ORIGINAL_ARTICLE
A Taghuchi based Multi Objective Time-Cost Constrained Scheduling for Resource Availability Cost Problem: A Case Study
In this paper, a new multi-objective time-cost constrained resource availability cost problem is proposed. The mathematical model is aimed to minimize resource availability cost by considering net present value of resource prices in order to evaluate the economic aspects of project to maximize the quality of project's resources to satisfy the expectations of stakeholders and to minimize the variation of resource usage during project. Since the problem is NP-hard, to deal with the problem a simulated annealing approach is applied, also to validate our results GAMS software is used in small size test problems. Due to the dependency of SA algorithm to its initial parameters a taghuchi method is used to find the best possible SA parameters combinations to reach near optimum solutions in large size problems.
https://www.riejournal.com/article_53423_5cc6627f3fa035e66b69404401cd2015.pdf
2017-12-01
269
282
10.22105/riej.2017.96349.1008
Constrained project scheduling
resource availability cost problem
Simulated Annealing Algorithm
Metaheuristic Algorithms
M.
Rabbani
mrabani@ut.ac.ir
1
Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran.
LEAD_AUTHOR
S.
Aghamohamadi
aghamohamadi.sor@ut.ac.ir
2
Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran.
AUTHOR
H.
Farrokhi-Asl
hamed.farrokhi@alumni.ut.ac.ir
3
Department of Industrial Engineering, Iran University of Science & Technology, Tehran, Iran.
AUTHOR
M.
Alavi mofrad
m.alavimofrad@ut.ac.ir
4
Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran.
AUTHOR
[1] Zhu, X., Ruiz, R., Li, S., & Li, X. (2017). An effective heuristic for project scheduling with resource availability cost. European journal of operational research, 257(3), 746-762.
1
[2] Möhring, R. H. (1984). Minimizing costs of resource requirements in project networks subject to a fixed completion time. Operations research, 32(1), 89-120.
2
[3] Hartmann, S. (1998). A competitive genetic algorithm for resource‐constrained project scheduling. Naval Research Logistics (NRL), 45(7), 733-750.
3
[4] Bouleimen, K. L. E. I. N., & Lecocq, H. O. U. S. N. I. (2003). A new efficient simulated annealing algorithm for the resource-constrained project scheduling problem and its multiple mode version. European journal of operational research, 149(2), 268-281.
4
[5] Brucker, P. (2002). Scheduling and constraint propagation. Discrete applied mathematics, 123(1), 227-256.
5
[6] Li, H., & Womer, N. K. (2015). Solving stochastic resource-constrained project scheduling problems by closed-loop approximate dynamic programming. European journal of operational research, 246(1), 20-33.
6
[7] Ma, W., Che, Y., Huang, H., & Ke, H. (2016). Resource-constrained project scheduling problem with uncertain durations and renewable resources. International journal of machine learning and cybernetics, 7(4), 613-621.
7
[8] Vanhoucke, M., & Coelho, J. (2016). An approach using SAT solvers for the RCPSP with logical constraints. European journal of operational research, 249(2), 577-591.
8
[9] Yassine, A. A., Mostafa, O., & Browning, T. R. (2017). Scheduling multiple, resource-constrained, iterative, product development projects with genetic algorithms. Computers & industrial engineering, 107, 39-56.
9
[10] Kreter, S., Rieck, J., & Zimmermann, J. (2016). Models and solution procedures for the resource-constrained project scheduling problem with general temporal constraints and calendars. European journal of operational research, 251(2), 387-403.
10
[11] Drexl, A., & Kimms, A. (2001). Optimization guided lower and upper bounds for the resource investment problem. Journal of the operational research society, 52(3), 340-351.
11
[12] Yamashita, D. S., Armentano, V. A., & Laguna, M. (2006). Scatter search for project scheduling with resource availability cost. European journal of operational research, 169(2), 623-637.
12
[13] Yamashita, D. S., Armentano, V. A., & Laguna, M. (2007). Robust optimization models for project scheduling with resource availability cost. Journal of scheduling, 10(1), 67-76.
13
[14] Ranjbar, M., Kianfar, F., & Shadrokh, S. (2008). Solving the resource availability cost problem in project scheduling by path relinking and genetic algorithm. Applied mathematics and computation, 196(2), 879-888.
14
[15] Van Peteghem, V., & Vanhoucke, M. (2013). An artificial immune system algorithm for the resource availability cost problem. Flexible services and manufacturing journal, 25(1-2), 122-144.
15
[16] Rad, M. S., Jamili, A., Tavakkoli-Moghaddam, R., & Paknahad, M. (2016, January). Resource constraint project scheduling to meet net present value and quality objectives of the program. Proceeding of 12th International Conference on Industrial Engineering (ICIE), 58-62. 10.1109/INDUSENG.2016.7519349
16
[17] Yassine, A. A., Mostafa, O., & Browning, T. R. (2017). Scheduling multiple, resource-constrained, iterative, product development projects with genetic algorithms. Computers & industrial engineering, 107, 39-56.
17
[18] Kreter, S., Rieck, J., & Zimmermann, J. (2016). Models and solution procedures for the resource-constrained project scheduling problem with general temporal constraints and calendars. European journal of operational research, 251(2), 387-403.
18
[19] Tavana, M., Abtahi, A. R., & Khalili-Damghani, K. (2014). A new multi-objective multi-mode model for solving preemptive time–cost–quality trade-off project scheduling problems. Expert systems with applications, 41(4), 1830-1846.
19
[20] Zhalechian, M., Tavakkoli-Moghaddam, R., & Rahimi, Y. (2017). A self-adaptive evolutionary algorithm for a fuzzy multi-objective hub location problem: An integration of responsiveness and social responsibility. Engineering applications of artificial intelligence, 62, 1-16.
20
[21] Torabi, S. A., & Hassini, E. (2008). An interactive possibilistic programming approach for multiple objective supply chain master planning. Fuzzy sets and systems, 159(2), 193-214.
21
[22] Verbeeck, C., Van Peteghem, V., Vanhoucke, M., Vansteenwegen, P., & Aghezzaf, E. H. (2017). A metaheuristic solution approach for the time-constrained project scheduling problem. OR spectrum, 39(2), 353-371.
22
[23] [24] Van Peteghem, V., & Vanhoucke, M. (2015). Heuristic methods for the resource availability cost problem. Handbook on project management and scheduling (pp. 339-359). Springer.
23
[24] Eshraghi, A. (2016). A new approach for solving resource constrained project scheduling problems using differential evolution algorithm. International journal of industrial engineering computations, 7(2), 205-216.
24
[25] Kreter, S., Rieck, J., & Zimmermann, J. (2016). Models and solution procedures for the resource-constrained project scheduling problem with general temporal constraints and calendars. European journal of operational research, 251(2), 387-403.
25
[26] Alhumrani, S. A., & Qureshi, R. J. (2016). Novel approach to solve Resource Constrained Project Scheduling Problem (RCPSP). International journal of modern education and computer science, 8(9), 60.
26
[27] Bilolikar, V. S., Jain, K., & Sharma, M. (2016). An adaptive crossover genetic algorithm with simulated annealing for multi mode resource constrained project scheduling with discounted cash flows. International journal of operational Research, 25(1), 28-46.
27
[28] Azadeh, A., Habibnejad-Ledari, H., Abdolhossein Zadeh, S., & Hosseinabadi Farahani, M. (2017). A single-machine scheduling problem with learning effect, deterioration and non-monotonic time-dependent processing times. International journal of computer integrated manufacturing, 30(2-3), 292-304.
28
ORIGINAL_ARTICLE
Scheduling Project Crashing Time Using Linear Programming Approach: Case Study
In today’s competitive environment completing a project within time and budget, is very challenging task for the project managers. This aim of this study is to develop a model that finds a proper trade-off between time and cost to expedite the execution process. Critical path method (CPM) is used to determine the longest duration and cost required for completing the project and then the time-cost trade–off problem (TCTP) is formulated as a linear programming model. Here, LINDO program is used to determine the solution of the model. To implement the proposed model, necessary data were collected through interviews and direct discussion with the project managers of Chowdhury Construction Company, Dhaka, Bangladesh. The analysis reveals that through proper scheduling of all activities, the project can be completed within 120 days from estimated duration of 140 days. Reduction of project duration by 17% is achieved by increasing cost by 3.73%, which is satisfactory.
https://www.riejournal.com/article_51839_2bbf6f065bd3b1c2b2f837778f822ea8.pdf
2017-12-01
283
292
10.22105/riej.2017.96572.1010
Linear Programming
critical path method
trade-off analysis
crashing
K.
Chitra
k.chitroleka@gmail.com
1
Department of Industrial and Production Engineering, Jessore University of Science and Technology, Jessore-7408, Bangladesh
LEAD_AUTHOR
P.
Halder
pobitra.halder@gmail.com
2
Department of Industrial and Production Engineering, Jessore University of Science and Technology, Jessore-7408, Bangladesh
AUTHOR
[1] Pour, N. S., Modarres, M., & Moghadam, R. T. (2012). Time-cost-quality trade-off in project scheduling with linguistic variables. World applied sciences journal, 18(3), 404-413.
1
[2] Kelley Jr, J. E. (1961). Critical-path planning and scheduling: Mathematical basis. Operations research, 9(3), 296-320.
2
[3] Mobini, M., Mobini, Z., & Rabbani, M. (2011). An Artificial Immune Algorithm for the project scheduling problem under resource constraints. Applied soft computing, 11(2), 1975-1982.
3
[4] Phillips Jr, S., & Dessouky, M. I. (1977). Solving the project time/cost tradeoff problem using the minimal cut concept. Management science, 24(4), 393-400.
4
[5] Shouman, М. Л., Abu El-Nour, A., & Elmehalawi, E. (1991). SCHEDULING NATURAL GAS PROJECTS IN CAIRO USING CPM AND TIME/COST TRADE-OFT. Alexandria engineering journal, 30(2), 157-166.
5
[6] Agarwal, A., Colak, S., & Erenguc, S. (2011). A neurogenetic approach for the resource-constrained project scheduling problem. Computers & operations research, 38(1), 44-50.
6
[7] Liu, L., Burns, S. A., & Feng, C. W. (1995). Construction time-cost trade-off analysis using LP/IP hybrid method. Journal of construction engineering and management, 121(4), 446-454.
7
[8] Hindelang, T. J., & Muth, J. F. (1979). A dynamic programming algorithm for decision CPM networks. Operations research, 27(2), 225-241.
8
[9] De, P., Dunne, E. J., Ghosh, J. B., & Wells, C. E. (1995). The discrete time-cost tradeoff problem revisited. European journal of operational research, 81(2), 225-238.
9
[10] Arauzo, J. A., Galán, J. M., Pajares, J., & López-Paredes, A. (2009). Multi-agent technology for scheduling and control projects in multi-project environments. An Auction based approach. Inteligencia artificial. Revista iberoamericana de inteligencia artificial, 13(42).
10
[11] Abbasimehr, H., & Alizadeh, S. (2013). A novel genetic algorithm based method for building accurate and comprehensible churn prediction models. International journal of research in industrial engineering, 2(4), 1.
11
[12] Feng, C. W., Liu, L., & Burns, S. A. (1997). Using genetic algorithms to solve construction time-cost trade-off problems. Journal of computing in civil engineering, 11(3), 184-189.
12
[13] Li, H., Cao, J. N., & Love, P. E. D. (1999). Using machine learning and GA to solve time-cost trade-off problems. Journal of construction engineering and management, 125(5), 347-353.
13
[14] Ponnambalam, S. G., Aravindan, P., & Rao, M. S. (2003). Genetic algorithms for sequencing problems in mixed model assembly lines. Computers & industrial engineering, 45(4), 669-690.
14
[15] Shahsavari-Pour, N., Modarres, M., Tavakoli-Moghadam, R., & Najafi, E. (2010). Optimizing a multi-objectives time-cost-quality trade-off problem by a new hybrid genetic algorithm. World applied science journal, 10(3), 355-363.
15
[16] Azaron, A., Perkgoz, C., & Sakawa, M. (2005). A genetic algorithm approach for the time-cost trade-off in PERT networks. Applied mathematics and computation, 168(2), 1317-1339.
16
[17] Azaron, A., Perkgoz, C., & Sakawa, M. (2005). A genetic algorithm approach for the time-cost trade-off in PERT networks. Applied mathematics and computation, 168(2), 1317-1339.
17
[18] El Razek, R. H. A., Diab, A. M., Hafez, S. M., & Aziz, R. F. (2010). Time-cost-quality trade-off software by using simplified genetic algorithm for typical repetitive construction projects. World academy of science, engineering and technology, 61, 312-321.
18
[19] Chua, D. K. H., Chan, W. T., & Govindan, K. (1997). A time-cost trade-off model with resource consideration using genetic algorithm. Civil engineering systems, 14(4), 291-311.
19
[20] Pathak, B. K., & Srivastava, S. (2007, September). MOGA-based time-cost tradeoffs: responsiveness for project uncertainties. Proceedings of congress on evolutionary computation, CEC. 3085-3092. IEEE.
20
[21] Pathak, B. K., Srivastava, S., & Srivastava, K. (2008). Pathak, B. K., Srivastava, S., & Srivastava, K. (2008). Neural network embedded multiobjective genetic algorithm to solve non-linear time-cost tradeoff problems of project scheduling. Journal of scientific and industrial research (JSIR), 67(2).
21
[22] Chen, W. N., Zhang, J., Chung, H. S. H., Huang, R. Z., & Liu, O. (2010). Optimizing discounted cash flows in project scheduling—An ant colony optimization approach. IEEE transactions on systems, man, and cybernetics, part C (applications and reviews), 40(1), 64-77.
22
[23] Zeinalzadeh, A. (2011). An application of mathematical model to time-cost tradeoff problem (case study). Australian journal of basic and applied sciences, 5(7), 208-214.
23
[24] Biswas, S. K., Karmaker, C. L., & Biswas, T. K. Time-Cost Trade-Off Analysis in a Construction Project Problem: Case Study.
24
[25] Mokhtari, H., Aghaie, A., Rahimi, J., & Mozdgir, A. (2010). Project time–cost trade-off scheduling: a hybrid optimization approach. The international journal of advanced manufacturing technology, 50(5-8), 811-822.
25
[26] Błaszczyk, T., & Nowak, M. (2009). The time‐cost trade‐off analysis in construction project using computer simulation and interactive procedure. Technological and economic development of economy, 15(4), 523-539.
26
[27] Hosseini-Nasab, H., Pourkheradmand, M., & Shahsavaripour, N. (2017). Solving multi-mode time-cost-quality trade-off problem in uncertainty condition using a novel genetic algorithm. International journal of management and fuzzy systems, 3(3), 32.
27
[28] Su, Z., Qi, J., & Wei, H. (2017). Simplifying the nonlinear continuous time-cost tradeoff problem. Journal of systems science and complexity, 30(4), 901-920.
28
[29] Zou, X., Fang, S. C., Huang, Y. S., & Zhang, L. H. (2016). Mixed-Integer linear programming approach for scheduling repetitive projects with time-cost trade-off consideration. Journal of computing in civil engineering, 31(3), 06016003.
29
ORIGINAL_ARTICLE
Modified Method for Solving Fully Fuzzy Linear Programming Problem with Triangular Fuzzy Numbers
The Fuzzy Linear Programming problem has been used as an important planning tool for the different disciplines such as engineering, business, finance, economics, etc. In this paper, we proposed a modified algorithm to find the fuzzy optimal solution of fully fuzzy linear programming problems with equality constraints. Recently, Ezzati et al. (Applied Mathematical Modelling, 39 (2015) 3183-3193) suggested a new algorithm to solve fully fuzzy linear programming problems. In this paper, we modified this algorithm and compare it with other existing methods. Furthermore, for illustration, some numerical examples and one real problem are used to demonstrate the correctness and usefulness of the proposed method.
https://www.riejournal.com/article_54494_2faa9c269f9829ea81e4d095ba532ab6.pdf
2017-12-01
293
311
10.22105/riej.2017.101594.1024
Linear programming problem
fully fuzzy linear programming
multi-objective linear programming
triangular fuzzy numbers
S. K.
Das
cool.sapankumar@gmail.com
1
Department of Mathematics, National Institute of Technology Jamshedpur, Jharkhand 831014,India.
LEAD_AUTHOR
[1] Zimmermann, H. J. (2011). Fuzzy set theory—and its applications. Springer Science & Business Media.
1
[2] Baykasoğlu, A., & Subulan, K. (2015). An analysis of fully fuzzy linear programming with fuzzy decision variables through logistics network design problem. Knowledge-Based systems, 90, 165-184.
2
[3] Dehghan, M., Hashemi, B., & Ghatee, M. (2006). Computational methods for solving fully fuzzy linear systems. Applied mathematics and computation, 179(1), 328-343.
3
[4] Ebrahimnejad, A., & Tavana, M. (2014). A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers. Applied mathematical modelling, 38(17), 4388-4395.
4
[5] Khan, I. U., Ahmad, T., & Maan, N. (2013). A simplified novel technique for solving fully fuzzy linear programming problems. Journal of optimization theory and applications, 159(2), 536-546.
5
[6] Khan, I. U., Ahmad, T., & Maan, N. (2013). A simplified novel technique for solving fully fuzzy linear programming problems. Journal of optimization theory and applications, 159(2), 536-546.
6
[7] Lotfi, F. H., Allahviranloo, T., Jondabeh, M. A., & Alizadeh, L. (2009). Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution. Applied mathematical modelling, 33(7), 3151-3156.
7
[8] Kumar, A., Kaur, J., & Singh, P. (2011). A new method for solving fully fuzzy linear programming problems. Applied mathematical modelling, 35(2), 817-823.
8
[9] Shamooshaki, M. M., Hosseinzadeh, A., & Edalatpanah, S. A. (2014). A new method for solving fully fuzzy linear programming with lr-type fuzzy numbers. International journal of data envelopment analysis and* Operations Research*, 1(3), 53-55.
9
[10] Veeramani, C., & Duraisamy, C. (2012). Solving fuzzy linear programming problem using symmetric fuzzy number approximation. International journal of operational research, 15(3), 321-336.
10
[11] Wu, H. C. (2008). Using the technique of scalarization to solve the multiobjective programming problems with fuzzy coefficients. Mathematical and computer modelling, 48(1), 232-248.
11
[12] Iskander, M. G. (2003). Using different dominance criteria in stochastic fuzzy linear multiobjective programming: A case of fuzzy weighted objective function. Mathematical and computer modelling, 37(1-2), 167-176.
12
[13] Mahdavi-Amiri, N., & Nasseri, S. H. (2006). Duality in fuzzy number linear programming by use of a certain linear ranking function. Applied mathematics and computation, 180(1), 206-216.
13
[14] Ezzati, R., Khorram, E., & Enayati, R. (2015). A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem. Applied mathematical modelling, 39(12), 3183-3193.
14
[15] Dubois, D. J. (1980). Fuzzy sets and systems: theory and applications (Vol. 144). Academic press.
15
[16] Kauffman, A., & Gupta, M. M. (1991). Introduction to fuzzy arithmetic: Theory and application. Mathematics of Computation.
16
[17] Liou, T. S., & Wang, M. J. J. (1992). Ranking fuzzy numbers with integral value. Fuzzy sets and systems, 50(3), 247-255.
17
[18] Das, S. K., Mandal, T., & Edalatpanah, S. A. (2017). A new approach for solving fully fuzzy linear fractional programming problems using the multi-objective linear programming. RAIRO-operations research, 51(1), 285-297.
18
[19] Das, S. K., Mandal, T., & Edalatpanah, S. A. (2017). A mathematical model for solving fully fuzzy linear programming problem with trapezoidal fuzzy numbers. Applied intelligence, 46(3), 509-519.
19
[20] Das, S. K., & Mandal, T. (2015). A single stage single constraints linear fractional programming problem: An approach. Operations research and applications: An international journal (ORAJ), 2(1), 9-14.
20
ORIGINAL_ARTICLE
Reviewing on Nanotechnology for Creating Antimicrobial for Chicken Feed: Max-Min Optimization Approach
Nanotechnology deals with studies of phenomena and manipulation on elements of matter at the atomic, molecular and macromolecular level (rangefrom1to100nm), where the properties of matter are significantly different from properties at larger scales of dimensions. Nanotechnology is science, engineering, and technology conducted at the nanoscale, which is about 1 to 100 nm where nano denotes the scale range of 10-9 and nanotechnology refers the properties of atoms and molecules measuring thoroughly 0.1 to 1000 nm. Nanotechnology is highly interdisciplinary as a field, and it requires knowledge drawn from a variety of scientific and engineering arenas. There are two main types of approaches to nanotechnology: the first approach is Top-down and another one is Bottom-up approach. The Top-down approach involves taking layer structures that are either reduced down size until they reach the nano-scale or deacon structured into their composite parts. This paper aims to deal with Top-down approach in order to utilize Biopolymer nanoparticles for Creating Antimicrobial for chicken feed so that the live average time of chicken will be increased noticeably by using max-min optimization approach. Finally, the applicability of the proposed approach and the solution methodologies are demonstrated in three steps.
https://www.riejournal.com/article_54469_00fe461c36f22a9aa76920eb7a460391.pdf
2017-12-01
312
327
10.22105/riej.2017.100933.1021
Nanotechnology
live average time
max-min approach
optimization
A. H.
Niknamfar
niknamfar@yahoo.com
1
Young Researchers and Elite Club, Qazvin Branch, Islamic Azad University, Qazvin, Iran.
LEAD_AUTHOR
H.
Esmaeili
h.esmaeili7227@gmail.com
2
Department of business management, Qazvin Branch, Islamic Azad University, Qazvin, Iran.
AUTHOR
[1] Kratica, J., Stanimirović, Z., Tošić, D., & Filipović, V. (2007). Two genetic algorithms for solving the uncapacitated single allocation p-hub median problem. European journal of operational research, 182(1), 15-28.
1
[2] Lin, C. C., & Chen, Y. C. (2003). The integration of Taiwanese and Chinese air networks for direct air cargo services. Transportation research part A: Policy and practice, 37(7), 629-647.
2
[3] Karimi, H., & Bashiri, M. (2011). Hub covering location problems with different coverage types. Scientia iranica, 18(6), 1571-1578.
3
[4] O’Kelly, M. E., & Bryan, D. L. (1998). Hub location with flow economies of scale. Transportation research part B: Methodological, 32(8), 605-616.
4
[5] Ebery, J., Krishnamoorthy, M., Ernst, A., & Boland, N. (2000). The capacitated multiple allocation hub location problem: Formulations and algorithms. European journal of operational research, 120(3), 614-631.
5
[6] Ebery, J. (2001). Solving large single allocation p-hub problems with two or three hubs. European journal of operational research, 128(2), 447-458.
6
[7] Pérez, M., Almeida, F., & Moreno-Vega, J. M. (2005, August). A hybrid GRASP-path relinking algorithm for the capacitated p–hub median problem. International workshop on hybrid metaheuristics (pp. 142-153). Springer, Berlin, Heidelberg.
7
[8] da Graça Costa, M., Captivo, M. E., & Clímaco, J. (2008). Capacitated single allocation hub location problem—A bi-criteria approach. Computers & operations research, 35(11), 3671-3695.
8
[9] de Camargo, R. S., Miranda, G. D., & Luna, H. P. (2008). Benders decomposition for the uncapacitated multiple allocation hub location problem. Computers & operations research, 35(4), 1047-1064.
9
[10] Contreras, I., Díaz, J. A., & Fernández, E. (2009). Lagrangean relaxation for the capacitated hub location problem with single assignment. OR spectrum, 31(3), 483-505.
10
[11] Sim, T., Lowe, T. J., & Thomas, B. W. (2009). The stochastic p-hub center problem with service-level constraints. Computers & operations research, 36(12), 3166-3177.
11
[12] Wei, G., Jinfu, Z., & Weiwei, W. (2010, August). A tree pruning algorithm for the capacitated p-hub median problems. Proceedings of international conference on information engineering (ICIE), 2010 WASE . 324-327). IEEE.
12
[13] Stanimirović, Z. (2012). A genetic algorithm approach for the capacitated single allocation p-hub median problem. Computing and informatics, 29(1), 117-132.
13
[14] Yang, K., Liu, Y., & Zhang, X. (2011). Stochastic p-hub center problem with discrete time distributions. Advances in neural networks–ISNN, 182-191.
14
[15] Yaman, H., & Elloumi, S. (2012). Star p-hub center problem and star p-hub median problem with bounded path lengths. Computers & operations research, 39(11), 2725-2732.
15
[16] Rabbani, M., Pakzad, F., & Kazemi, S. M. (2013, April). A new modelling for p-hub median problem by considering flow-dependent costs. Proceedings of 5th international conference on modeling, simulation and applied optimization (ICMSAO). 1-4. IEEE.
16
[17] Yang, K., Liu, Y. K., & Yang, G. Q. (2013). Solving fuzzy p-hub center problem by genetic algorithm incorporating local search. Applied soft computing, 13(5), 2624-2632.
17
[18] Bashiri, M., Mirzaei, M., & Randall, M. (2013). Modeling fuzzy capacitated p-hub center problem and a genetic algorithm solution. Applied mathematical modelling, 37(5), 3513-3525.
18
[19] Farahani, R. Z., Hekmatfar, M., Arabani, A. B., & Nikbakhsh, E. (2013). Hub location problems: A review of models, classification, solution techniques, and applications. Computers & industrial engineering, 64(4), 1096-1109.
19
[20] Eiselt, H. A., & Laporte, G. (1997). Sequential location problems. European journal of operational research, 96(2), 217-231.
20
[21] Eiselt, H. A., Laporte, G., & Thisse, J. F. (1993). Competitive location models: A framework and bibliography. Transportation science, 27(1), 44-54.
21
[22] Marianov, V., Serra, D., & ReVelle, C. (1999). Location of hubs in a competitive environment. European journal of operational research, 114(2), 363-371.
22
[23] Marianov, V., Serra, D., & ReVelle, C. (1999). Location of hubs in a competitive environment. European journal of operational research, 114(2), 363-371.
23
[24] Lüer-Villagra, A., & Marianov, V. (2013). A competitive hub location and pricing problem. European journal of operational research, 231(3), 734-744.
24
[25] Sasaki, M., Campbell, J. F., Krishnamoorthy, M., & Ernst, A. T. (2014). A Stackelberg hub arc location model for a competitive environment. Computers & operations research, 47, 27-41.
25
[26] Tavakkoli-Moghaddam, R., Gholipour-Kanani, Y., & Shahramifar, M. (2013). A multi-objective imperialist competitive algorithm for a capacitated single-allocation hub location problem. International journal of engineering-transactions C: Aspects, 26(6), 605-612.
26
[27] Chou, C. C. (2010). Application of FMCDM model to selecting the hub location in the marine transportation: A case study in southeastern Asia. Mathematical and computer modelling, 51(5), 791-801.
27
[28] Michael, R. G., & David, S. J. (1979). Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman & Co. New York, NY, USA.
28
[29] [Tizhoosh, H. R. (2005, November). Opposition-based learning: a new scheme for machine intelligence. Proceedings of international conference on intelligent agents, web technologies and internet commerce,computational intelligence for modelling, control and automation. 695-701. IEEE.
29
[30] Tizhoosh,H. R.(2009). Opposite fuzzy sets with applications in image processing. Proceedings of international fuzzy systems association world congress. 36-41. Lisbon, Postugal.
30
[31] Ventresca, M., & Tizhoosh, H. R. (2006, July). Improving the convergence of backpropagation by opposite transfer functions. Proceedings of international joint conference on neural networks, IJCNN'06. 4777-4784. IEEE.
31
[32] Shokri, M., Tizhoosh, H. R., & Kamel, M. S. (2007, April). Opposition-based Q (λ) with non-markovian update. Proceedings of international symposium on approximate dynamic programming and reinforcement learning, ADPRL. 288-295. IEEE.
32
[33] Rahnamayan, S., Tizhoosh, H. R., & Salama, M. M. (2008). Opposition-based differential evolution. IEEE transactions on evolutionary computation, 12(1), 64-79.
33
[34] Ergezer, M., Simon, D., & Du, D. (2009, October). Oppositional biogeography-based optimization. proceedings of International Conference on Systems, Man and Cybernetics, SMC. 1009-1014. IEEE.
34
[35] Wang, H., Wu, Z., Rahnamayan, S., Liu, Y., & Ventresca, M. (2011). Enhancing particle swarm optimization using generalized opposition-based learning. Information sciences, 181(20), 4699-4714.
35
[36] Malisia, A. (2008). Improving the exploration ability of ant-based algorithms. Oppositional concepts in computational intelligence, 121-142.
36
[37] Ventresca, M., & Tizhoosh, H. R. (2007, April). Simulated annealing with opposite neighbors. Proceedings of symposium on foundations of computational intelligence, FOCI. 186-192. IEEE.
37
[38] Seif, Z., & Ahmadi, M. B. (2015). Opposition versus randomness in binary spaces. Applied soft computing, 27, 28-37.
38
[39] Pasandideh, S. H. R., Niaki, S. T. A., & Niknamfar, A. H. (2014). Lexicographic max–min approach for an integrated vendor-managed inventory problem. Knowledge-Based systems, 59, 58-65.
39
[40] Kubotani, H., & Yoshimura, K. (2003). Performance evaluation of acceptance probability functions for multi-objective SA. Computers & operations research, 30(3), 427-442.
40
[41] Simon, D. (2008). Biogeography-based optimization. IEEE transactions on evolutionary computation, 12(6), 702-713.
41
[42] Ramezanian, R., Rahmani, D., & Barzinpour, F. (2012). An aggregate production planning model for two phase production systems: Solving with genetic algorithm and tabu search. Expert systems with applications, 39(1), 1256-1263.
42
[43] Fei, C., & Li, J. P. (2012, December). Multi-focus image fusion based on nonsubsampled contourlet transform and multi-objective optimization. Proceedings of international conference on wavelet active media technology and information processing (ICWAMTIP). 189-192. IEEE.
43
[44] Sarrafha, K., Rahmati, S. H. A., Niaki, S. T. A., & Zaretalab, A. (2015). A bi-objective integrated procurement, production, and distribution problem of a multi-echelon supply chain network design: A new tuned MOEA. Computers & operations research, 54, 35-51.
44
[45] Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. A. M. T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE transactions on evolutionary computation, 6(2), 182-197.
45
[46] Sivanandam, S. N., & Deepa, S. N. (2007). Introduction to genetic algorithms. Springer Science & Business Media.
46
[47] Deb, K. (2001). Multi-objective optimization using evolutionary algorithms (Vol. 16). John Wiley & Sons.
47
[48] Zitzler, E., & Thiele, L. (1998, September). Multiobjective optimization using evolutionary algorithms—a comparative case study. Proceedings of international conference on parallel problem solving from nature. 292-301. Springer, Berlin, Heidelberg.
48
[49] Schott, J. R. (1995). Fault tolerant design using single and multicriteria genetic algorithm optimization. Difference technical information center.
49
Retrieved from http://www.dtic.mil/dtic/tr/fulltext/u2/a296310.pdf
50
[50] Peace, G. S. (1993). Taguchi methods: a hands-on approach. Addison Wesley Publishing Company.
51
[51] Rahmati, S. H. A., Hajipour, V., & Niaki, S. T. A. (2013). A soft-computing Pareto-based meta-heuristic algorithm for a multi-objective multi-server facility location problem. Applied soft computing, 13(4), 1728-1740.
52
[52] Tawarmalani, M., & Sahinidis, N. V. (2005). A polyhedral branch-and-cut approach to global optimization. Mathematical programming, 103(2), 225-249.
53
[53] Tawarmalani, M., & Sahinidis, N. V. (2004). Global optimization of mixed-integer nonlinear programs: A theoretical and computational study. Mathematical programming, 99(3), 563-591.
54
[54] Tzeng, G. H., & Huang, J. J. (2011). Multiple attribute decision making: methods and applications. CRC press.
55
ORIGINAL_ARTICLE
Produced Water Treatment Using a New Designed Electroflotation Cell
A novel continuous electroflotation cell, about 0.6 liter volume capacity, using aluminum electrodes was designed for oil produced water treatment. The treating performance of a novel continuous electroflotation cell for oil produced water was investigated. The pH, current density, and feed water flow rate as affecting parameters of electroflotation process were studied. The results show that the removal efficiency decreased with increasing feed flow rate. However, it increased with increasing current density. The AC current was preferred because DC current causes passivation of the anode with time. The maximum removal for all types of pollutants is achieved at pH6. The designed electroflotation cell could remove different constituents of oil produced water with range 87.5 - 99.5 % at 25°C, 5V, pH7 and AC current density of 80A/m2 through a bipolar connection of the 8 electrodes with feed water flow rate of 60ml/min (3.6l/hr). The energy consumption was about 1.38Kwh/m3 and the operating cost (cost/m3) was about 0.3US$/m3 for the produced water treatment.
https://www.riejournal.com/article_54400_eab8868f1d961352c0b14b1fb28f2e29.pdf
2017-12-01
328
338
10.22105/riej.2017.100959.1022
Electroflotation
Produced Water
Water Treatment
cell design
M.
Abdel Khalek
kalekma@yahoo.com
1
Department of Central Metallurgical Research and Development Institute (CMRDI), Cairo, Egypt.
LEAD_AUTHOR
F.
El-Hosiny
fouadelhosiny57@gmail.com
2
Department of Chemistry, Faculty of Science, Ain Shams University, Cairo, Egypt.
AUTHOR
K.
Selim
k2selem@yahoo.com
3
Department of Central Metallurgical Research and Development Institute (CMRDI), Cairo, Egypt.
AUTHOR
I.
Osama
gigi_abdelatif@yahoo.com
4
Department of Central Metallurgical Research and Development Institute (CMRDI), Cairo, Egypt.
AUTHOR
[1] Veil, J. A., Puder, M. G., Elcock, D., & Redweik Jr, R. J. (2004). A white paper describing produced water from production of crude oil, natural gas, and coal bed methane (No. ANL/EA/RP-112631). Argonne National Lab., IL (US).
1
[2] Reynolds Rodney, R. (2003). Produced water and associated issues: A manual for independent operator. South-Midcontinent Region, PTTC
2
[3] Arthur, J. D., Langhus, B. G., & Patel, C. (2005). Technical summary of oil & gas produced water treatment technologies. All Consulting, LLC, Tulsa, OK.
3
[4] Lee, R., Seright, R., Hightower, M., Sattler, A., Cather, M., McPherson, B., ... & Whitworth, M. (2002, October). Strategies for produced water handling in New México. Proceedings of conference on ground water protection council produced water. 16-17. Boston, Massachsetts.
4
[5] Knudsen, B. L., Hjelsvold, M., Frost, T. K., Svarstad, M. B. E., Grini, P. G., Willumsen, C. F., & Torvik, H. (2004, January). Meeting the zero discharge challenge for produced water. Proceedings of SPE international conference on health, safety, and environment in oil and gas exploration and production. Society of petroleum engineers.
5
[6] Nandi, B. K., & Patel, S. (2013). Effects of operational parameters on the removal of brilliant green dye from aqueous solutions by electrocoagulation. Arabian Journal of Chemistry, 10, S2961-S2968.
6
[7] Chaturvedi, S. I. (2013). Electro-coagulation: a novel wastewater treatment method. International journal of modern engineering research, 3(1), 93-100.
7
[8] Sahu, O., Mazumdar, B., & Chaudhari, P. K. (2014). Treatment of wastewater by electrocoagulation: a review. Environmental science and pollution research, 21(4), 2397-2413.
8
[9] Holt, P. K., Barton, G. W., Wark, M., & Mitchell, C. A. (2002). A quantitative comparison between chemical dosing and electrocoagulation. Colloids and surfaces A: Physicochemical and engineering aspects, 211(2), 233-248.
9
[10] Al-abdalaali, A. A. (2010). Removal of boron from simulated iraqi surface water by electrocoagulation method. Academic scientific journals, 18(11), 1266-1284.
10
[11] Comninellis, C., & Chen, G. (Eds.). (2010). Electrochemistry for the Environment (Vol. 2015). New York: Springer.
11
[12] Rice, E. W., Baird, R. B., Eaton, A. D., & Clesceri, L. S. (2012). Standard methods for the examination of water and wastewater. American public health association, american water works association, and water environment federation.
12
[13] Federation, W. E., & American Public Health Association. (2005). Standard methods for the examination of water and wastewater. American public health association (APHA): Washington, DC, USA.
13
[14] Kim, H. C., & Lee, K. (2009). Significant contribution of dissolved organic matter to seawater alkalinity. Geophysical research letters, 36(20).
14
[15] Selim, K.A., El-Hosiny, F.I., Abdel-Khalek, M.A., & Osama, I. (2017). Kinetics and Thermo-dynamics of Some Heavy Metals Removal from Industrial Effluents Through Electro-Flotation Process. Colloid and Surface Science, 2(2) 47-53.
15
[16] El-Hosiny, F. I., Abdel-Khalek, M. A., Selim, K. A., & Osama, I. (2017). Physicochemical study of dye removal using electro-coagulation-flotation process. Physicochemical problems of mineral processing, 54(1), 1-14.
16
[17] Parsa, J. B., Vahidian, H. R., Soleymani, A. R., & Abbasi, M. (2011). Removal of Acid Brown 14 in aqueous media by electrocoagulation: Optimization parameters and minimizing of energy consumption. Desalination, 278(1), 295-302.
17
[18] Jiang, J. Q., Graham, N., André, C., Kelsall, G. H., & Brandon, N. (2002). Laboratory study of electro-coagulation–flotation for water treatment. Water research, 36(16), 4064-4078.
18
[19] Mouedhen, G., Feki, M., Wery, M. D. P., & Ayedi, H. F. (2008). Behavior of aluminum electrodes in electrocoagulation process. Journal of hazardous materials, 150(1), 124-135.
19
[20] Holt, P. K., Barton, G. W., Wark, M., & Mitchell, C. A. (2002). A quantitative comparison between chemical dosing and electrocoagulation. Colloids and surfaces A: Physicochemical and engineering aspects, 211(2), 233-248.
20
[21] Kim, M. S., Dockko, S., Myung, G., & Kwak, D. H. (2015). Feasibility study of high-rate dissolved air flotation process for rapid wastewater treatment. Journal of water supply: Research and technology-aqua, 64(8), 927-936.
21
[22] Kwak, D. H., & Kim, M. S. (2013). Feasibility of carbon dioxide bubbles as a collector in flotation process for water treatment. Journal of water supply: Research and technology-aqua, 62(1), 52-65.
22
[23] Al-Qodah, Z., & Al-Shannag, M. (2017). Heavy metal ions removal from wastewater using electrocoagulation processes: A comprehensive review. Separation science and technology, 52(17), 2649-2676.
23
[24] Edzwald, J. K. (2010). Dissolved air flotation and me. Water research, 44(7), 2077-2106.
24
[25] Dalvand, A., Gholami, M., Joneidi, A., & Mahmoodi, N. M. (2011). Dye removal, energy consumption and operating cost of electrocoagulation of textile wastewater as a clean process. Clean–soil, air, water, 39(7), 665-672.
25
[26] Ghosh, D., Medhi, C. R., & Purkait, M. K. (2011). Techno-economic analysis for the electrocoagulation of fluoride-contaminated drinking water. Toxicological & environ chemistry, 93(3), 424-437.
26
[27] Zheng, T. (2017). Treatment of oilfield produced water with electrocoagulation: improving the process performance by using pulse current. Journal of water reuse and desalination, 7(3), 378-386.
27
ORIGINAL_ARTICLE
A Barzilai Borwein Adaptive Trust-Region Method for Solving Systems of Nonlinear Equation
In this paper, we introduce a new adaptive trust-region approach to solve systems of nonlinear equations. In order to improve the efficiency of adaptive radius strategy proposed by Esmaeili and Kimiaei [8], Barzilai Borwein technique (BB) [3] with low memory is used which can truly control the trust-region radius. In addition, the global convergence of the new approach is proved. Computational experience suggests that the new approach is more effective in practice in comparison with other adaptive trust-region algorithms.
https://www.riejournal.com/article_53509_a52488bdea4d2e4f67c6c8869f470c60.pdf
2017-12-01
339
349
10.22105/riej.2017.101854.1027
Nonlinear equations
trust-region framework
adaptive radius
two-point gradient technique
F.
Rahpeymaii
rahpeyma_83@yahoo.com
1
Department of Mathematics, Payame Noor University, PO BOX 19395--3697, Tehran, Iran.
AUTHOR
M.
Kimiaei
kimiaeim83@univie.ac.at
2
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria.
LEAD_AUTHOR
[1] Ahookhosh, M., & Amini, K. (2010). A nonmonotone trust region method with adaptive radius for unconstrained optimization problems. Computers & mathematics with applications, 60(3), 411-422.
1
[2] Ahookhosh, M., Amini, K., & Peyghami, M. R. (2012). A nonmonotone trust-region line search method for large-scale unconstrained optimization. Applied mathematical modelling, 36(1), 478-487.
2
[3] Barzilai, J., & Borwein, J. M. (1988). Two-point step size gradient methods. IMA journal of numerical analysis, 8(1), 141-148.
3
[4] Bouaricha, A., & Schnabel, R. B. (1998). Tensor methods for large sparse systems of nonlinear equations. Mathematical programming, 82(3), 377-400.
4
[5] Broyden, C. G. (1971). The convergence of an algorithm for solving sparse nonlinear systems. Mathematics of computation, 25(114), 285-294.
5
[6] Conn, A. R., Gould, N. I., & Toint, P. L. (2000). Trust region methods. Society for Industrial and Applied Mathematics.
6
[7] Dolan, E. D., & Moré, J. J. (2002). Benchmarking optimization software with performance profiles. Mathematical programming, 91(2), 201-213.
7
[8] Esmaeili, H., & Kimiaei, M. (2014). A new adaptive trust-region method for system of nonlinear equations. Applied mathematical modelling, 38(11), 3003-3015.
8
[9] Fan, J. (2006). Convergence rate of the trust region method for nonlinear equations under local error bound condition. Computational optimization and applications, 34(2), 215-227.
9
[10] Fan, J., & Pan, J. (2011). An improved trust region algorithm for nonlinear equations. Computational optimization and applications, 48(1), 59-70.
10
[11] Fan, J., & Pan, J. (2010). A modified trust region algorithm for nonlinear equations with new updating rule of trust region radius. International journal of computer mathematics, 87(14), 3186-3195.
11
[12] Grippo, L., & Sciandrone, M. (2007). Nonmonotone derivative-free methods for nonlinear equations. Computational optimization and applications, 37(3), 297-328.
12
[13] La Cruz, W., Martínez, J., & Raydan, M. (2006). Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Mathematics of computation, 75(255), 1429-1448.
13
[14] Li, D. H., & Fukushima, M. (2000). A derivative-free line search and global convergence of Broyden-like method for nonlinear equations. Optimization methods and software, 13(3), 181-201.
14
[15] Lukšan, L., & Vlček, J. (1999). Sparse and partially separable test problems for unconstrained and equality constrained optimization. Technical report, 767.
15
[16] Moré, J. J., Garbow, B. S., & Hillstrom, K. E. (1981). Testing unconstrained optimization software. ACM transactions on mathematical software (TOMS), 7(1), 17-41.
16
[17] Nesterov, Y. (2007). Modified Gauss–Newton scheme with worst case guarantees for global performance. Optimisation methods and software, 22(3), 469-483.
17
[18] Bonnans, J. F., Gilbert, J. C., Lemaréchal, C., & Sagastizábal, C. A. (2006). Numerical optimization: Theoretical and practical aspects. Springer Science & Business Media.
18
[19] Toint, P. L. (1986). Numerical solution of large sets of algebraic nonlinear equations. Mathematics of computation, 46(173), 175-189.
19
[20] Yuan, G., Lu, S., & Wei, Z. (2011). A new trust-region method with line search for solving symmetric nonlinear equations. International journal of computer mathematics, 88(10), 2109-2123.
20
[21] Yuan, G., Wei, Z., & Lu, X. (2011). A BFGS trust-region method for nonlinear equations. Computing, 92(4), 317-333.
21
[22] Yuan, Y. X. (1994). Trust region algorithms for nonlinear equations. Hong Kong Baptist University, Department of Mathematics.
22
[23] Zhang, J. L., & Wang, Y. (2003). A new trust region method for nonlinear equations. Mathematical methods of operations research, 58(2), 283-298.
23
ORIGINAL_ARTICLE
CAS Wavelet Function Method for Solving Abel Equations with Error Analysis
In this paper we use a computational method based on CAS wavelets for solving nonlinear fractional order Volterra integral equations. We solve particularly Abel equations. An operational matrix of fractional order integration for CAS wavelets is used. Block Pulse Functions (BPFs) and collocation method are employed to derive a general procedure for forming this matrix. The error analysis of proposed numerical scheme is studied theoretically. Finally, comparison of numerical results with exact solution are shown.
https://www.riejournal.com/article_54493_78da80d54334fe864a5d5dc0f111f61b.pdf
2017-12-01
350
364
10.22105/riej.2017.100538.1017
Abel integral equations
CAS wavelet
fractional order volterra integral equations
operational matrix
Error analysis
R.
Ezzati
ezati@kiau.ac.ir
1
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.
LEAD_AUTHOR
K.
Maleknejad
maleknejad@iust.ac.ir
2
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.
AUTHOR
E.
Fathizadeh
einollahfathizade@ymail.com
3
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.
AUTHOR
[1] He, J. H. (1998). Nonlinear oscillation with fractional derivative and its applications. International Proceedings of international conference on vibrating engineering, 288-291.
1
[2] Mainardi, F. (2012). Fractional calculus: some basic problems in continuum and statistical mechanics. arXiv preprint arXiv:1201.0863.
2
[3] Rossikhin, Y. A., & Shitikova, M. V. (1997). Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Applied mechanics reviews, 50, 15-67.
3
[4] Baillie, R. T. (1996). Long memory processes and fractional integration in econometrics. Journal of econometrics, 73(1), 5-59.
4
[5] Evans, R. M., Katugampola, U. N., & Edwards, D. A. (2017). Applications of fractional calculus in solving Abel-type integral equations: Surface–volume reaction problem. Computers & mathematics with applications, 73(6), 1346-1362.
5
[6] Chen, C. F., & Hsiao, C. H. (1975). Design of piecewise constant gains for optimal control via Walsh functions. IEEE transactions on automatic control, 20(5), 596-603.
6
[7] [7] D. S. Shih, F. C. Kung, C. M. Chao,( 1986). Laguerre series approach to the analysis of a linear control system incorporation observers, International Journal of Control, vol. 43 (1986), pp.123-128.
7
[8] Paraskevopoulos, P. N., Sparis, P. D., & Mouroutsos, S. G. (1985). The Fourier series operational matrix of integration. International journal of systems science, 16(2), 171-176.
8
[9] Podlubny, I. (1997). The Laplace transform method for linear differential equations of the fractional order. arXiv preprint funct-an/9710005.
9
[10] Sadri, K., Amini, A., & Cheng, C. (2018). A new operational method to solve Abel’s and generalized Abel’s integral equations. Applied mathematics and computation, 317, 49-67.
10
[11] Li, Y., & Zhao, W. (2010). Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Applied mathematics and computation, 216(8), 2276-2285.
11
[12] Babaaghaie, A., & Maleknejad, K. (2017). Numerical solutions of nonlinear two-dimensional partial Volterra integro-differential equations by Haar wavelet. Journal of computational and applied mathematics, 317, 643-651.
12
[13] Fathizadeh, E., Ezzati, R., & Maleknejad, K. (2017). Hybrid rational haar wavelet and block pulse functions method for solving population growth model and abel integral equations. Mathematical problems in engineering, 2017.
13
[14] ur Rehman, M., & Khan, R. A. (2011). The Legendre wavelet method for solving fractional differential equations. Communications in nonlinear science and numerical simulation, 16(11), 4163-4173.
14
[15] Heydari, M. H., Hooshmandasl, M. R., Ghaini, F. M., & Fereidouni, F. (2013). Two-dimensional Legendre wavelets for solving fractional Poisson equation with Dirichlet boundary conditions. Engineering analysis with boundary elements, 37(11), 1331-1338.
15
[16] Heydari, M. H., Hooshmandasl, M. R., Cattani, C., & Li, M. (2013). Legendre wavelets method for solving fractional population growth model in a closed system. Mathematical problems in engineering, http://dx.doi.org/10.1155/2013/161030
16
[17] Wang, Y., & Zhu, L. (2017). Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method. Advances in difference equations, 2017(1), 27.
17
[18] Wang, Y., & Zhu, L. (2017). Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method. Advances in difference equations, 2017(1), 27.
18
[19] Wang, Y., & Fan, Q. (2012). The second kind Chebyshev wavelet method for solving fractional differential equations. Applied mathematics and computation, 218(17), 8592-8601.
19
[20] Fathizadeh, E., Ezzati, R., & Maleknejad, K. (2017). The construction of operational matrix of fractional integration using the fractional chebyshev polynomials. International journal of applied and computational mathematics, 1-23.
20
[21] Kilicman, A., & Al Zhour, Z. A. A. (2007). Kronecker operational matrices for fractional calculus and some applications. Applied mathematics and computation, 187(1), 250-265.
21
[22] Darani, M. R. A., Adibi, H., & Lakestani, M. (2010). Numerical solution of integro-differential equations using flatlet oblique multiwavelets. Dynamics of continuous, discrete & impulsive systems. series A, 17(1), 55-74.
22
[23] Darani, M. A., Adibi, H., Agarwal, R. P., & Saadati, R. (2008). Flatlet oblique multiwavelet for solving integro-differential equations. Dynamics of continuous, discrete and impulsive systems, series A: Matematical analysis, 15, 755-768.
23
[24] Lakestani, M., Dehghan, M., & Irandoust-Pakchin, S. (2012). The construction of operational matrix of fractional derivatives using B-spline functions. Communications in nonlinear science and numerical simulation, 17(3), 1149-1162.
24
[25] Daubechies, I. (1992). Ten lectures on wavelets. Society for industrial and applied mathematics.
25
[26] Keinert, F. (2003). Wavelets and multiwavelets. CRC Press.
26
[27] Yousefi, S., & Banifatemi, A. (2006). Numerical solution of Fredholm integral equations by using CAS wavelets. Applied mathematics and computation, 183(1), 458-463.
27
[28] Saeedi, H., Moghadam, M. M., Mollahasani, N., & Chuev, G. N. (2011). A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order. Communications in nonlinear science and numerical simulation, 16(3), 1154-1163.
28
[29] Maleknejad, K., & Ebrahimzadeh, A. (2014). Optimal control of volterra integro-differential systems based on Legendre wavelets and collocation method. World academy of science, engineering and technology, international journal of mathematical, computational, physical, electrical and computer engineering, 8(7), 1040-1044.
29
[30] Adibi, H., & Assari, P. (2009). Using CAS wavelets for numerical solution of Volterra integral equations of the second kind. Dynamics of continuous, discrete and impulsive systems series A: Mathematical analysis, 16, 673-685.
30
[31] Danfu, H., & Xufeng, S. (2007). Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration. Applied mathematics and computation, 194(2), 460-466.
31
[32] Abualrub, T., Sadek, I., & Abukhaled, M. (2009). Optimal control systems by time-dependent coefficients using cas wavelets. Journal of applied mathematics, 2009.
32
[33] Yi, M., & Huang, J. (2015). CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel. International journal of computer mathematics, 92(8), 1715-1728.
33
[34] Podlubny, I. (1998). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Vol. 198). Academic press.
34
[35] Kilicman, A., & Al Zhour, Z. A. A. (2007). Kronecker operational matrices for fractional calculus and some applications. Applied mathematics and computation, 187(1), 250-265.
35
[36] Wazwaz A. M. (1997). A first course in integral equations. World Scientific Publishing.
36
[37] Folland, G. B. (2013). Real analysis: modern techniques and their applications. John Wiley & Sons.
37
[38] Barzkar, A., Assari, P., & Mehrpouya, M. A. (2012). Application of the cas wavelet in solving fredholm-hammerstein integral equations of the second kind with error analysis. World applied sciences journal. 18. 1695-1704. 10.5829/idosi.wasj.2012.18.12.467
38