Random Matrix Generators for Optimizing a Fuzzy Biofuel Supply Chain System

. Complex industrial systems often contain various uncertainties. Hence sophisticated fuzzy optimization (metaheuristics) techniques have become commonplace; and are currently indispensable for e(cid:27)ective design, maintenance and operations of such systems. Unfortunately, such state-of-the-art techniques suffer several drawbacks when applied to large-scale problems. In line of improving the performance of metaheuristics in those, this work proposes the fuzzy random matrix theory (RMT) as an add-on to the cuckoo search (CS) technique for solving the fuzzy large-scale multiobjective (MO) optimization problem; biofuel supply chain. The fuzzy biofuel supply chain problem accounts for uncertainties resulting from (cid:29)uctu-ations in the annual electricity generation output of the biomass power plant [kWh/year]. The details of these investigations are presented and analyzed.


Introduction
Industrial optimization often involves systems containing various complexities and uncertainties -thus requiring heavy computational eort when performing optimization. In such scenarios metaheuristics play a prominent role (Ganesan et al. ). Decision makers are globally facing various optimization challenges when optimizing supply chains -this is attributed to its large-scale and complex structure. Currently various state-of-the-art tools have been developed to overcome these challenges where they have been used to: • Model these supply chains (Seuring [55]; Brandenburg et al. [12]; Ahi and Searcy [3]) • ciently optimize the decision making process (Ogunbanwo et al. [47]; Mastrocinque et al. [43]) Fuel supply chains have broad applications spanning across diverse industrial sectors. For instance in Lin et al. [38], the annual biomassethanol production cost in a fuel supply chain was minimized. In that work, the large-scale c 2020 Journal of Advanced Engineering and Computation (JAEC) 33 supply chain model consisted of: stacking, in-eld preprocessing, transportation, transportation, biomass harvesting, packing/storage, ethanol production and ethanol distribution. Aiming to reduce the cost of production in a biorenery (to approximately 62%), the researchers used the mixed integer programming technique. Another interesting work on a switchgrass-based bioethanol supply chain (located in North Dakota, U.S) was presented in the work of Zhang et al. [71]. In that work the supply chain system was modeled and optimized using mixed integer linear programming to attain the optimal utilization of marginal land for switchgrass production. The end goal for that work was to establish an economical and sustainable harvest of bioethanol. In Osmani and Zhang [49], a sustainable dual feedstock bioethanol supply chain (large-scale) was optimized in a stochastic environment. The optimization problem considered the following factors: biomass purchase and sales price; as well as biomass supply and demand. In addition to a mixed integer linear programming approach, the authors used a decomposition method: sample average approximation algorithm. Similar research eorts utilizing the mixed integer linear/nonlinear programming methodology could be observed in Osmani and Zhang [48] and Gao and You [29]. An exception to those works is seen in Marufuzzaman et al. [42] where the authors employed a combined L-shaped techniques and Lagrangian Relaxation approaches instead of a mixed integer programming methodology.
In that work, insights on carbon regulatory mechanisms and other uncertainties observed in biofuel supply chains was provided. A holistic review on metaheuristic techniques implemented to bioenergy supply chains could be seen in De Meyer et al. [20] and Castillo-Villar [14].
Due to uncertain variables in biofuel supply chains, recent works have integrated fuzzy formulations into these supply chain models. A very interesting fuzzy methodology for modeling supply chains was introduced in Kozarevi¢, S. and Pu²ka [35]. In that work the authors proposed a method for data processing and measurement of practices and performances of supply chains. This is done by conducting applying transformation of the obtained linguistic val-ues (using appropriate fuzzy methods) into crisp values of research variable dimensions. A more practical work could be seen in Babazadeh [6]. In that work, the author developed a novel fuzzy framework for a bioenergy supply chain: the possibilistic programming model based on possibilistic mean and absolute deviation of fuzzy numbers. The model performance was evaluated by using data from a real-world case study and it was shown that the proposed method performed better than a pure possibilistic programming model. A similar work can be seen in Lin et al. [37]. In that work the uncertain factors considered were the demand of biomass energy (due to unstable price of fossil fuels) and the number of job oer opportunities springing up from the energy facilities. To account for these uncertainties the authors employed a fuzzy multiple objective linear programming to solve the problem. Another eective implementation of fuzzy framework for biofuel supply chains could be seen in the work of Balaman et al. [9]. In that work, a hybrid solution strategy combining fuzzy set theory and epsilon-constraint method was proposed. The proposed methodology was applied to handle system-specic uncertainties during the optimization of the supply chain and transportation network (entire West Midlands (WM) region of the UK). Fuzzy optimization has also been employed to model the design of renewable energy supply chains (integrated with district heating systems) (Balaman et al. [10]). In the work of Balaman et al. [10], the authors developed a novel decision model to obtain the optimal supply chain conguration and district heating system to meet the thermal demand of a certain locality. To this end, the authors formulated and validated a Fuzzy Mixed Integer Linear Programming (MILP) which consists of multiple types of biomass and systemic uncertainties.
Cuckoo search (CS) technique has been eciently employed for optimizing real-world supply chains. A series of metaheuristics including CS was applied to a supply chain (consumerpackaged goods industry) (Mattos et al. [44]). The performance as well as the results generated by the techniques employed was presented in that work. Similar CS implementations on supply chains is given in Srivastav 34 c 2020 Journal of Advanced Engineering and Computation (JAEC) and Agrawal [58] and Abdelsalam and Elassal [1]. Supply chains models often contain many variables (large-scale) -where these variables and expressions are interlinked in a complex way. The mathematical structure (universality) of such supply chains often resemble those observed in the nuclei of heavy atoms (e.g. gold, rhodium and platinum • Quantum chromodynamics (Akemann [4]) • Transport optimization (Krbálek and Seba [36]) • Big Data (Qiu and Antonik [52]) • Finance It is important to note that key characteristics of supply chain networks are highly similar to those mentioned complex systems. Hence the premise: that supply chains may naturally contain universality. Following this chain of thought, the RMT framework was utilized to improve the the conventional Cuckoo Search method (CS) in this work. This was carried out by performing certain modications to the stochastic generator component of the algorithm. In this work the conventional Gaussian stochastic generator is replaced with a RMT-based generator.
This work targets to solve the complex MO fuzzy biofuel supply chain model. The previous approaches to solve this problem uses conventional linear and nonlinear programming approaches which do not account for the complexity of the large-scale problem at hand (Ghaderi et al. This paper is organized as follows: Section 2 presents the fuzzy formulation of the MO biofuel supply chain model. In Section 3 the conventional CS approach is presented while an overview of RMT and its role in the development of stochastic generators is described in Section 4. Section 5 presents the results and discussion followed by the nal section; conclusions and potential directions for future work.

Biofuel supply chain: fuzzy formulation
The fuel supply chain formulation utilized in this work was developed in Tan et al. [61].
In that work only two objective functions were considered: prot (Pr) and social welfare (SW). The environmental benets objective was incorporated into the SW function. In this work, the environmental benets function was isolated from the SW function and taken as an independent objective function (denoted Env). Various factors inuence electricity generation output of biomass power plants. For instance, plant system repairs, maintenance, inspections which involve turnaround periods and downtime inuence the electricity generation output. Since biomass plant type considered in this model involves various types of fuel sources (e.g. sugarcane, wheat straw, bean straw, rice husk, corncobs, branches, bark, and wood chips), the biomass plant would have to be frequently tuned to maintain robustness in the face of fuel heterogeneity. Such tuning would incur downtime which could heavily inuence the power generation output. To account for these uncertain-ties, the fuzzy formulation was employed -where the annual electricity generation output of the biomass power plant [kWh/year] is fuzzied with its respective constraint: where the uncertainty in the annual electricity generation output of the biomass power plant is assumed to contain a variation of approximately 30% from the mean. The optimization formulation of the biofuel supply chain problem is then redened in the fuzzy environment with the elaborated structure as follows: Minimize (objective functions: Pr, SW, Env) subject to fuzzy constraints: and Crisp (Non-fuzzy) constraints.
The left side of i th fuzzy constraint in (2), n j=1ã ij x j is aggregated as a fuzzy set -utilizing Zadeh's extension principle. Assuming a credibility level ε, 0 < ε < B 1+C chosen by the decision maker, as a risk is taken and all the membership degrees smaller than ε levels are ignored (Rommelfanger et al. [53]). All fuzzy where α = d/j.
The fuzzy coecients B = 1, C = 0.1 and the α ∈ (0, 1). The following points are considered when we replace a crisp system by a fuzzy system (Atanu et al. [73]): (i) Specication of fuzzy inequality relations and methodology to obtain its crisp equivalents.
(ii) The interpretation`minimization' in logistic type objective functions.
Therefore the fuzzy fuel supply chain model in this chapter consists of three objective functions to be maximized along with associated inequality constraints (see equation (9)). The objective functions are shown in equations (1)- (3): such that, The fuzzy constraints are as follows: 36 c 2020 Journal of Advanced Engineering and Computation (JAEC) VOLUME: 4 | ISSUE: 1 | 2020 | March The crisp constraints for the biofuel supply chain model are given below: where The decision parameters are: q t , IQ i,t , SQ i,k,t , P Q i,k,t and BR i,t .
Details on the parameter settings of the biofuel supply chain model used in this work could be obtained in Tan et al. [61]. Joshi et al. [33]). It was initially inspired by brood parasitism which was often found among certain species of cuckoo birds. This parasitism occurs when the cuckoo birds lay their eggs in the nests of other bird species (non-cuckoo birds). The heavy-tailed random walk probability distribution, Lévy ights was used as a stochastic generator for the CS technique. The iterative expression at iteration, t for the candidate solution i for the CS technique is: such that the Lévy distribution is given as follows: wherett is the random variable, β > 0 is the relaxation factor (which is modied based on the problem at hand) and λ ∈ (1, 3] is the Lévy ight step-size. With t ≥ 1, λ is related to the fractal dimension and the Lévy distribution becomes a specic sort of Pareto distribution. The CS algorithm is based on a few fundamental philosophies. For instance each cuckoo bird lays a single egg at one time and randomly places the egg in a selected nest. The second being: via tness screening, the best egg (candidate solution) is carried forward into the next iteration. The worst solutions are discarded from further iterations. The nests represent the objective space (or the optimization problem landscape). The parameter setting for the CS technique used in this work is shown in Tab. 1 while its respective algorithm is given in Algorithm 1: Step 1: Initialize algorithmic parameters; y i , β, λ, N Step 2: Dene parameters in the constraints and decision variable Step 3: Via Lévy ights randomly lay a cuckoo egg in a nest Step 4: Dene tness function based on solution selection criteria Step 5: Screen eggs and evaluate candidate solution IF: tness criteria is satised Select candidate solution (egg) to be considered in the next iteration, n + 1

ELSE: tness criteria is not satised
Discard candidate solution (egg) from further iterations Step 6: Rank the best solutions obtained during tness screening Step 7: If the tness criterion is satised and t = T max halt and print solutions,else proceed to Step 3.

Random matrix theory & stochastic generators
Random Matrix Theory (RMT) is a robust mathematical framework which is very eective for describing behavior of complex systems. RMT is known to exhibit universality a property of global symmetries shared by many systems within a certain symmetry class. Details on the application of RMT on a non-fuzzy (crisp) biofuel supply chain model could be seen in Ganesan et al. [28]. In RMT there exists two probability distributions describing: the random matrix entries and the eigenvalue spread. The nearest neighbor spacing probability distribution of eigenvalues is given by Wigner's Surmise: where s is the eigenvalue spacing, A i and B i are constant parameters. The normalized spacing, s and the mean spacing s is as follows: such that s = λ n+1 − λ n , where λ n is the n th eigenvalue sequentially such that λ 1 < ... < λ n < λ n+1 . The rst type of random matrices are those that are modeled based on complex quantum systems (which have chaotic classical counterparts). RMT consists of four major ensembles to determine the spacing distributions of the eigenvalues: the Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE) and Gaussian Symplectic Ensemble (GSE). In this work, the GUE distribution is considered: These ensembles describe the probability density functions governing the random matrix entries. The constants, A i and B i are selected such that the following averaging properties are respected: , it was seen that variations in the type of stochastic generators have an inuence on the optimization results. Therefore in this work the RMT is employed as the stochastic generator to solve the fuzzy biofuel supply chain problem. Essentially RMT deals with systems with a complex network of many interlinked and interacting components -which are often encountered in real-world settings. The proposed algorithmic framework for developing a random matrix generator is as follows: Algorithm 2: Random Matrix Generator Step 1: Generate random eigenvalue spacings, s from a GUE Step 2: Determine the average eigenvalue spacing, ∆λ Step 3: Set initial eigenvalue, λ 0 38 c 2020 Journal of Advanced Engineering and Computation (JAEC) Step 4: Set initial n × n matrix, H ij Step 5: Determine consequent eigenvalues Step 6: Determine n × 1 eigenvector, E i : Step 7: Generate random variables from a Gaussian probability distribution function endowed eigenvector as the variance, σ 2 = E i :

Results and discussion
The following frameworks have been introduced in the past for tackling MO optimization problems: Strength Pareto Evolutionary Algorithm (SPEA-2) (Zhao et al. [69]), Weighted sum approach (Naidu et al. [46]), Normal-Boundary Intersection (NBI) (Ahmadi et al. [2]; Ganesan et al. [23]) and Non-Dominated Sorting Genetic Algorithm (NSGA-II) (Mousavi et al. [45]). Scalarization and NBI approaches involve the aggregation of multiple target objectives. Eectively transforming the MO problem into a single objective one reduces its complexity to a high degree -making it easier to solve. In this work, the objective functions of the fuzzy MO biofuel supply chain problem was combined into a single function using the weighted sum approach (Kalita et al. [34]). This procedure eectively transforms the fuzzy MO problem into a single-objective optimization problem which can be solved for dierent weight values. The computational experiments employed in this work was done using the C++ programming language on a computer using a 64-bit Win 10 platform with an Intel Core i5-7200U CPU (2.50 GHz).
Due to the stochastic nature of the algorithms employed in this work, the computational technique was executed multiple times (3 executions) and the best solution was taken. 28 solutions were obtained for a variation of weights. These  where z 1 , z 2 and z 3 are individual candidate solutions. The ranked weighted individual solutions obtained using the fuzzy CS approach with random matrix generators (fuzzy RMT-CS) is given in Tab. 2. The entire Pareto frontier constructed using the fuzzy RMT-CS technique is shown in Fig. 1. In this analysis (Tab. 2 and Fig. 1), the values for the fuzzy membership function in equation (3)  In Fig. 1, some prevalent trends could be observed in the distribution of the solution points in the objective space. One of these trends is the high concentration of solution points in specic regions of the objective space: P R ∈ (3.0 × 10 8 , 3.8 × 10 8 ), SW ∈ (1, 004, 400, 1, 004, 800) and Env ∈ (0, 8 × 10 4 ). This high concentration could be attributed to the technique iteratively reaching the most optimal (local or near optimal) region of the objective space. Despite those high concentrations, some solution points could be observed to exist beyond those optimal regions. This shows that the proposed technique generates a sparse distribution of solutions and hence has good exploration capabilities -where the fuzzy RMT-CS approach explores regions in the objective space in search of other local optima.
The variation of the objective function, PR with respect to the parameters in the fuzzy membership function is given in Fig. 2: The median value of PR is 327,626,500 and it falls between µ = 0.05 and µ = 0.1. The minimum value of PR (259, 954, 000) is obtained at µ = 0.0667. The objective function, SW plotted with respect to the parameters in the fuzzy membership function is given in Fig. 3: The maximum and minimum value of the SW objective function shown in Fig. 3  The fuzzy RMT-CS approach produced feasible solutions; where the constraints in the fuzzy biofuel supply chain model was not broken. The computations performed during the numerical experiments were stable and the algorithm achieved convergence every time during execution. In terms of robustness, the fuzzy RMT-CS method performed stable computations and converged towards a feasible solution during each variation of the fuzzy membership function. The random matrix segment successfully complemented the CS technique to navigate through the objective space of the complex and multivariate biofuel supply chain problem.
Nevertheless the proposed technique does have some disadvantages. The rst is the algorithmic complexity -where the addition of the RMT segment into the CS technique signicantly increases the complexity of the algorithm. This in eect may considerably impact the computational time of the optimization process. Additionally, this technique only considers the GUE ensemble distribution and not the GOE as well as the GSE distributions for the RMT. This may adversely impact the performance of the proposed technique. Finally this work considers the uncertainty in the annual electricity generation output of the biomass power plant [kWh/year] to be of type-1 fuzzy uncertainty. It is high possible that uctuations in the monthly/weekly electricity generation output of the biomass power plant may more precisely capture the mentioned uncertainties -making the model more realistic.

Conclusion and recommendations
In this work, the proposed MO biofuel supply chain problem was reformulated by taking into account uctuations in the annual electricity generation output of the biomass power plant [kWh/year]. This eectively converts the problem into a fuzzy MO problem; which is nonlinear, nonconvex and multivariate. To deal with this MO problem the CS technique was retrotted with the RMT approach to boost its performance when faced with high levels of complexity. The proposed approach was eectively applied and Pareto ecient solutions were attained. The dominance of these solutions were gauged using the HVI.
Further computational tests could be done by using the GSE and GOE ensembles for the RMT segment in the proposed approach. In addition, RMT -based generators could also be employed to complement other metaheuristic techniques such as PSO (Mousavi et al. [45]), dierential evolution (Ganesan et al. [24]) as well as other computational approaches (Ganesan et al.[25]). The accuracy of the fuzzy formulation proposed in this work could be further improved by ac-counting for monthly/weekly uctuations; by reformulating the MO biofuel supply chain problem by utilizing a type-2 fuzzy framework. This work can also be extended by exploring other approaches for handling uncertainty such as: robust optimization, (Bairamzadeh et al. [7]; Kara et al. [31]), stochastic optimal control (Vinod et al. [63]) and chance constraint optimization (Cheng et al. [17]). This extension could include emerging areas of applications such as complex networks in alternative energy systems (Syahputra et al. [60]; Lot et al. [39]), social networks, gene networks (Youseph et al. [67]) and pharmaceutical supply chain networks (Zahiri et al. [68]).   (CNN) and Bi-directional Recurrent Neural Networks (RNN) for image tagging with embedded natural language processing, End to end steering learning systems and GAN. His work involves experimental research with software prototypes and mathematical modelling and design He is an editorial board member for the Journal of Energy Optimization and Engineering (IJEOE), and invited guest editor for Journal of Visual Languages Communication (JVLC-Elsevier). He has published more than 30 papers in leading international conference proceedings and peer reviewed journals.
Igor LITVINCHEV received his M.Sc.