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Buffer allocation in unreliable production lines using a knowledge based system
PII:
Computers Ops Res. Vol. 25, No. 12, pp. 1055±1067, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0305-0548(98)00034-3 0305-0548/98 $19.00 + 0.00
BUFFER ALLOCATION IN UNRELIABLE PRODUCTION LINES USING A KNOWLEDGE BASED SYSTEM G. A. Vouros1 and H. T. Papadopoulos2{ Department of Mathematics, University of the Aegean, 832 00 Karlovassi, Samos, Greece and 2 Department of Business Administration, University of the Aegean, 821 00 Chios, Greece
1
(Received April 1997; accepted January 1998) Scope and PurposeÐOne of the most important but dicult optimization problems concerns the optimal allocation of buers in a production system (line) with stochastic inputs and outputs. The buer allocation problem is a non-linear problem with integer variables and there exists no closed-form solution for the objective function. AbstractÐThe optimization of production lines performance is a problem of great complexity and, therefore, of signi®cant research interest. The problem may involve the optimization of many con¯icting objectives, such as increasing throughput and reducing work-in-process time. The majority of existing studies have used various heuristics and search methods based on operations research. These methods have been proved to be computationally inecient, especially for large production lines. This paper presents ASBA2, a knowledge based system that determines near optimal buer allocation plans, with the objective of maximising production lines throughput. The allocation plan is calculated subject to a given amount of total buer slots, in a computationally ecient way. ASBA2 operates in close cooperation with a simulation method, which provides ASBA2 with performance measures concerning production line behaviour. Moreover, to evaluate results provided by ASBA2, we have utilized an exact numerical algorithm for calculating the throughput of unreliable production lines. # 1998 Elsevier Science Ltd. All rights reserved Key words: Buer allocation, optimization, production lines, knowledge-based system
1. INTRODUCTION AND LITERATURE REVIEW
Over the years a large amount of research has been devoted to the analysis and modelling of production lines. For a systematic classi®cation of the relevant works the interested reader is addressed to a review paper by Papadopoulos and Heavey [2]. The allocation of buer units in production lines is a major optimization problem faced by manufacturing systems designers as well as by researchers. It has to do with devising an allocation plan for distributing a certain amount of buer space among the intermediate buers of a production line. The aim is to achieve performance objectives such as maximum throughput and/or minimum work-in-process (WIP) time. The problem is complicated and very critical since it introduces computational complexity and involves trade-os between the constraints and the objectives posed by the problem itself. Computational complexity is due to the increasingly large number of feasible allocations with respect to the total number of buer slots and the number of stations in a production line. Although several researchers have studied the problem of optimizing buer allocation to maximize the eciency of an unreliable production line, there is no method that can handle this problem for large production lines in a computationally ecient way. Hillier and So [4] and Hillier et al. [5] studied the buer allocation problem for short production lines. This method cannot solve the problem for large production lines. It is based mainly on comprehensive studies for characterizing the optimal buer allocation pattern. Authors have provided extensive numerical results for balanced lines with up to 6 stations and limited results for lines with up to 9 stations. Other relevant studies are those by Conway et al. [3] who conducted simulation experiments to study the buer allocation problem, Powell [6] who studied the buer allocation problem for {To whom all correspondence should be addressed. E-mail: [email protected] 1055
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unbalanced production lines and So [7] who dealt with the buer allocation problem in a production line, having as an objective minimization of the average work-in-process, provided a minimum required throughput is attained. Trade-os between the objectives of devising a buer allocation plan make the buer allocation problem more complicated. For instance, let us consider the trade-o between throughput and average WIP. WIP plays an important role carrying a signi®cant cost for the whole production line. Usually, the objective is to minimize WIP subject to a minimum required throughput. A speci®c buer allocation plan may help in achieving a required minimum throughput, but it may have the undesirable eect of increasing the average WIP, or vice-versa. In terms of the constraints posed on the total buer space, increase of the available total buer space results in increase of WIP, while reduction of it may result in reduction of throughput. On the other hand, increasing the amount of total buer slots may result in increasing the maximum throughput of the system, but it may be against economy. In other words, there is a trade-o between the cost of maintaining buer slots and the attainable maximum throughput of a manufacturing system. Of course, the problem becomes more complicated when more realistic constraints are posed. For instance, the additional requirement of keeping the number of buer slots in a station lower than (or greater than) a given number may result in decreasing (increasing) the maximum throughput attainable by the manufacturing system. This paper presents ASBA2, a knowledge based system for solving the buer allocation problem in unreliable, balanced and unbalanced, production lines. The objective of ASBA2 is to increase the throughput of the production line given a total amount of buer slots. ASBA2 is an extension of a previous system, called the advisor system for buer allocation (ASBA) which was developed for solving this problem for reliable production lines, i.e. lines with machines that were not allowed to fail ( [1]). ASBA2 is a generative model that works in close cooperation with an evaluative-simulation model. This paper is organized as follows. Section 2 states the problem and assumptions of the model, whereas, Section 3 presents major aspects of the knowledge based system. Detailed description of the types of knowledge exploited by the knowledge based system is given in Appendix A. In Section 4, we provide numerical results obtained from the system. Finally, Section 5 concludes the paper and some future research directions are suggested.
2. THE MODEL AND THE BUFFER ALLOCATION PROBLEM
In asynchronous production lines, when a workstation has completed its processing and the next buer has available space, then the processed part is passed on. Then, the workstation starts processing a new part that is taken from its buer. In case the buer has no parts, the workstation remains idle until a new part is placed in the buer. This type of line is subject to manufacturing blocking (or blocking after service, as it is known in the literature) and starving. 2.1. Assumptions of the model It is assumed that the ®rst station is never starved and the last station is never blocked. The processing (service) times at each workstation are assumed to be independent random variables following either the exponential or Erlang-k (k r2) distribution, with mean service rates, mi, i = 1, 2, . . ., K. The workstations of the line are subject to random failures according to a Poisson distribution. That is, the times to failures are assumed to follow the exponential distribution with mean failure rates bi, i = 1, 2, . . ., K. The repair times are also assumed to follow the exponential or Erlang-m distribution (m r1), with mean repair rates ri, i = 1, 2, . . ., K. Although the assumption concerned with the distribution of processing times is not so realistic, it has been used by many researchers as it facilitates modelling a production line by the well-known Markovian state method. According to this method, the line is modelled as a series or tandem queueing network with K single-server stations that correspond to the K workstations of the production line. This allows the manufacturing system designer to get insight into performance measures of the production line, such as the mean throughput, the average sojourn time and the mean queue lengths. These measures, at the preliminary design stage, help the manufacturing designer to evaluate and ®ne-tune the performance of a production line. At the
Buer allocation in unreliable production lines
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Fig. 1. A K-station production line with K ÿ 1 intermediate buers.
operational level, simulation and other techniques have to be utilized to cope with the on-line decisions that must be taken. The K-station line has K ÿ 1 intermediate locations for buers, labelled B2, B3, . . ., BK, in Fig. 1. The basic performance measures in the analysis of production lines are the throughput (or mean production rate) and the average work-in-process (WIP) or equivalently the average production (sojourn) time. So Ref. [7] dealt with the optimal buer allocation problem of production lines, having as target the minimization of the average WIP. The object of the current paper is the buering of asynchronous, unreliable production lines with the assumptions given above. The objective is the maximization of line throughput, subject to a given total buer space. 2.1.1. The optimal buer allocation (OBA) problem. In mathematical terms, our problem (P) could be stated as follows: Find B = (B2, B3, . . ., BK) so as to max XK(B) subject to: aK i = 2Bi=N, Bir0 Bi integer (i = 2, 3, . . . , K) where N is a ®xed nonnegative integer, denoting the total buer space available in the production line. B = (B2, B3, . . . , BK) is the ``buer vector'', i.e. a vector with elements the buer capacities of the K ÿ 1 buers. XK denotes the throughput of the K-station line. This is a function of the mean service rates of the K stations, mi (i = 1, 2, . . ., K), of the coecients of variation, CVi, of the service times and the buer capacities, Bi. It is also a function of higher moments of the service time distribution, but these are usually ignored, for simplicity. Besides, it has been experimentally, by simulation, proved that these higher moments have a minor eect on the throughput of the production lines [10]. 2.2. Methodology of investigation To solve the optimal buer allocation problem (P), we have performed the following steps: . (S1) We modeled the queueing process of the production line as a ®nite state, continuous time Markov chain, due to the assumption of the Erlang-k (k r1) distribution for the processing times. . (S2) To calculate the exact throughput of a production line we have utilized an algorithm developed by Heavey et al. [8]. The algorithm solves the sparse system of the steady-state probabilities of the resulting Markov chain. It gives the exact throughput for any K-station line with ®nite intermediate buers and phase-type service times. The number of states and the number of feasible allocations of N buer slots among the K ÿ 1 intermediate buer locations increase greatly with N and K. The latter is given by the formula: N 1 N 2 . . . N K ÿ 2 K ÿ 2!
1
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. (S2') As an alternative method to the above mentioned exact numerical methods, we developed a simulation algorithm that calculates the throughput of the production line as well as the idle, blocking and the mean waiting time in each workstation. . (S3) To ®nd a near optimal buer allocation (OBA) for maximizing the throughput of the line, we utilized the knowledge based system ASBA2, that has been developed speci®cally for solving this problem. ASBA2 is a generative system that works in close cooperation with the simulation algorithm. Speci®cally, each time ASBA2 submits a request to the simulation algorithm it provides a buer allocation plan. Simulation, on the other hand, provides ASBA2 with the throughput of the production line as well as with the idle, blocking and average waiting time in each station. These performance measures help ASBA2 to evaluate the proposed plan and adjust it accordingly. Figure 2 depicts in detail the communication between ASBA2 and the simulation algorithm. Given that exact methods fail to provide the optimal buer allocation plans for large production lines and for a large number of slots in intermediate buers, we measured the suboptimality of ASBA2 only for these cases where the exact solution is known. Suboptimality is measured by the absolute percentage dierence between the throughput achieved by ASBA2 and the maximum throughput. Obtained numerical results and remarks are given in Section 5.
3. THE KNOWLEDGE BASED SYSTEM
Complexity, criticality and the experience intensity of allocating buers in a production line justify the need for an advisory system. In Ref. [1] we have investigated the use of a speci®c framework for representation and reasoning, according to which we have structured and encoded knowledge for solving the following problem: given a total buer size and a minimum required throughput, ®nd a buer allocation con®guration that minimizes the average work-inprocess subject to a minimum required throughput for a production line with reliable machines. The knowledge based system proposed, called ASBA, had been designed so as to enable the incorporation of realistic constraints concerning the allocation of buer slots. Such a constraint may be the lower or greater number of buer slots that must be assigned in a buer. The system proved to be ecient for achieving a near optimum solution to the stated problem. However, it deals only with reliable machines. This is considered to be a critical relaxation. The objective is to extend ASBA to deal eectively with a more general case, i.e. with unreliable unbalanced and balanced, production lines. As it has been stated previously, no computationally ecient technique exists for the solution of this problem, especially for large lines. This section, in conjunction with Appendix A, describes our investigation of extending ASBA to solve the problem of devising buer allocation plans for production lines with unreliable machines. ASBA2 solves the buer allocation problem stated in Section 2: it calculates a near
Fig. 2. Communication between ASBA2 and the simulation algorithm.
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Table 1. ASBA2's results in buer allocation of 3-station exponential lines with m1=1, m2=1.2, m3=1.4, r1=r2=r3=0.5, b1=0.05, b2=0.05, b3=0.05 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0) (1±1) (2±1) (3±1) (3±2) (3±3) (4±3) (5±3) (6±3) (6±4)
0.6341 0.6744 0.7113 0.7361 0.7587 0.7716 0.7911 0.8060 0.8178 0.8266
(1±0) (1±1) (2±1) (3±1) (3±2) (4±2) (5±2) (5±3) (6±3) (7±3)
0.6341 0.6744 0.7113 0.7361 0.7587 0.7777 0.7922 0.8060 0.8178 0.8274
0.00 0.00 0.00 0.00 0.00 0.78 0.13 0.00 0.00 0.09
Table 2. ASBA2's results in buer allocation of 3-station exponential lines with m1=1, m2=1.2, m3=1.4, r1=r2=r3=0.5, b1=0.05, b2=0.01, b3=0.25 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0) (1±1) (1±2) (2±2) (3±2) (2±4) (4±3) (3±5) (4±5) (5±5)
0.5748 0.6204 0.6486 0.6783 0.6982 0.7141 0.7334 0.7465 0.7610 0.7720
(1±0) (1±1) (2±1) (2±2) (2±3) (3±3) (3±4) (4±4) (4±5) (5±5)
0.5748 0.6204 0.6493 0.6783 0.6987 0.7188 0.7343 0.7489 0.7610 0.7720
0.00 0.00 1.10 0.00 0.07 0.65 0.12 0.32 0.00 0.00
optimal buer allocation that maximizes throughput subject to the constraint on the total buer space. To solve this problem, ASBA2 performs the following actions: (1) Allocates buer slots to each station. (2) Forms strategic decisions concerning goals to pursue and constraints to be relaxed. With respect to the goals and constraints, the system decides where to allocate buer slots (target stations), from where these slots must be taken (source station) and the number of buer slots to be transferred. (3) Transfers buer slots from source to target buer and frees buer slots when these are not in use. (4) Submits requests to a simulation model, to calculate the throughput of the production line, as well as the idle, blocking and average waiting time in each station, for a given buer allocation plan. The types of knowledge utilized by ASBA2 are described in detail in Appendix A. In particular, Appendix A describes strategic, operational, control and heuristic types of knowledge and provides examples for each type. 4. NUMERICAL RESULTS
In this section, we present numerical results that have been obtained by running ASBA2. Results are split into two classes: (a) for exponential lines and (b) for Erlangian lines. Table 3. ASBA2's results in buer allocation of 3-station exponential lines with m1=m2=m3=1, r1=r2=r3=0.5, b1=0.05, b2=0.01, b3=0.25 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0) (1±1) (2±1) (2±2) (3±2) (3±3) (5±2) (2±6) (6±3) (5±5)
0.4700 0.5137 0.5306 0.5583 0.5687 0.5878 0.5801 0.6109 0.6028 0.6232
(0±1) (1±1) (1±2) (1±3) (2±3) (2±4) (2±5) (3±5) (3±6) (3±7)
0.4814 0.5137 0.5409 0.5600 0.5773 0.5914 0.6022 0.6122 0.6204 0.6271
2.36 0.00 1.90 0.30 1.48 0.60 3.66 0.21 2.83 0.62
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Table 4. ASBA2's results in buer allocation of 4-station exponential lines with m1=m2=m3=m4=1, r1=r2=r3=r4=0.5, b1=0.05, b2=0.02, b3=0.01, b4=0.001 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(0±1±0) (1±1±0) (1±1±1) (2±1±1) (2±1±2) (2±2±2) (3±2±2) (3±3±2) (5±2±2) (5±3±2)
0.5250 0.5577 0.5916 0.6128 0.6307 0.6563 0.6721 0.6890 0.6899 0.7076
(0±1±0) (1±1±0) (1±1±1) (1±2±1) (2±2±1) (2±2±2) (2±3±2) (3±3±2) (3±4±2) (4±3±3)
0.5250 0.5577 0.5916 0.6155 0.6378 0.6563 0.6725 0.6890 0.7007 0.7129
0.00 0.00 0.00 0.43 1.11 0.00 0.05 0.00 1.54 0.74
Table 5. ASBA2's results in buer allocation of 4-station exponential lines with m1=1.6, m2=1.4, m3=1.2, m4=1, r1=r2=r3=r4=0.5, b1=0.05, b2=0.02, b3=0.01, b4=0.005 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(0±1±0) (1±1±0) (1±1±1) (2±1±1) (1±3±1) (1±1±4) (2±3±2) (3±2±3) (4±2±3) (3±2±5)
0.6476 0.6652 0.7289 0.7394 0.7670 0.8013 0.8166 0.8348 0.8390 0.8672
(0±0±1) (0±1±1) (0±1±2) (1±1±2) (1±2±2) (1±2±3) (1±3±3) (1±3±4) (2±3±4) (2±3±5)
0.6572 0.7064 0.7361 0.7673 0.7938 0.8173 0.8346 0.8514 0.8660 0.8796
1.46 5.83 0.97 3.63 3.37 1.95 2.15 1.95 3.11 1.41
Speci®cally, Tables 1±7, provide numerical results concerning the buer allocation in production lines with exponentially distributed service and repairing times, whereas, Tables 8±14, provide analogous numerical results for production lines with service and repairing times following the Erlangian distribution. Recall that the target goal of ASBA2 is to maximize the throughput of a production line. The achieved throughput is denoted by XK in Tables 1±14 and is given by column 3. Respectively, column 5 gives the maximum throughput denoted by XK*. This is the throughput achieved by the optimal buer allocation (OBA) given in column 4. Columns 1 and 2 give, respectively, the total buer size, denoted by N, and the near optimum buer allocation obtained by ASBA2, Table 6. ASBA2's results in buer allocation of 5-station exponential lines with m1=m2=m3=m4=m5=1, r1=r2=r3=r4=r5=0.5, b1=0.1, b2=0.2, b3=0.25, b4=0.3, b5=0.35 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0±0±0) (2±0±0±0) (2±1±0±0) (1±1±1±1) (1±2±1±1) (2±2±1±1) (3±1±1±2) (2±2±2±2) (3±2±2±2) (3±1±3±3)
0.2848 0.2870 0.3005 0.3420 0.3512 0.3538 0.3603 0.3833 0.3852 0.3956
(0±0±1±0) (0±0±1±1) (0±1±1±1) (0±1±2±1) (0±1±2±2) (0±2±2±2) (1±2±2±2) (1±2±3±2) (1±2±3±3) (1±3±3±3)
0.2997 0.3158 0.3340 0.3471 0.3579 0.3699 0.3794 0.3894 0.3990 0.4066
4.97 9.12 10.00 1.47 1.87 4.35 5.03 1.56 3.45 2.70
Table 7. ASBA2's results in buer allocation of 5-station exponential lines with m1=1, m2=1.1, m3=1.2, m4=1.3, m5=1.4, r1=r2=r3=r4=r5=0.5, b1=b2=b3=b4=b5=0.05 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(0±1±0±0) (1±1±0±0) (1±1±1±0) (1±1±1±1) (2±1±1±1) (2±2±1±1) (1±2±2±2) (1±3±2±2) (3±2±2±2) (2±4±2±2)
0.5213 0.5514 0.5824 0.5976 0.6196 0.6422 0.6380 0.6519 0.6829 0.6914
(0±1±0±0) (1±1±0±0) (1±1±1±0) (1±2±1±0) (2±2±1±0) (2±2±1±1) (2±2±2±1) (3±2±2±1) (3±3±2±1) (3±3±3±1)
0.5213 0.5514 0.5824 0.6027 0.6213 0.6422 0.6585 0.6744 0.6894 0.7005
0.00 0.00 0.00 0.84 0.27 0.00 3.11 3.33 0.94 1.30
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Table 8. ASBA2's results in buer allocation of 3-station Erlangian lines with m1=1, m2=1.2, m3=1.4, r1=r2=r3=0.5, b1=0.05, b2=0.05, b3=0.05 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0) (1±1) (2±1) (2±2) (2±3) (3±3) (4±3) (4±4) (5±4) (7±3)
0.7083 0.7489 0.7857 0.8026 0.8109 0.8347 0.8507 0.8555 0.8669 0.8776
(1±0) (1±1) (2±1) (3±1) (3±2) (4±2) (5±2) (6±2) (6±3) (7±3)
0.7083 0.7489 0.7857 0.8086 0.8261 0.8421 0.8538 0.8627 0.8709 0.8776
0.00 0.00 0.00 0.74 1.84 0.87 0.36 0.83 0.46 0.00
Table 9. ASBA2's results in buer allocation of 3-station Erlangian lines with m1=1, m2=1.2, m3=1.4, r1=r2=r3=0.5, b1=0.05, b2=0.01, b3=0.25 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0) (1±1) (1±2) (3±1) (3±2) (3±3) (4±3) (4±4) (6±3) (5±5)
0.6377 0.6896 0.7192 0.7354 0.7650 0.7841 0.7961 0.8092 0.8124 0.8273
(1±0) (1±1) (1±2) (2±2) (2±3) (3±3) (3±4) (4±4) (4±5) (5±5)
0.6377 0.6896 0.7192 0.7473 0.7670 0.7841 0.7978 0.8092 0.8191 0.8273
0.00 0.00 0.00 1.60 0.26 0.00 0.21 0.00 0.81 0.00
denoted by BA. Lastly, column 6 of the tables gives D, the absolute percentage dierence between the throughput achieved from ASBA2 and the maximum throughput, i.e. D = (vXK* ÿ XKv)/XK* 100. This expresses the failure of our system to achieve the required throughput. Although ASBA2 uses a simulation method to calculate the throughput of the production line, the throughput XK of the near optimal buer allocation (given in column 3) is calculated using the exact method developed by Heavey et al. [8]. This allows having a standard method with which to measure the failure of our system to achieve the maximum throughput. ASBA2, as well as its predecessor ASBA, is able to solve the problem for large production lines, allowing large buer capacities for each buer. However, to have a measure of ASBA2 Table 10. ASBA2's results in buer allocation of 3-station Erlangian lines with m1=m2=m3=1, r1=r2=r3=0.5, b1=0.05, b2=0.01, b3=0.25 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0) (1±1) (1±2) (3±1) (3±2) (4±2) (4±3) (4±4) (5±4) (5±5)
0.5177 0.5678 0.5947 0.5892 0.6159 0.6203 0.6356 0.6452 0.6475 0.6534
(0±1) (1±1) (1±2) (1±3) (2±3) (2±4) (2±5) (2±6) (3±6) (3±7)
0.5349 0.5678 0.5947 0.6119 0.6247 0.6354 0.6429 0.6483 0.6529 0.6563
3.22 0.00 0.00 3.71 1.41 2.37 1.13 0.47 0.83 0.44
Table 11. ASBA2's results in buer allocation of 4-station Erlangian lines with m1=m2=m3=m4=1, r1=r2=r3=r4=0.5, b1=0.05, b2=0.02, b3=0.01, b4=0.001 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0±0) (2±0±0) (1±1±1) (2±0±2) (3±0±2) (2±2±2) (3±2±2) (4±2±2) (4±2±3) (4±3±3)
0.5936 0.6074 0.6792 0.6608 0.6720 0.7422 0.7580 0.7680 0.7776 0.7926
(0±1±0) (1±1±0) (1±1±1) (1±2±1) (2±2±1) (2±2±2) (3±2±2) (3±3±2) (4±3±2) (4±3±3)
0.6024 0.6407 0.6792 0.7019 0.7258 0.7422 0.7580 0.7723 0.7828 0.7926
1.46 5.20 0.00 5.86 7.41 0.00 0.00 0.56 0.66 0.00
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Table 12. ASBA2's results in buer allocation of 4-station Erlangian lines with m1=1.6, m2=1.4, m3=1.2, m4=1, r1=r2=r3=r4=0.5, b1=0.05, b2=0.02, b3=0.01, b4=0.005 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(0±0±1) (0±0±2) (1±1±1) (3±0±1) (2±2±1) (2±2±2) (3±1±3) (2±4±2) (3±2±4) (4±2±4)
0.7512 0.7780 0.8263 0.7914 0.8512 0.8919 0.9002 0.9068 0.9342 0.9367
(0±0±1) (0±1±1) (0±1±2) (1±1±2) (1±1±3) (1±2±3) (1±2±4) (1±3±4) (1±3±5) (2±3±5)
0.7512 0.8029 0.8339 0.8627 0.8838 0.9056 0.9199 0.9315 0.9415 0.9497
0.00 3.10 0.91 8.26 3.69 1.51 2.14 2.65 0.78 1.37
Table 13. ASBA2's results in buer allocation of 5-station Erlangian lines with m1=m2=m3=m4=m5=1, r1=r2=r3=r4=r5=0.5, b1=0.1, b2=0.2, b3=0.25, b4=0.3, b5=0.35 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0±0±0) (1±1±0±0) (2±1±0±0) (1±1±1±1) (2±1±1±1) (3±1±1±1) (3±2±1±1) (3±2±1±2) (3±2±2±2) (4±2±3±3)
0.3192 0.3337 0.3348
(0±0±1±0) (0±0±1±1) (0±1±1±1)
0.3378 0.3578 0.3782
5.51 6.74 11.50
success, we have studied its behaviour for a number of cases whose optimal buer allocation can be calculated using exact methods. Therefore, we have studied ASBA2's performance mainly for production lines (balanced and unbalanced) with a small number of stations and small number of total buer units, i.e. for 3-station, 4-station and 5-station production lines. The ®rst buer in all cases is considered to be saturated (containing always more than 0 jobs at the ®rst station). 4.1. Remarks From the results given in Tables 1±14, we can make the following observations: (1) Results produced by ASBA2 are very close to the optimal solutions. However, we sacri®ce the optimal solution for computational eciency: ASBA systems do not need to enumerate all the possible solutions for producing the near-optimum solution. They rather exploit speci®c knowledge about allocating buer storage. Particularly, as it can be noticed, ASBA2's failure to achieve the maximum throughput in the cases where service and repairing times are considered to be exponential is on average close to 1.43% and only in 4 cases out of 70, in Tables 5 and 6, its measure of failure exceeds 5%. It must be pointed that the sub-optimality of ASBA2 increases as the number of stations increases. Speci®cally, for the exponential cases, for threestation production lines, the failure of ASBA2 is in average close to 0.5%, for four-station production lines it is close to 1% and ®nally, for ®ve-station production lines it is, on average, close to 2.5%. Table 14. ASBA2's results in buer allocation of 5-station Erlangian lines with m1=1, m2=1.1, m3=1.2, m4=1.3, m5=1.4, r1=r2=r3=r4=r5=0.5, b1=b2=b3=b4=b5=0.05 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(0±0±1±0) (1±0±1±0) (1±1±1±0) (1±1±1±1) (1±1±2±1) (2±1±1±2) (3±1±2±1) (1±3±2±2) (2±3±2±2) (3±2±3±2)
0.5825 0.6265 0.6673
(0±1±0±0) (1±1±0±0) (1±1±1±0)
0.5968 0.6338 0.6673
2.40 1.15 0.00
Buer allocation in unreliable production lines
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Moreover, ASBA2's failure to achieve the maximum throughput in cases where service and breakdown times are considered to be Erlang is on average close to 1.73% and in 7 cases out of 56, in Tables 11±13, its measure of failure exceeds 5%. We must notice that in Tables 13 and 14, ASBA2 has succeeded in calculating a buer allocation plan, in cases where exact techniques have failed to provide the optimal buer allocation. We should notice that for these cases, the exact throughput of the production line for the near optimal buer allocation cannot be calculated by the exact methods utilized. Again, the sub-optimality of ASBA2 increases as the number of stations increases. Speci®cally, for the Erlangian cases, for three-station production lines, the failure of ASBA2 is on average close to 0.7%, for four-station production lines it is close to 2% and ®nally, for ®ve-station production lines it is, on average, close to 4.5%. (2) Calculations provided by ASBA2 can be explained in terms of strategic, operational and heuristic knowledge used. This enables us to investigate the utilization of various heuristic rules which intuitively seem promising in calculating optimal buer allocations. However, since ASBA2 is combined with a simulation method, it must be stated that in speci®c cases, promising rules fail, due to simulation weakness to provide results that justify making progress towards maximizing throughput. This is a major drawback of ASBA2. (3) The total time needed by the overall system (ASBA2 and the simulation algorithm) to compute a solution is, on average, 1 min. The system runs on a Pentium in 200 MHz. The total time needed is the time that simulation needs to compute the performance measures plus the time that ASBA2 needs to take a decision, make the appropriate adjustments in the production line and monitor the progress achieved. It must be pointed out that approximately 90% of the time is consumed by the simulation module. 5. CONCLUDING REMARKS
The paper describes ASBA2, a knowledge based system that solves the well-known problem for buer allocation in production lines. The system is an extension of ASBA, which authors have described in an earlier paper. ASBA allocates buer space in reliable production lines, aiming at reducing WIP, subject to a given total buer space and a required throughput. ASBA computes near optimal buer allocations for reliable, balanced and unbalanced, production lines, whereas ASBA2 aims to extend the functionality of ASBA to unreliable, balanced and unbalanced, production lines. The paper describes how ASBA2 computes a near optimal buer allocation for unreliable lines, aiming at increasing throughput subject to given total buer space. The paper gives results for speci®c cases where the optimal buer allocation can be computed from exact methods and draws conclusions for ASBA2's behavior. Results, compared with exact methods, are encouraging. The system, in a computationally ecient way, using speci®c types of knowledge, computes results very close to the optimal ones. However, as the number of stations increase, the sub-optimality of ASBA2 increases as well. This result suggests that more strategies, operators and heuristics must be checked. ASBA2 provides the framework in which dierent strategies, operators and heuristics can be presented and evaluated. We plan to extend ASBA2 to deal with WIP as well, i.e. with the problem that ASBA dealt with, but for unreliable production lines. Our ®rst step in this direction will be to design a simulation algorithm that computes performance measures concerning the behavior of a production line that do not deviate much from computations that would be given from exact methods. This will also help us to provide results that are closer to the optimal ones. AcknowledgementsÐThis work has been supported by the Research Committee of the University of the Aegean, the members of which we sincerely thank. Without their support this research work would not have been completed. We highly appreciate the invaluable help of Mr M. Vidalis (Ph.D. student in the Department of Mathematics, University of the Aegean) for the development and running of the exact methods.
REFERENCES 1. Papadopoulos, H. T. and Vouros, G. A., A model management system (MMS) for the design and operation of production lines. International Journal of Production Research, 1997, 35(8), 2213±2236. 2. Papadopoulos, H. T. and Heavey, C., Queueing theory in manufacturing systems analysis and design: A classi®cation of models for production and transfer lines. European Journal of Operational Research, 1996, 92, 1±27. 3. Conway, R. W., Maxwell, W. L., McClain, J. O. and Thomas, L. J., The role of work-in-process inventories in series production lines. Operations Research, 1988, 36, 229±241.
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4. Hillier, F. S. and So, K. C., The eect of the coecient of variation of operation times on the allocation of storage space in production line system. IIE Transactions, 1991, 23, 198±206. 5. Hillier, F. S., So, K. C. and Boling, R. W., Notes: Toward characterizing the optimal allocation of storage space in production line systems with variable processing times. Management Science, 1993, 39(1), 126±133. 6. Powell, S. G., Buer allocation in unbalanced serial lines. Working Paper No. 289. The Amos Tuck School of Business Administration, Dartmouth College, 1992. 7. So, K. C., Optimal buer allocation strategy for minimizing work-in-process inventory in unpaced production lines. Working Paper presented at the ORSA/TIMS meeting in Detroit, U.S.A., Oct. 1994. 8. Heavey, C., Papadopoulos, H. T. and Browne, J., The throughput rate of multistation unreliable production lines. European Journal of Operational Research, 1993, 68, 69±89. 9. Sardo Pascual, L., A knowledge-based system for the optimal buer allocation in unreliable production lines. Diploma Thesis, Department of Mathematics, University of the Aegean, 1996. 10. Alka, A., The throughput rate of multistation production lines. Ph.D. Thesis, Department of Industrial and Systems Engineering, University of Florida, Gainesville, 1986.
APPENDIX A This section presents the types of knowledge encoded in ASBA2 according to the speci®c framework developed for ASBA. These are the strategic, the operational, the control and the heuristic knowledge. 7.1. Strategic Knowledge This is knowledge about generic procedures followed during problem solving and about forming speci®c strategic decisions in problematic situations that arise. As already described in Ref. [1], such knowledge is used to decide which goals to pursue, which constraints to relax, whether certain progress is made in achieving a speci®c goal and to resolve trade-os between objectives and constraints. Figure 3 depicts the most generic strategy followed by ASBA2. According to this strategy, the major steps that ASBA2 performs are as follows:
Fig. 3. The major strategy of ASBA2.
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Fig. 4. Decision frame format. * ASBA2 constructs an initial state. During that step it calculates an initial distribution of buer slots to the intermediate buers in the production line and it submits a request to the simulation model. * It frees buer slots that are not used. As depicted in Fig. 2, the number of buer slots per station that are not utilized are provided by the simulation algorithm. * ASBA2 forms a strategic decision concerning how to proceed. Such a decision comprises the current goal of the system, as well as what constraints and objectives must be relaxed. Relaxation of constraints and objectives help the system to avoid lines of reasoning that do not achieve any progress towards system's objectives. Goals that the system pursues during reasoning include reducing the blocking time of a machine, allocating the total buer size and increasing throughput. When the system monitors that it cannot achieve any progress towards maximizing throughput, then it halts and outputs the proposed buer allocation plan. Strategic decisions are recorded in decision frames and comprise the internal state of ASBA2. Figure 4 depicts the format of a decision frame. It must be noticed that a history of decision frames is kept with respect to the beginning of the ASBA2 operation. Decisions recorded in a decision frame guide subsequent strategic decisions, guide and justify the execution of particular operations. Therefore a decision frame is gradually instantiated. * The system proceeds to decide on the speci®c operations that must be performed in order to achieve the goal speci®ed in the decision frame. Therefore, it decides the station in which the number of buer slots must be increased (target station), the number of buer slots that must be added and where the new slots shall be taken from (source station). In case the source station is unspeci®ed, then ASBA2 allocates the non-allocated buer slots. As Fig. 4 depicts, these decisions are also recorded in the decision frame. * ASBA2 performs the operation resulting from the decision. * Submits a request to the simulation algorithm. * Finally, it proceeds to monitor whether it has achieved any progress towards the goal speci®ed in the decision frame. For instance, in achieving the goal ``reduce the blocking time of the 2nd station'', the system performs a concrete action. To decide whether that action is successful, the system must monitor that the blocking time of the 2nd station has been reduced. Moreover, the throughput of the production line has to be increased. The system proceeds until it cannot identify any violated constraint, i.e. it cannot pursue any goal, or it cannot make any progress towards any goal and towards increasing throughput. Strategic knowledge in ASBA2 is implemented by PROLOG rules. These rules follow the generic structure: (preconditions, strategy body, post conditions). Preconditions check the state of the ASBA2 for strategy execution. This state comprises the physical con®guration of the production line and the ASBA2 internal state. The physical con®guration is the current buer allocation plan. As already explained, ASBA2 internal state is recorded in decision frames. The body of a strategy represents a sequence of strategic steps. A strategic step can be a strategy or a speci®c operation. In particular, each of the major steps identi®ed in Fig. 3 can be realized either by another strategy or by an operation. For example, the calculation of an initial distribution of buer slots to stations can be done by a number of alternative strategies. A possible strategy is to assign credits to stations. Credits are proportional to stations' eective rates and to their positioning in the production line [9]. An alternative strategy would be to omit this step and let ASBA2 decide on an initial distribution by pursuing the goal ``allocate the total buer size''. Steps, such as submitting a request to the simulation algorithm, are implemented directly by operators. Post conditions are procedures that check whether the execution of the strategy has achieved the target goal described in the ASBA2 internal state (decision frame). Of particular interest here is the ``progress'' procedure that helps monitor progress towards a goal state. Strategies, during execution, are instantiated to the speci®c circumstances to which they apply. Strategy instantiation results in a sequence of concrete actions that are performed to the physical con®guration of the production line. 7.2. Operational Knowledge This is knowledge about operators. Operators are actions that are executed under speci®c circumstances and aect the physical con®guration of the system. Operators are invoked by strategies to achieve the goals speci®ed in decision frames. Operators can be categorized according to the context, i.e. the goal, in which they can be used: * Operators for reducing the blocking time of a machine. There are three such operators. An example of such an operator is the following one:
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G. A. Vouros and H. T. Papadopoulos if the goal is to ``reduce the blocking time'' of the station M, then transfer one buer unit from the source buer to the buer next to the station M.
* Operators for allocating the total buer size. There are seven such operators. An example of such an operator is: if the goal is ``allocation of the total buer size'' and the station with the maximum blocking time (speci®ed in the decision frame) needs more buer units, then allocate one buer unit to the buer next to the station with the maximum blocking time. * Operators for increasing throughput. Although all the operators aim at increasing throughput, there are three operators that aim speci®cally to that goal. An example of such an operator is: if the goal is to increase throughput and there is not a signi®cant dierence between the average waiting times of two adjacent stations, then ®nd a source buer and transfer one unit to each buer of the adjacent stations. Operators, although speci®ed as PROLOG rules in the prototype system developed, have the form: (preconditions, eects). Preconditions check for particular conditions in the physical con®guration of the line. Eects are procedures that aect the physical con®guration. Such procedures allocate units to a buer and transfer speci®c number of buer units from one station to another. 7.3. Control Knowledge This kind of knowledge comprises the mechanisms for exploitation of the knowledge categories described in Section A.2. It is a generic mechanism that does not refer to the problem of buer allocation. Its generality allows extending the system eectively by incorporating more strategies, operators and/or heuristics in the system. Particularly, the control mechanism . Monitors the execution of strategies and operators. . Searches for alternative ways of executing strategies and operators. . Searches for alternative strategies and operators to satisfy the current goals. The control task is a top-down one. Given a goal, the system determines what changes in the physical con®guration should be done, in order for the target goal to be achieved. The changes are made by particular operators invoked by speci®c strategies. The control task is implemented in PROLOG. The major steps performed by the control mechanism are the following: (a) Checks the preconditions of a strategy. (b) Instantiates the strategy by executing the strategic steps speci®ed in the strategy body. (c) During strategy instantiation, strategic decisions are recorded in the decision frames which drive and justify the execution of subsequent strategic steps. (d) When the strategy has been executed, post conditions are checked. In case one of these fails, then alternative ways of executing strategic steps are tested. (e) If all the alternatives have been checked, then the strategy instantiation fails and the strategic decisions recorded in the decision frame are withdrawn. 7.4. Heuristic Knowledge This kind of knowledge comprises mainly conditions that check for the occurrence of speci®c con®gurations in the production line. Such conditions drive the execution of strategies and operators. For example, deciding on the credits that should be assigned to each station for an initial distribution of buer slots, a heuristic rule concerns the positioning of the buers in the production line. In this way, buers closer to the middle station are assigned more credits than buers towards the two ends. It must be pointed out that heuristic rules are mostly utilized in order for ASBA2 to decide the source and the target buer as well as the number of buer slots that should be transferred. A major heuristic used in these cases is the following one: The target buer is the one next to the station with the greatest blocking time. To identify the source buer, the system uses heuristics concerning the blocking and average waiting times in stations.
AUTHORS' BIOGRAPHIES George A. Vouros is a Lecturer in the area of Arti®cial Intelligence at the Department of Mathematics, University of the Aegean, Greece. He received his Ph.D. from the University of Athens in 1992. He has participated in many E.U. funded projects and his current research interests include expert systems and intelligent multimedia systems. He is member of AAAI, IEEE and member of the board of directors of the Hellenic Society for Arti®cial Intelligence. H. T. Papadopoulos (Chrissoleon) is an Associate Professor in the area of Production and Operations Management at the Department of Business Administration of the University of the Aegean, Chios island. He received his B.Sc. (in Mathematics) from the Aristotle University of Thessaloniki, Greece, his M.Sc. (in Operations Research and Informatics) from the National Kapodistrian University of Athens and his Ph.D. in Operations Research and Industrial Engineering
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from the Department of Industrial Engineering of the University College Galway, Ireland in 1989. Before joining the University of the Aegean, he was the Logistics and Administration Manager and Customer Services Sales Manager at Digital Equipment Corporation (DEC) Hellas. His research interests include stochastic modelling, design and analysis of manufacturing systems, production and operations management, optimization of queueing systems, Logistics and Purchasing management and development of Decision Support Systems. He is co-author of a book on the analysis and design of manufacturing systems, published by Chapman and Hall. He is a member of INFORMS (former ORSA/ TIMS).
Computers Ops Res. Vol. 25, No. 12, pp. 1055±1067, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0305-0548(98)00034-3 0305-0548/98 $19.00 + 0.00
BUFFER ALLOCATION IN UNRELIABLE PRODUCTION LINES USING A KNOWLEDGE BASED SYSTEM G. A. Vouros1 and H. T. Papadopoulos2{ Department of Mathematics, University of the Aegean, 832 00 Karlovassi, Samos, Greece and 2 Department of Business Administration, University of the Aegean, 821 00 Chios, Greece
1
(Received April 1997; accepted January 1998) Scope and PurposeÐOne of the most important but dicult optimization problems concerns the optimal allocation of buers in a production system (line) with stochastic inputs and outputs. The buer allocation problem is a non-linear problem with integer variables and there exists no closed-form solution for the objective function. AbstractÐThe optimization of production lines performance is a problem of great complexity and, therefore, of signi®cant research interest. The problem may involve the optimization of many con¯icting objectives, such as increasing throughput and reducing work-in-process time. The majority of existing studies have used various heuristics and search methods based on operations research. These methods have been proved to be computationally inecient, especially for large production lines. This paper presents ASBA2, a knowledge based system that determines near optimal buer allocation plans, with the objective of maximising production lines throughput. The allocation plan is calculated subject to a given amount of total buer slots, in a computationally ecient way. ASBA2 operates in close cooperation with a simulation method, which provides ASBA2 with performance measures concerning production line behaviour. Moreover, to evaluate results provided by ASBA2, we have utilized an exact numerical algorithm for calculating the throughput of unreliable production lines. # 1998 Elsevier Science Ltd. All rights reserved Key words: Buer allocation, optimization, production lines, knowledge-based system
1. INTRODUCTION AND LITERATURE REVIEW
Over the years a large amount of research has been devoted to the analysis and modelling of production lines. For a systematic classi®cation of the relevant works the interested reader is addressed to a review paper by Papadopoulos and Heavey [2]. The allocation of buer units in production lines is a major optimization problem faced by manufacturing systems designers as well as by researchers. It has to do with devising an allocation plan for distributing a certain amount of buer space among the intermediate buers of a production line. The aim is to achieve performance objectives such as maximum throughput and/or minimum work-in-process (WIP) time. The problem is complicated and very critical since it introduces computational complexity and involves trade-os between the constraints and the objectives posed by the problem itself. Computational complexity is due to the increasingly large number of feasible allocations with respect to the total number of buer slots and the number of stations in a production line. Although several researchers have studied the problem of optimizing buer allocation to maximize the eciency of an unreliable production line, there is no method that can handle this problem for large production lines in a computationally ecient way. Hillier and So [4] and Hillier et al. [5] studied the buer allocation problem for short production lines. This method cannot solve the problem for large production lines. It is based mainly on comprehensive studies for characterizing the optimal buer allocation pattern. Authors have provided extensive numerical results for balanced lines with up to 6 stations and limited results for lines with up to 9 stations. Other relevant studies are those by Conway et al. [3] who conducted simulation experiments to study the buer allocation problem, Powell [6] who studied the buer allocation problem for {To whom all correspondence should be addressed. E-mail: [email protected] 1055
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unbalanced production lines and So [7] who dealt with the buer allocation problem in a production line, having as an objective minimization of the average work-in-process, provided a minimum required throughput is attained. Trade-os between the objectives of devising a buer allocation plan make the buer allocation problem more complicated. For instance, let us consider the trade-o between throughput and average WIP. WIP plays an important role carrying a signi®cant cost for the whole production line. Usually, the objective is to minimize WIP subject to a minimum required throughput. A speci®c buer allocation plan may help in achieving a required minimum throughput, but it may have the undesirable eect of increasing the average WIP, or vice-versa. In terms of the constraints posed on the total buer space, increase of the available total buer space results in increase of WIP, while reduction of it may result in reduction of throughput. On the other hand, increasing the amount of total buer slots may result in increasing the maximum throughput of the system, but it may be against economy. In other words, there is a trade-o between the cost of maintaining buer slots and the attainable maximum throughput of a manufacturing system. Of course, the problem becomes more complicated when more realistic constraints are posed. For instance, the additional requirement of keeping the number of buer slots in a station lower than (or greater than) a given number may result in decreasing (increasing) the maximum throughput attainable by the manufacturing system. This paper presents ASBA2, a knowledge based system for solving the buer allocation problem in unreliable, balanced and unbalanced, production lines. The objective of ASBA2 is to increase the throughput of the production line given a total amount of buer slots. ASBA2 is an extension of a previous system, called the advisor system for buer allocation (ASBA) which was developed for solving this problem for reliable production lines, i.e. lines with machines that were not allowed to fail ( [1]). ASBA2 is a generative model that works in close cooperation with an evaluative-simulation model. This paper is organized as follows. Section 2 states the problem and assumptions of the model, whereas, Section 3 presents major aspects of the knowledge based system. Detailed description of the types of knowledge exploited by the knowledge based system is given in Appendix A. In Section 4, we provide numerical results obtained from the system. Finally, Section 5 concludes the paper and some future research directions are suggested.
2. THE MODEL AND THE BUFFER ALLOCATION PROBLEM
In asynchronous production lines, when a workstation has completed its processing and the next buer has available space, then the processed part is passed on. Then, the workstation starts processing a new part that is taken from its buer. In case the buer has no parts, the workstation remains idle until a new part is placed in the buer. This type of line is subject to manufacturing blocking (or blocking after service, as it is known in the literature) and starving. 2.1. Assumptions of the model It is assumed that the ®rst station is never starved and the last station is never blocked. The processing (service) times at each workstation are assumed to be independent random variables following either the exponential or Erlang-k (k r2) distribution, with mean service rates, mi, i = 1, 2, . . ., K. The workstations of the line are subject to random failures according to a Poisson distribution. That is, the times to failures are assumed to follow the exponential distribution with mean failure rates bi, i = 1, 2, . . ., K. The repair times are also assumed to follow the exponential or Erlang-m distribution (m r1), with mean repair rates ri, i = 1, 2, . . ., K. Although the assumption concerned with the distribution of processing times is not so realistic, it has been used by many researchers as it facilitates modelling a production line by the well-known Markovian state method. According to this method, the line is modelled as a series or tandem queueing network with K single-server stations that correspond to the K workstations of the production line. This allows the manufacturing system designer to get insight into performance measures of the production line, such as the mean throughput, the average sojourn time and the mean queue lengths. These measures, at the preliminary design stage, help the manufacturing designer to evaluate and ®ne-tune the performance of a production line. At the
Buer allocation in unreliable production lines
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Fig. 1. A K-station production line with K ÿ 1 intermediate buers.
operational level, simulation and other techniques have to be utilized to cope with the on-line decisions that must be taken. The K-station line has K ÿ 1 intermediate locations for buers, labelled B2, B3, . . ., BK, in Fig. 1. The basic performance measures in the analysis of production lines are the throughput (or mean production rate) and the average work-in-process (WIP) or equivalently the average production (sojourn) time. So Ref. [7] dealt with the optimal buer allocation problem of production lines, having as target the minimization of the average WIP. The object of the current paper is the buering of asynchronous, unreliable production lines with the assumptions given above. The objective is the maximization of line throughput, subject to a given total buer space. 2.1.1. The optimal buer allocation (OBA) problem. In mathematical terms, our problem (P) could be stated as follows: Find B = (B2, B3, . . ., BK) so as to max XK(B) subject to: aK i = 2Bi=N, Bir0 Bi integer (i = 2, 3, . . . , K) where N is a ®xed nonnegative integer, denoting the total buer space available in the production line. B = (B2, B3, . . . , BK) is the ``buer vector'', i.e. a vector with elements the buer capacities of the K ÿ 1 buers. XK denotes the throughput of the K-station line. This is a function of the mean service rates of the K stations, mi (i = 1, 2, . . ., K), of the coecients of variation, CVi, of the service times and the buer capacities, Bi. It is also a function of higher moments of the service time distribution, but these are usually ignored, for simplicity. Besides, it has been experimentally, by simulation, proved that these higher moments have a minor eect on the throughput of the production lines [10]. 2.2. Methodology of investigation To solve the optimal buer allocation problem (P), we have performed the following steps: . (S1) We modeled the queueing process of the production line as a ®nite state, continuous time Markov chain, due to the assumption of the Erlang-k (k r1) distribution for the processing times. . (S2) To calculate the exact throughput of a production line we have utilized an algorithm developed by Heavey et al. [8]. The algorithm solves the sparse system of the steady-state probabilities of the resulting Markov chain. It gives the exact throughput for any K-station line with ®nite intermediate buers and phase-type service times. The number of states and the number of feasible allocations of N buer slots among the K ÿ 1 intermediate buer locations increase greatly with N and K. The latter is given by the formula: N 1 N 2 . . . N K ÿ 2 K ÿ 2!
1
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G. A. Vouros and H. T. Papadopoulos
. (S2') As an alternative method to the above mentioned exact numerical methods, we developed a simulation algorithm that calculates the throughput of the production line as well as the idle, blocking and the mean waiting time in each workstation. . (S3) To ®nd a near optimal buer allocation (OBA) for maximizing the throughput of the line, we utilized the knowledge based system ASBA2, that has been developed speci®cally for solving this problem. ASBA2 is a generative system that works in close cooperation with the simulation algorithm. Speci®cally, each time ASBA2 submits a request to the simulation algorithm it provides a buer allocation plan. Simulation, on the other hand, provides ASBA2 with the throughput of the production line as well as with the idle, blocking and average waiting time in each station. These performance measures help ASBA2 to evaluate the proposed plan and adjust it accordingly. Figure 2 depicts in detail the communication between ASBA2 and the simulation algorithm. Given that exact methods fail to provide the optimal buer allocation plans for large production lines and for a large number of slots in intermediate buers, we measured the suboptimality of ASBA2 only for these cases where the exact solution is known. Suboptimality is measured by the absolute percentage dierence between the throughput achieved by ASBA2 and the maximum throughput. Obtained numerical results and remarks are given in Section 5.
3. THE KNOWLEDGE BASED SYSTEM
Complexity, criticality and the experience intensity of allocating buers in a production line justify the need for an advisory system. In Ref. [1] we have investigated the use of a speci®c framework for representation and reasoning, according to which we have structured and encoded knowledge for solving the following problem: given a total buer size and a minimum required throughput, ®nd a buer allocation con®guration that minimizes the average work-inprocess subject to a minimum required throughput for a production line with reliable machines. The knowledge based system proposed, called ASBA, had been designed so as to enable the incorporation of realistic constraints concerning the allocation of buer slots. Such a constraint may be the lower or greater number of buer slots that must be assigned in a buer. The system proved to be ecient for achieving a near optimum solution to the stated problem. However, it deals only with reliable machines. This is considered to be a critical relaxation. The objective is to extend ASBA to deal eectively with a more general case, i.e. with unreliable unbalanced and balanced, production lines. As it has been stated previously, no computationally ecient technique exists for the solution of this problem, especially for large lines. This section, in conjunction with Appendix A, describes our investigation of extending ASBA to solve the problem of devising buer allocation plans for production lines with unreliable machines. ASBA2 solves the buer allocation problem stated in Section 2: it calculates a near
Fig. 2. Communication between ASBA2 and the simulation algorithm.
Buer allocation in unreliable production lines
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Table 1. ASBA2's results in buer allocation of 3-station exponential lines with m1=1, m2=1.2, m3=1.4, r1=r2=r3=0.5, b1=0.05, b2=0.05, b3=0.05 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0) (1±1) (2±1) (3±1) (3±2) (3±3) (4±3) (5±3) (6±3) (6±4)
0.6341 0.6744 0.7113 0.7361 0.7587 0.7716 0.7911 0.8060 0.8178 0.8266
(1±0) (1±1) (2±1) (3±1) (3±2) (4±2) (5±2) (5±3) (6±3) (7±3)
0.6341 0.6744 0.7113 0.7361 0.7587 0.7777 0.7922 0.8060 0.8178 0.8274
0.00 0.00 0.00 0.00 0.00 0.78 0.13 0.00 0.00 0.09
Table 2. ASBA2's results in buer allocation of 3-station exponential lines with m1=1, m2=1.2, m3=1.4, r1=r2=r3=0.5, b1=0.05, b2=0.01, b3=0.25 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0) (1±1) (1±2) (2±2) (3±2) (2±4) (4±3) (3±5) (4±5) (5±5)
0.5748 0.6204 0.6486 0.6783 0.6982 0.7141 0.7334 0.7465 0.7610 0.7720
(1±0) (1±1) (2±1) (2±2) (2±3) (3±3) (3±4) (4±4) (4±5) (5±5)
0.5748 0.6204 0.6493 0.6783 0.6987 0.7188 0.7343 0.7489 0.7610 0.7720
0.00 0.00 1.10 0.00 0.07 0.65 0.12 0.32 0.00 0.00
optimal buer allocation that maximizes throughput subject to the constraint on the total buer space. To solve this problem, ASBA2 performs the following actions: (1) Allocates buer slots to each station. (2) Forms strategic decisions concerning goals to pursue and constraints to be relaxed. With respect to the goals and constraints, the system decides where to allocate buer slots (target stations), from where these slots must be taken (source station) and the number of buer slots to be transferred. (3) Transfers buer slots from source to target buer and frees buer slots when these are not in use. (4) Submits requests to a simulation model, to calculate the throughput of the production line, as well as the idle, blocking and average waiting time in each station, for a given buer allocation plan. The types of knowledge utilized by ASBA2 are described in detail in Appendix A. In particular, Appendix A describes strategic, operational, control and heuristic types of knowledge and provides examples for each type. 4. NUMERICAL RESULTS
In this section, we present numerical results that have been obtained by running ASBA2. Results are split into two classes: (a) for exponential lines and (b) for Erlangian lines. Table 3. ASBA2's results in buer allocation of 3-station exponential lines with m1=m2=m3=1, r1=r2=r3=0.5, b1=0.05, b2=0.01, b3=0.25 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0) (1±1) (2±1) (2±2) (3±2) (3±3) (5±2) (2±6) (6±3) (5±5)
0.4700 0.5137 0.5306 0.5583 0.5687 0.5878 0.5801 0.6109 0.6028 0.6232
(0±1) (1±1) (1±2) (1±3) (2±3) (2±4) (2±5) (3±5) (3±6) (3±7)
0.4814 0.5137 0.5409 0.5600 0.5773 0.5914 0.6022 0.6122 0.6204 0.6271
2.36 0.00 1.90 0.30 1.48 0.60 3.66 0.21 2.83 0.62
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Table 4. ASBA2's results in buer allocation of 4-station exponential lines with m1=m2=m3=m4=1, r1=r2=r3=r4=0.5, b1=0.05, b2=0.02, b3=0.01, b4=0.001 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(0±1±0) (1±1±0) (1±1±1) (2±1±1) (2±1±2) (2±2±2) (3±2±2) (3±3±2) (5±2±2) (5±3±2)
0.5250 0.5577 0.5916 0.6128 0.6307 0.6563 0.6721 0.6890 0.6899 0.7076
(0±1±0) (1±1±0) (1±1±1) (1±2±1) (2±2±1) (2±2±2) (2±3±2) (3±3±2) (3±4±2) (4±3±3)
0.5250 0.5577 0.5916 0.6155 0.6378 0.6563 0.6725 0.6890 0.7007 0.7129
0.00 0.00 0.00 0.43 1.11 0.00 0.05 0.00 1.54 0.74
Table 5. ASBA2's results in buer allocation of 4-station exponential lines with m1=1.6, m2=1.4, m3=1.2, m4=1, r1=r2=r3=r4=0.5, b1=0.05, b2=0.02, b3=0.01, b4=0.005 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(0±1±0) (1±1±0) (1±1±1) (2±1±1) (1±3±1) (1±1±4) (2±3±2) (3±2±3) (4±2±3) (3±2±5)
0.6476 0.6652 0.7289 0.7394 0.7670 0.8013 0.8166 0.8348 0.8390 0.8672
(0±0±1) (0±1±1) (0±1±2) (1±1±2) (1±2±2) (1±2±3) (1±3±3) (1±3±4) (2±3±4) (2±3±5)
0.6572 0.7064 0.7361 0.7673 0.7938 0.8173 0.8346 0.8514 0.8660 0.8796
1.46 5.83 0.97 3.63 3.37 1.95 2.15 1.95 3.11 1.41
Speci®cally, Tables 1±7, provide numerical results concerning the buer allocation in production lines with exponentially distributed service and repairing times, whereas, Tables 8±14, provide analogous numerical results for production lines with service and repairing times following the Erlangian distribution. Recall that the target goal of ASBA2 is to maximize the throughput of a production line. The achieved throughput is denoted by XK in Tables 1±14 and is given by column 3. Respectively, column 5 gives the maximum throughput denoted by XK*. This is the throughput achieved by the optimal buer allocation (OBA) given in column 4. Columns 1 and 2 give, respectively, the total buer size, denoted by N, and the near optimum buer allocation obtained by ASBA2, Table 6. ASBA2's results in buer allocation of 5-station exponential lines with m1=m2=m3=m4=m5=1, r1=r2=r3=r4=r5=0.5, b1=0.1, b2=0.2, b3=0.25, b4=0.3, b5=0.35 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0±0±0) (2±0±0±0) (2±1±0±0) (1±1±1±1) (1±2±1±1) (2±2±1±1) (3±1±1±2) (2±2±2±2) (3±2±2±2) (3±1±3±3)
0.2848 0.2870 0.3005 0.3420 0.3512 0.3538 0.3603 0.3833 0.3852 0.3956
(0±0±1±0) (0±0±1±1) (0±1±1±1) (0±1±2±1) (0±1±2±2) (0±2±2±2) (1±2±2±2) (1±2±3±2) (1±2±3±3) (1±3±3±3)
0.2997 0.3158 0.3340 0.3471 0.3579 0.3699 0.3794 0.3894 0.3990 0.4066
4.97 9.12 10.00 1.47 1.87 4.35 5.03 1.56 3.45 2.70
Table 7. ASBA2's results in buer allocation of 5-station exponential lines with m1=1, m2=1.1, m3=1.2, m4=1.3, m5=1.4, r1=r2=r3=r4=r5=0.5, b1=b2=b3=b4=b5=0.05 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(0±1±0±0) (1±1±0±0) (1±1±1±0) (1±1±1±1) (2±1±1±1) (2±2±1±1) (1±2±2±2) (1±3±2±2) (3±2±2±2) (2±4±2±2)
0.5213 0.5514 0.5824 0.5976 0.6196 0.6422 0.6380 0.6519 0.6829 0.6914
(0±1±0±0) (1±1±0±0) (1±1±1±0) (1±2±1±0) (2±2±1±0) (2±2±1±1) (2±2±2±1) (3±2±2±1) (3±3±2±1) (3±3±3±1)
0.5213 0.5514 0.5824 0.6027 0.6213 0.6422 0.6585 0.6744 0.6894 0.7005
0.00 0.00 0.00 0.84 0.27 0.00 3.11 3.33 0.94 1.30
Buer allocation in unreliable production lines
1061
Table 8. ASBA2's results in buer allocation of 3-station Erlangian lines with m1=1, m2=1.2, m3=1.4, r1=r2=r3=0.5, b1=0.05, b2=0.05, b3=0.05 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0) (1±1) (2±1) (2±2) (2±3) (3±3) (4±3) (4±4) (5±4) (7±3)
0.7083 0.7489 0.7857 0.8026 0.8109 0.8347 0.8507 0.8555 0.8669 0.8776
(1±0) (1±1) (2±1) (3±1) (3±2) (4±2) (5±2) (6±2) (6±3) (7±3)
0.7083 0.7489 0.7857 0.8086 0.8261 0.8421 0.8538 0.8627 0.8709 0.8776
0.00 0.00 0.00 0.74 1.84 0.87 0.36 0.83 0.46 0.00
Table 9. ASBA2's results in buer allocation of 3-station Erlangian lines with m1=1, m2=1.2, m3=1.4, r1=r2=r3=0.5, b1=0.05, b2=0.01, b3=0.25 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0) (1±1) (1±2) (3±1) (3±2) (3±3) (4±3) (4±4) (6±3) (5±5)
0.6377 0.6896 0.7192 0.7354 0.7650 0.7841 0.7961 0.8092 0.8124 0.8273
(1±0) (1±1) (1±2) (2±2) (2±3) (3±3) (3±4) (4±4) (4±5) (5±5)
0.6377 0.6896 0.7192 0.7473 0.7670 0.7841 0.7978 0.8092 0.8191 0.8273
0.00 0.00 0.00 1.60 0.26 0.00 0.21 0.00 0.81 0.00
denoted by BA. Lastly, column 6 of the tables gives D, the absolute percentage dierence between the throughput achieved from ASBA2 and the maximum throughput, i.e. D = (vXK* ÿ XKv)/XK* 100. This expresses the failure of our system to achieve the required throughput. Although ASBA2 uses a simulation method to calculate the throughput of the production line, the throughput XK of the near optimal buer allocation (given in column 3) is calculated using the exact method developed by Heavey et al. [8]. This allows having a standard method with which to measure the failure of our system to achieve the maximum throughput. ASBA2, as well as its predecessor ASBA, is able to solve the problem for large production lines, allowing large buer capacities for each buer. However, to have a measure of ASBA2 Table 10. ASBA2's results in buer allocation of 3-station Erlangian lines with m1=m2=m3=1, r1=r2=r3=0.5, b1=0.05, b2=0.01, b3=0.25 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0) (1±1) (1±2) (3±1) (3±2) (4±2) (4±3) (4±4) (5±4) (5±5)
0.5177 0.5678 0.5947 0.5892 0.6159 0.6203 0.6356 0.6452 0.6475 0.6534
(0±1) (1±1) (1±2) (1±3) (2±3) (2±4) (2±5) (2±6) (3±6) (3±7)
0.5349 0.5678 0.5947 0.6119 0.6247 0.6354 0.6429 0.6483 0.6529 0.6563
3.22 0.00 0.00 3.71 1.41 2.37 1.13 0.47 0.83 0.44
Table 11. ASBA2's results in buer allocation of 4-station Erlangian lines with m1=m2=m3=m4=1, r1=r2=r3=r4=0.5, b1=0.05, b2=0.02, b3=0.01, b4=0.001 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0±0) (2±0±0) (1±1±1) (2±0±2) (3±0±2) (2±2±2) (3±2±2) (4±2±2) (4±2±3) (4±3±3)
0.5936 0.6074 0.6792 0.6608 0.6720 0.7422 0.7580 0.7680 0.7776 0.7926
(0±1±0) (1±1±0) (1±1±1) (1±2±1) (2±2±1) (2±2±2) (3±2±2) (3±3±2) (4±3±2) (4±3±3)
0.6024 0.6407 0.6792 0.7019 0.7258 0.7422 0.7580 0.7723 0.7828 0.7926
1.46 5.20 0.00 5.86 7.41 0.00 0.00 0.56 0.66 0.00
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Table 12. ASBA2's results in buer allocation of 4-station Erlangian lines with m1=1.6, m2=1.4, m3=1.2, m4=1, r1=r2=r3=r4=0.5, b1=0.05, b2=0.02, b3=0.01, b4=0.005 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(0±0±1) (0±0±2) (1±1±1) (3±0±1) (2±2±1) (2±2±2) (3±1±3) (2±4±2) (3±2±4) (4±2±4)
0.7512 0.7780 0.8263 0.7914 0.8512 0.8919 0.9002 0.9068 0.9342 0.9367
(0±0±1) (0±1±1) (0±1±2) (1±1±2) (1±1±3) (1±2±3) (1±2±4) (1±3±4) (1±3±5) (2±3±5)
0.7512 0.8029 0.8339 0.8627 0.8838 0.9056 0.9199 0.9315 0.9415 0.9497
0.00 3.10 0.91 8.26 3.69 1.51 2.14 2.65 0.78 1.37
Table 13. ASBA2's results in buer allocation of 5-station Erlangian lines with m1=m2=m3=m4=m5=1, r1=r2=r3=r4=r5=0.5, b1=0.1, b2=0.2, b3=0.25, b4=0.3, b5=0.35 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(1±0±0±0) (1±1±0±0) (2±1±0±0) (1±1±1±1) (2±1±1±1) (3±1±1±1) (3±2±1±1) (3±2±1±2) (3±2±2±2) (4±2±3±3)
0.3192 0.3337 0.3348
(0±0±1±0) (0±0±1±1) (0±1±1±1)
0.3378 0.3578 0.3782
5.51 6.74 11.50
success, we have studied its behaviour for a number of cases whose optimal buer allocation can be calculated using exact methods. Therefore, we have studied ASBA2's performance mainly for production lines (balanced and unbalanced) with a small number of stations and small number of total buer units, i.e. for 3-station, 4-station and 5-station production lines. The ®rst buer in all cases is considered to be saturated (containing always more than 0 jobs at the ®rst station). 4.1. Remarks From the results given in Tables 1±14, we can make the following observations: (1) Results produced by ASBA2 are very close to the optimal solutions. However, we sacri®ce the optimal solution for computational eciency: ASBA systems do not need to enumerate all the possible solutions for producing the near-optimum solution. They rather exploit speci®c knowledge about allocating buer storage. Particularly, as it can be noticed, ASBA2's failure to achieve the maximum throughput in the cases where service and repairing times are considered to be exponential is on average close to 1.43% and only in 4 cases out of 70, in Tables 5 and 6, its measure of failure exceeds 5%. It must be pointed that the sub-optimality of ASBA2 increases as the number of stations increases. Speci®cally, for the exponential cases, for threestation production lines, the failure of ASBA2 is in average close to 0.5%, for four-station production lines it is close to 1% and ®nally, for ®ve-station production lines it is, on average, close to 2.5%. Table 14. ASBA2's results in buer allocation of 5-station Erlangian lines with m1=1, m2=1.1, m3=1.2, m4=1.3, m5=1.4, r1=r2=r3=r4=r5=0.5, b1=b2=b3=b4=b5=0.05 N
BA
XK
OBA
XK*
D
1 2 3 4 5 6 7 8 9 10
(0±0±1±0) (1±0±1±0) (1±1±1±0) (1±1±1±1) (1±1±2±1) (2±1±1±2) (3±1±2±1) (1±3±2±2) (2±3±2±2) (3±2±3±2)
0.5825 0.6265 0.6673
(0±1±0±0) (1±1±0±0) (1±1±1±0)
0.5968 0.6338 0.6673
2.40 1.15 0.00
Buer allocation in unreliable production lines
1063
Moreover, ASBA2's failure to achieve the maximum throughput in cases where service and breakdown times are considered to be Erlang is on average close to 1.73% and in 7 cases out of 56, in Tables 11±13, its measure of failure exceeds 5%. We must notice that in Tables 13 and 14, ASBA2 has succeeded in calculating a buer allocation plan, in cases where exact techniques have failed to provide the optimal buer allocation. We should notice that for these cases, the exact throughput of the production line for the near optimal buer allocation cannot be calculated by the exact methods utilized. Again, the sub-optimality of ASBA2 increases as the number of stations increases. Speci®cally, for the Erlangian cases, for three-station production lines, the failure of ASBA2 is on average close to 0.7%, for four-station production lines it is close to 2% and ®nally, for ®ve-station production lines it is, on average, close to 4.5%. (2) Calculations provided by ASBA2 can be explained in terms of strategic, operational and heuristic knowledge used. This enables us to investigate the utilization of various heuristic rules which intuitively seem promising in calculating optimal buer allocations. However, since ASBA2 is combined with a simulation method, it must be stated that in speci®c cases, promising rules fail, due to simulation weakness to provide results that justify making progress towards maximizing throughput. This is a major drawback of ASBA2. (3) The total time needed by the overall system (ASBA2 and the simulation algorithm) to compute a solution is, on average, 1 min. The system runs on a Pentium in 200 MHz. The total time needed is the time that simulation needs to compute the performance measures plus the time that ASBA2 needs to take a decision, make the appropriate adjustments in the production line and monitor the progress achieved. It must be pointed out that approximately 90% of the time is consumed by the simulation module. 5. CONCLUDING REMARKS
The paper describes ASBA2, a knowledge based system that solves the well-known problem for buer allocation in production lines. The system is an extension of ASBA, which authors have described in an earlier paper. ASBA allocates buer space in reliable production lines, aiming at reducing WIP, subject to a given total buer space and a required throughput. ASBA computes near optimal buer allocations for reliable, balanced and unbalanced, production lines, whereas ASBA2 aims to extend the functionality of ASBA to unreliable, balanced and unbalanced, production lines. The paper describes how ASBA2 computes a near optimal buer allocation for unreliable lines, aiming at increasing throughput subject to given total buer space. The paper gives results for speci®c cases where the optimal buer allocation can be computed from exact methods and draws conclusions for ASBA2's behavior. Results, compared with exact methods, are encouraging. The system, in a computationally ecient way, using speci®c types of knowledge, computes results very close to the optimal ones. However, as the number of stations increase, the sub-optimality of ASBA2 increases as well. This result suggests that more strategies, operators and heuristics must be checked. ASBA2 provides the framework in which dierent strategies, operators and heuristics can be presented and evaluated. We plan to extend ASBA2 to deal with WIP as well, i.e. with the problem that ASBA dealt with, but for unreliable production lines. Our ®rst step in this direction will be to design a simulation algorithm that computes performance measures concerning the behavior of a production line that do not deviate much from computations that would be given from exact methods. This will also help us to provide results that are closer to the optimal ones. AcknowledgementsÐThis work has been supported by the Research Committee of the University of the Aegean, the members of which we sincerely thank. Without their support this research work would not have been completed. We highly appreciate the invaluable help of Mr M. Vidalis (Ph.D. student in the Department of Mathematics, University of the Aegean) for the development and running of the exact methods.
REFERENCES 1. Papadopoulos, H. T. and Vouros, G. A., A model management system (MMS) for the design and operation of production lines. International Journal of Production Research, 1997, 35(8), 2213±2236. 2. Papadopoulos, H. T. and Heavey, C., Queueing theory in manufacturing systems analysis and design: A classi®cation of models for production and transfer lines. European Journal of Operational Research, 1996, 92, 1±27. 3. Conway, R. W., Maxwell, W. L., McClain, J. O. and Thomas, L. J., The role of work-in-process inventories in series production lines. Operations Research, 1988, 36, 229±241.
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4. Hillier, F. S. and So, K. C., The eect of the coecient of variation of operation times on the allocation of storage space in production line system. IIE Transactions, 1991, 23, 198±206. 5. Hillier, F. S., So, K. C. and Boling, R. W., Notes: Toward characterizing the optimal allocation of storage space in production line systems with variable processing times. Management Science, 1993, 39(1), 126±133. 6. Powell, S. G., Buer allocation in unbalanced serial lines. Working Paper No. 289. The Amos Tuck School of Business Administration, Dartmouth College, 1992. 7. So, K. C., Optimal buer allocation strategy for minimizing work-in-process inventory in unpaced production lines. Working Paper presented at the ORSA/TIMS meeting in Detroit, U.S.A., Oct. 1994. 8. Heavey, C., Papadopoulos, H. T. and Browne, J., The throughput rate of multistation unreliable production lines. European Journal of Operational Research, 1993, 68, 69±89. 9. Sardo Pascual, L., A knowledge-based system for the optimal buer allocation in unreliable production lines. Diploma Thesis, Department of Mathematics, University of the Aegean, 1996. 10. Alka, A., The throughput rate of multistation production lines. Ph.D. Thesis, Department of Industrial and Systems Engineering, University of Florida, Gainesville, 1986.
APPENDIX A This section presents the types of knowledge encoded in ASBA2 according to the speci®c framework developed for ASBA. These are the strategic, the operational, the control and the heuristic knowledge. 7.1. Strategic Knowledge This is knowledge about generic procedures followed during problem solving and about forming speci®c strategic decisions in problematic situations that arise. As already described in Ref. [1], such knowledge is used to decide which goals to pursue, which constraints to relax, whether certain progress is made in achieving a speci®c goal and to resolve trade-os between objectives and constraints. Figure 3 depicts the most generic strategy followed by ASBA2. According to this strategy, the major steps that ASBA2 performs are as follows:
Fig. 3. The major strategy of ASBA2.
Buer allocation in unreliable production lines
1065
Fig. 4. Decision frame format. * ASBA2 constructs an initial state. During that step it calculates an initial distribution of buer slots to the intermediate buers in the production line and it submits a request to the simulation model. * It frees buer slots that are not used. As depicted in Fig. 2, the number of buer slots per station that are not utilized are provided by the simulation algorithm. * ASBA2 forms a strategic decision concerning how to proceed. Such a decision comprises the current goal of the system, as well as what constraints and objectives must be relaxed. Relaxation of constraints and objectives help the system to avoid lines of reasoning that do not achieve any progress towards system's objectives. Goals that the system pursues during reasoning include reducing the blocking time of a machine, allocating the total buer size and increasing throughput. When the system monitors that it cannot achieve any progress towards maximizing throughput, then it halts and outputs the proposed buer allocation plan. Strategic decisions are recorded in decision frames and comprise the internal state of ASBA2. Figure 4 depicts the format of a decision frame. It must be noticed that a history of decision frames is kept with respect to the beginning of the ASBA2 operation. Decisions recorded in a decision frame guide subsequent strategic decisions, guide and justify the execution of particular operations. Therefore a decision frame is gradually instantiated. * The system proceeds to decide on the speci®c operations that must be performed in order to achieve the goal speci®ed in the decision frame. Therefore, it decides the station in which the number of buer slots must be increased (target station), the number of buer slots that must be added and where the new slots shall be taken from (source station). In case the source station is unspeci®ed, then ASBA2 allocates the non-allocated buer slots. As Fig. 4 depicts, these decisions are also recorded in the decision frame. * ASBA2 performs the operation resulting from the decision. * Submits a request to the simulation algorithm. * Finally, it proceeds to monitor whether it has achieved any progress towards the goal speci®ed in the decision frame. For instance, in achieving the goal ``reduce the blocking time of the 2nd station'', the system performs a concrete action. To decide whether that action is successful, the system must monitor that the blocking time of the 2nd station has been reduced. Moreover, the throughput of the production line has to be increased. The system proceeds until it cannot identify any violated constraint, i.e. it cannot pursue any goal, or it cannot make any progress towards any goal and towards increasing throughput. Strategic knowledge in ASBA2 is implemented by PROLOG rules. These rules follow the generic structure: (preconditions, strategy body, post conditions). Preconditions check the state of the ASBA2 for strategy execution. This state comprises the physical con®guration of the production line and the ASBA2 internal state. The physical con®guration is the current buer allocation plan. As already explained, ASBA2 internal state is recorded in decision frames. The body of a strategy represents a sequence of strategic steps. A strategic step can be a strategy or a speci®c operation. In particular, each of the major steps identi®ed in Fig. 3 can be realized either by another strategy or by an operation. For example, the calculation of an initial distribution of buer slots to stations can be done by a number of alternative strategies. A possible strategy is to assign credits to stations. Credits are proportional to stations' eective rates and to their positioning in the production line [9]. An alternative strategy would be to omit this step and let ASBA2 decide on an initial distribution by pursuing the goal ``allocate the total buer size''. Steps, such as submitting a request to the simulation algorithm, are implemented directly by operators. Post conditions are procedures that check whether the execution of the strategy has achieved the target goal described in the ASBA2 internal state (decision frame). Of particular interest here is the ``progress'' procedure that helps monitor progress towards a goal state. Strategies, during execution, are instantiated to the speci®c circumstances to which they apply. Strategy instantiation results in a sequence of concrete actions that are performed to the physical con®guration of the production line. 7.2. Operational Knowledge This is knowledge about operators. Operators are actions that are executed under speci®c circumstances and aect the physical con®guration of the system. Operators are invoked by strategies to achieve the goals speci®ed in decision frames. Operators can be categorized according to the context, i.e. the goal, in which they can be used: * Operators for reducing the blocking time of a machine. There are three such operators. An example of such an operator is the following one:
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G. A. Vouros and H. T. Papadopoulos if the goal is to ``reduce the blocking time'' of the station M, then transfer one buer unit from the source buer to the buer next to the station M.
* Operators for allocating the total buer size. There are seven such operators. An example of such an operator is: if the goal is ``allocation of the total buer size'' and the station with the maximum blocking time (speci®ed in the decision frame) needs more buer units, then allocate one buer unit to the buer next to the station with the maximum blocking time. * Operators for increasing throughput. Although all the operators aim at increasing throughput, there are three operators that aim speci®cally to that goal. An example of such an operator is: if the goal is to increase throughput and there is not a signi®cant dierence between the average waiting times of two adjacent stations, then ®nd a source buer and transfer one unit to each buer of the adjacent stations. Operators, although speci®ed as PROLOG rules in the prototype system developed, have the form: (preconditions, eects). Preconditions check for particular conditions in the physical con®guration of the line. Eects are procedures that aect the physical con®guration. Such procedures allocate units to a buer and transfer speci®c number of buer units from one station to another. 7.3. Control Knowledge This kind of knowledge comprises the mechanisms for exploitation of the knowledge categories described in Section A.2. It is a generic mechanism that does not refer to the problem of buer allocation. Its generality allows extending the system eectively by incorporating more strategies, operators and/or heuristics in the system. Particularly, the control mechanism . Monitors the execution of strategies and operators. . Searches for alternative ways of executing strategies and operators. . Searches for alternative strategies and operators to satisfy the current goals. The control task is a top-down one. Given a goal, the system determines what changes in the physical con®guration should be done, in order for the target goal to be achieved. The changes are made by particular operators invoked by speci®c strategies. The control task is implemented in PROLOG. The major steps performed by the control mechanism are the following: (a) Checks the preconditions of a strategy. (b) Instantiates the strategy by executing the strategic steps speci®ed in the strategy body. (c) During strategy instantiation, strategic decisions are recorded in the decision frames which drive and justify the execution of subsequent strategic steps. (d) When the strategy has been executed, post conditions are checked. In case one of these fails, then alternative ways of executing strategic steps are tested. (e) If all the alternatives have been checked, then the strategy instantiation fails and the strategic decisions recorded in the decision frame are withdrawn. 7.4. Heuristic Knowledge This kind of knowledge comprises mainly conditions that check for the occurrence of speci®c con®gurations in the production line. Such conditions drive the execution of strategies and operators. For example, deciding on the credits that should be assigned to each station for an initial distribution of buer slots, a heuristic rule concerns the positioning of the buers in the production line. In this way, buers closer to the middle station are assigned more credits than buers towards the two ends. It must be pointed out that heuristic rules are mostly utilized in order for ASBA2 to decide the source and the target buer as well as the number of buer slots that should be transferred. A major heuristic used in these cases is the following one: The target buer is the one next to the station with the greatest blocking time. To identify the source buer, the system uses heuristics concerning the blocking and average waiting times in stations.
AUTHORS' BIOGRAPHIES George A. Vouros is a Lecturer in the area of Arti®cial Intelligence at the Department of Mathematics, University of the Aegean, Greece. He received his Ph.D. from the University of Athens in 1992. He has participated in many E.U. funded projects and his current research interests include expert systems and intelligent multimedia systems. He is member of AAAI, IEEE and member of the board of directors of the Hellenic Society for Arti®cial Intelligence. H. T. Papadopoulos (Chrissoleon) is an Associate Professor in the area of Production and Operations Management at the Department of Business Administration of the University of the Aegean, Chios island. He received his B.Sc. (in Mathematics) from the Aristotle University of Thessaloniki, Greece, his M.Sc. (in Operations Research and Informatics) from the National Kapodistrian University of Athens and his Ph.D. in Operations Research and Industrial Engineering
Buer allocation in unreliable production lines
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from the Department of Industrial Engineering of the University College Galway, Ireland in 1989. Before joining the University of the Aegean, he was the Logistics and Administration Manager and Customer Services Sales Manager at Digital Equipment Corporation (DEC) Hellas. His research interests include stochastic modelling, design and analysis of manufacturing systems, production and operations management, optimization of queueing systems, Logistics and Purchasing management and development of Decision Support Systems. He is co-author of a book on the analysis and design of manufacturing systems, published by Chapman and Hall. He is a member of INFORMS (former ORSA/ TIMS).