A DYNAMIC PROGRAMMING ALGORITHM FOR THE BUFFER ALLOCATION PROBLEM IN HOMOGENEOUS ASYMPTOTICALLY RELIABLE SERIAL PRODUCTION LINES

In this study, the buffer allocation problem (BAP) in homogeneous, asymptotically reliable serial production lines is considered. A known aggregation method, given by Lim, Meerkov, and Top (1990), for the performance evaluation (i.e., estimation of throughput) of this type of production lines when the buffer allocation is known, is used as an evaluative method in conjunction with a newly developed dynamic programming (DP) algorithm for the BAP. The proposed algorithm is applied to production lines where the number of machines is varying from four up to a hundred machines. The proposed algorithm is fast because it reduces the volume of computations by rejecting allocations that do not lead to maximization of the line’s throughput. Numerical results are also given for large production lines.


Introduction and literature review
The analysis of production or flow lines and, more generally, of manufacturing systems has been the object of numerous studies.Usually, production lines are modeled as serial queuing networks and analyzed either analytically as Markovian models or via approximate decomposition methods and simulation.
The literature on the modeling of production lines is vast and a large amount of research over the years has been devoted to this area.One of the first most complete analyses of such systems was published in 1962 by Sevastyanov [20].The large amount of research allows us to review only the most directly relevant studies here.
For a systematic classification of the relevant works on the stochastic modeling of production lines, the interested reader is referred to the books by Buzacott and Shanthikumar [3], Papadopoulos et al. [17], Gershwin [7], Altiok [2], and Helber [10], and the review papers by Dallery and Gershwin [5] and Papadopoulos and Heavey [16], among others.
Hillier and Boling [11] and Heavey et al. [9], analyzed serial production lines using Markovian models, whereas Gershwin [6] and Dallery et al. [4] utilized decomposition approaches to evaluate the performance of the same type of production lines.The former method is practically applicable only to short lines due to the curse of the dimensionality problem arising in the Markovian models.The latter method can be applied to large production lines at the expense of the accuracy of the numerical results.Altiok [2], among others, studied the phase-type distributions and their use to approximate any general distribution of service or interarrival times in the stochastic modeling of production lines.
Meerkov and his colleagues have developed performability, a quite interesting theory and method of analysis of production lines.Lim et al. [14] presented an asymptotic analysis technique for a model of a serial production line, gave estimates of its accuracy, and formulated a convergence theorem for their solution algorithm.
One of the most interesting questions that designers face in a serial production line is the buffer allocation problem (BAP), that is, how much buffer storage to allow and where to place it within the line.This is an important question because buffers can have a great impact on the efficiency of the production line.Buffers compensate for the blocking and the starving of the line's stations.However, buffer storage is expensive both due to its direct cost and due to the increase of the work-in-process (WIP) inventories.In [17], both evaluative and generative (optimization) models are given for modeling the various types of manufacturing systems.
One of the optimization methods for solving the BAP is the dynamic programming (DP) method.Several authors have employed the DP method for solving the BAP (see Jafari and Shanthikumar [12], Kubat and Sumita [13], and Yamashita and Altiok [23], among others).However, this method was employed in the case of a production line with synchronous transfer as defined in [17], among others, where the steady-state throughput can be approximated in a closed recursive form.Another classification of the research work relevant to the BAP is based on whether the lines under study are balanced or unbalanced.A line is called balanced (or unbalanced) if the mean processing times at each station are equal (or unequal).Powell [19] provided a literature review according to this scheme.
In this paper, a DP algorithm for the BAP is developed.This algorithm makes use of an aggregation procedure to approximate various performance measures of the production line developed by Lim et al. [14].
The remainder of this paper is structured as follows.Section 2 presents the problem and the model.Section 3 presents the proposed DP algorithm with an application example.Section 4 gives numerical results for several production lines, both short and large, and Section 5 concludes the paper and proposes some areas for further research.Finally, the appendix gives the aggregation method for the approximation of throughput for a given buffer allocation, as taken by Lim et al. [14].

The model and the problem definition
The main assumptions of our model are the same as those made by Lim et al. [14], as follows.
(1) A serial production line (see Figure 2.1) consists of M machines M i , i = 1,...,M, and M − 1 buffers B i of finite capacity at least one (this assumption is dictated by (A.1), as otherwise function Q(a,N) is not defined).Assuming that each buffer stores at least 1 unit and N is the total buffer capacity that can be allocated, it is obvious that for a production line consisting of M machines and M − 1 buffers, the maximum buffer capacity that each buffer can store is up to N − M + 2 units.(2) An inexhaustive supply of workpieces is available upstream machine M 1 , and an unlimited storage area is present downstream machine M M .Thus the first machine is never starved and the last machine is never blocked.(3) All machines have equal and constant service times.Time is scaled so that this machine cycle takes one time unit.Thus processing times are assumed to be deterministic and identical for all machines and are taken as the time unit.(4) Machine M i , being not blocked and not starved, takes part in production during any time slot with probability δ i and fails to do so with probability 1 − δ i (production lines in which machines have this property are called homogeneous by Lim et al. [14], that is, a homogeneous line is characterized by machines with one parameter only, δ i , instead of the two parameters p i and r i usually used in the literature, where p i and r i denote, respectively, the failure rate and the repair rate of machine M i ).It is assumed that blocked and/or starved machines do not fail.(5) It is assumed that the production lines under consideration are homogeneous, asymptotically reliable, that is, δ i = 1 − εΛ i , where 0 < ε 1 and Λ i , i = 1,...,M, is independent of ε.The Λ i 's are known as the loss parameters, as defined by Lim et al. [14].
Following the lines of Lim et al. [14], denote by L i the cumulative losses in the ith operation (jobs/h) and by AX i the actual throughput, then δ i = 1 − L i /AX i , and since δ i = 1 − εΛ i , where 0 < ε 1, it is obvious that L i AX i .Thus the loss parameter Λ i refers to the fraction of the cumulative losses in the ith operation divided by the actual throughput of machine M i .
Major decisions in designing production lines involve the workload allocation and the BAPs with respect to an objective function such as profit maximization or throughput maximization for a given total buffer capacity.In this study, the latter has been chosen to deal with; namely, our objective is to find the optimal buffer allocation for a given total buffer capacity in order to maximize the average production rate of the production line.The above problem may be expressed mathematically as follows.
Find the optimal vector B = (N 1 ,...,N M−1 ) that maximizes {X M } given that M−1 j=1 N j = N, where X M is the mean production rate (throughput) of a production line consisting of M machines and M − 1 buffers.N is the total buffer capacity and N i is the capacity of buffer i, i = 1,...,M − 1.

The dynamic programming algorithm
Before expressing the DP algorithm mathematically, we introduce the following symbols.
Z i is the buffer capacity that the DP algorithm allocates to buffer i and to all buffers upstream buffer i.
is the value of the aggregated loss parameter Λ f j , defined in the appendix, when buffer space N j−1 is allocated to buffer j − 1, where From the various values of parameter Λ f j−1 , that one with the maximum value is selected.
The DP algorithm consists of four steps which are summarized below.
Step 1. Calculate the forward pass loss parameters by using expression (A.9) of Lim et al. [14].
Step 2. The fundamental recursion equation.Execute the following recursive equations: (3.1) Step 3. Termination of the algorithm.The algorithm terminates when the value of Step 4. Determination of the optimal buffer allocation.The optimal buffer allocation is given by vector (T 1 ,T 2 ,...,T M−1 ), with its elements obtained as follows.
(1) Allocate T M−1 units of buffer space to buffer B M−1 , where is obtained, and so forth.(4) Allocate T 2 and T 1 units of buffer capacity to buffers B 2 and B 1 , respectively, where T 2 and T 1 are the values for which 3.1.Application of the algorithm: an example.In this section, an application of the algorithm is given for a production line with 4 machines and loss parameters Λ 1 = 3.4,Λ 2 = 2.1, Λ 3 = 4.3, and Λ 4 = 1.1, respectively.The total buffer capacity that is to be allocated is 10 units.Step 1.In this step, we calculate the forward pass loss parameters 8.These parameters are presented in Table 3.1 and will be used throughout the algorithm.
Step 2. In DP, computations are carried out in stages by breaking down the problem into subproblems.Each subproblem is then considered separately with the objective of reducing the volume of computations.However, since the subproblems are interdependent, a procedure must be devised to link the computations in a manner that guarantees that a feasible solution for each stage is also feasible for the entire problem.
For 5) are used and  6) are used and  7) are used and   .Therefore Step 3. The algorithm terminates as f 3 (Z 3 ) has been calculated.

Numerical results
In this section, numerical results are presented showing buffer allocations obtained using the proposed DP algorithm for four, five, six stations, and large production lines with up to a hundred stations (the latter are given in Section 4.1).For the short lines, by the enumeration method, all possible buffer allocations of a given total buffer capacity were tested and the optimal buffer allocation, namely, that one giving the maximum throughput, was obtained.The DP algorithm was implemented in PASCAL and in a very slow old PC486 system.Tables 4.1, 4.2, and 4.3 present the buffer allocations, for given total buffer capacities, obtained by the DP algorithm for short lines with four, five, and six stations, respectively.
Comment.We have applied enumeration for all total buffer capacities given in Table 4.1, that is, for 4 to 10 units of total buffer capacities.Comparing the results from the enumeration method with those obtained by the DP algorithm, we have found that in all cases the results were identical.Also notice that the run time is very small and lies between 0.04 and 0.19 seconds even in a slow old PC486 system.In Tables 4.2 and 4.3, for the cases where the results from the enumeration method differ from those obtained by the proposed algorithm, a percentage error has been introduced.The error has been calculated using the following formula: where X M,enum and X M,DP denote the throughput of the buffer configuration obtained by enumeration and the proposed DP algorithm, respectively.

4.1.
Numerical results for large production lines.In this section, numerical results are presented, showing buffer allocations obtained using the proposed DP algorithm in production lines with many stations M, 10 ≤ M ≤ 100.Tables 4.4, 4.5, 4.6, and 4.7 present the buffer allocations obtained by the DP algorithm for given total buffer capacities, for large production lines with ten, fifty, eighty, and one hundred stations, respectively.
Unfortunately, we cannot compare these results with those obtained from enumeration because it is impossible to use enumeration in large production lines (because of the huge number of states that should be examined).

Conclusions and further research
In this study, we present a dynamic programming algorithm that solves the buffer allocation problem (BAP) of N units of total buffer capacity in a homogeneous asymptotically reliable serial production line consisting of M machines and M − 1 buffers.The main conclusions are as follows.
(1) The proposed dynamic programming algorithm for short (with M < 10 stations) production lines found, in almost all cases, the optimal solution for the BAP.In the cases where the algorithm did not give the optimal solution, it gave a nearoptimal solution.(2) The algorithm is quite fast and in all cases where we applied it, we did not encounter any bugs and the algorithm always converged to a solution.The run time in all cases was quite small.(3) The DP algorithm can be applied in large production lines to effectively (rapidly and accurately) find a near-optimal solution to the BAP.Even in large systems, the proposed algorithm worked quite effectively.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Table 4 .
4. Application of the DP algorithm in a ten-station production line with M = 10, N = 20.The production rate for this specific buffer allocation is 0.9382.

Table 4 .
5. Application of the DP algorithm in a fifty-station production line with M = 50, N = 90.The production rate for this specific buffer allocation is 0.8651.

Table 4 .
6. Application of the algorithm in an eighty-station production line with M = 80, N = 200.The run time is 6 minutes and 0.44 second in a PC486.The production rate for this specific buffer allocation is 0.8810.

Table 4 .
7. Application of the DP algorithm in a hundred-station production line with M = 100, N = 400.The run time is 24 minutes and 6.81 seconds in a PC486.The production rate for this specific buffer allocation is 0.9112.