Abstract
The objective of a fixed-cost transportation problem (FCTP) is to minimize the total transportation cost (or to maximize the total profit). FCTPs become more complex due to the presence of uncertain factors like suppliers’ capacity, customers’ demand, fixed cost and amount of goods transported. In order to deal with its complexity, three mathematical models, namely the expected value, chance-constrained and measure chance models, are basically available in the literature. This paper aims to solve the crisp equivalent models of an uncertain two-echelon FCTP by employing a robust hybrid evolutionary algorithm. The framework of the designed algorithm comprises of parameter-free Jaya algorithm and the efficient quadratic approximation operator. Initially, its efficiency is validated on the 24 constrained benchmark functions from CEC 2006 (Liang et al. in Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization, 2006) and 30 unconstrained benchmark functions (Rao in Int J Ind Eng Comput 7:19–34, 2016). Friedman rank test and Iman–Davenport test are conducted for both constrained and unconstrained test instances. Additionally, five real-life engineering design problems are solved to further analyze the flexibility of the proposed algorithm. The computational results and convergence graphs justify the robustness of the designed algorithm over state-of-the-art algorithms. Finally, its application is demonstrated on chance-constrained and measure chance two-echelon FCTP models. Near-optimal results are contradicted by genetic algorithm and particle swarm optimization in solving both these models. Further, a few large-scale instances of FCTP are solved by the proposed algorithm.
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References
Adlakha V, Kowalski K (2004) A simple algorithm for the source induced fixed charge transportation problem J. Oper Res Soc 55(12):1275–1280
Adlakha V, Kowalski K, Wang S, Lev B, Shen W (2014) On approximation of the fixed charge transportation problem. Omega 43:64–70
Akbari M, Molla-Alizadeh-Zavardehi S, Niroomand S (2017) Meta-heuristic approaches for fixed-charge solid transportation problem in two-stage supply chain network. Oper Res 20(1):447–471. https://doi.org/10.1007/s12351-017-0332-7
Angulo G, Vyve M (2017) Fixed-charge transportation problems on trees. Oper Res Lett 45:275–281
Aslan M, Gunduz M, Kiran MS (2019) JayaX: Jaya algorithm with xor operator for binary optimization. Appl Soft Comput 82:105576
Balaji A, Nilakantan J, Nielsen I, Jawahar N, Ponnambalam S (2019) Solving fixed charge transportation problem with truck load constraint using meta-heuristics. Ann Oper Res 273(1–2):207–236
Bansal J, Deep K (2010) Quadratic approximation PSO for economic dispatch problems with valve-point effects. In: International conference on swarm, evolutionary, and memetic computing. LNCS, 6466, pp 460–467
Bertazzi L, Maggioni F (2018) A stochastic multi-stage fixed charge transportation problem: worst-case analysis of the rolling horizon approach. Eur J Oper Res 267(2):555–569
Calvete H, Galé C, Iranzo J, Toth P (2018) A matheuristic for the two-stage fixed-charge transportation problem. Comput Oper Res 95:113–122
Cui Q, Sheng Y (2013) Uncertain programming model for solid transportation problem. Information 15(2):342–348
Dalman H (2018a) Uncertain programming model for multi-item solid transportation problem. Int J Mach Learn Cybern 9(4):559–567
Dalman H (2018b) Entropy-based multi-item solid transportation problems with uncertain variables. Soft Comput 23(14):5931–5943. https://doi.org/10.1007/s00500-018-3255-1
Dalman H (2018c) Uncertain random programming models for fixed charge multi-item solid transportation problem. New Trends Math Sci 6(1):37–51
Dalman H (2018d) A simulation algorithm with uncertain random variables. Int J Optim Control Theor Appl 8(2):195–200
Dalman H, Sivri M (2017) Multi-objective solid transportation problem in uncertain environment. Iran J Sci Technol Trans A Sci 41(2):505–514
Dalman H, SivriM (2018) A fuzzy logic based approach to solve interval multi-objective nonlinear transportation problem. In: Proceedings of the national academy of sciences, India section A: physical sciences. https://doi.org/10.1007/s40010-017-0469-z
Deep K, Das K (2009) Performance improvement of real-coded genetic algorithm with quadratic approximation based hybridization. Int J Intell Defence Support Syst 2(4):319–334
Demsar J (2006) statistical comparison of classifiers over multiple data sets. J Mach Learn Res 7:1–30
Fan Z, Fang Y, Li W, Yuan Y, Wang Z, Bian X (2018) LSHADE44 with an improved € constraint-handling method for solving constrained single-objective optimization problems. In: 2018 IEEE congress on evolutionary computation (CEC). IEEE, pp 1–8
Glover F, Amini M, Kochenberger G (2005) Parametric ghost image processes for fixed charge problems: a study of transportation networks. J Heuristics 11(4):307–336
Gunduz M, Aslan M (2021) DJAYA: a discrete Jaya algorithm for solving traveling salesman problem. Appl Soft Comput 105:107275
He F (2012) A stochastic programming model and algorithm for transportation problem. In: International conference on computer science and information processing, pp 24–26
Hellwig M, Beyer HG (2018) A matrix adaptation evolution strategy for constrained real-parameter optimization. In: 2018 IEEE congress on evolutionary computation (CEC). IEEE, pp 1–8
Hirsch W, Dantzig G (1968) The fixed charge problem. Nav Res Logist 15:413–424
Hitchcock F (1941) The distribution of a product from several sources to numerous localities. J Math Phys 20:224–230
Klose A (2006) Single-sink fixed-charge transportation: applications and exact solution algorithms. Working papers, Department of Mathematical Sciences, University of Aarhus, p 5
Kowalski K, Lev B, Shen W, Tu Y (2014) A fast and simple branching algorithm for solving small scale fixed-charge transportation problem. Oper Res Perspect 1(1):1–5
Kumar A, Wu G, Ali MZ, Mallipeddi R, Suganthan PN, Das S (2020) A test-suite of non-convex constrained optimization problems from the real-world and some baseline results. Swarm Evol Comput 56:100693
Liang J, Runarsson T, Montes E, Clerc M, Suganthan P, Coello C, Deb K (2006) Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization. Technical report. Nanyang Technological University http://www.ntu.edu.sg/home/EPNSugan
Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin
Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10
Liu B (2010a) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, Berlin
Majumder S, Kundu P, Kar S, Pal T (2019) Uncertain multi-objective multi-item fixed charge solid transportation problem with budget constraint. Soft Comput 23(10):3279–3301
Mohan C, Shanker K (1994) A random search technique for global optimization based on quadratic approximation. Asia Pac J Oper Res 11:93–101
Mou D, Zhou W, Chang X (2013) A transportation problem with uncertain truck times and unit costs. Ind Eng Manag Syst 12(1):30–35
Rao R (2016) Jaya: A simple and new optimization algorithm for solving constrained and unconstrained optimization problems. Int J Ind Eng Comput 7:19–34
Safi M, Razmjoo A (2013) Solving fixed charge transportation problem with interval parameters. Appl Math Model 37:8341–8347
Shen J, Zhu K (2020) An uncertain two-echelon fixed charge transportation problem. Soft Comput 24:3529–3541
Shirazi N, Esfahani M, Soleimani H (2015) Modeling and solving a three-stage fixed charge transportation problem considering stochastic demand and price. J Ind Eng Manag Stud 2(1):27–40
Trivedi A, Sanyal K, Verma P, Srinivasan D (2017) A unified differential evolution algorithm for constrained optimization problems. In: 2017 IEEE congress on evolutionary computation (CEC). IEEE, pp 1231–1238
Warid W, Hizam H, Mariun N, Abdul-Wahab NI (2016) Optimal power flow using the Jaya algorithm. Energies 9(9):678
Wolpert DH, Macready WG (1995) No free lunch theorems for search. Technical report SFI-TR-95-02-010 Vol. 10 Citeseer
Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1:67–82
Yang L, Feng Y (2007) A bicriteria solid transportation problem with fixed charge under stochastic environment. Appl Math Modell 31(12):2668–2683
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Das, R., Das, K.N. & Mallik, S. An improved quadratic approximation-based Jaya algorithm for two-echelon fixed-cost transportation problem under uncertain environment. Soft Comput 26, 10301–10320 (2022). https://doi.org/10.1007/s00500-022-07344-w
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DOI: https://doi.org/10.1007/s00500-022-07344-w