼ 0, (A.10)
pðN � 1;1;0Þ ¼ CXN�1Y1, (A.11)
pðN � 1;1;1Þ ¼CXN�1
p1
r1 þ r2 � r1r2 � p1r2
p1 þ p2 � p1p2 � p1r2, (A.12)
pðN;0;0Þ ¼ 0, (A.13)
pðN;0;1Þ ¼ 0, (A.14)
pðN;1;0Þ ¼ CXN�1 r1 þ r2 � r1r2 � p1r2
p1r2, (A.15)
pðN;1;1Þ ¼ 0, (A.16)
pðn;a1;a2Þ ¼ CXnYa1
1 Ya2
2 ,
2 � n � N � 2; a1 ¼ 0;1; a2 ¼ 0;1, (A.17)
where C is a normalizing constant and
Y1 ¼r1 þ r2 � r1r2 � r1p2
p1 þ p2 � p1p2 � p1r2, (A.18)
Y2 ¼r1 þ r2 � r1r2 � p1r2
p1 þ p2 � p1p2 � r1p2, (A.19)
X ¼Y2
Y1. (A.20)
For convenience, we define internal states as thosestates in which 2 � n � N � 2; while boundary states asthose states in which n ¼ 0, 1, N � 1, or N. We further letpB be the summation of the steady-state probabilities of
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all boundary states and pI be the summation of thesteady-state probabilities of all internal states. Then,
pB ¼X
all boundary states
pðn;a1;a2Þ
¼ CXr1 þ r2 � r1r2 � r1p2
r1p2
þ CX þ CXY2
þCX
p2
r1 þ r2 � r1r2 � r1p2
p1 þ p2 � p1p2 � r1p2
þ CXN�1þ CXN�1Y1 þ
CXN�1
p1
r1 þ r2 � r1r2 � p1r2
p1 þ p2 � p1p2 � p1r2
þ CXN�1 r1 þ r2 � r1r2 � p1r2
p1r2. (A.21)
In addition, we calculate pI in the continuous variableversion by
pI ¼X
all internal states
pðn;a1;a2Þ
¼XN�2
n¼2
CXnð1þ Y1Þð1þ Y2Þ
¼
CXN�1
� X2
X � 1if jX � 1j4d;
ð1þ Y1Þð1þ Y2Þ
CðN � 3Þð1þ Y1Þð1þ Y2Þ if jX � 1j � d;
8>>>>>><>>>>>>:
(A.22)
where d is a very small non-zero positive value. Let pT bethe summation of all steady-state probabilities of bothboundary states and internal states, then
pT ¼ pB þ pI ¼ 1. (A.23)
To find C, let LB be pB=C and LI be pI=C. LB and LI can becalculated directly by the machine parameters of the lineand the buffer size N. Since pB and pI must sum to 1, then
C ¼1
LB þ LI. (A.24)
The production rate of the line can be calculated by
P ¼r1
r1 þ p1
ð1� pbÞ
¼r1
r1 þ p1ð1� pðN;1;0ÞÞ
¼r1
r1 þ p11� CXN�1 r1 þ r2 � r1r2 � p1r2
p1r2
� �, (A.25)
where pb ¼ pðN;1;0Þ is the probability of blocking.Finally, let us consider how to calculate the average
inventory for the line. We compute the average inventoryfor the internal states, denoted by nI, as
nI ¼X
all internal states
npðn;a1;a2Þ
¼XN�2
n¼2
CnXnð1þ Y1Þð1þ Y2Þ
¼
CXð�2X þ ðN � 1ÞXN�2
�XN�1
� X2
X � 1Þ
X � 1if jX � 1j4d;
ð1þ Y1Þð1þ Y2Þ
1
2CNðN � 3Þð1þ Y1Þð1þ Y2Þ if jX � 1j � d;
8>>>>>>>><>>>>>>>>:
(A.26)
where d is a very small non-zero positive value. Then, tocalculate the average inventory for the line, we need toconsider both internal states and boundary states whosesteady-state probability is non-zero and n is non-zero.Then, the average inventory of the line is calculated as
n ¼X
all states
npðn;a1;a2Þ
¼ pð1;0;0Þ þ pð1;0;1Þ þ pð1;1;1Þ þ nI
þ ðN � 1ÞðpðN � 1;0;0Þ þ pðN � 1;1;0Þ þ pðN � 1;1;1ÞÞ
þ NpðN;1;0Þ. (A.27)
Appendix B
Here, we provide the proofs of our assertion for thecase in which some N%
i ¼ Nmin. Recall that the assertion is:
Assertion. The constrained problem
max JðN1; . . . ;Nk�1Þ
¼ A0PðN1; . . . ;Nk�1Þ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini
s.t. PðN1; . . . ;Nk�1Þ � P%,
Ni � Nmin; 8i ¼ 1; . . . ; k� 1
has the same solution for all A0 in which the solution of theunconstrained problem (5) has P0oP%.
Let B be fijN%
i ¼ Nming. Hence, N%
i ¼ Nmin; 8i 2 B, whileN%
i 4Nmin; 8ieB. In this case, those N%
i equal to Nmin are onthe boundary of the feasible region of the optimalsolution. By condition (15), we know that m%
i ¼ 0; 8ieB
and ia0. Hence, we simplify the KKT conditions (12)–(15)to
�
@JðN%Þ
@N1
@JðN%Þ
@N2
..
.
@JðN%Þ
@Nk�1
0BBBBBBBBBBB@
1CCCCCCCCCCCA
� m%
0
@PðN%Þ
@N1
@PðN%Þ
@N2
..
.
@PðN%Þ
@Nk�1
0BBBBBBBBBBB@
1CCCCCCCCCCCA
�Xi2B
m%
i ei ¼
0
0
..
.
0
0BBBB@
1CCCCA, (B.1)
where
ei ¼
0
..
.
1
..
.
0
0BBBBBBB@
1CCCCCCCA
in which the 1 is in the ith position.
m%
0 ðP%� PðN%
ÞÞ ¼ 0, (B.2)
m%
i ðNmin � N%
i Þ ¼ 0; 8i ¼ 1; . . . ; k� 1, (B.3)
where m%
i � 0; 8i ¼ 0; . . . ; k� 1. We first argue that not allN%
i ; ieB satisfy @JðN%Þ=@Ni ¼ 0. This is because if
@JðN%Þ=@Ni ¼ 0 for all N%
i ; ieB and all other N%
i ¼ Nmin;
i 2 B, then the solution of the constrained problemðN%
1 ; . . . ;N%
k�1Þ is equivalent to the solution of the
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15 20
25 30
35 40
40 45 50 55 60 65 70
70
75
80
85
90
N3
P* surfaceThe optimal solution
N1
N2
Fig. C1. P% surface of the example.
C. Shi, S.B. Gershwin / Int. J. Production Economics 122 (2009) 725–740 739
unconstrained problem, since it is just constrained bythe buffer size constraint. This is a conflict. Therefore, itis not possible for all N%
i ; ieB to satisfy @JðN%Þ=@Ni ¼ 0.
So, 9i for which N%
i4Nmin and @JðN%
Þ=@Nia0. Since
N%
i4Nmin, by condition (B.3), we know m%
i¼ 0. Thus,
m%
0 a0 since otherwise condition (B.1) would be violated.Hence, by condition (B.2), we know that the optimalsolution N% satisfies PðN%
Þ ¼ P%. In this case, there aremore than one active inequality constraints: the pro-duction rate constraint g0ðNÞ and buffer size constraintsgiðNÞ; 8i 2 B. It is not hard to see that rg0ðN
%Þ and
rgiðN%Þ; 8i 2 B are linearly independent so the optimal
solution is regular.As before, replacing m%
0 by m040 and m%
i ;8i 2 B bymi40; 8i 2 B in constraint (B.1) gives
�rJðNÞ þ m0rðP%� PðNÞÞ þ
Xi2B
mirðNmin � NiÞ ¼ 0, (B.4)
where N is the unique solution. Note that N is exactly theoptimal solution of the following optimization problem:
min � JðNÞ ¼ �JðNÞ þ m0ðP%� PðNÞÞ þ
Xi2B
miðNmin � NiÞ
s.t. Nmin � Ni � 0; 8i ¼ 1; . . . ; k� 1,
Ni ¼ Nmin; 8i 2 B (B.5)
which is equivalent to
max JðNÞ ¼ JðNÞ � m0ðP%� PðNÞÞ
s.t. Nmin � Ni � 0; 8i ¼ 1; . . . ; k� 1,
Ni ¼ Nmin; 8i 2 B (B.6)
or
max JðNÞ ¼ ðAþ m0ÞPðNÞ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini
s.t. Ni � Nmin; 8i ¼ 1; . . . ; k� 1,
Ni ¼ Nmin; 8i 2 B (B.7)
or, finally,
max JðNÞ ¼ A0PðNÞ �Xk�1
i¼1
biNi �Xk�1
i¼1
cini
s.t. Ni � Nmin; 8i ¼ 1; . . . ; k� 1,
Ni ¼ Nmin; 8i 2 B, (B.8)
where A0 ¼ Aþ m0. Again, this is exactly our unconstrainedproblem in which A is replaced by A0, and N is its optimalsolution, except that in the optimal solution, Ni ¼ Nmin;
8i 2 B. A similar argument as in Section 3.1 could proveour assertion for this case.
Appendix C
We provide a brief discussion about the P% surfacesearch here. All buffer size allocations, N, such that PðNÞ ¼P% compose the P% surface. For instance, consider a four-machine line with parameters r1 ¼ 0:11, p1 ¼ 0:008,r2 ¼ 0:12, p2 ¼ 0:01, r3 ¼ 0:1, p3 ¼ 0:01, r4 ¼ 0:09, andp4 ¼ 0:01. P%
¼ 0:88. A portion of the P% surface of thisexample is shown in Fig. C1. It means that all points onthis surface satisfy PðNÞ ¼ P%. We further let the profit isJðNÞ ¼ 2000PðNÞ �
P3i¼1Ni �
P3i¼1ni. Then, the optimal
solution achieved by our algorithm is also shown in thisfigure. To verify the optimal solution, we conduct a searchon this P% surface. It is obvious that the P% surface isboundless and it is impractical to search the wholesurface, even for the simplest three-machine line. There-fore, in our implementation, we only search around theoptimal solution achieved by our algorithm. Since there isonly one maximum, searching around the optimal solu-tion is accurate enough.
In the P% surface search, we first need to find allfeasible points on the surface around the optimal solutiongained by our algorithm. To do this, we set ranges for allbuffers and then at each time we vary one buffer size by asmall step size (0.01 or 0.1) while keeping other buffersunchanged. Therefore, we can calculate the productionrates and the profits for all combinations of thosebuffers. All those points, whose production rate P satisfyjP � P%
j � d where d is a very small non-zero positivevalue, compose the P% surface. Then we compare theprofits of those points. The point having the maximalprofit in the P% search becomes the optimal solution bythis method. Then, we compare our optimal solution withthis one and calculate the error.
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