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Parameter optimization using the surface response technique in automated guided vehicles
Chapter 8
Parameter optimization using the surface response technique in automated guided vehicles Arzu Eren Senaras ¸ Department of Econometrics, Uludag˘ University, Go¨ru¨kle, Turkey
Chapter Outline 8.1 Introduction 8.2 A short literature review 8.3 Methodology 8.3.1 Response surface methodology 8.3.2 Discrete event simulation 8.4 Developed model
187 188 189 189 189 190
8.4.1 The developed simulation model 190 8.4.2 Optimization using response surface methodology 190 8.5 Conclusion 196 References 196
8.1 Introduction An automated guided vehicle (AGV) consists of a mobile robot used for transportation and automatic material handling, for example, for finished goods, raw materials, and products in process. The design and operation of AGV systems are highly complex due to high levels of randomness and the large number of variables involved. This complexity makes simulation an extremely useful technique in modeling these systems (Negahban & Smith, 2014). The purpose of this study is to analyze AGV parameters in a manufacturing workshop by using the surface response technique and the simulation method. The simulation model is developed in the Arena software package. The surface response technique is applied in Minitab 14.0. The AGV helpful utilization rate, the expected average number of pieces, and the system requirements have been satisfied, and the tested model has been run for different criteria to observe the results.
Sustainable Engineering Products and Manufacturing Technologies. DOI: https://doi.org/10.1016/B978-0-12-816564-5.00008-6 © 2019 Elsevier Inc. All rights reserved.
187
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SECTION | II Sustainable Engineering Products
8.2 A short literature review In this literature survey, several simulation studies concerning AGV systems are discussed. Kabir and Suzuki (2018) investigated how the duration of battery charging for AGVs can be varied to increase the flexibility of a manufacturing system. Chawla, Kumar, and Surjit (2018) investigated the fleet size optimization of AGVs in different layouts of a flexible manufacturing system (FMS) through the application of the analytical method, and the gray wolf optimization (GWO) algorithm was used. The results from the analytical method and gray wolf optimization algorithm are compared and validated for the different sizes of FMS layouts with computational experiments. Mahalakshmi and Murugesan (2017) studied the job shop environment problem for multiple AGVs in the manufacturing process. As the earliness of the AGV results in waiting and tardiness results in temporary storage of the products on the shop floor, it is essential to minimize the earliness and tardiness of AGVs. The authors therefore propose a mathematical optimization program to minimize the total earliness and total tardiness of AGVs in the manufacturing system. Since it is difficult to solve the mathematical program through a conventional method, an optimization technique called the artificial immune system algorithm is used to obtain the optimal solution. The proposed algorithm is tested with numerical examples and also compared with the other methods in the literature. Vladim´ır, Milan, and Patrik (2017) studied methods for the determination of the number of AGVs and choosing the optimal internal company logistics track. The simulation results of the logistics system were various in terms of increasing the use of operation areas, optimized material supply, and a created layout that would be able to flexibly respond to future company requirements. Demesure, Defoort, Bekrar, Damien, and Djemaı¨mohamed (2017) proposed motion planning and the scheduling of AGVs in an FMS. Numerical and experimental results are provided to show the pertinence and the feasibility of the proposed strategy. Wang, Chen, Chiang, and Pan (2016) studied a scheduling problem in the FMS in which orders require the completion of different parts in various quantities. The orders arrive randomly and continuously, and they all have predetermined due dates. Two scheduling decisions were considered in this study: launching parts into the system for production and determining the order sequence for collecting the completed parts. Neradilov´a and Fedorko (2016) aimed to present the possibilities of computer simulation methods for obtaining data for a full-scale economic analysis implementation. Leite, Esposito, Vieira, and Lima (2015) investigated the utilization rate of an AGV system in an industrial environment and evaluated the advantages
Parameter optimization using the surface response technique Chapter | 8
189
and disadvantages of the project. They used the simulation software Promodel 7.0 to develop a model. Their model aims to analyze and optimize the use of AGVs. Problems were identified and solutions were adopted by the authors according to the results obtained from the simulations. Udhayakumar and Kumanan (2010) studied to find the near optimum schedule for two AGVs based on the balanced workload and the minimum traveling time for maximum utilization. Kesen and Baykoc¸ (2007) developed a simulation model of a hypothetical system using an AGV. The model has a job shop environment and was based on the JIT philosophy.
8.3 Methodology In this study, a discrete event simulation (DES) is created. DES is a tool for analyzing a situation in which a designer needs a tool to optimize system input to satisfy system constraints and maximize (or minimize) targets. At this point, we propose the surface response technique as a tool for maximization.
8.3.1
Response surface methodology
Response surface methodology (RSM) is a tool that was introduced in the early 1950s by Box and Wilson (1951). RSM is a collection of mathematical and statistical techniques that is useful for the approximation and optimization of stochastic models. The objective function associated with such models is subject to random noise and is referred to as noisy or stochastic objective function (Myers, Khuri, & Carter, 1989). It also has important applications in the design, development, and formulation of new products, as well as in the improvement of existing product designs (Myers, Montgomery Dougles, & Anderson-Cook Christine, 2016: 1).
8.3.2
Discrete event simulation
Simulation is one of the most powerful analysis tools available to those responsible for the design and operation of complex process or systems. In an increasingly competitive world, simulation has become a very powerful tool for the planning, design, and control of systems (Pegden, 1990: 3). Discrete event system simulation is the modeling of systems in which the state variable changes only at a discrete set of points in time (Banks, Carson, Nelson, & Nicol, 2004: 13). In a discrete model, however, change can occur only at separated points in time, such as a manufacturing system with parts arriving and leaving at specific times, machines going down and coming back up at specific times, and breaks for workers (Kelton, Sadowski, & Sadowski, 2004: 9).
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SECTION | II Sustainable Engineering Products
8.4 Developed model In this study, production number is considered as a dependent output variable, and various combinations of AGVs and their parameters are considered as an independent variable. First, DES will be briefly considered with its assumptions, and then the surface response technique will be implemented to provide parameter optimization.
8.4.1
The developed simulation model
A simulation model was developed by using ARENA simulation software package. Model assumptions are as follows: 1. Demanded production rate is 35 U/h, so cycle time is 1.71. 2. There are four stations, namely, entrance station 1, station 2, and sortie. 3. In entrance station, machine time is 1.15; in station 1, machine time is 1.3; and in station 2, machine time 1.32 minute. 4. When entity enters in the system, first process 1 is realized then it waits for AGV arriving and AGV transport entity to the first station and there process 2 is realized but AGV waits for entity and after process finishes, AGV transports entity to station 2 and process 3. Same in station 1, AGV waits entity. When process is finished, AGV transports entity to the sortie station. There entity leaves AGV. 5. Model is running 300 and 50 minutes is assumed to be warm up period.
8.4.2
Optimization using response surface methodology
BoxBehnken design is used for surface design experience. First, these factors are chosen for experiments. These factors and their rates are shown in Table 8.1. According to the design table, below scenarios are tested, and related number of production (No. of Prd) are obtained as shown in Table 8.2. The response surface analyze is applied using BoxBehnken design. Replicate number is one. Design of experiment table is randomized, and TABLE 8.1 Factors and rates. 21
0
1
Number of AGV in system
6
10
14
AGV velocity
15
25
35
AGV acceleration
1
3
5
AGV, Automated guided vehicle.
Parameter optimization using the surface response technique Chapter | 8
191
TABLE 8.2 Coded. Scenarios
Velocity
Acceleration
Number of AGV
Results
1
1
21
0
51
2
0
1
21
65
3
21
21
0
51
4
1
0
21
45
5
0
21
1
68
6
21
0
21
45
7
0
0
0
76
8
1
0
1
98
9
1
1
0
105
10
0
21
21
31
11
21
0
1
98
12
0
0
0
76
13
0
0
0
76
14
0
1
1
72
15
21
1
0
105
AGV, Automated guided vehicle.
when 15 experiments are realized, it is obtained. Residual analysis is performing the model (Fig. 8.1). The residual plots do not indicate problem. The residual plots do not reveal any major violations (Table 8.3): R2 5 85:70% R2 ðadjÞ 5 59:95% where R2 is reasonable for fitting uniformity. Estimated regression coefficients for model shown below: Let X1, X2, X3 show velocity, acceleration, number of AGV, respectively, as coded. Analyze of variance for model is shown in Table 8.4. Y 5 76 1 2:05E 2 15X1 1 18:25X2 1 18:75X3 1 7:25X1 3 X1 2 5:25X2 3 X2 2 11:75X3 3 X3 1 3:29E 2 17X1 3 X2 2 6:35E 2 15X1 3 X3 2 7:5X2 3 X3 In the results, curvature is displayed as a significant factor in which interaction is not very effective but square is important coefficients. This
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SECTION | II Sustainable Engineering Products
Residual plots for sonuc Versus fits
99
–20
90
–20
Residual
Percent
Normal probability plot
50 10 1
–20 –20 –20
–20
–10
5 Residual
10
20
20
Histogram
100
Versus order
3.6
Residual
Frequency
80
20
4.8
2.4 1.2 0.0
40 60 Fitted value
10 0 –10 –20
–15 –10
–5 0 5 Residual
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Observation order
–15
FIGURE 8.1 Residual plots for results: normal probability plot, versus fits, histogram, versus order.
TABLE 8.3 Coefficients and P values. Term Constant X1
Coefficients
SE coefficients
T
P
76
8.536
8.903
0
0
5.227
0
1
X2
18.25
5.227
3.491
.017
X3
18.75
5.227
3.587
.016
X1 3 X1
7.25
7.694
0.942
.389
X2 3 X2
2 5.25
7.694
2 0.682
.525
X3 3 X3
2 11.75
7.694
2 1.527
.187
X1 3 X2
0
7.393
0
1
X1 3 X3
0
7.393
0
1
X2 3 X3
2 7.5
7.393
2 1.015
.357
indicates that the first order model is not an adequate experiment and a higher order model is required. For this model below graphic are obtained as Figs. 8.28.6.
193
Parameter optimization using the surface response technique Chapter | 8
TABLE 8.4 Analyze of variance for model. F
P
DF
Seq SS
Adj SS
Adj Ms
9
6549.4
6549.4
727.71
3.33
.099
Linear
3
5477
5477
1825.67
8.35
.022
X1
1
X2
1
2664.5
2664.5
2664.5
12.19
.017
X3
1
2812.5
2812.5
2812.5
12.87
.016
Square
3
847.4
847.4
282.47
1.29
.373
X1 3 X1
1
267.47
194.08
194.08
0.89
.389
X2 3 X2
1
70.16
101.77
101.77
0.47
.525
X3 3 X3
1
509.77
509.77
509.77
2.33
.187
Interaction
3
225
225
75
X1 3 X2
1
0
0
0
0
1
X1 3 X3
1
0
0
0
0
1
X2 3 X3
1
225
225
225
Residual error
5
1093
1093
218.6
Lack-of-fit
3
1093
1093
364.33
Pure error
2
0
0
Total
14
0
0
0
0
1
0.34
.796
1.03
.357
0
7642.4
80 No of Prd
60 40
1
20
0 –1
0 Acceleration
1
Number of AGV
–1
FIGURE 8.2 No. of Prd, acceleration and number of AGV. AGV, Automated guided vehicle; No. of Prd, Number of production.
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SECTION | II Sustainable Engineering Products
100
Number of Prd
75 1 0
50 –1
0 Acceleration
Velocity
–1
1
FIGURE 8.3 No. of Prd, acceleration and velocity. No. of Prd, Number of production.
100 80 Number of Prd
60
1
40 0 –1
0 Velocity
1
Number of AGV
–1
FIGURE 8.4 No. of Prd, number of AGV and velocity. AGV, Automated guided vehicle; No. of Prd, Number of production.
Desirability function: For ith response is assigned a desirability function, di, where the value of di varies between 0 and 1. The function, di, is defined differently based on the objective of the response. For example, if target production number is assumed to be 95, desirability function is 1 for production number 98 where number of AGV coded is 0.4747 (number of AGV as uncoded is 11.89).
Parameter optimization using the surface response technique Chapter | 8
195
Contour plots of number of production –1.0
Acceleration × velocity
–0.5
0.0
0.5
Number of AGV × velocity
1.0 Number of production
1.0
< 20 20 – 40 40 – 60 60 – 80 > 80
0.5 0.0 –0.5
1.0
Number of AGV × acceleration
–1.0
Hold values 0 Velocity 0 Acceleration Number of AGV 0
0.5 0.0 –0.5 –1.0 –1.0
–0.5
0.0
0.5
1.0
FIGURE 8.5 Contour plots of No. of Prd. No. of Prd, Number of production.
FIGURE 8.6 Acceleration, number of AGV and velocity. AGV, Automated guided vehicle.
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SECTION | II Sustainable Engineering Products
8.5 Conclusion Satisfying target No. of Prd is crucial for any organization. On the other hand, it is very important to invest necessary required number of AGV due to financial constraint. System designer should determine right number of AGV. A simulation is a tool for testing scenarios (what if analyze). On the other hand, the model designer needs a tool to optimize system input to satisfy the system constraint and maximize (or minimize) the target. At this point, we propose surface response technique as a tool for maximization. In this study, production number is considered as dependent output variable, various combinations of AGV and their parameter are considered independent variable. Following model is obtained: Y 5 76 1 2:05E 2 15X1 1 18:25X2 1 18:75X3 1 7:25X1 3 X1 2 5:25X2 3 X2 2 11:75X3 3 X3 1 3:29E 2 17X1 3 X2 2 6:35E 2 15X1 3 X3 2 7:5X2 3 X3 And for the target No. of Prd, required number of AGV is calculated. Optimization and comparison of multiple variables has been successfully carried out in the current study using RSM. The experimental optimum was found to be 11.89 for number of AGV. The experimental optimum matches closely with the predicted optimum.
References Banks, J., Carson, J. S., Nelson, B. L., & Nicol, D. M. (2004). Discrete-event system simulation (4th ed.). Upper Saddle River, NJ: Prentice-Hall, Inc. Box, G. E. P., & Wilson, K. B. (1951). “On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society, Series B, 13, 138, Discussion: 3845. Chawla, V. K., Kumar, C. A., & Surjit, A. (2018). Automatic guided vehicles fleet size optimization for flexible manufacturing system by grey wolf optimization algorithm. Management Science Letters, 8(2), 7990. Available from https://doi.org/10.5267/j.msl.2017.12.004, 2018. Demesure, G., Defoort, M., Bekrar, A., Damien, T., & Djemaı¨, M. (2017). Decentralized motion planning and scheduling of AGVs in FMS. Transactions on Industrial Informatics. Available from https://doi.org/10.1109/TII.2017.2749520. Kabir, Q. S., & Suzuki, Y. (2018). Increasing manufacturing flexibility through battery management of automated guided vehicles,. Computers & Industrial Engineering, 117, 225236. Kelton, W. D., Sadowski, R. P., & Sadowski, D. A. (2004). Simulation with arena. McGrawHill. Kesen, S. E., & Baykoc¸, F. (2007). Simulation of automated guided vehicle (AGV) systems based on just-in-time (JIT) philosophy in a job-shop environment. Simulation Modeling Practice and Theory, 15, 272284. Leite, L. F. V., Esposito, R. M. A., Vieira, A. P., & Lima, F. (2015). Simulation of a production line with automated guided vehicle: A case study. Independent Journal of Management & Production, 6(n.2). Mahalakshmi., & Murugesan, R. (2017). Optimization algorithm for minimizing the earliness/ tardiness of automated guided vehicles using artificial immune system. International Journal of Applied Engineering Research, 12(22), 1191311919. (2017)ISSN 09734562.
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Myers, R. H., Khuri, A. I., & Carter, W. H., Jr (1989). Response surface methodology: 19661988. Technometrics, 31(2), 137157. Myers, R. H., Montgomery Dougles, C., & Anderson-Cook Christine, M. (2016). Response surface methodology process and product optimization using designed experiments (4th ed.). Hoboken, NJ: John Wiley & Sons. Inc. Negahban, A., & Smith, J. S. (2014). Simulation for manufacturing system design and operation: Literature review and analysis. Journal of Manufacturing Systems, 33, 241261. Neradilov´a, H., & Fedorko, G. (2016). The use of computer simulation methods to reach data for economic analysis of automated logistic systems. Economic Analysis of Automated Logistic Systems, 6, 700710. Pegden, C. D. (1990). Introduction to simulation using Siman (2nd ed.). New York: McgrawHill Inc. Udhayakumar, P., & Kumanan, S. (2010). Task scheduling of AGV in FMS using nontraditional optimization techniques. International Journal of Simulation, 9(1), 2839. Vladim´ır, V., Milan, G., & Patrik, G. (2017). Computer simulation as a tool for the optimization of logistics using automated guided vehicles. Procedia Engineering, 192, 923928. (2017). Wang, Y.-C., Chen, T., Chiang, H., & Pan, H.-C. (2016). A simulation analysis of part launching and order collection decisions for a flexible manufacturing system. Simulation Modelling Practice and Theory, 69, 8091, December.
Parameter optimization using the surface response technique in automated guided vehicles Arzu Eren Senaras ¸ Department of Econometrics, Uludag˘ University, Go¨ru¨kle, Turkey
Chapter Outline 8.1 Introduction 8.2 A short literature review 8.3 Methodology 8.3.1 Response surface methodology 8.3.2 Discrete event simulation 8.4 Developed model
187 188 189 189 189 190
8.4.1 The developed simulation model 190 8.4.2 Optimization using response surface methodology 190 8.5 Conclusion 196 References 196
8.1 Introduction An automated guided vehicle (AGV) consists of a mobile robot used for transportation and automatic material handling, for example, for finished goods, raw materials, and products in process. The design and operation of AGV systems are highly complex due to high levels of randomness and the large number of variables involved. This complexity makes simulation an extremely useful technique in modeling these systems (Negahban & Smith, 2014). The purpose of this study is to analyze AGV parameters in a manufacturing workshop by using the surface response technique and the simulation method. The simulation model is developed in the Arena software package. The surface response technique is applied in Minitab 14.0. The AGV helpful utilization rate, the expected average number of pieces, and the system requirements have been satisfied, and the tested model has been run for different criteria to observe the results.
Sustainable Engineering Products and Manufacturing Technologies. DOI: https://doi.org/10.1016/B978-0-12-816564-5.00008-6 © 2019 Elsevier Inc. All rights reserved.
187
188
SECTION | II Sustainable Engineering Products
8.2 A short literature review In this literature survey, several simulation studies concerning AGV systems are discussed. Kabir and Suzuki (2018) investigated how the duration of battery charging for AGVs can be varied to increase the flexibility of a manufacturing system. Chawla, Kumar, and Surjit (2018) investigated the fleet size optimization of AGVs in different layouts of a flexible manufacturing system (FMS) through the application of the analytical method, and the gray wolf optimization (GWO) algorithm was used. The results from the analytical method and gray wolf optimization algorithm are compared and validated for the different sizes of FMS layouts with computational experiments. Mahalakshmi and Murugesan (2017) studied the job shop environment problem for multiple AGVs in the manufacturing process. As the earliness of the AGV results in waiting and tardiness results in temporary storage of the products on the shop floor, it is essential to minimize the earliness and tardiness of AGVs. The authors therefore propose a mathematical optimization program to minimize the total earliness and total tardiness of AGVs in the manufacturing system. Since it is difficult to solve the mathematical program through a conventional method, an optimization technique called the artificial immune system algorithm is used to obtain the optimal solution. The proposed algorithm is tested with numerical examples and also compared with the other methods in the literature. Vladim´ır, Milan, and Patrik (2017) studied methods for the determination of the number of AGVs and choosing the optimal internal company logistics track. The simulation results of the logistics system were various in terms of increasing the use of operation areas, optimized material supply, and a created layout that would be able to flexibly respond to future company requirements. Demesure, Defoort, Bekrar, Damien, and Djemaı¨mohamed (2017) proposed motion planning and the scheduling of AGVs in an FMS. Numerical and experimental results are provided to show the pertinence and the feasibility of the proposed strategy. Wang, Chen, Chiang, and Pan (2016) studied a scheduling problem in the FMS in which orders require the completion of different parts in various quantities. The orders arrive randomly and continuously, and they all have predetermined due dates. Two scheduling decisions were considered in this study: launching parts into the system for production and determining the order sequence for collecting the completed parts. Neradilov´a and Fedorko (2016) aimed to present the possibilities of computer simulation methods for obtaining data for a full-scale economic analysis implementation. Leite, Esposito, Vieira, and Lima (2015) investigated the utilization rate of an AGV system in an industrial environment and evaluated the advantages
Parameter optimization using the surface response technique Chapter | 8
189
and disadvantages of the project. They used the simulation software Promodel 7.0 to develop a model. Their model aims to analyze and optimize the use of AGVs. Problems were identified and solutions were adopted by the authors according to the results obtained from the simulations. Udhayakumar and Kumanan (2010) studied to find the near optimum schedule for two AGVs based on the balanced workload and the minimum traveling time for maximum utilization. Kesen and Baykoc¸ (2007) developed a simulation model of a hypothetical system using an AGV. The model has a job shop environment and was based on the JIT philosophy.
8.3 Methodology In this study, a discrete event simulation (DES) is created. DES is a tool for analyzing a situation in which a designer needs a tool to optimize system input to satisfy system constraints and maximize (or minimize) targets. At this point, we propose the surface response technique as a tool for maximization.
8.3.1
Response surface methodology
Response surface methodology (RSM) is a tool that was introduced in the early 1950s by Box and Wilson (1951). RSM is a collection of mathematical and statistical techniques that is useful for the approximation and optimization of stochastic models. The objective function associated with such models is subject to random noise and is referred to as noisy or stochastic objective function (Myers, Khuri, & Carter, 1989). It also has important applications in the design, development, and formulation of new products, as well as in the improvement of existing product designs (Myers, Montgomery Dougles, & Anderson-Cook Christine, 2016: 1).
8.3.2
Discrete event simulation
Simulation is one of the most powerful analysis tools available to those responsible for the design and operation of complex process or systems. In an increasingly competitive world, simulation has become a very powerful tool for the planning, design, and control of systems (Pegden, 1990: 3). Discrete event system simulation is the modeling of systems in which the state variable changes only at a discrete set of points in time (Banks, Carson, Nelson, & Nicol, 2004: 13). In a discrete model, however, change can occur only at separated points in time, such as a manufacturing system with parts arriving and leaving at specific times, machines going down and coming back up at specific times, and breaks for workers (Kelton, Sadowski, & Sadowski, 2004: 9).
190
SECTION | II Sustainable Engineering Products
8.4 Developed model In this study, production number is considered as a dependent output variable, and various combinations of AGVs and their parameters are considered as an independent variable. First, DES will be briefly considered with its assumptions, and then the surface response technique will be implemented to provide parameter optimization.
8.4.1
The developed simulation model
A simulation model was developed by using ARENA simulation software package. Model assumptions are as follows: 1. Demanded production rate is 35 U/h, so cycle time is 1.71. 2. There are four stations, namely, entrance station 1, station 2, and sortie. 3. In entrance station, machine time is 1.15; in station 1, machine time is 1.3; and in station 2, machine time 1.32 minute. 4. When entity enters in the system, first process 1 is realized then it waits for AGV arriving and AGV transport entity to the first station and there process 2 is realized but AGV waits for entity and after process finishes, AGV transports entity to station 2 and process 3. Same in station 1, AGV waits entity. When process is finished, AGV transports entity to the sortie station. There entity leaves AGV. 5. Model is running 300 and 50 minutes is assumed to be warm up period.
8.4.2
Optimization using response surface methodology
BoxBehnken design is used for surface design experience. First, these factors are chosen for experiments. These factors and their rates are shown in Table 8.1. According to the design table, below scenarios are tested, and related number of production (No. of Prd) are obtained as shown in Table 8.2. The response surface analyze is applied using BoxBehnken design. Replicate number is one. Design of experiment table is randomized, and TABLE 8.1 Factors and rates. 21
0
1
Number of AGV in system
6
10
14
AGV velocity
15
25
35
AGV acceleration
1
3
5
AGV, Automated guided vehicle.
Parameter optimization using the surface response technique Chapter | 8
191
TABLE 8.2 Coded. Scenarios
Velocity
Acceleration
Number of AGV
Results
1
1
21
0
51
2
0
1
21
65
3
21
21
0
51
4
1
0
21
45
5
0
21
1
68
6
21
0
21
45
7
0
0
0
76
8
1
0
1
98
9
1
1
0
105
10
0
21
21
31
11
21
0
1
98
12
0
0
0
76
13
0
0
0
76
14
0
1
1
72
15
21
1
0
105
AGV, Automated guided vehicle.
when 15 experiments are realized, it is obtained. Residual analysis is performing the model (Fig. 8.1). The residual plots do not indicate problem. The residual plots do not reveal any major violations (Table 8.3): R2 5 85:70% R2 ðadjÞ 5 59:95% where R2 is reasonable for fitting uniformity. Estimated regression coefficients for model shown below: Let X1, X2, X3 show velocity, acceleration, number of AGV, respectively, as coded. Analyze of variance for model is shown in Table 8.4. Y 5 76 1 2:05E 2 15X1 1 18:25X2 1 18:75X3 1 7:25X1 3 X1 2 5:25X2 3 X2 2 11:75X3 3 X3 1 3:29E 2 17X1 3 X2 2 6:35E 2 15X1 3 X3 2 7:5X2 3 X3 In the results, curvature is displayed as a significant factor in which interaction is not very effective but square is important coefficients. This
192
SECTION | II Sustainable Engineering Products
Residual plots for sonuc Versus fits
99
–20
90
–20
Residual
Percent
Normal probability plot
50 10 1
–20 –20 –20
–20
–10
5 Residual
10
20
20
Histogram
100
Versus order
3.6
Residual
Frequency
80
20
4.8
2.4 1.2 0.0
40 60 Fitted value
10 0 –10 –20
–15 –10
–5 0 5 Residual
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Observation order
–15
FIGURE 8.1 Residual plots for results: normal probability plot, versus fits, histogram, versus order.
TABLE 8.3 Coefficients and P values. Term Constant X1
Coefficients
SE coefficients
T
P
76
8.536
8.903
0
0
5.227
0
1
X2
18.25
5.227
3.491
.017
X3
18.75
5.227
3.587
.016
X1 3 X1
7.25
7.694
0.942
.389
X2 3 X2
2 5.25
7.694
2 0.682
.525
X3 3 X3
2 11.75
7.694
2 1.527
.187
X1 3 X2
0
7.393
0
1
X1 3 X3
0
7.393
0
1
X2 3 X3
2 7.5
7.393
2 1.015
.357
indicates that the first order model is not an adequate experiment and a higher order model is required. For this model below graphic are obtained as Figs. 8.28.6.
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Parameter optimization using the surface response technique Chapter | 8
TABLE 8.4 Analyze of variance for model. F
P
DF
Seq SS
Adj SS
Adj Ms
9
6549.4
6549.4
727.71
3.33
.099
Linear
3
5477
5477
1825.67
8.35
.022
X1
1
X2
1
2664.5
2664.5
2664.5
12.19
.017
X3
1
2812.5
2812.5
2812.5
12.87
.016
Square
3
847.4
847.4
282.47
1.29
.373
X1 3 X1
1
267.47
194.08
194.08
0.89
.389
X2 3 X2
1
70.16
101.77
101.77
0.47
.525
X3 3 X3
1
509.77
509.77
509.77
2.33
.187
Interaction
3
225
225
75
X1 3 X2
1
0
0
0
0
1
X1 3 X3
1
0
0
0
0
1
X2 3 X3
1
225
225
225
Residual error
5
1093
1093
218.6
Lack-of-fit
3
1093
1093
364.33
Pure error
2
0
0
Total
14
0
0
0
0
1
0.34
.796
1.03
.357
0
7642.4
80 No of Prd
60 40
1
20
0 –1
0 Acceleration
1
Number of AGV
–1
FIGURE 8.2 No. of Prd, acceleration and number of AGV. AGV, Automated guided vehicle; No. of Prd, Number of production.
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SECTION | II Sustainable Engineering Products
100
Number of Prd
75 1 0
50 –1
0 Acceleration
Velocity
–1
1
FIGURE 8.3 No. of Prd, acceleration and velocity. No. of Prd, Number of production.
100 80 Number of Prd
60
1
40 0 –1
0 Velocity
1
Number of AGV
–1
FIGURE 8.4 No. of Prd, number of AGV and velocity. AGV, Automated guided vehicle; No. of Prd, Number of production.
Desirability function: For ith response is assigned a desirability function, di, where the value of di varies between 0 and 1. The function, di, is defined differently based on the objective of the response. For example, if target production number is assumed to be 95, desirability function is 1 for production number 98 where number of AGV coded is 0.4747 (number of AGV as uncoded is 11.89).
Parameter optimization using the surface response technique Chapter | 8
195
Contour plots of number of production –1.0
Acceleration × velocity
–0.5
0.0
0.5
Number of AGV × velocity
1.0 Number of production
1.0
< 20 20 – 40 40 – 60 60 – 80 > 80
0.5 0.0 –0.5
1.0
Number of AGV × acceleration
–1.0
Hold values 0 Velocity 0 Acceleration Number of AGV 0
0.5 0.0 –0.5 –1.0 –1.0
–0.5
0.0
0.5
1.0
FIGURE 8.5 Contour plots of No. of Prd. No. of Prd, Number of production.
FIGURE 8.6 Acceleration, number of AGV and velocity. AGV, Automated guided vehicle.
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8.5 Conclusion Satisfying target No. of Prd is crucial for any organization. On the other hand, it is very important to invest necessary required number of AGV due to financial constraint. System designer should determine right number of AGV. A simulation is a tool for testing scenarios (what if analyze). On the other hand, the model designer needs a tool to optimize system input to satisfy the system constraint and maximize (or minimize) the target. At this point, we propose surface response technique as a tool for maximization. In this study, production number is considered as dependent output variable, various combinations of AGV and their parameter are considered independent variable. Following model is obtained: Y 5 76 1 2:05E 2 15X1 1 18:25X2 1 18:75X3 1 7:25X1 3 X1 2 5:25X2 3 X2 2 11:75X3 3 X3 1 3:29E 2 17X1 3 X2 2 6:35E 2 15X1 3 X3 2 7:5X2 3 X3 And for the target No. of Prd, required number of AGV is calculated. Optimization and comparison of multiple variables has been successfully carried out in the current study using RSM. The experimental optimum was found to be 11.89 for number of AGV. The experimental optimum matches closely with the predicted optimum.
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