Development of genetic fuzzy logic controllers for complex production systems

Computers & Industrial Engineering 57 (2009) 1247–1257

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Development of genetic fuzzy logic controllers for complex production systems Seyed Mahdi Homayouni *, Sai Hong Tang, Napsiah Ismail Department of Mechanical and Manufacturing Engineering, Engineering Faculty, University Putra Malaysia, Selangor, Malaysia

a r t i c l e

i n f o

Article history: Received 24 July 2007 Received in revised form 22 April 2009 Accepted 9 June 2009 Available online 14 June 2009 Keywords: Genetic algorithm Fuzzy logic controller Genetic fuzzy logic controller Complex production systems

a b s t r a c t Complex production systems can produce more than one part type. For these systems, production rate and priority of production for each part type is determined by production controllers. In this paper, genetic fuzzy logic control (GFLC) methodology is used to develop two production control architectures namely ‘‘genetic distributed fuzzy” (GDF), and ‘‘genetic supervisory fuzzy” (GSF) controllers. Previously these controllers have been applied to single-part-type production systems. In the new approach the GDF and GSF controllers are developed to control complex production systems. The methodology is illustrated and evaluated using two test cases; two-part-type production line and re-entrant production systems. Genetic algorithm is used to tune the membership functions of input variables of GSF or GDF controllers. The objective function of the GSF controller minimizes the production cost based on workin-process (WIP) and backlog costs, while surplus minimization is considered by GDF controller. The results show that GDF and GSF controllers can improve the performance of production systems. GSF controllers decrease the WIP level and its variations. GDF controllers show their abilities in reducing the backlog level but generally, production cost for GDF controller is greater than GSF controller. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Controlling the production rates of manufacturing systems is notoriously difficult, since such systems are dynamic, uncertain and non-linear (Mok & Porter, 2006). The production control is at the heart of the whole manufacturing process. Three main policies to control the production systems are token-based, time-based, and surplus-based (Gershwin, 2000). In surplus-based control systems the differences between cumulative production and cumulative demand in a specific period of time is controlled. The main objective of this control policy is to produce smoothly while total demand is satisfied; it can also keeps work-in-process (WIP) to be as low as possible and reduce surplus or backlog. The WIP level is highly related to the fluctuations of demand. WIP is accumulated when the actual production rate is higher than demand. Surplusbased control policy contains ‘‘bang–bang” (Tsourveloudis, Dretoulakis, & Ioannidis, 2000), ‘‘base-stock” (Duri, Frein, & Di Mascolo, 2000) and ‘‘hedging point” (Sharifnia, 1988) control architectures which are based on surplus and backlog. In this control method the production is controlled to its maximum rate whenever inventory is below a critical level (hedging point) and set to zero whenever inventory is above that level (Bai & Gershwin, 1994). Mok and Porter (2006) implemented hedging point control architecture and

* Corresponding author. Tel.: +60 389466332; fax: +60 386567122. E-mail addresses: [email protected] (S.M. Homayouni), saihong@eng. upm.edu.my (S.H. Tang), [email protected] (N. Ismail). 0360-8352/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2009.06.002

used genetic algorithm (GA) to optimize the performance of this control architecture. The main objective in establishing control systems for production system is to minimize the long term average cost of manufacturing system, i.e. to be competitive on price (Monfared & Steiner, 2000). Since 1990s fuzzy logic controllers (FLCs) (Michels, Klawonn, Kruse, & Nürnberger, 2006) have been implemented to improve the performance of different control architectures in production systems. Custodio, Bispo, and Sentieiro (1992) suggested application of fuzzy controllers to solve short range planning and scheduling problems. Tsourveloudis et al. (2000) developed a fuzzy control architecture, called heuristic distributed fuzzy (HDF) controllers, which outperformed the conventional hedging point controller proposed by Bai and Gershwin (1994). The main objective of HDF controllers is to keep the WIP as low as possible, and concurrently to maintain high machine utilization and throughput. These controllers were called distributed, because the control modules are distributed in production processes and control each machine separately. The inputs for HDF controllers are WIP level in upstream and downstream buffers of the machine, current machine status, and production surplus of the machine. The output is the production rate of the machine. The main fuzzy rule for HDF controller for a single-part-type transfer line is expressed as (Tsourveloudis et al., 2000):

If Bði1Þ;i is LBðkÞ and Bi;ðiþ1Þ is LBðkÞ and Si is LMSðkÞ ðkÞ

and X i is LX i

then Ri is LRðkÞ

ð1Þ

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The k is rule number (k = 1, . . ., n), i is number of machine/workstation, B(i1),i is the level of upstream buffer while Bi,(i+1) is the level of downstream buffer. LB is a linguistic value of the variable buffer level with term set B = {Empty, Almost Empty, OK, Almost Full, and Full}. LMS is the linguistic variable for Si which denotes state of machine Mi, which can be either 1 (operative) or 0 (stopped) and subsequently S = {0, and 1}. LX represents value of surplus Xi, with term set X = {Negative, OK, and Positive}. The production rate Ri takes linguistic values LR from the term set R = {Zero, Low, Normal, and High}. Tsourveloudis et al. (2000) introduced the fuzzy rules for assembly and disassembly modules, as well. They claimed that every production system can be modeled by using these major production modules. Ioannidis, Tsourveloudis, and Valavanis (2004) developed a heuristic supervisory fuzzy (HSF) controller to tune the HDF control modules. The overall production control system contains two levels of surplus-based control system. The objectives are to keep the WIP and cycle time as low as possible while maintaining quality of service by keeping backlog at low levels. The production rate in each production stage (with HDF controllers) is controlled to satisfy demand, avoid overloading and eliminating machine starvation or blockage, while HSF provides overall control strategy for the whole production line of each part type. The input variables of HSF controllers are: mean surplus of end product, the difference between the surplus of end product and initial lower bound of surplus Il, and the relative WIP error ew which is

ew ¼

WIPðtÞ  WIPðtÞ WIPðtÞ

ð2Þ

where WIP(t) is the current level of WIP in all the buffers in the production system and WIPðtÞ is the mean WIP of the production system until time t. The outputs of HSF controller are the production surplus’ upper and lower bound correction factors (uc and lc), where 1 6 lc, uc 6 1. It is noteworthy that each part type is controlled individually by using HSF controllers. Production surplus is divided into three areas. If the surplus is lower than a lower surplus bound lb, then machine produces at maximum rate. If the surplus is above the upper bound ub, then production is stopped. If the surplus is between these bounds the production rate is decided in relation with the adjacent buffer levels and machine state. The production surplus bounds are modified according to the following mechanism:

ub ¼ Iu þ uc nu þ minðxe ; 0Þ

ð3Þ

lb ¼ min½ðIl þ lc nl Þ; ub 

ð4Þ

where Iu and Il are the initial upper and lower surplus bounds, respectively, xe is the surplus of end product, and nu, nl are constants chosen in such a way that lb would never exceeds ub when xe is positive. The rule base of the HSF controller contains rules of the following form (Ioannidis et al., 2004):

If mxe is LMX ðkÞ and ex is LEX ðkÞ and ew is LEW ðkÞ ðkÞ then uc is LU ðkÞ c and lc is LLc :

ð5Þ

where, k is the rule number (k = 1, . . ., 29), LMXe is a linguistic value of the variable mean final product surplus (mxe) with term set MX = {Negative Big, Negative Small, Zero, Positive Small, and Positive Big}, ex denotes the error of end product surplus which is the difference between surplus x and the lower bound of surplus; the term set of the corresponding linguistic value is EX = {Negative, Zero, and Positive}. LEw represents the relative deviation of WIP from its mean value, and it is chosen from the term set EW = {Negative, Zero, and Positive}. The upper surplus bound correction factor takes linguistic values LUc from the term set Uc = {Negative, Negative Zero, Zero, Positive Zero, and Positive} and the lower surplus bound correction factor takes linguistic values LLc from the term set Lc = {Negative, Negative Zero, Zero, Positive Zero, and Positive}.

Tsourveloudis, Doitsidis, and Ioannidis (2007) used GA to optimize the membership functions (MFs) of FLCs in HDF and HSF control architectures. They implemented their methods for simple production systems. The current research is an extension to their work which develops these controllers for complex production systems. The complex production system may be defined as a production system which produce more than one part type (multi-parttype production systems), and/or the part types may meet one machine more than one time in various stages of their process flow (re-entrant production systems). The remaining contents of this paper are organized as follows. Second section provides a review on literature in the area of genetic fuzzy logic controllers (GFLC), its principles and applications. In third section, the methodology of GFLC is clarified for the complex production systems. Section 4 describes the principles of the complex production system, especially the two test cases of the current research, two-part-type and re-entrant production systems. Section 5 lists down the experiments done to show the improvement in the performance of GSF and GDF controllers in comparison with the HSF and HDF controllers. Finally, Section 6 concludes the paper, and suggests the opportunities for further studies. 2. Genetic fuzzy logic controllers While fuzzy logic controllers exhibit their abilities in different kind of manufacturing problems, in 1990s a growing interest on enhancement of performance of the FLCs using learning approaches appeared (Cordon, Gomide, Herrera, Hoffman, & Magdalena, 2004). FLCs can be considered as knowledge-based systems, incorporating human knowledge into their ‘‘knowledge base” (KB) through fuzzy rules, fuzzy membership functions, and selection of scaling factors (Seng, Khalid, & Yusof, 1999). Due to their learning capabilities, artificial neural networks (Magdalena, 1995) and evolutionary algorithms (Hoffmann, 2001) have been used to improve the performance of FLCs and reducing the difficulty for designing their rule base. Genetic algorithms (Haupt & Haupt, 2004) are the most important family member of evolutionary algorithms (EAs). Genetic algorithms provide robust search capabilities in complex spaces, and thereby tender an effective approach for optimizing the FLCs. This is addressed as ‘‘genetic fuzzy logic controller” (Cordon & Herrera, 1995 and Herrera, 1997). GAs are introduced as a powerful tool for automating definition of the KB (Cordon & Herrera, 1995). According to Cordon et al. (2004) one may optimize the MFs with fixed fuzzy rules or fuzzy rules with fixed MFs, or both of them concurrently. The current research focuses on optimization of MFs of input variables with fixed rules. To adapt the MFs, firstly one has to parameterize the shape of the MFs. These parameters are evolved by using GA operators. Triangular (Cooper & Vidal, 1993) and trapezoidal (Karr & Gentry, 1993) shapes for MFs have been more common in past literatures. Moreover Shimojima, Fukuda, and Hasegawa (1995) as well as Liska and Melsheimer (1994) used more complex MFs such as Gaussian function. Both binary (Cooper & Vidal, 1993) and real (Liska & Melsheimer, 1994 and Phark, Kandel, & Langholz, 1994) encoding methods have been applied to represent the chromosomes. Gurocak (1999) mentioned that the initial rule base and the selected shape of the rule base have a great role in the success of the GFLC. Herrera, Lozano, and Verdegay (1998) reviewed the properties of real coding genetic algorithm and the main operators available for this coding scheme. Achiche, Baron, and Balazinski (2004) proposed a new encoding scheme in fuzzy decision support system. They stated that both binary and real encoding schemes are efficient for such applications, however, the real coding systems shows more satisfactory results. Wong and Hamouda (2002) applied the GA-based fuzzy rules in selection of cutting tool problem. They stated that improvement in

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Genetic Algorithm Optimization Module

MFs Definition

Production System

M1 HDF Controller

BI,1

Performance Indices

fitness values in first generations of the GA is much more than improvements between last generations. This means that GA converges to the optimal fitness value in few generations. Tedford and Lowe (2003) researched on genetic adapted fuzzy scheduling for a flexible production system. The GA was implemented to optimize the available solution from fuzzy logic-based scheduling system. Chiou and Lan (2005) provided a new encoding method for GFLCs. Franke and Lepping (2006) used simulation to show that scheduling with GFLC significantly improve the achieved quality of the schedules. The results showed an improvement of about 10% for the objective function with the GFLC scheduling system. Tsourveloudis et al. (2007) proposed the application of GA to improve HDF and HSF controllers for simple production systems. They called these new controllers as evolutionary distributed fuzzy (EDF) controller and evolutionary supervisory fuzzy (ESF) controller. The results showed that in most of the cases GA can improve the performance of production control systems significantly. These two control architecture are shown in Figs. 1 and 2.

M2 HDF Controller

B1,2

M1

M3 HDF Controller

B2,3

M2

B3,F

M3

Production Control Module Fig. 2. EDF control architecture (Tsourveloudis et al., 2007).

Initial Parameters for Fuzzy Controllers

GA Optimization Module

Termination, Optimized Parameters for MFs

3. GFLCs for complex production systems Knowledge acquisition for heuristic controllers is quite imprecise. Normally, learning processes are applied to ensure that heuristic controllers perform properly (Tedford & Lowe, 2003 and Tsourveloudis et al., 2007). EDF and ESF controllers (Tsourveloudis et al., 2007) are introduced as genetic distributed fuzzy (GDF) and genetic supervisory fuzzy (GSF) controllers, respectively, in current paper. The methodologies for GDF and GSF controllers in complex production systems are similar. In order to reduce the complexity of problem, only optimization of MFs of input variables of FLCs are considered in this paper. The efficiency of FLCs is highly dependent upon accuracy of their MFs. Consequently, if the selection of MFs is not based on a systematic optimization procedure, then it cannot guarantee a minimum objective function value. GA creates MFs that fit best to scheduling objectives. For designing GDF or GSF controllers, a set of possible MFs are considered as the search space and initial population. Fig. 3 shows the overall methodology for GDF or GSF controllers. Three main modules of this methodology are illustrated in Fig. 3. The GA and FLC modules are developed in MATLABÒ software (Mathworks Inc., 2008). Simulation models of proposed complex production systems are developed in SimulinkÒ software (Mathworks Inc., 2006). 3.1. Construction of chromosome A well-established chromosome is essential in order to optimize the MFs of GDF or GSF controllers. The chromosome is implemented to evolve during the process GA. Parameters of MFs for Genetic Algorithm Optimization Module

Supervisory Controller Production System

Tuning Parameters

HDF

Performance Indices

MFs Definition

Simulation Results (Fitness Values)

Update the Parameters of MFs for Fuzzy Controllers

Fuzzy Logic Controller Module

Simulation Model

Updated Fuzzy Controller are Used for Simulation

Fig. 3. Main framework for tuning the fuzzy controllers.

GSF or GDF controllers construct the chromosome. The MFs are chosen to be of trapezoidal and sigmoid shapes. Two and four parameters are required to define sigmoid and trapezoidal shapes of MFs. Fig. 4 shows the sigmoid and trapezoidal MF Shapes and their required parameters. In this research a chromosome consist of all the required parameters for all the MFs which define the input variables of the GDF or GSF controllers. Each chromosome is defined as a string of some genes. Each gene is a parameter of an MF for a linguistic variable. The GDF controllers enhance the performance of the HDF controller. The linguistic variables of GDF control architecture are the same as the HDF controller (Eq. (1)). In GDF controller, there are three input variables. While ‘‘machine state” input variable has fixed limitations (0 and 1) only ‘‘buffer capacity” and ‘‘surplus level” are considered to be optimized. The same MFs define the upstream and downstream buffer capacity variable. Buffer capacity has five MFs and surplus level has three MFs. Thus, eight MFs should be optimized to construct the optimum GDF controller. First and last MFs of each variable are of sigmoid shape; and trapezoidal shape is used to define the remaining MFs of variables. Gathering all the parameters for all the MFs constitute the whole chromosome for GDF controller which contains 24 genes, as illustrated in Fig. 5. The GSF controller is developed to enhance the performance of the HSF controller (described in Section 1). Hence, the same linguistic variables and MFs define the FLC in GSF controller. As de-

µ

Sigmoid MF

Trapezoidal MF

1

Bj,i

Mi

Bi,l

Production Control Module Fig. 1. ESF control architecture (Tsourveloudis et al., 2007).

a1

b1

a2

b2

c2

d2

Fig. 4. Parameters for sigmoid and trapezoidal shape of MFs.

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scribed in Section 1, HDF controllers are applied to control the machines locally and HSF controls each product in the overall production system. The same structure is implemented for GSF controller. It means that only the fuzzy rule of GSF controller is considered to be optimized. The GSF controller (Eq. (5)) has three linguistic variables (MX, EX and EW). MX has five MFs, EX and EW has three MFs; thus, eleven different MFs are used to define the input variables for the GSF controller. The sigmoid shape is used to define the first and the last MF of each linguistic variable. Gathering all the parameters for all the MFs constitute the whole number of parameters which should be determined by the GA, for defining the MFs of the FLC. A 32-gene chromosome can define the input variables of the GSF controller. Fig. 6 illustrates the required parameters in detail. The input variables for GDF and GSF controllers are normalized to be implemented in knowledge base of the FLC. Hence, the parameters of MFs can be in range of [0, 1]. As illustrated in Figs. 5 and 6, there are two parameters for the first sigmoid MF of each input variable. The first one is defined as the difference with the zero point of the interval and the second one is defined as the difference of the second parameter and the first parameter. For trapezoidal shape of MFs, the first parameter is defined as the differences with the last parameter of the previous MF, and the following parameters are defined as the differences with the previous parameter of the current MF. The actual chromosome for GA in GSF and GDF controllers is a 32-gene and 24-gene chromosomes, respectively, which is filled up with real numbers in the range of (0.000, 0.050). The chromosome defines the parameters of MFs by using an extra formula which is defined by the author. The formula observes the technical constraints to provide feasible parameters for the MFs. For the input variables which contain five MFs, the genes are converted to a real number in the range of (0.050, 0.100). For determining input variables which contain three MFs, the genes are converted to a real number in the range of (0.100, 0.200). These numbers are selected by the authors to produce feasible random MFs. Generally two constraint is required to define feasible parameters of MFs. Firstly the parameters of an MF should provide a trapezoidal or sigmoid shape (either ai 6 bi for sigmoid shape MFs or ai 6 bi 6 ci 6 di for trapezoidal shape MFs); and secondly the MFs of the input variables should be appeared in a sequence (ai 6 ai+1). Figs. 5 and 6 illus-

trate the conversion method for GDF and GSF methods, respectively. Sample chromosomes are used to define feasible MFs of both controllers. 3.2. Methodology of GFLC The GA as the central module in the methodology of GFLC is introduced in Fig. 3. The overall genetic algorithm proposed for GDF and GSF controllers are the same, however, the chromosome and fitness functions are different. GA creates the initial population, performs the crossover and mutation operator, and finally finds the optimal definition for the MFs of the input variables of GDF and GSF controllers. To evaluate the chromosomes, an FLC construction module updates the parameters of MFs in GDF or GSF controllers. Finally, based on the designed test cases the updated controller is applied in simulation model of the production system. The simulation module calculates the fitness function for the chromosomes and returns the results to the GA module. This process is illustrated in an algorithm shown in Fig. 7. The characteristics of GA which optimizes the fuzzy controllers are selected as follows: (1) the population size is 40, (2) the mutation rate is 0.01, (3) the 20 fittest individuals are qualified for the next generation, (4) each individual is evaluated by the results of a simulation run for 500 time units, (5) roulette wheel operator is used for choosing parents, (6) the GA is run for 100 times. As described in previous section the 24-gene and 32-gene chromosomes are designed for GDF and GSF controllers, respectively. Thus, the initial population with the size of 40 is filled with a random number in the range of (0.000, 0.050). This is to simplify the process of various operators in the GA module. Fitness functions for GSF and GDF control architecture are proposed by Tsourveloudis et al. (2007). For GSF controller the focus is on WIP level and reducing backlog levels. Therefore, the fitness function (which should be minimized) for GSF controllers is:

Fig. 5. The parameters and sample chromosomes to define MFs of input variables for GDF controller.

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Fig. 6. The parameters and sample chromosomes to define MFs of input variables for GSF controller.

F ¼ C I WIP þ C b BL

ð6Þ

where, WIP is the mean of work-in-process and BL is the mean of backlog. CI and Cb represent the unit costs of inventory and backlog, respectively. The key note in complex production systems is that for each part type, fitness function is calculated separately by simulation model in SimulinkÒ software. GDF controller focuses on the surplus level in each time instant. GDF minimizes the surplus level of each part type in all of the machines. The fitness function is chosen in a way that keeps the surplus as close to zero as possible. A system surplus close to zero suggests that the system satisfy demand by keeping backlog in low levels while the finished items inventory level is also kept low. The fitness function for GDF controller is defined as:

F¼

n X ½xi 2

ð7Þ

i¼1

where xi is the level of surplus in each time instant, and n is the number of simulation runs. When all of the chromosomes of current generation are evaluated, and if the stop criterion for GA module (100 numbers of generations) is not achieved, then the next generation is produced. Chromosomes are ranked based on their fitness value. In each generation, 20 of the fittest chromosomes are selected for the next generation. Roulette wheel selection method selects 10 pairs of parents to breed offspring by using ‘‘Arithmetical crossover” operator described by Herrera et al. (1998). Suppose that X 1 ¼ ðx11 ; . . . ; x1n Þ and X 2 ¼ ðx21 ; . . . ; x2n Þ are two parents selected for mating. Two offspring Ok ¼ ðok1 ; . . . ; okn Þ k ¼ 1; 2 are generated where oli ¼ ax1i þ ð1  aÞx2i and o2i ¼ ax2i þ ð1  aÞx1i , a is constant which is selected by the authors to be 0.5. Offspring and fittest chromosomes comprise the new population. Mutation operator is used to ensure that all the subspaces are subject to be selected in GA. Due to ‘‘elitism” operator the first chromosome of new population which is the fittest one, does not involve in mutation operator. Suppose xi is the random gene from the selected chromosome which should be mutated, then the new content of this gene equals to xi ¼ 0:05  xi where xi is the new content of the mutated gene. After mutation operator new generation is created. Once again the population is used to update the GSF or GDF controllers and this algorithm is run for 100 times. Finally, the fittest chromosome of the last population is used to construct the optimal or near optimal GSF or GDF controllers. These controllers can be implemented in real implementations.

4. Complex production systems Tsourveloudis et al. (2007) evaluated the performance of GSF and GDF control systems for single-part-type production systems (line and network). The performance of GDF and GSF in complex production systems is considered in this paper. In such type of production system, the priority of production for various part types on each machine has to be determined, while specifying the production rate (for more information on the priority of job assignment, refer to (Ioannidis et al., 2004)). The overall objective of production control in complex production systems is to satisfy final demand for all the part types while minimizing WIP level and surplus/backlog. The performance of HDF and HSF control architecture for multi-part-type production line and HSF control architecture for reentrant production system have been evaluated by Tsourveloudis et al. (2000) and Ioannidis et al. (2004). Two-part-type production line is illustrated in Fig. 8. In this figure, Bi,j shows the specific buffer for part type j which is performed by machine i. In multi-part-type production systems, each machine (Mi) performs kth operations on the jth part type. In re-entrant production system each part type can be processed more than one time on each machine in its process flow. Bai and Gershwin (1994) introduced a new approach for controlling multi-part-type production systems. In this method machines are virtually divided into as many sub-machines as the number of operations to be performed. Sub-machine Mi,j,k represents the controller of kth operation on jth part type in machine Mi. Dotted lines show the submachines (see Figs. 8 and 9). Each part type is divided into one items according to the specific operation on each machine. For example, in Fig. 9, first part type has two items in machine two. Each item of part type has a specific buffer. Controllers for each sub-machine regulates the operation on each part type. With this formation, a multi-part-type production network is decomposed into as many single-part-type networks as the number of part types produced. Moreover, the structure of GSF control system is shown in Fig. 8. As mentioned in the GSF controller, only the MFs of the supervisory controller are optimized, by using the GA operators. In order to control the machines locally HDF controllers are applied as well. Fig. 9 shows the re-entrant production system test case introduced by Ioannidis et al. (2004). Seven machines and three part types comprised this production system. As it is obvious from

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GA Optimization Module

Fuzzy Logic Controller Module

Model Simulation

Create initial population based on definition of MFs

Call GSF or GDF construction module

Construct GDF or GSF controller

Update GDF and GSF for all chromosomes

Run simulation of production system model for each chromosome

Calculate fitness function for each chromosome

Rank all of the chromosomes based on fitness value

Using roulette wheel operator to choose parents

Create offspring by using crossover operator

Using mutation operator for offsprings and parents except elites

Create next generation by parents and offsprings

NO

Control stop criterion YES Use optimized definition of MFs to construct optimized GDF and GSF

Fig. 7. Different steps of GFLCs for production control architecture.

Fig. 9, four machines assembled two part type into one new part type, and three of them disassemble one part into two new part types. It is assumed that all the assembly and disassembly factors are equal to one. In this figure there is a specific and dedicated buffer for every item of each part type. Thus Bj,1 is the buffer of item one of part type j. The processing times for various part types and sub-machines are shown in Table 1. Ioannidis and Tsourveloudis (2006) described the simulation model for simple production systems, in this paper the same simulation model is modified to be used for two proposed test cases. The typical assumptions for simulation of both GDF and GSF controllers are selected as follows: 1. machine (Mi) fails randomly with a failure rate pi, 2. machine (Mi) is repaired randomly with rate rri. Unlimited repair working personnel is assumed, which means there is always somebody to repair a failed machine, 3. time to failure and time to repair are exponentially distributed, 4. demand is constant with rate di, 5. all machines operate at known rates (ri), 6. the initial buffers are infinite sources of raw material,

7. the final buffer for each part type is infinite. It means that all the final products are assumed to be sold, 8. buffers between two machine series have finite capacities, 9. setup times, transportation times and change over times are negligible and will be included in the processing times.

5. Results and discussion The proposed methodology for controlling the two-part-type and/or re-entrant production systems are evaluated through two main test cases. While the GSF controller is assumed to optimize the performance of HSF controllers, the performance of the GSF controller for two-part-type production system is compared with the HSF control scheme (Ioannidis et al., 2004). The processing time for all of the machines in all of the tests is equal to 0.325. The first test is to compare the WIP level for GSF and HSF controllers. Four different demand value and four different buffer capacities (BC) are used to simulate the production system. The results of WIP level for the first part type with various demands (parts per time unit) and buffer capacities are shown in Fig. 10. In this test

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Genetic Algorithm Optimizer

Fitness function

Supervisory Controller

M1

M4

M3

M2

HDF M1,1

HDF M3,1

HDF M2,1

HDF M4,1

BS,1

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M4,1

BF,1

BS,2

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M2,2

B2,2

M3,2

B3,2

M4,2

BF,2

HDF M1,2

HDF M2,2

HDF M3,2

HDF M4,2

Supervisory Controller

Fitness function

Genetic Algorithm Optimizer

Fig. 8. Two-part-type production system and GSF control architecture.

M2

B1,0

M3

M2,1,2

B1,7

M3,1,2

B1,8

M2,3,1

B3,8

M3,3,1

B3,9

M2,1,1

B1,1

M3,1,1

B1,4

B1,2

M4

M6 B3,10

M6,3,2

M4,,3,2 B2,7

M4,2,2 B2,1

M1

B3,1

B3,11 B2,8

M6,2,2 M6,2,1

M4,2,1

B2,3

M4,3,1

M6,3,1

M7

B2,4

M7,1,1

B1,6

B3,4

M7,2,1

B2,6

M7,3,1

B3,6

B3,3

M5

B1,0

M1,1,1

B1,3

M5,1,1

B1,5

B2,0

M1,2,1

B2,2

M5,2,1

B2,5

B3,0

B3,2

M5,3,1

B3,5

M1,3,1

M5,3,2

B3,7

Fig. 9. Re-entrant multi-part-type production test case.

case the demand is constant in each test, the failure and repair rate for all machines is 0.5. It should be noted that the WIP level is calculated as the mean of summation of WIP level in all the internal buffers through the production line for each part type. Second test is to evaluate the effects of implementing GSF controller on backlog level in multi-part-type production system. The results are compared to those for HSF controller from

Ioannidis et al. (2004), where the backlog level is plotted against four various demand values, and for four different buffer capacities. Fig. 11 illustrates the results of this test. The failure and repair rates are constant in all the tests and equal to 0.5. The results of the second test illustrate that GSF controllers can decrease the amount of backlog level in comparison with HSF controllers.

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Table 1 Processing time for different machines in re-entrant production test case. Part type

Operation

Machine 1

2

3

4

5

6

7

1

1 2

0.2 –

0.2 –

0.2 –

– –

0.15 –

– –

0.2 –

2

1 2

0.2 –

0.2 –

0.2 –

0.15 0.15

0.15 –

0.15 0.15

0.2 –

3

1 2

0.2 –

0.2 –

0.2 –

0.15 0.15

0.15 0.15

0.15 0.15

0.2 –

8

HSF

7

BC=2

5

BC=4

BC=4

4

BC=6

BC=6

BC=8

BC=8

6

WIP level

GSF

BC=2

3 2 1 0 0.6

0.7

0.8

0.9

Demand (Part Per Time Unit) Fig. 10. WIP level for various buffer capacities of first part type.

Based on the above tests, for first test case, the GA shows its robustness in improving the performance of HSF controllers in multi-part-type production systems. A better view of the significance of the results can be shown by a concise cost analysis. While the backlog and WIP levels are concurrently important for the companies, one needs to consider them by using an integrated index. This index introduced by using the unit costs of holding WIP and unit cost caused by penalties of backlog. The overall production cost for the proposed control architecture consists of inventory and backlog costs. The capital invested to purchase the raw material contributed to the inventory costs. In addition, processing the parts through production system makes added value for the parts; which can be assumed as the inventory costs. It is assumed that inventory cost is independent from the stage of process. It is assumed that WIP is not suitable for cost assessment. The backlog cost is so important to be calculated correctly. The backlog costs

are not as straight as the WIP holding costs. Backlog causes in reducing the confidence of the customers to have new orders. Moreover, the backlog need to be completed in nearest time period to the current period, this may cause in difficulty of scheduling and more tardiness of subsequent orders. The production cost based on the unit cost of mean WIP and mean backlog level in production system evaluate the GDF control architecture. The GDF control architecture is tested for two-part-type production line. Various unit costs for WIP and backlog are tested. Table 2 shows the effects of various CI and Cb on the total production cost. In this test, the buffer capacity is equal to 10, and the failure and repair rates are equal to 0.5. Total production cost is calculated by Eq. (6) which is equal to the fitness function of GSF control architecture. Table 3 compares total costs for GSF and HSF control systems, for various WIP and backlog unit costs (CI and Cb). In this test the failure and repair rate

6

Backlog Level

5

HSF 4

GSF

BC=2

BC=2

3

BC=4

BC=4

2

BC=6

BC=6

BC=8

BC=8

1 0

0.6

0.7

0.8

0.9

Demand (Part Per Time Unit) Fig. 11. Backlog level for various buffer capacities of first part type.

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S.M. Homayouni et al. / Computers & Industrial Engineering 57 (2009) 1247–1257 Table 2 Cost analysis for various demands, WIP and backlog unit costs for GDF and HDF control architectures. Demand

CI

Cb

HDF

GDF

WIP

BL

C

WIP

BL

C

0.5

0.75 0.5 0.25

0.25 0.5 0.75

7.59 8.51 10.07

2.15 1.89 1.24

6.23 5.2 3.45

5.12 4.06 3.76

1.18 0.48 0.17

4.14 2.27 1.07

1

0.75 0.5 0.25

0.25 0.5 0.75

8.81 10.08 12.54

2.87 1.61 0.92

7.33 5.85 3.83

5.15 6.78 7.22

2.2 1.12 0.36

4.41 3.95 2.08

Table 3 Cost analysis for various demands, WIP and backlog unit costs for GSF and HSF control architectures. Demand

CI

Cb

HSF

GSF

WIP

BL

C

WIP

BL

C

0.5

0.75 0.5 0.25

0.25 0.5 0.75

5.48 6.87 7.29

1.62 0.63 0.18

4.52 3.75 1.96

3.62 2.69 3.18

1.16 0.38 0.12

3.01 1.54 0.89

1

0.75 0.5 0.25

0.25 0.5 0.75

5.71 7.14 8.71

3.34 1.52 0.32

5.12 4.33 2.42

3.52 4.28 5.57

3.46 1.62 0.41

3.51 2.95 1.70

are constant in entire experiments and equal to 0.5. The capacity of all buffers in the experiments is equal to 10. The results show that a significant reduction of WIP can be obtained by using GA. More than 30% decrease in overall production cost based on mean WIP and mean backlog is achieved in most of the cases. Additionally, based on results presented in Table 3, it is obvious that if the unit cost of holding WIP is small and backlog cost is much greater, then the differences between HSF and GSF are not significant. The subsequent experiment is a comparison between GDF and GSF control architectures for various WIP and backlog unit costs. This test is done for buffer capacity of 10. Moreover, failure and repair rates are equal to 0.5. Processing time for all machines and for both part types is equal to 0.325. Table 4 shows the comparative results. To evaluate the performance of GSF controller against HSF controller in re-entrant production systems, four various buffer capacities and four various demand value are implemented to the simulation model. Overall, sixteen different experiments have been done. First test for re-entrant production system is developed to evaluate the effects of various buffer capacities on WIP level of the first part type versus various demands. This is shown in Fig. 12. In all the cases the failure and repair rates of all machines are 0.1 and 0.5, respectively. Apparently, from Fig. 12 GSF controller reduces the variation of WIP level, this is due to its goals, which force it to keep the mean surplus of final product close to zero and WIP close to its mean value, while GDF focuses on reducing surplus in all machines. In second test for the re-entrant production system, GDF and GSF controllers are compared to each other. In this test, the WIP level for all three part types is calculated. The buffer capacity for all the buffers is 10. The failure and repair rates are 0.1

Table 4 Comparative production cost for GDF and GSF control architectures. Demand

CI

Cb

GDF C

GSF C

0.5

0.75 0.5 0.25

0.25 0.5 0.75

4.14 2.27 1.07

3.01 1.54 0.89

1

0.75 0.5 0.25

0.25 0.5 0.75

4.41 3.95 2.08

3.51 2.95 1.70

and 0.5, respectively. Fig. 13 shows the comparative results for WIP level versus four various demand values for GDF and GSF controllers. The main advantage of using fuzzy logic controllers in controlling the multi-part-type production systems is that they can approximates the way human operators adjust the processing rate so as to minimize idle periods due to starving or blockage. Two possible drawbacks can be identified. Firstly, the implementation of the fuzzy production controller is not easy in real applications. On-time regulation of processing rate requires online controlling of buffer levels and production surplus. This might be unrealistic in practice. The second comment is related with the decision space complexity of the fuzzy production control. Genetic fuzzy logic controllers add more complexity to the fuzzy production control space, but the designer can be sure that the obtained fuzzy controller is the optimized one. Furthermore the process of designing membership function in fuzzy production control which is a time consuming and non-precise process can be ignored by implementing GAs for the fuzzy production control. Results obtained from test cases shows the overall applicability of GFLCs for complex production systems. Fig. 10 shows that GSF controller can decrease the WIP level rather than HSF control architecture. It is shown that various buffer capacities do not have any significant effect on WIP level. Fig. 11 shows the backlog level for GSF controller which increases exponentially when the demand is high. The GSF cannot decrease the backlog level. In all cases, the GSF control architecture reduces WIP when demand is low and increases backlog when demand is high. WIP and backlog are two measures of the production systems. If one would like to reduce the backlog level, the yield of the system has to be increased. Therefore the WIP inside the production system will be increased. The backlog level is at low level when the demand can be easily satisfied. In these cases a noticeable decrease of WIP is more important than a small increase in backlog level. The backlog is more important in cases that demand is high, in these cases the backlog level is maintain at low levels and therefore WIP level will be increased (see Tables 2 and 3). Based on the obtained results, the following observations are made. The GSF controller achieves a substantial reduction of WIP when demand rate is low. Based on the results of Tables 2 and 3 it can be understood that GDF control approach gives lower backlog level rather than GSF controller.

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30 25

HSF

WIP level

20

GSF

BC=2

BC=2

15

BC=4

BC=4

10

BC=6

BC=6

5

BC=8

BC=8

0 0.6

0.7

0.8

0.9

Demand (Part Per Time Unit) Fig. 12. WIP level for first part type in re-entrant production system.

35 30

WIP level

25

GDF

20 15

GSF

part 1

part 1

part 2

part 2

part 3

part 3

10 5 0 0.75

1

1.25

1.5

Demand (Part Per Time Unit) Fig. 13. WIP level for various part types in re-entrant production system for GDF and GSF controllers.

On the other hand, when demand is close to the system’s capacity the GSF control approach gives lower production costs. Table 4 shows that the overall production costs for GSF control architecture is less than the cost for GDF controllers. 6. Conclusions GSF and GDF controllers have been presented for complex production systems which can produce more than one part type and may use the re-entrants in the process flow of the products. In this methodology the GA selects the membership functions for the fuzzy controllers to minimize the amount of WIP and backlog concurrently. Two test cases are proposed: two-part-type production line and re-entrant production system. The simulation results showed a significant improvement in the performance of supervisory control system with implementing the GA strategies. Since the fitness function for the GSF controller is a contribution of mean WIP and mean backlog levels, the results show that based on the importance of each WIP or backlog, one may decide to apply GSF or HSF control system. When the backlog cost is much more than WIP holding costs it is easier to use HSF rather than GSF, because GSF need a simulation model of the desired production system to find the optimal MFs while HSF can be applied without any simulations. Furthermore GDF and GSF controllers are compared to each other. The results show that both of these controllers can be used in special situations. When the backlog cost is much more than WIP costs, GDF can obtain less amount of backlog, while GSF try

to minimize the WIP levels. In this case the GDF is more useful. Moreover re-entrant production system test case shows that GSF and GDF can be used easily for more complex production systems. The only problem in these cases is to model the production systems. Nevertheless, it is obvious that the GA shows its added advantage for choosing the MFs of fuzzy control system by improving the overall performance of the production system. For further researches one may consider the other patterns for demand (such as seasonal demand) or more complicated production systems. Other optimization methods such as neural networks and EAs can also be considered for further researches in this area. Moreover, application of GA to optimize other components of HDF and HSF control systems especially the sequence of fuzzy rules, the number of MFs for each linguistic variable, and choosing the integration function are proposed. References Achiche, S., Baron, L., & Balazinski, M. (2004). Real/binary-like coded versus binary coded genetic algorithms to automatically generate fuzzy knowledge bases: A comparative study. Engineering Applications of Artificial Intelligence, 17, 313–325. Bai, S. X., & Gershwin, S. B. (1994). Scheduling manufacturing systems with work– in–process inventory control: Multiple-part-type systems. International Journal of Production Research, 32(2), 365–386. Chiou, Y. C., & Lan, L. W. (2005). Genetic fuzzy logic controller: An iterative evolution algorithm with new encoding method. Fuzzy Sets and Systems, 152(3), 617–635. Cooper, M., & Vidal, J. (1993). Genetic design of fuzzy controllers. In Proceedings of 2nd international conference on fuzzy theory and technology.

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