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Multi-objective optimisation of dynamic scheduling in robotic flexible assembly cells via fuzzy-based Taguchi approach
Accepted Manuscript Multi-Objective Optimisation of Dynamic Scheduling in Robotic Flexible Assembly Cells via Fuzzy-Based Taguchi Approach Khalid Abd, Kazem Abhary, Romeo Marian PII: DOI: Reference:
S0360-8352(16)30263-7 http://dx.doi.org/10.1016/j.cie.2016.07.028 CAIE 4425
To appear in:
Computers & Industrial Engineering
Received Date: Accepted Date:
29 November 2013 27 July 2016
Please cite this article as: Abd, K., Abhary, K., Marian, R., Multi-Objective Optimisation of Dynamic Scheduling in Robotic Flexible Assembly Cells via Fuzzy-Based Taguchi Approach, Computers & Industrial Engineering (2016), doi: http://dx.doi.org/10.1016/j.cie.2016.07.028
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Multi-Objective Optimisation of Dynamic Scheduling in Robotic Flexible Assembly Cells via Fuzzy-Based Taguchi Approach
*Khalid Abd School of Engineering, University of South Australia, Adelaide 5095, South Australia. Tel.: +61-8-830-23552; E-mail address: [email protected]
Kazem Abhary School of Engineering, University of South Australia, Adelaide 5095, South Australia. Tel.: +61-8-830-23475; fax: +61 8 830 23380. E-mail address: [email protected]
Romeo Marian School of Engineering, University of South Australia, Adelaide 5095, South Australia. Tel.: +61-8-830-25275; fax: +61 8 830 23380. E-mail address: [email protected]
*Corresponding author
Multi-Objective Optimisation of Dynamic Scheduling in Robotic Flexible Assembly Cells via Fuzzy-Based Taguchi Approach
Khalid Abd, Kazem Abhary and Romeo Marian
Abstract This paper presents Taguchi method coupled with fuzzy logic for dealing with multi-objective optimization problems for dynamic scheduling in robotic flexible assembly cells (RFACs). This is the first study to address these particular problems. In this study, Taguchi optimization method has been applied to reduce the number of experiments required for scheduling RFACs. The experiments are implemented with four different scheduling factors, namely sequencing rule, dispatching rule, cell utilisation and due date tightness. These factors are difficult to optimise considering the objectives of multiple functions instead of a single objective. Therefore, a multiple performance characteristics index (MPCI) based fuzzy logic approach has been developed to derive the optimal solution. The predicted results of MPCIs have been verified via a confirmation test. Results of the confirmation test show significant improvement in MPCI using the optimal levels of the scheduling factors.
Keywords: Taguchi's method, Fuzzy logic, Scheduling, Robotics, Assembly cells
1. Introduction Flexible manufacturing systems have attracted significant attention in recent years, due to their flexibility and dexterity in dealing with unexpected events. One class of such systems is called robotic flexible assembly cells (RFACs). RFACs are highly modern systems, structured with industrial robot(s), assembly stations and an automated material handling system, all monitored by computer numerical control (Manivannan, 1993; Marian, Kargas, Luong, & Abhary, 2003; Sawik, 1999). The design of RFACs with multi robots leads to increased productivity in a shorter cycle time and with lower production costs (Xidias, Zacharia, & Aspragathos, 2010). However, there are certain difficulties that have arisen with this design concept. For example, more than one robot operating simultaneously in the same work environment requires a complex control system to prevent collisions between robots (Nof & Chen, 2003), and also to prevent deadlock problems (Lee & Lee, 2002). Moreover, industrial robots must be employed as effectively as possible due to high cost of the robots (Xidias, et al., 2010). To overcome the above difficulties, efficient scheduling of RFACs is required. Few studies have been devoted to scheduling RFACs. These studies may be categorised according to the approaches adopted. In the first category are those studies which applied heuristic approaches to solve scheduling problems such as (Jiang, Seneviratne, & Earles, 1998; Lee & Lee, 2002; Lin, Egbelu, & Wu, 1995; Nof & Drezner, 1993; Pelagagge, Cardarelli, & Palumbo, 1995; Rabinowitz, Mehrez, & Samaddar., 1991; Sawik, 1995). The studies in the second category which investigated simulation as an approach to scheduling RFACs, for instance, (Basran, Petriu, & Petriu, 1997), (Gilbert, Coupez, Peng, & Delchambre, 1990) and (Hsu & Fu, 1995). There are only two studies in the third category, by (Del Valle & Camacho, 1996) and (Van Brussel, Cottrez, & Valckenaers, 1990) who implemented expert systems approaches to solve scheduling problems. The major limitation of all the above studies concentrated only on assembly of one type of product at a time. Scheduling RFACs in a multi-product assembly environment is presented in four recent studies: first, a scheduling scheme of RFACs ( Abd, Abhary, & Marian, 2011a); second, a strategy for scheduling RFACs using simple scheduling rules (Abd, Abhary, & Marian, 2011b); third, a methodology to select the best scheduling rule of RFACs using a multiple criteria decision-making method (Abd, Abhary, & Marian, 2011c); and fourth, a new
scheduling rule based on Fuzzy Logic for scheduling RFACs and then validating the performance of the suggested rule using a simulation program (Abd, Abhary, & Marian, 2012a; Abd, Abhary, & Marian, 2102b). Even though these recent studies have been devoted to scheduling RFACs in a multi-product assembly environment, they concentrated only on the static scheduling of RFACs. The dynamic scheduling of RFACs adds more complexity to finding optimal solutions to the multi-objective optimisation problems. As far as we know, this is the first study that addresses these particular problems. Therefore, the main contribution of this paper is to apply an intelligence approach to deal with multi-objective optimisation scheduling problems for RFACs in a dynamic environment. In the last few years, the application of Taguchi based fuzzy logic approach, to solve optimisation problems in a wide range of applications, has attracted significant attention, for example, in engineering design (Sun, Fang, & Hsueh, 2012; Lin & Kuo, 2011), the electronics industry (Tsai, 2011) and material cutting (Gupta, Singh, & Aggarwal, 2011; Hsiang, Lin, & Lai, 2012; Pandey, & Dubey, 2012; Sharma, Chattopadhyaya, & Hloch, 2011). In this study, a Taguchi approach incorporated with a fuzzy logic approach is applied to multi-objective optimisation problems of scheduling RFACs.
2.
Fuzzy logic based Taguchi optimisation methodology
The Taguchi approach has been developed as an optimisation technique by Genichi Taguchi since the 1950s. The potential benefit of the Taguchi method is its ability to solve complex problems by drastically reducing the number of experiments to be performed, accordingly reducing the cost of experiments (Bendell, Disney, & Pridmore, 1989; Ross, 1988). Taguchi designed special orthogonal arrays based on the number of factors affecting the decision and their levels. These arrays determine the number of necessary experiments. Taguchi defined a performance measure called signal-to-noise (S/N) ratio. The S/N ratio characteristic is classified into three categories depending on the goal of the problem: the smaller the better, the larger the better and the nominal the best (Lee, 2000). According to Taguchi, The S/N ratio results can be analysed using the analysis of mean (ANOM) and analysis of variance (ANOVA), to determine the optimal conditions affecting the performance characteristics (Mori, 1990; Taguchi, 1993; Taguchi, Elsayed, & Hsaing, 1989).
For a single performance measure, in the Taguchi method, the optimum level of the process parameters is the one having the highest S/N ratio. Multi-performance measure optimisation is not as straightforward as that of a single process response optimization. An overall evaluation of S/N ratios is required for the optimisation of the multi-process response due to the fact that a higher S/N ratio for one process response may correspond to a lower S/N ratio for another process response (Rao, 2011). To overcome this problem, a multiple performance characteristics index (MPCI) based fuzzy logic approach is developed to derive the optimal solution. Fuzzy logic (FL) was introduced first by Zadeh (1965). FL is a nonlinear mapping of an input data vector into a scalar output. In general, a fuzzy logic system (FLS) consists of four components (Mendel, 1992; Zadeh, 1976): knowledge base, fuzzification, inference engine and defuzzification. The knowledge base stores both membership functions (MF) and fuzzy rules. A membership function (MF) embodies a fuzzy set à graphically. The values of the MF are between 0 and 1, denoted by µÃ (x) where x is an element of Ã; these values are called degree of membership. Triangular and trapezoidal, as shown in Figure 1, are the most well-known of membership functions shapes (Mendel, 1992). A fuzzy rule is structured to control the output variable, such rules reflect a human reasoning mechanism. A fuzzy rule has two parts; the antecedent and the consequent IF THEN . µÃ(x)
Ã
1
Degree of membership
µÃ(x)
Triangular
0.5
0.5
0
Trapezoidal Ã
1
a
b
c
0
x a
b
d
e
Figure 1. Two examples of fuzzy numbers, triangular and trapezoidal
Fuzzification represents the process of converting the S/N ratios obtained by the Taguchi method into fuzzy inputs, using the membership functions. The inference engine maps from fuzzy input to fuzzy output, using IF-THEN type fuzzy rules. Defuzzification translates the fuzzy output into a MPCI, using the membership functions of the output variable. In this study, the Taguchi approach is coupled with fuzzy logic for multi-objective optimisation problems of scheduling RFACs. The proposed optimisation methodology comprises five steps as depicted in Figure 2.
i.
Identifying the scheduling problems’ characteristics, and then determining scheduling factors and the number of levels for each factor.
ii.
Choosing the appropriate orthogonal array (L9) to conduct the experiments, and transforming the analytical results of the experiments into S/N ratio for each objective function. The experimental runs will be conducted under different combinations of scheduling factors using simulation software package SIMPROCESS (Swegles, 1997).
iii.
Converting the S/N results to a value range between 0 and 1. This process is called normalisation.
iv.
Applying fuzzy approach, using fuzzified, inference engine and defuzzified operation to obtain the multiple performance characteristics index (MPCI).
v.
Analysing and verifying the results statistically, using a variance analysis (ANOVA) and confirmation test respectively.
Initialisation Start
Identify the Control Factors and Their Levels Determine Level Number of Each Factor
Fuzzy Logic Approach
Statistical Analysis
Fuzzification
Determine the Optimal Level of Each Factor
Fuzzy Input
Inference Engine Fuzzy output
Knowledge Base
Define RFACs Scheduling Problems and Objectives
vi. vii. viii. Select the Appropriate Taguchi Orthogonal Array ix. x. Conduct Experiments xi. Calculate the objective values xii. Compute the S/N Ratio forxiii. Each of the Objectives xiv. xv. Normalisation of the S/N xvi. Values xvii. Taguchi Approach
Defuzzification Crisp Output (MCPI)
Determine the Optimal Factor Verify the experimental results
End
Figure 2. Proposed methodology
3 Outline of the RFACs model 3.1 RFACs description The present RFACs model consists of six resources: two robots (R 1 and R2) for fetching the assembled parts and placing them at assembly stations (AS1, AS2 and AS3), parts feeder (PF) for supplying parts to the cell, gripper changing station (GC), input conveyors (IC 1 & IC2) for supplying the base parts, and output conveyor (OC) for conveying out a final product when assembly processes are completed. Figure 3 shows the configuration of the RFACs to be studied.
The presented RFACs model is assumed to assemble
product types (P1, P2, … Pn).
Each product is considered as an independent job. In this study, six products are taken as an example. Table 1 shows the details of required stations along with assembly operations time for each product type. This table also includes the time of activities that are carried out by utilising robots to assemble products.
Figure 3. A robotic flexible assembly cell
Table 1. Assembly operations requirements Time of Assembly operations
Assembly Station
P1
P2
P3
P4
P5
P6
Insert lens on front cover
S1
4
3
3
4
3
4
Insert Keypad on Front Cover
S1
5
4
5
6
4
6
Assemble PC Board with Front Cover
S2
6
8
10
9
8
9
Insert Antenna on Back Cover
S3
9
0
0
9
0
0
Assemble Back Cover with Front Cover
S2
7
11
10
11
7
10
Robot movement time (Sec.)
23
17
17
23
17
17
Robot gripper pickup & release time (Sec.)
6
4
4
6
4
4
Total Processing time for each product (Sec.)
60
47
49
68
43
50
Description
In order to simulate the RFACs model, six customers’ orders are considered. These orders are labelled as Order1, Order2… Order6, as shown in Table 2. Batch size of each product type is also given in Table 2.
Table 2. Data of customer orders Product type
Batch size P3 P4 25 -
Order1
P1 20
P2 25
P5 20
P6 25
Order2
30
40
50
40
-
30
Order3
-
25
40
30
20
35
Order4
25
-
20
25
-
20
Order5
30
-
30
30
30
40
Order6
45
20
35
35
50
20
3.2 Definition of problem The scheduling of the RFAC requires finding a way to determine how to use the cell resources in an optimal manner to assemble multi-products. Let us consider that customer orders {O = 1,2,….m} are processed in RFACs. Each order consists of a set of products {P=1, 2,….n}. These products go through specific assembly operations {op = op1n, op2n,… opin} that have to be implemented by a set of robots {R = 1,2,….t}. The robots have multipurpose end effectors. Each robot can perform only one job at a time and each job can be processed by only one robot. Robots are continuously available, which means that no robot breaks down in the assembly cell. The RFACs scheduling problem is subject to three classes of constraints, namely robot motion constraints, robot access constraints and tooling resource constraints. Robot motion constraints: robot arms cannot move from one place to another directly. The reason for this is to avoid collision with the other robot arms. This is achieved by assigning control points in the cell. Control points {C1, C2,… C4} are set to simplify path planning and avoid collision. For example, R1 cannot move from AS1 to PF directly. To move from AS1 to PF, R1 should move via control point C1, as shown in Figure 4.
Figure 4. Robot move constraints
Robot access constraints: to prevent collisions between robots in a shared area, more than one robot cannot access the same resource simultaneously. For instance, just one robot, R1 or R2, can access OC or AS1 or GC or PF or AS2 at a time. Tooling resource constraints: to fetch and assemble parts, the hand of each robot should be equipped with the right tool; however, a specific tool may not be available for the two robots at the same time, due to the restricted number of available tools. 3.3 Objectives functions Three common objective functions, namely makespan (Cmax), total tardiness (TD) and number of tardy jobs (NT), are to be minimised. The objectives are described using the following equations:
where CPO and DPO denote the completion time and due date of product P in order O respectively, and UP is the indicator for whether product P is tardy or not. 4. Experimental setup The scheduling of any flexible manufacturing system contains a number of decision points that affect the system’s performance. These decision points can be identified via scheduling stages. At each decision point, different rules can be utilized for decision making. For example, the decision making for job selection usually includes the smallest number of jobs in the system’s queue. In this study, order selection and job-robot selection are considered as scheduling factors. Figure 5 shows the decision points in the RFACs. Decision making 2
Decision making 1
Order Selection
Order n
Order
Job n Job n Job n
Job-Robot Selection
Robot 1 Robot 2 Robot R
Figure 5. The decision points in the RFACs
Sequencing rules: When more than one order is waiting for processing, the orders will be sequenced, from the highest order priority to the lowest order priority, using sequencing rules. Three types of sequencing rules are applied (Chan, Chan, & Lau, 2002; Ozbayrak & Bell, 2003), as shown in Table 3. Table 3. Sequencing rules for the cell to select the order of jobs waiting for processing Sequencing rule
Description
FCFS
Select the order according to the rule First Come First Serve The order with total shortest processing time will be selected The order with total longest processing time will be chosen
TSPT TLPT
Dispatching rules: After selecting and loading an appropriate order into the system, the order of jobs has to be dispatched to the available robots. Three types of dispatching rules are used (Chan, Chan, & Lau, 2002; Chan, Chan, & Kazerooni, 2003), as shown in Table 4.
Table 4. Dispatching rules for the robot to select the next job Dispatching rule SNQ SIO WINQ
Description Job will be selected according to the smallest number in the queue The job with the Shortest Imminent Operation time will be chosen Select the job in the queue which requires the least work
The real context of manufacturing is often dynamic and faces disruptions - referred to as real-time events - which can change system performance and status (Gholami, Zandieh, & Alem-Tabriz, 2009). Real time events are categorised into two types (Vieira, Hermann, & Lin, 2003): the first is resource-related such as resource breakdown, tool failure, loading limits, shortage of material; the second is job-related, such as due date changing, early or late arrival time of jobs, changing of processing time, urgency of jobs, job cancellation and so on. Lu & Liu (2010) and Vinod & Sridharan (2008) indicate that the due date and the arrival time of jobs are the most significant events that affect job shop scheduling. Therefore, in this study, arrival time and due date are considered as the main as another two factors. Arrival time: The arrival time is defined as the degree of workload in the system. A low workload means a long arrival time for jobs. Conversely, a high workload means a short
arrival time. In recent studies of job shop scheduling, the arrival times have been assumed to follow an exponential distribution (Lu & Liu, 2010; Nie, Gao, Li, & Li, 2012). The mean inter-arrival time of jobs is calculated using the following formula:
where
is mean inter-arrival time for jobs,
is mean number of operations per job,
is the mean processing time per job,
is shop utilisation; and
is number of machines
in the system. In the present study, the Equation 4 is used by replacing
with
(number of
robots in the system). Due date: In a dynamic scheduling environment, the total work content (TWK) rule has been extensively used for due date assignment (Ramasesh, 1990). For the TWK rule, the due date of each job is set using the following equation:
where
and
are the due date and arrival time of job
total processing time of job , and
is the
is due date tightness factor. Usually, the range of
between 1 and 10. A high value of Conversely, a low value of
respectively,
is
means the required time to finish the job is looser.
means the required time to finish the job is tighter (Lu & Liu,
2010). In this study, the four factors, sequencing rule (SR), dispatching rule (DR), cell utilisation (U) and due date tightness (K), which influence the scheduling of RFACs, are considered. These factors are set with different levels to explore the effect of the proposed model. The scheduling factors are summarised in Table 5. Table 5. Factors and their control levels Factor
Symbol
Level
Type
Sequencing Rules
A
Dispatching Rules
B
Cell Utilisation
C
Due Date Tightness
D
A(1) A(2) A(3) B(1) B(2) B(3) C(1) C(2) C(3) D(1) D(2) D(3)
FCFS TSPT TLPT SNQ SIO WINQ Low – 75% Moderate – 85% High – 95% Loose – 6 Moderate – 4 Tight – 2
5. Experimental design and results 5.1 Taguchi’s orthogonal array selection Taguchi has developed a pattern of tabulated orthogonal arrays based on the number of factors and their levels. The orthogonal array determines the number of possible experiments. In this investigation, four scheduling factors are taken into consideration with three levels of each factor. Consequently, a total of 81 (3×3×3×3) different combinations are considered. However, according to Taguchi method, the number of experiments can be reduce from 81 to 9 using the standard orthogonal array L9 (34). Table 6 illustrates the arrangement of the experimental design of this study which corresponds to orthogonal array L9. Table 6. Standard L9 (34) orthogonal array Trial No. 1 2 3 4 5 6 7 8 9
A A(1) A(1) A(1) A(2) A(2) A(2) A(3) A(3) A(3)
Levels of control factors B C D B(1) C(1) D(1) B(2) C(2) D(2) B(3) C(3) D(3) B(1) C(2) D(3) B(2) C(3) D(1) B(3) C(1) D(2) B(1) C(3) D(2) B(2) C(1) D(3) B(3) C(2) D(1)
5.2 Calculation of the signal-to-noise (S/N) ratio The present study uses a robust design criterion called the signal-to-noise (S/N) ratio. The S/N ratio is considered as the key step in Taguchi method and reflects the factor performance. Usually, the S/N ratio characteristics are classified into three types: the smaller the better, the larger the better and nominal the best. Each type has a specific formula for calculating the S/N ratio. In scheduling problems, nearly all objective functions are classified in the smaller the better type; their corresponding S/N ratio is as follows (Phadke, 1989):
Table 7 shows the experimental results of the makespan (Cmax), the total tardiness (TD) and the number of tardy jobs (NT), with their robust design criteria being based on the combinations of experimental factors as shown in Table 7 and Figure 6.
Table 7. Details of the combinations of different levels and their results Trial
Levels of control factors B C D
A
Makespan Cmax S/N
Total Tardiness TD S/N
Number of tardy jobs NTD S/N
1
FCFS
SNQ
Low
Loose
38499
-91.71
10070
-80.061
4
-12.041
2
FCFS
SIO
Moderate
Moderate
34736
-90.82
19124
-85.632
12
-21.584
3
FCFS
WINQ
High
Tight
31571
-89.99
50161
-94.007
22
-26.848
4
TSPT
SNQ
Moderate
Tight
35387
-90.98
28904
-89.219
18
-25.105
5
TSPT
SIO
High
Loose
32488
-90.23
5666.0
-75.066
2
-6.0210
6
TSPT
WINQ
Low
Moderate
42356
-92.54
9909.0
-79.921
7
-16.902
7
TLPT
SNQ
High
Moderate
33003
-90.37
2994.0
-69.525
6
-15.563
8
TLPT
SIO
Low
Tight
39999
-92.04
31792
-90.046
19
-25.575
9
TLPT
WINQ
Moderate
Loose
36310
-91.20
542.00
-54.680
2
-6.0210
-70
-90
S/N of TD
S/N of Cmax
-80
-91
A
-92 1
B
C
D
A
-90
2
B
1
3
C 2
D 3
-5
S/N of NT
-15
A
-25 1
B
C 2
D 3
Figure 6. Factors responses graph for makespan (Cmax), total tardiness (TD) and number of tardy jobs (NT)
S/N ratio ranges for the three responses, makespan (Cmax), total tardiness (TD) and number of tardy jobs (NT) are all different. To avoid the different ranges, the values of the S/N ratio results must be normalised to values between 0 and 1. In this case, since the objective is the ‘smaller the better’ type, 0 means the best performance while 1 denotes the worst. The normalisation is determined as follows:
where uik is the normalised S/N value for the ki response in the ith trial; ƞi(k) is the value of S/N; Min ƞi(k) and Max ƞi(k) are the smallest value of ƞi(k) and the largest value of ƞi(k) respectively. The normalisation data are shown in Table 8. Table 8. Normalisation results of S/N ratio
Trial
S/N of Cmax
μ Cmax
S/N of TD
μ TD
S/N of NT
μ NT
1 2 3 4 5 6 7 8 9
-91.71 -90.82 -89.99 -90.98 -90.23 -92.54 -90.37 -92.04 -91.20
0.32 0.67 1.00 0.61 0.90 0.00 0.85 0.19 0.52
-80.061 -85.632 -94.007 -89.219 -75.066 -79.921 -69.525 -90.046 -54.680
0.35 0.21 0.00 0.12 0.48 0.36 0.62 0.10 1.00
-12.041 -21.584 -26.848 -25.105 -6.021 -16.902 -15.563 -25.575 -6.021
0.71 0.25 0.00 0.08 1.00 0.48 0.54 0.06 1.00
6. Fuzzy logic implementation and results for MCPI In order to find out the optimum scheduling for multi-objective problems in RFACs, fuzzy logic is used. In the present study, the fuzzy inference system contains three input parameters and one output parameter. The input parameters are the S/N ratio of Cmax, TD and NT, while MPCI is the output parameter. Three steps are required for the fuzzy approach implementation to generate MPCI. Firstly, the fuzzy sets for the inputs and output are determined. In this study, the S/N ratios of Cmax, TD and NT are break down into a set of linguistic values for the inputs: low (L), medium (M), and high (H); while the output is set into seven linguistic values: tiny (T), very small (VS), small (S), medium (M), large (L), very large (VL) and huge (H). Table 9 shows the linguistic values of inputs/output. Table 9. Input and output variables with their fuzzy values System variable
Linguistic variables
Inputs
S/N ratios for, Cmax, TD and NT
Output
MPCI
Linguistic Value
Term Set
Numerical Range
Low Medium High Tiny Very Small Small Medium Large Very Large Huge
L M H T VS S M L VL H
[0– 0.25] [0.25–0.75] [0.75–1] [0– 0.167] [0– 0.334] [0.167 – 0.5] [0.334 – 0.667] [0.5 – 0.834] [0.667 – 1] [0.834 –1]
Secondly, the membership functions for inputs/output are plotted. There are different types of membership functions’ shapes such as triangular, trapezoidal, Gaussian, singleton, etc. The triangular shape is the common membership functions shape and a powerful way to approach the convex function (Pedrycz, 1994). The membership plot for the inputs/output parameters using the triangular shape is shown in Figure 7.
Degree of membership
0.5
T VS S M L VL H
1
Degree of membership
L M H
1
0.5
0
0 0
0. 5
1
0
0. 5
1
Output variable
Input variable
Figure 7. Input / output of Memberships functions
Finally, to control the output parameter (MPCI), fuzzy rules are structured. Fuzzy rules are derived directly based on the formula (nm), where
and
denote input parameters
3
and their linguistic values. Thus, the number of fuzzy rules is 3 = 27, as illustrated in Table 10. Each rule is mathematically evaluated through a process named implication. In this study, Mamdani implication is applied (Klir & Yuan, 2005). Table 10. Fuzzy rules table S/N of Cmax
S/N ratio for S/N of TD
Low
Low
Medium High
Medium
Low
Medium High
High
Low Medium High
S/N of NTD Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High
MPCI Tiny Very Small Small Very Small Small Medium Small Medium Large Very Small Small Medium Small Medium Large Medium Large Very Large Small Medium Large Medium Very Large Very Large Large Very Large Huge
The evaluation values of each fuzzy rule are combined into one single output value via the defuzzification process. In this study, the well-known method for the defuzzification process which is called the centre of gravity (COG) (Mendel, 1992) is used to transform the fuzzy inference output µÃ (x) into the non-fuzzy value x'. The non-fuzzy value x' is called MPCI, based on the following equation:
In this study, the fuzzy logic toolbox in MATLAB software is used to construct the fuzzy inference system of the MPCI. Figure 8 shows the fuzzified rule viewer for MPCI, which can accept any value of the S/N ratio for the three input parameters: Cmax, TD and NT. The output (MPCI) in this figure can be interpreted easily, for example, as in the following: IF the S/N of Cmax is (0.32), the S/N of TD is (0.35) and S/N of NTD is (0.71) THEN MCPI will be (0.449). Figure 9 presents the surface view graphs corresponding to the particular fuzzy rules of the implemented fuzzy system. This Figure shows the effects of the combination of inputs parameters on the MPCI. Table 11 shows the MPCI values corresponding to each experimental run obtained by using the fuzzy system.
S/N of Cmax = 0.32
S/N of TD = 0.35
S/N of NTD = 0.71
Figure 8. Final output of fuzzy rules
MCPI = 0.449
Figure 9. 3D surface plots of the inputs/output
Table 11. MPCI values obtained by fuzzy system Trial
μ Cmax
μ TD
μ NT
MCPI
1 2 3 4 5 6 7 8 9
0.32 0.67 1.00 0.61 0.90 0.00 0.85 0.19 0.52
0.35 0.21 0.00 0.12 0.48 0.36 0.62 0.10 1.00
0.71 0.25 0.00 0.08 1.00 0.48 0.54 0.06 1.00
0.449 0.419 0.50 0.383 0.833 0.267 0.840 0.058 0.834
Trial number 7 gives the highest MPCI value among the nine trials, as shown in Table 11. Thus, A3B3C3 and D2 (refer to table 6) are selected to be the initial levels for the scheduling factors. 7. Analytical results and discussion 7.1 Effect of scheduling factors on MPCI The main factor effect can be assessed from the mean of each factor level, using the following mathematical expression (Phadke, 1989; Ranjitk, 2001), and summarised in Table 12.
where MPCIji is the mean of factor j at level i; k is the number of levels in factor j; and ƞji is the value of MPCI with factor
at level i. The highest value of MPCI among all
combinations of the factors denotes the optimum level for each factor. As shown in Table 12, it is clear that the optimal levels of A (Sequencing Rule), B (Dispatching Rule), C (Cell
Utilisation) and D (Due Date Tightness) are 3, 1, 3 and 3 respectively, due to their MPCI value. The significant scheduling factor also can be determined by calculating the value of difference between the maximum and minimum for each factor. The greatest difference is the most significant factor. It is noticed, that factor C (Cell Utilisation) has the strongest effect on the scheduling of RFACs, followed by factor D (Due Date Tightness). It is also seen that factor B (Dispatching Rule) has weakest impact on the scheduling. Table 12. Response table of the MPCI values Factor
MPCI
Max-Min
Rank
0.122
3
0.120
4
0.466
1
0.391
2
A(1)
(ƞ1 + ƞ2 + ƞ3 ) / 3
0.456
A(2)
(ƞ4 + ƞ5 + ƞ6 ) / 3
0.494
A(3)
(ƞ7 + ƞ8 + ƞ9 ) / 3
0.578
B(1)
(ƞ1 + ƞ4 + ƞ7 ) / 3
0.557
B(2)
(ƞ2 + ƞ5 + ƞ8 ) / 3
0.437
B(3)
(ƞ3 + ƞ6 + ƞ9 ) / 3
0.534
C(1)
(ƞ1 + ƞ6 + ƞ8 ) / 3
0.258
C(2)
(ƞ2 + ƞ4 + ƞ9 ) / 3
0.545
C(3)
(ƞ3 + ƞ5 + ƞ7 ) / 3
0.724
D(1)
(ƞ1 + ƞ5 + ƞ9 ) / 3
0.705
D(2)
(ƞ2 + ƞ6 + ƞ7 ) / 3
0.509
D(3)
(ƞ3 + ƞ4 + ƞ8 ) / 3
0.314
The relative effect among the scheduling factors for the MPCI can also be verified using analysis of variance (ANOVA) as summarised in Table 13. From ANOVA results, the effect of each scheduling factor on the MPCI becomes apparent. The factors C (Cell Utilisation) and D (Due Date Tightness) contribute just over 92% of the total variance relating to the MPCI. Factors B (Dispatching Rule) and A (Sequencing Rule) contribute nearly 8%. The ANOVA results are similar to the ‘Mix-Min’ analysis described in Table 12. Table 13. ANOVA of MPCI Factor
DOF
SS
MS
PC%
A
2
0.023
0.012
3.78%
B
2
0.025
0.012
4.02%
C
2
0.332
0.166
54.46%
D
2
0.230
0.115
37.74%
Error
0
0.000
0.000
0.00%
total
8
0.610
0.076
100.00%
7.2 Confirmation test The final step is to predict and verify the improvement of the performance characteristics using the optimal levels of the scheduling factors. The best value of the MPCI can be predicted using the following equation (Montgomery, 2004):
where ƞm is the total mean of the MPCI; ƞi is the mean of the MPCI at the optimal level and q is the number of scheduling factors. Based on equation 10, the predicted MPCI for initial and optimal levels of scheduling factors can then be calculated as shown below: For the initial levels, as mentioned earlier, A3B3C3 and D2 are selected to be the initial levels for the scheduling factors: MPCI Initial = 0.509 + (0.578 - 0.509) + (0.557 - 0.509) + (0.724 - 0.509) + (0.509 - 0.509) = 0.841 For optimal levels A3B1C3 and D1 according to Table 12, the MPCI is estimated in the same way: MPCI Optimal = 0.509 + (0.578 - 0.509) + (0.557 - 0.509) + (0.724 - 0.509) + (0.705 - 0.509) = 1.037 MPCIconfirmed is also determined for both the initial and optimal levels of the scheduling factors. Table 5 shows that trial 7 provides the largest MPCI value which represents MPCIInitial, while MPCIoptimal can be graphically calculated by obtaining the value of the optimal levels with the fuzzy logic rule. The percentage prediction error (PPE) is calculated in order to check the closeness of the predicted value to that obtained by the actual experimental value (Pandey & Dubey, 2012).
The results of the confirmation run for the initial and optimal levels of the scheduling factors are summarised in Table 14. The results indicate that the optimal levels of scheduling factors produce the best MPCI among all trials. In other words, it can be seen that the MPCI value of the optimal level of scheduling factors is improved by approximately 0.11 compared to the initial level of scheduling factors. The results also show that the PPE is acceptable (less than 10%). In summary, it can be said that the experimental results are confirmed.
Table 14. Predicted results for Initial and Optimal condition Initial condition
Optimal condition
Level
A3B1C3D2
A3B1C3D1
MPCI prediction
0.841
1.037
MPCI confirmed
0.840
0.948
PPE
0.001%
9.38%
Improvement MPCI = 11%
8. Conclusion In this study, a hybrid approach obtained by integrating the Taguchi method with fuzzy logic theory has been utilised to achieve multi-objective optimisation of dynamic scheduling in RFACs. The experimental results are analysed using different statistical analysis techniques to find the best combination of scheduling factors to improve the MPCI. The results are then verified through a confirmation test. The main findings of the study are given below: 1- The selected hybrid approach is efficient and effective for finding the optimal scheduling of RFACs with fewer experiments compared to full factorial experimental methods. 2- The optimum values of different factors have been found as to be sequencing rule TLPT, dispatching rule SNQ, cell utilisation 95% and due date tightness 6. 3- The most significant factors affecting the scheduling strategy have been identified as factor C (Cell Utilisation) and factor D (Due Date Tightness), which account for nearly 92%. 4- A percentage predication error of less than 10% has been obtained. This percentage indicates that the suggested hybrid approach can be successfully used in the multiobjective optimisation of scheduling in RFACs. 5- The confirmed MPCI value of optimal scheduling factors produces the best result (0.948) against the initial setting (0.84). Therefore, the MPCI increased by 11% compared to the initial setting. For future work, expand the proposed study to be more applicable to real scheduling problems, by taking into account robot breakdowns during the scheduling of RFACs. Three parameters may be considered in the scope of this area of study: (1) the number of robots subject to breakdown (2) breakdown frequency which represents how many times each robot will break down during the scheduling horizon (3) repair duration which represents the mean time required to repair the robot after its breakdown.
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Highlights
To our knowledge, this is first study that addresses dynamic scheduling in RFACs.
Cell utilisation and Due Date Tightness are the significant scheduling factors.
Percentage of prediction error (9%) confirms the efficient of proposed methodology.
The Results of confirmation run improved by 11% compared to initial experiments.
S0360-8352(16)30263-7 http://dx.doi.org/10.1016/j.cie.2016.07.028 CAIE 4425
To appear in:
Computers & Industrial Engineering
Received Date: Accepted Date:
29 November 2013 27 July 2016
Please cite this article as: Abd, K., Abhary, K., Marian, R., Multi-Objective Optimisation of Dynamic Scheduling in Robotic Flexible Assembly Cells via Fuzzy-Based Taguchi Approach, Computers & Industrial Engineering (2016), doi: http://dx.doi.org/10.1016/j.cie.2016.07.028
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Multi-Objective Optimisation of Dynamic Scheduling in Robotic Flexible Assembly Cells via Fuzzy-Based Taguchi Approach
*Khalid Abd School of Engineering, University of South Australia, Adelaide 5095, South Australia. Tel.: +61-8-830-23552; E-mail address: [email protected]
Kazem Abhary School of Engineering, University of South Australia, Adelaide 5095, South Australia. Tel.: +61-8-830-23475; fax: +61 8 830 23380. E-mail address: [email protected]
Romeo Marian School of Engineering, University of South Australia, Adelaide 5095, South Australia. Tel.: +61-8-830-25275; fax: +61 8 830 23380. E-mail address: [email protected]
*Corresponding author
Multi-Objective Optimisation of Dynamic Scheduling in Robotic Flexible Assembly Cells via Fuzzy-Based Taguchi Approach
Khalid Abd, Kazem Abhary and Romeo Marian
Abstract This paper presents Taguchi method coupled with fuzzy logic for dealing with multi-objective optimization problems for dynamic scheduling in robotic flexible assembly cells (RFACs). This is the first study to address these particular problems. In this study, Taguchi optimization method has been applied to reduce the number of experiments required for scheduling RFACs. The experiments are implemented with four different scheduling factors, namely sequencing rule, dispatching rule, cell utilisation and due date tightness. These factors are difficult to optimise considering the objectives of multiple functions instead of a single objective. Therefore, a multiple performance characteristics index (MPCI) based fuzzy logic approach has been developed to derive the optimal solution. The predicted results of MPCIs have been verified via a confirmation test. Results of the confirmation test show significant improvement in MPCI using the optimal levels of the scheduling factors.
Keywords: Taguchi's method, Fuzzy logic, Scheduling, Robotics, Assembly cells
1. Introduction Flexible manufacturing systems have attracted significant attention in recent years, due to their flexibility and dexterity in dealing with unexpected events. One class of such systems is called robotic flexible assembly cells (RFACs). RFACs are highly modern systems, structured with industrial robot(s), assembly stations and an automated material handling system, all monitored by computer numerical control (Manivannan, 1993; Marian, Kargas, Luong, & Abhary, 2003; Sawik, 1999). The design of RFACs with multi robots leads to increased productivity in a shorter cycle time and with lower production costs (Xidias, Zacharia, & Aspragathos, 2010). However, there are certain difficulties that have arisen with this design concept. For example, more than one robot operating simultaneously in the same work environment requires a complex control system to prevent collisions between robots (Nof & Chen, 2003), and also to prevent deadlock problems (Lee & Lee, 2002). Moreover, industrial robots must be employed as effectively as possible due to high cost of the robots (Xidias, et al., 2010). To overcome the above difficulties, efficient scheduling of RFACs is required. Few studies have been devoted to scheduling RFACs. These studies may be categorised according to the approaches adopted. In the first category are those studies which applied heuristic approaches to solve scheduling problems such as (Jiang, Seneviratne, & Earles, 1998; Lee & Lee, 2002; Lin, Egbelu, & Wu, 1995; Nof & Drezner, 1993; Pelagagge, Cardarelli, & Palumbo, 1995; Rabinowitz, Mehrez, & Samaddar., 1991; Sawik, 1995). The studies in the second category which investigated simulation as an approach to scheduling RFACs, for instance, (Basran, Petriu, & Petriu, 1997), (Gilbert, Coupez, Peng, & Delchambre, 1990) and (Hsu & Fu, 1995). There are only two studies in the third category, by (Del Valle & Camacho, 1996) and (Van Brussel, Cottrez, & Valckenaers, 1990) who implemented expert systems approaches to solve scheduling problems. The major limitation of all the above studies concentrated only on assembly of one type of product at a time. Scheduling RFACs in a multi-product assembly environment is presented in four recent studies: first, a scheduling scheme of RFACs ( Abd, Abhary, & Marian, 2011a); second, a strategy for scheduling RFACs using simple scheduling rules (Abd, Abhary, & Marian, 2011b); third, a methodology to select the best scheduling rule of RFACs using a multiple criteria decision-making method (Abd, Abhary, & Marian, 2011c); and fourth, a new
scheduling rule based on Fuzzy Logic for scheduling RFACs and then validating the performance of the suggested rule using a simulation program (Abd, Abhary, & Marian, 2012a; Abd, Abhary, & Marian, 2102b). Even though these recent studies have been devoted to scheduling RFACs in a multi-product assembly environment, they concentrated only on the static scheduling of RFACs. The dynamic scheduling of RFACs adds more complexity to finding optimal solutions to the multi-objective optimisation problems. As far as we know, this is the first study that addresses these particular problems. Therefore, the main contribution of this paper is to apply an intelligence approach to deal with multi-objective optimisation scheduling problems for RFACs in a dynamic environment. In the last few years, the application of Taguchi based fuzzy logic approach, to solve optimisation problems in a wide range of applications, has attracted significant attention, for example, in engineering design (Sun, Fang, & Hsueh, 2012; Lin & Kuo, 2011), the electronics industry (Tsai, 2011) and material cutting (Gupta, Singh, & Aggarwal, 2011; Hsiang, Lin, & Lai, 2012; Pandey, & Dubey, 2012; Sharma, Chattopadhyaya, & Hloch, 2011). In this study, a Taguchi approach incorporated with a fuzzy logic approach is applied to multi-objective optimisation problems of scheduling RFACs.
2.
Fuzzy logic based Taguchi optimisation methodology
The Taguchi approach has been developed as an optimisation technique by Genichi Taguchi since the 1950s. The potential benefit of the Taguchi method is its ability to solve complex problems by drastically reducing the number of experiments to be performed, accordingly reducing the cost of experiments (Bendell, Disney, & Pridmore, 1989; Ross, 1988). Taguchi designed special orthogonal arrays based on the number of factors affecting the decision and their levels. These arrays determine the number of necessary experiments. Taguchi defined a performance measure called signal-to-noise (S/N) ratio. The S/N ratio characteristic is classified into three categories depending on the goal of the problem: the smaller the better, the larger the better and the nominal the best (Lee, 2000). According to Taguchi, The S/N ratio results can be analysed using the analysis of mean (ANOM) and analysis of variance (ANOVA), to determine the optimal conditions affecting the performance characteristics (Mori, 1990; Taguchi, 1993; Taguchi, Elsayed, & Hsaing, 1989).
For a single performance measure, in the Taguchi method, the optimum level of the process parameters is the one having the highest S/N ratio. Multi-performance measure optimisation is not as straightforward as that of a single process response optimization. An overall evaluation of S/N ratios is required for the optimisation of the multi-process response due to the fact that a higher S/N ratio for one process response may correspond to a lower S/N ratio for another process response (Rao, 2011). To overcome this problem, a multiple performance characteristics index (MPCI) based fuzzy logic approach is developed to derive the optimal solution. Fuzzy logic (FL) was introduced first by Zadeh (1965). FL is a nonlinear mapping of an input data vector into a scalar output. In general, a fuzzy logic system (FLS) consists of four components (Mendel, 1992; Zadeh, 1976): knowledge base, fuzzification, inference engine and defuzzification. The knowledge base stores both membership functions (MF) and fuzzy rules. A membership function (MF) embodies a fuzzy set à graphically. The values of the MF are between 0 and 1, denoted by µÃ (x) where x is an element of Ã; these values are called degree of membership. Triangular and trapezoidal, as shown in Figure 1, are the most well-known of membership functions shapes (Mendel, 1992). A fuzzy rule is structured to control the output variable, such rules reflect a human reasoning mechanism. A fuzzy rule has two parts; the antecedent and the consequent IF
Ã
1
Degree of membership
µÃ(x)
Triangular
0.5
0.5
0
Trapezoidal Ã
1
a
b
c
0
x a
b
d
e
Figure 1. Two examples of fuzzy numbers, triangular and trapezoidal
Fuzzification represents the process of converting the S/N ratios obtained by the Taguchi method into fuzzy inputs, using the membership functions. The inference engine maps from fuzzy input to fuzzy output, using IF-THEN type fuzzy rules. Defuzzification translates the fuzzy output into a MPCI, using the membership functions of the output variable. In this study, the Taguchi approach is coupled with fuzzy logic for multi-objective optimisation problems of scheduling RFACs. The proposed optimisation methodology comprises five steps as depicted in Figure 2.
i.
Identifying the scheduling problems’ characteristics, and then determining scheduling factors and the number of levels for each factor.
ii.
Choosing the appropriate orthogonal array (L9) to conduct the experiments, and transforming the analytical results of the experiments into S/N ratio for each objective function. The experimental runs will be conducted under different combinations of scheduling factors using simulation software package SIMPROCESS (Swegles, 1997).
iii.
Converting the S/N results to a value range between 0 and 1. This process is called normalisation.
iv.
Applying fuzzy approach, using fuzzified, inference engine and defuzzified operation to obtain the multiple performance characteristics index (MPCI).
v.
Analysing and verifying the results statistically, using a variance analysis (ANOVA) and confirmation test respectively.
Initialisation Start
Identify the Control Factors and Their Levels Determine Level Number of Each Factor
Fuzzy Logic Approach
Statistical Analysis
Fuzzification
Determine the Optimal Level of Each Factor
Fuzzy Input
Inference Engine Fuzzy output
Knowledge Base
Define RFACs Scheduling Problems and Objectives
vi. vii. viii. Select the Appropriate Taguchi Orthogonal Array ix. x. Conduct Experiments xi. Calculate the objective values xii. Compute the S/N Ratio forxiii. Each of the Objectives xiv. xv. Normalisation of the S/N xvi. Values xvii. Taguchi Approach
Defuzzification Crisp Output (MCPI)
Determine the Optimal Factor Verify the experimental results
End
Figure 2. Proposed methodology
3 Outline of the RFACs model 3.1 RFACs description The present RFACs model consists of six resources: two robots (R 1 and R2) for fetching the assembled parts and placing them at assembly stations (AS1, AS2 and AS3), parts feeder (PF) for supplying parts to the cell, gripper changing station (GC), input conveyors (IC 1 & IC2) for supplying the base parts, and output conveyor (OC) for conveying out a final product when assembly processes are completed. Figure 3 shows the configuration of the RFACs to be studied.
The presented RFACs model is assumed to assemble
product types (P1, P2, … Pn).
Each product is considered as an independent job. In this study, six products are taken as an example. Table 1 shows the details of required stations along with assembly operations time for each product type. This table also includes the time of activities that are carried out by utilising robots to assemble products.
Figure 3. A robotic flexible assembly cell
Table 1. Assembly operations requirements Time of Assembly operations
Assembly Station
P1
P2
P3
P4
P5
P6
Insert lens on front cover
S1
4
3
3
4
3
4
Insert Keypad on Front Cover
S1
5
4
5
6
4
6
Assemble PC Board with Front Cover
S2
6
8
10
9
8
9
Insert Antenna on Back Cover
S3
9
0
0
9
0
0
Assemble Back Cover with Front Cover
S2
7
11
10
11
7
10
Robot movement time (Sec.)
23
17
17
23
17
17
Robot gripper pickup & release time (Sec.)
6
4
4
6
4
4
Total Processing time for each product (Sec.)
60
47
49
68
43
50
Description
In order to simulate the RFACs model, six customers’ orders are considered. These orders are labelled as Order1, Order2… Order6, as shown in Table 2. Batch size of each product type is also given in Table 2.
Table 2. Data of customer orders Product type
Batch size P3 P4 25 -
Order1
P1 20
P2 25
P5 20
P6 25
Order2
30
40
50
40
-
30
Order3
-
25
40
30
20
35
Order4
25
-
20
25
-
20
Order5
30
-
30
30
30
40
Order6
45
20
35
35
50
20
3.2 Definition of problem The scheduling of the RFAC requires finding a way to determine how to use the cell resources in an optimal manner to assemble multi-products. Let us consider that customer orders {O = 1,2,….m} are processed in RFACs. Each order consists of a set of products {P=1, 2,….n}. These products go through specific assembly operations {op = op1n, op2n,… opin} that have to be implemented by a set of robots {R = 1,2,….t}. The robots have multipurpose end effectors. Each robot can perform only one job at a time and each job can be processed by only one robot. Robots are continuously available, which means that no robot breaks down in the assembly cell. The RFACs scheduling problem is subject to three classes of constraints, namely robot motion constraints, robot access constraints and tooling resource constraints. Robot motion constraints: robot arms cannot move from one place to another directly. The reason for this is to avoid collision with the other robot arms. This is achieved by assigning control points in the cell. Control points {C1, C2,… C4} are set to simplify path planning and avoid collision. For example, R1 cannot move from AS1 to PF directly. To move from AS1 to PF, R1 should move via control point C1, as shown in Figure 4.
Figure 4. Robot move constraints
Robot access constraints: to prevent collisions between robots in a shared area, more than one robot cannot access the same resource simultaneously. For instance, just one robot, R1 or R2, can access OC or AS1 or GC or PF or AS2 at a time. Tooling resource constraints: to fetch and assemble parts, the hand of each robot should be equipped with the right tool; however, a specific tool may not be available for the two robots at the same time, due to the restricted number of available tools. 3.3 Objectives functions Three common objective functions, namely makespan (Cmax), total tardiness (TD) and number of tardy jobs (NT), are to be minimised. The objectives are described using the following equations:
where CPO and DPO denote the completion time and due date of product P in order O respectively, and UP is the indicator for whether product P is tardy or not. 4. Experimental setup The scheduling of any flexible manufacturing system contains a number of decision points that affect the system’s performance. These decision points can be identified via scheduling stages. At each decision point, different rules can be utilized for decision making. For example, the decision making for job selection usually includes the smallest number of jobs in the system’s queue. In this study, order selection and job-robot selection are considered as scheduling factors. Figure 5 shows the decision points in the RFACs. Decision making 2
Decision making 1
Order Selection
Order n
Order
Job n Job n Job n
Job-Robot Selection
Robot 1 Robot 2 Robot R
Figure 5. The decision points in the RFACs
Sequencing rules: When more than one order is waiting for processing, the orders will be sequenced, from the highest order priority to the lowest order priority, using sequencing rules. Three types of sequencing rules are applied (Chan, Chan, & Lau, 2002; Ozbayrak & Bell, 2003), as shown in Table 3. Table 3. Sequencing rules for the cell to select the order of jobs waiting for processing Sequencing rule
Description
FCFS
Select the order according to the rule First Come First Serve The order with total shortest processing time will be selected The order with total longest processing time will be chosen
TSPT TLPT
Dispatching rules: After selecting and loading an appropriate order into the system, the order of jobs has to be dispatched to the available robots. Three types of dispatching rules are used (Chan, Chan, & Lau, 2002; Chan, Chan, & Kazerooni, 2003), as shown in Table 4.
Table 4. Dispatching rules for the robot to select the next job Dispatching rule SNQ SIO WINQ
Description Job will be selected according to the smallest number in the queue The job with the Shortest Imminent Operation time will be chosen Select the job in the queue which requires the least work
The real context of manufacturing is often dynamic and faces disruptions - referred to as real-time events - which can change system performance and status (Gholami, Zandieh, & Alem-Tabriz, 2009). Real time events are categorised into two types (Vieira, Hermann, & Lin, 2003): the first is resource-related such as resource breakdown, tool failure, loading limits, shortage of material; the second is job-related, such as due date changing, early or late arrival time of jobs, changing of processing time, urgency of jobs, job cancellation and so on. Lu & Liu (2010) and Vinod & Sridharan (2008) indicate that the due date and the arrival time of jobs are the most significant events that affect job shop scheduling. Therefore, in this study, arrival time and due date are considered as the main as another two factors. Arrival time: The arrival time is defined as the degree of workload in the system. A low workload means a long arrival time for jobs. Conversely, a high workload means a short
arrival time. In recent studies of job shop scheduling, the arrival times have been assumed to follow an exponential distribution (Lu & Liu, 2010; Nie, Gao, Li, & Li, 2012). The mean inter-arrival time of jobs is calculated using the following formula:
where
is mean inter-arrival time for jobs,
is mean number of operations per job,
is the mean processing time per job,
is shop utilisation; and
is number of machines
in the system. In the present study, the Equation 4 is used by replacing
with
(number of
robots in the system). Due date: In a dynamic scheduling environment, the total work content (TWK) rule has been extensively used for due date assignment (Ramasesh, 1990). For the TWK rule, the due date of each job is set using the following equation:
where
and
are the due date and arrival time of job
total processing time of job , and
is the
is due date tightness factor. Usually, the range of
between 1 and 10. A high value of Conversely, a low value of
respectively,
is
means the required time to finish the job is looser.
means the required time to finish the job is tighter (Lu & Liu,
2010). In this study, the four factors, sequencing rule (SR), dispatching rule (DR), cell utilisation (U) and due date tightness (K), which influence the scheduling of RFACs, are considered. These factors are set with different levels to explore the effect of the proposed model. The scheduling factors are summarised in Table 5. Table 5. Factors and their control levels Factor
Symbol
Level
Type
Sequencing Rules
A
Dispatching Rules
B
Cell Utilisation
C
Due Date Tightness
D
A(1) A(2) A(3) B(1) B(2) B(3) C(1) C(2) C(3) D(1) D(2) D(3)
FCFS TSPT TLPT SNQ SIO WINQ Low – 75% Moderate – 85% High – 95% Loose – 6 Moderate – 4 Tight – 2
5. Experimental design and results 5.1 Taguchi’s orthogonal array selection Taguchi has developed a pattern of tabulated orthogonal arrays based on the number of factors and their levels. The orthogonal array determines the number of possible experiments. In this investigation, four scheduling factors are taken into consideration with three levels of each factor. Consequently, a total of 81 (3×3×3×3) different combinations are considered. However, according to Taguchi method, the number of experiments can be reduce from 81 to 9 using the standard orthogonal array L9 (34). Table 6 illustrates the arrangement of the experimental design of this study which corresponds to orthogonal array L9. Table 6. Standard L9 (34) orthogonal array Trial No. 1 2 3 4 5 6 7 8 9
A A(1) A(1) A(1) A(2) A(2) A(2) A(3) A(3) A(3)
Levels of control factors B C D B(1) C(1) D(1) B(2) C(2) D(2) B(3) C(3) D(3) B(1) C(2) D(3) B(2) C(3) D(1) B(3) C(1) D(2) B(1) C(3) D(2) B(2) C(1) D(3) B(3) C(2) D(1)
5.2 Calculation of the signal-to-noise (S/N) ratio The present study uses a robust design criterion called the signal-to-noise (S/N) ratio. The S/N ratio is considered as the key step in Taguchi method and reflects the factor performance. Usually, the S/N ratio characteristics are classified into three types: the smaller the better, the larger the better and nominal the best. Each type has a specific formula for calculating the S/N ratio. In scheduling problems, nearly all objective functions are classified in the smaller the better type; their corresponding S/N ratio is as follows (Phadke, 1989):
Table 7 shows the experimental results of the makespan (Cmax), the total tardiness (TD) and the number of tardy jobs (NT), with their robust design criteria being based on the combinations of experimental factors as shown in Table 7 and Figure 6.
Table 7. Details of the combinations of different levels and their results Trial
Levels of control factors B C D
A
Makespan Cmax S/N
Total Tardiness TD S/N
Number of tardy jobs NTD S/N
1
FCFS
SNQ
Low
Loose
38499
-91.71
10070
-80.061
4
-12.041
2
FCFS
SIO
Moderate
Moderate
34736
-90.82
19124
-85.632
12
-21.584
3
FCFS
WINQ
High
Tight
31571
-89.99
50161
-94.007
22
-26.848
4
TSPT
SNQ
Moderate
Tight
35387
-90.98
28904
-89.219
18
-25.105
5
TSPT
SIO
High
Loose
32488
-90.23
5666.0
-75.066
2
-6.0210
6
TSPT
WINQ
Low
Moderate
42356
-92.54
9909.0
-79.921
7
-16.902
7
TLPT
SNQ
High
Moderate
33003
-90.37
2994.0
-69.525
6
-15.563
8
TLPT
SIO
Low
Tight
39999
-92.04
31792
-90.046
19
-25.575
9
TLPT
WINQ
Moderate
Loose
36310
-91.20
542.00
-54.680
2
-6.0210
-70
-90
S/N of TD
S/N of Cmax
-80
-91
A
-92 1
B
C
D
A
-90
2
B
1
3
C 2
D 3
-5
S/N of NT
-15
A
-25 1
B
C 2
D 3
Figure 6. Factors responses graph for makespan (Cmax), total tardiness (TD) and number of tardy jobs (NT)
S/N ratio ranges for the three responses, makespan (Cmax), total tardiness (TD) and number of tardy jobs (NT) are all different. To avoid the different ranges, the values of the S/N ratio results must be normalised to values between 0 and 1. In this case, since the objective is the ‘smaller the better’ type, 0 means the best performance while 1 denotes the worst. The normalisation is determined as follows:
where uik is the normalised S/N value for the ki response in the ith trial; ƞi(k) is the value of S/N; Min ƞi(k) and Max ƞi(k) are the smallest value of ƞi(k) and the largest value of ƞi(k) respectively. The normalisation data are shown in Table 8. Table 8. Normalisation results of S/N ratio
Trial
S/N of Cmax
μ Cmax
S/N of TD
μ TD
S/N of NT
μ NT
1 2 3 4 5 6 7 8 9
-91.71 -90.82 -89.99 -90.98 -90.23 -92.54 -90.37 -92.04 -91.20
0.32 0.67 1.00 0.61 0.90 0.00 0.85 0.19 0.52
-80.061 -85.632 -94.007 -89.219 -75.066 -79.921 -69.525 -90.046 -54.680
0.35 0.21 0.00 0.12 0.48 0.36 0.62 0.10 1.00
-12.041 -21.584 -26.848 -25.105 -6.021 -16.902 -15.563 -25.575 -6.021
0.71 0.25 0.00 0.08 1.00 0.48 0.54 0.06 1.00
6. Fuzzy logic implementation and results for MCPI In order to find out the optimum scheduling for multi-objective problems in RFACs, fuzzy logic is used. In the present study, the fuzzy inference system contains three input parameters and one output parameter. The input parameters are the S/N ratio of Cmax, TD and NT, while MPCI is the output parameter. Three steps are required for the fuzzy approach implementation to generate MPCI. Firstly, the fuzzy sets for the inputs and output are determined. In this study, the S/N ratios of Cmax, TD and NT are break down into a set of linguistic values for the inputs: low (L), medium (M), and high (H); while the output is set into seven linguistic values: tiny (T), very small (VS), small (S), medium (M), large (L), very large (VL) and huge (H). Table 9 shows the linguistic values of inputs/output. Table 9. Input and output variables with their fuzzy values System variable
Linguistic variables
Inputs
S/N ratios for, Cmax, TD and NT
Output
MPCI
Linguistic Value
Term Set
Numerical Range
Low Medium High Tiny Very Small Small Medium Large Very Large Huge
L M H T VS S M L VL H
[0– 0.25] [0.25–0.75] [0.75–1] [0– 0.167] [0– 0.334] [0.167 – 0.5] [0.334 – 0.667] [0.5 – 0.834] [0.667 – 1] [0.834 –1]
Secondly, the membership functions for inputs/output are plotted. There are different types of membership functions’ shapes such as triangular, trapezoidal, Gaussian, singleton, etc. The triangular shape is the common membership functions shape and a powerful way to approach the convex function (Pedrycz, 1994). The membership plot for the inputs/output parameters using the triangular shape is shown in Figure 7.
Degree of membership
0.5
T VS S M L VL H
1
Degree of membership
L M H
1
0.5
0
0 0
0. 5
1
0
0. 5
1
Output variable
Input variable
Figure 7. Input / output of Memberships functions
Finally, to control the output parameter (MPCI), fuzzy rules are structured. Fuzzy rules are derived directly based on the formula (nm), where
and
denote input parameters
3
and their linguistic values. Thus, the number of fuzzy rules is 3 = 27, as illustrated in Table 10. Each rule is mathematically evaluated through a process named implication. In this study, Mamdani implication is applied (Klir & Yuan, 2005). Table 10. Fuzzy rules table S/N of Cmax
S/N ratio for S/N of TD
Low
Low
Medium High
Medium
Low
Medium High
High
Low Medium High
S/N of NTD Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High
MPCI Tiny Very Small Small Very Small Small Medium Small Medium Large Very Small Small Medium Small Medium Large Medium Large Very Large Small Medium Large Medium Very Large Very Large Large Very Large Huge
The evaluation values of each fuzzy rule are combined into one single output value via the defuzzification process. In this study, the well-known method for the defuzzification process which is called the centre of gravity (COG) (Mendel, 1992) is used to transform the fuzzy inference output µÃ (x) into the non-fuzzy value x'. The non-fuzzy value x' is called MPCI, based on the following equation:
In this study, the fuzzy logic toolbox in MATLAB software is used to construct the fuzzy inference system of the MPCI. Figure 8 shows the fuzzified rule viewer for MPCI, which can accept any value of the S/N ratio for the three input parameters: Cmax, TD and NT. The output (MPCI) in this figure can be interpreted easily, for example, as in the following: IF the S/N of Cmax is (0.32), the S/N of TD is (0.35) and S/N of NTD is (0.71) THEN MCPI will be (0.449). Figure 9 presents the surface view graphs corresponding to the particular fuzzy rules of the implemented fuzzy system. This Figure shows the effects of the combination of inputs parameters on the MPCI. Table 11 shows the MPCI values corresponding to each experimental run obtained by using the fuzzy system.
S/N of Cmax = 0.32
S/N of TD = 0.35
S/N of NTD = 0.71
Figure 8. Final output of fuzzy rules
MCPI = 0.449
Figure 9. 3D surface plots of the inputs/output
Table 11. MPCI values obtained by fuzzy system Trial
μ Cmax
μ TD
μ NT
MCPI
1 2 3 4 5 6 7 8 9
0.32 0.67 1.00 0.61 0.90 0.00 0.85 0.19 0.52
0.35 0.21 0.00 0.12 0.48 0.36 0.62 0.10 1.00
0.71 0.25 0.00 0.08 1.00 0.48 0.54 0.06 1.00
0.449 0.419 0.50 0.383 0.833 0.267 0.840 0.058 0.834
Trial number 7 gives the highest MPCI value among the nine trials, as shown in Table 11. Thus, A3B3C3 and D2 (refer to table 6) are selected to be the initial levels for the scheduling factors. 7. Analytical results and discussion 7.1 Effect of scheduling factors on MPCI The main factor effect can be assessed from the mean of each factor level, using the following mathematical expression (Phadke, 1989; Ranjitk, 2001), and summarised in Table 12.
where MPCIji is the mean of factor j at level i; k is the number of levels in factor j; and ƞji is the value of MPCI with factor
at level i. The highest value of MPCI among all
combinations of the factors denotes the optimum level for each factor. As shown in Table 12, it is clear that the optimal levels of A (Sequencing Rule), B (Dispatching Rule), C (Cell
Utilisation) and D (Due Date Tightness) are 3, 1, 3 and 3 respectively, due to their MPCI value. The significant scheduling factor also can be determined by calculating the value of difference between the maximum and minimum for each factor. The greatest difference is the most significant factor. It is noticed, that factor C (Cell Utilisation) has the strongest effect on the scheduling of RFACs, followed by factor D (Due Date Tightness). It is also seen that factor B (Dispatching Rule) has weakest impact on the scheduling. Table 12. Response table of the MPCI values Factor
MPCI
Max-Min
Rank
0.122
3
0.120
4
0.466
1
0.391
2
A(1)
(ƞ1 + ƞ2 + ƞ3 ) / 3
0.456
A(2)
(ƞ4 + ƞ5 + ƞ6 ) / 3
0.494
A(3)
(ƞ7 + ƞ8 + ƞ9 ) / 3
0.578
B(1)
(ƞ1 + ƞ4 + ƞ7 ) / 3
0.557
B(2)
(ƞ2 + ƞ5 + ƞ8 ) / 3
0.437
B(3)
(ƞ3 + ƞ6 + ƞ9 ) / 3
0.534
C(1)
(ƞ1 + ƞ6 + ƞ8 ) / 3
0.258
C(2)
(ƞ2 + ƞ4 + ƞ9 ) / 3
0.545
C(3)
(ƞ3 + ƞ5 + ƞ7 ) / 3
0.724
D(1)
(ƞ1 + ƞ5 + ƞ9 ) / 3
0.705
D(2)
(ƞ2 + ƞ6 + ƞ7 ) / 3
0.509
D(3)
(ƞ3 + ƞ4 + ƞ8 ) / 3
0.314
The relative effect among the scheduling factors for the MPCI can also be verified using analysis of variance (ANOVA) as summarised in Table 13. From ANOVA results, the effect of each scheduling factor on the MPCI becomes apparent. The factors C (Cell Utilisation) and D (Due Date Tightness) contribute just over 92% of the total variance relating to the MPCI. Factors B (Dispatching Rule) and A (Sequencing Rule) contribute nearly 8%. The ANOVA results are similar to the ‘Mix-Min’ analysis described in Table 12. Table 13. ANOVA of MPCI Factor
DOF
SS
MS
PC%
A
2
0.023
0.012
3.78%
B
2
0.025
0.012
4.02%
C
2
0.332
0.166
54.46%
D
2
0.230
0.115
37.74%
Error
0
0.000
0.000
0.00%
total
8
0.610
0.076
100.00%
7.2 Confirmation test The final step is to predict and verify the improvement of the performance characteristics using the optimal levels of the scheduling factors. The best value of the MPCI can be predicted using the following equation (Montgomery, 2004):
where ƞm is the total mean of the MPCI; ƞi is the mean of the MPCI at the optimal level and q is the number of scheduling factors. Based on equation 10, the predicted MPCI for initial and optimal levels of scheduling factors can then be calculated as shown below: For the initial levels, as mentioned earlier, A3B3C3 and D2 are selected to be the initial levels for the scheduling factors: MPCI Initial = 0.509 + (0.578 - 0.509) + (0.557 - 0.509) + (0.724 - 0.509) + (0.509 - 0.509) = 0.841 For optimal levels A3B1C3 and D1 according to Table 12, the MPCI is estimated in the same way: MPCI Optimal = 0.509 + (0.578 - 0.509) + (0.557 - 0.509) + (0.724 - 0.509) + (0.705 - 0.509) = 1.037 MPCIconfirmed is also determined for both the initial and optimal levels of the scheduling factors. Table 5 shows that trial 7 provides the largest MPCI value which represents MPCIInitial, while MPCIoptimal can be graphically calculated by obtaining the value of the optimal levels with the fuzzy logic rule. The percentage prediction error (PPE) is calculated in order to check the closeness of the predicted value to that obtained by the actual experimental value (Pandey & Dubey, 2012).
The results of the confirmation run for the initial and optimal levels of the scheduling factors are summarised in Table 14. The results indicate that the optimal levels of scheduling factors produce the best MPCI among all trials. In other words, it can be seen that the MPCI value of the optimal level of scheduling factors is improved by approximately 0.11 compared to the initial level of scheduling factors. The results also show that the PPE is acceptable (less than 10%). In summary, it can be said that the experimental results are confirmed.
Table 14. Predicted results for Initial and Optimal condition Initial condition
Optimal condition
Level
A3B1C3D2
A3B1C3D1
MPCI prediction
0.841
1.037
MPCI confirmed
0.840
0.948
PPE
0.001%
9.38%
Improvement MPCI = 11%
8. Conclusion In this study, a hybrid approach obtained by integrating the Taguchi method with fuzzy logic theory has been utilised to achieve multi-objective optimisation of dynamic scheduling in RFACs. The experimental results are analysed using different statistical analysis techniques to find the best combination of scheduling factors to improve the MPCI. The results are then verified through a confirmation test. The main findings of the study are given below: 1- The selected hybrid approach is efficient and effective for finding the optimal scheduling of RFACs with fewer experiments compared to full factorial experimental methods. 2- The optimum values of different factors have been found as to be sequencing rule TLPT, dispatching rule SNQ, cell utilisation 95% and due date tightness 6. 3- The most significant factors affecting the scheduling strategy have been identified as factor C (Cell Utilisation) and factor D (Due Date Tightness), which account for nearly 92%. 4- A percentage predication error of less than 10% has been obtained. This percentage indicates that the suggested hybrid approach can be successfully used in the multiobjective optimisation of scheduling in RFACs. 5- The confirmed MPCI value of optimal scheduling factors produces the best result (0.948) against the initial setting (0.84). Therefore, the MPCI increased by 11% compared to the initial setting. For future work, expand the proposed study to be more applicable to real scheduling problems, by taking into account robot breakdowns during the scheduling of RFACs. Three parameters may be considered in the scope of this area of study: (1) the number of robots subject to breakdown (2) breakdown frequency which represents how many times each robot will break down during the scheduling horizon (3) repair duration which represents the mean time required to repair the robot after its breakdown.
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Highlights
To our knowledge, this is first study that addresses dynamic scheduling in RFACs.
Cell utilisation and Due Date Tightness are the significant scheduling factors.
Percentage of prediction error (9%) confirms the efficient of proposed methodology.
The Results of confirmation run improved by 11% compared to initial experiments.