Random matrix generators for optimizing a fuzzy biofuel supply chain system

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Complex industrial systems often contain various uncertainties. Hence sophisticated fuzzy optimization (metaheuristics) techniques have become commonplace; and are currently indispensable for e ective design, maintenance and operations of such systems. Unfortunately, such state-of-the-art techniques suffer several drawbacks when applied to largescale 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 uctuations in the annual electricity generation output of the biomass power plant [kWh/year]. The details of these investigations are presented and analyzed.

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Nội dung Text: Random matrix generators for optimizing a fuzzy biofuel supply chain system

VOLUME: 4 | ISSUE: 1 | 2020 | March Random Matrix Generators for Optimizing a Fuzzy Biofuel Supply Chain System 1 2,∗ 1 Timothy GANESAN , Pandian VASANT , Pratik SANGHVI , 3 4 Joshua THOMAS , Igor LITVINCHEV Royal Bank of Canada, Canada 1 Universiti Teknologi Petronas, Malaysia 2 3 UOW Malaysia, KDU Penang University College, Malaysia 4 Nuevo Leon State University, Mexico *Corresponding Author:Pandian VASANT (Email: pvasant@gmail.com) (Received: 12-Nov-2019; accepted: 3-Feb-2020; published: 31-Mar-2020) DOI: http://dx.doi.org/10.25073/jaec.202041.268 1. Introduction Abstract. Complex industrial systems of- ten contain various uncertainties. Hence sophis- Industrial optimization often involves systems ticated fuzzy optimization (metaheuristics) tech- containing various complexities and uncertain- niques have become commonplace; and are cur- ties - thus requiring heavy computational eort rently indispensable for eective design, main- when performing optimization. In such scenar- tenance and operations of such systems. Un- ios metaheuristics play a prominent role (Gane- fortunately, such state-of-the-art techniques suf- san et al.[25]; Ganesan et al.[26]; Yang [66]; fer several drawbacks when applied to large- Ganesan et al. [24]; Ganesan et al. [27]; Hong scale problems. In line of improving the per- et al.[32]; Dong et al. [21]). Decision mak- formance of metaheuristics in those, this work ers are globally facing various optimization chal- proposes the fuzzy random matrix theory (RMT) lenges when optimizing supply chains - this is as an add-on to the cuckoo search (CS) tech- attributed to its large-scale and complex struc- nique for solving the fuzzy large-scale multiobjec- ture. Currently various state-of-the-art tools tive (MO) optimization problem; biofuel supply have been developed to overcome these chal- chain. The fuzzy biofuel supply chain problem lenges where they have been used to: accounts for uncertainties resulting from uctu- ations in the annual electricity generation out- • Model these supply chains (Seuring [55]; put of the biomass power plant [kWh/year]. The Brandenburg et al. [12]; Ahi and Searcy details of these investigations are presented and [3]) analyzed. • ciently optimize the decision making pro- cess (Ogunbanwo et al. [47]; Mastrocinque et al. [43]) Keywords Fuel supply chains have broad applications span- Random matrix theory, fuzzy framework, ning across diverse industrial sectors. For in- cuckoo search, biofuel supply chain, stance in Lin et al. [38], the annual biomass- multiobjective (MO), large-scale opti- ethanol production cost in a fuel supply chain mization. was minimized. In that work, the large-scale c 2020 Journal of Advanced Engineering and Computation (JAEC) 33
VOLUME: 4 | ISSUE: 1 | 2020 | March supply chain model consisted of: stacking, ues (using appropriate fuzzy methods) into crisp in-eld preprocessing, transportation, trans- values of research variable dimensions. A more portation, biomass harvesting, packing/stor- practical work could be seen in Babazadeh [6]. age, ethanol production and ethanol distribu- In that work, the author developed a novel fuzzy tion. Aiming to reduce the cost of produc- framework for a bioenergy supply chain: the tion in a biorenery (to approximately 62%), possibilistic programming model based on pos- the researchers used the mixed integer program- sibilistic mean and absolute deviation of fuzzy ming technique. Another interesting work on a numbers. The model performance was evalu- switchgrass-based bioethanol supply chain (lo- ated by using data from a real-world case study cated in North Dakota, U.S) was presented in and it was shown that the proposed method per- the work of Zhang et al. [71]. In that work formed better than a pure possibilistic program- the supply chain system was modeled and op- ming model. A similar work can be seen in Lin timized using mixed integer linear programming et al. [37]. In that work the uncertain factors to attain the optimal utilization of marginal land considered were the demand of biomass energy for switchgrass production. The end goal for (due to unstable price of fossil fuels) and the that work was to establish an economical and number of job oer opportunities springing up sustainable harvest of bioethanol. In Osmani from the energy facilities. To account for these and Zhang [49], a sustainable dual feedstock uncertainties the authors employed a fuzzy mul- bioethanol supply chain (large-scale) was opti- tiple objective linear programming to solve the mized in a stochastic environment. The opti- problem. Another eective implementation of mization problem considered the following fac- fuzzy framework for biofuel supply chains could tors: biomass purchase and sales price; as well be seen in the work of Balaman et al. [9]. In as biomass supply and demand. In addition to that work, a hybrid solution strategy combining a mixed integer linear programming approach, fuzzy set theory and epsilon-constraint method the authors used a decomposition method: sam- was proposed. The proposed methodology was ple average approximation algorithm. Similar applied to handle system-specic uncertainties research eorts utilizing the mixed integer lin- during the optimization of the supply chain and ear/nonlinear programming methodology could transportation network (entire West Midlands be observed in Osmani and Zhang [48] and Gao (WM) region of the UK). Fuzzy optimization and You [29]. An exception to those works is has also been employed to model the design of seen in Marufuzzaman et al. [42] where the au- renewable energy supply chains (integrated with thors employed a combined L-shaped techniques district heating systems) (Balaman et al. [10]). and Lagrangian Relaxation approaches instead In the work of Balaman et al. [10], the authors of a mixed integer programming methodology. developed a novel decision model to obtain the In that work, insights on carbon regulatory optimal supply chain conguration and district mechanisms and other uncertainties observed in heating system to meet the thermal demand of biofuel supply chains was provided. A holistic a certain locality. To this end, the authors for- review on metaheuristic techniques implemented mulated and validated a Fuzzy Mixed Integer to bioenergy supply chains could be seen in De Linear Programming (MILP) which consists of Meyer et al. [20] and Castillo-Villar [14]. multiple types of biomass and systemic uncer- tainties. Due to uncertain variables in biofuel supply chains, recent works have integrated fuzzy for- Cuckoo search (CS) technique has been e- mulations into these supply chain models. A ciently employed for optimizing real-world sup- very interesting fuzzy methodology for model- ply chains. A series of metaheuristics including ing supply chains was introduced in Kozarevi¢, CS was applied to a supply chain (consumer- S. and Pu²ka [35]. In that work the authors packaged goods industry) (Mattos et al. [44]). proposed a method for data processing and mea- The performance as well as the results gen- surement of practices and performances of sup- erated by the techniques employed was pre- ply chains. This is done by conducting apply- sented in that work. Similar CS implemen- ing transformation of the obtained linguistic val- tations on supply chains is given in Srivastav 34 c 2020 Journal of Advanced Engineering and Computation (JAEC)
VOLUME: 4 | ISSUE: 1 | 2020 | March and Agrawal [58] and Abdelsalam and Elassal generation output of the biomass power plant [1]. Supply chains models often contain many [kWh/year]. This work contributes to the eld variables (large-scale) - where these variables by addressing both concerns: (1) by reformulat- and expressions are interlinked in a complex ing the MO biofuel supply chain problem as a way. The mathematical structure (universality) fuzzy problem to account for uncertainties aris- of such supply chains often resemble those ob- ing from the annual electricity generation output served in the nuclei of heavy atoms (e.g. gold, of the biomass power plant [kWh/year]. (2)The rhodium and platinum). Random matrices were complex MO fuzzy biofuel supply chain prob- developed to specically model complex systems lem is solved by using the RMT-based method which contain universality - i.e. large and com- - which has been observed to be very eective plex systems with highly interconnected com- for solving highly complex large-scale problems ponents (Che [16]). Random Matrix Theory (Ganesan et al. [28]). (RMT) has been utilized to model such systems This paper is organized as follows: Section 2 in: presents the fuzzy formulation of the MO bio- fuel supply chain model. In Section 3 the con- • Solid state physics (Verbaarschot [62]; ventional CS approach is presented while an Beenakker [11]) overview of RMT and its role in the development • Quantum information theory (Collins and of stochastic generators is described in Section Nechita [18]) 4. Section 5 presents the results and discussion followed by the nal section; conclusions and po- • Quantum chromodynamics (Akemann [4]) tential directions for future work. • Transport optimization (Krbálek and Seba 2. Biofuel supply chain: [36]) • Big Data (Qiu and Antonik [52]) • Finance fuzzy formulation The fuel supply chain formulation utilized in It is important to note that key characteristics of this work was developed in Tan et al. [61]. supply chain networks are highly similar to those In that work only two objective functions were mentioned complex systems. Hence the premise: considered: prot (Pr) and social welfare (SW). that supply chains may naturally contain univer- The environmental benets objective was incor- sality. Following this chain of thought, the RMT porated into the SW function. In this work, framework was utilized to improve the the con- the environmental benets function was isolated ventional Cuckoo Search method (CS) in this from the SW function and taken as an indepen- work. This was carried out by performing cer- dent objective function (denoted Env ). Various tain modications to the stochastic generator factors inuence electricity generation output of component of the algorithm. In this work the biomass power plants. For instance, plant sys- conventional Gaussian stochastic generator is re- tem repairs, maintenance, inspections which in- placed with a RMT-based generator. volve turnaround periods and downtime inu- This work targets to solve the complex MO ence the electricity generation output. Since fuzzy biofuel supply chain model. The previ- biomass plant type considered in this model in- ous approaches to solve this problem uses con- volves various types of fuel sources (e.g. sug- ventional linear and nonlinear programming ap- arcane, wheat straw, bean straw, rice husk, proaches which do not account for the complex- corncobs, branches, bark, and wood chips), the ity of the large-scale problem at hand (Ghaderi biomass plant would have to be frequently tuned et al. [30], Chávez et al.[15], Roni et al.[54], to maintain robustness in the face of fuel het- Bairamzadeh et al.[8]). Additionally current erogeneity. Such tuning would incur downtime works and tackling this problem do not con- which could heavily inuence the power gener- sider the uncertainty in the annual electricity ation output. To account for these uncertain- c 2020 Journal of Advanced Engineering and Computation (JAEC) 35
VOLUME: 4 | ISSUE: 1 | 2020 | March ties, the fuzzy formulation was employed - where (i) Specication of fuzzy inequality relations the annual electricity generation output of the and methodology to obtain its crisp equiv- biomass power plant [kWh/year] is fuzzied with alents. its respective constraint: (ii) The interpretation `minimization' in logistic type objective functions. X X qt ∈ [Qmin , Qmax ] → ˜ min , Q q˜t ∈ [Q ˜ max ] t t (1) Therefore the fuzzy fuel supply chain model in this chapter consists of three objective functions where to be maximized along with associated inequal- ˜ min = [1260000, 2340000], Q ity constraints (see equation (9)). The objective ˜ max = [1714000, 3182000]. functions are shown in equations (1)-(3): Q X where the uncertainty in the annual electricity Pr = P (1 − EC) × qt − (4) generation output of the biomass power plant  t  is assumed to contain a variation of approxi- F Cp    mately 30% from the mean. The optimization   GC · qt        formulation of the biofuel supply chain problem     X ! is then redened in the fuzzy environment with X X   + + SC · IQi,t + SQi,k,t · P Pi   the elaborated structure as follows:      t  i k         Minimize (objective functions: Pr, SW, Env) +Y1t · extraY1 + Y2t · extraY2   subject to fuzzy constraints: X n X SW = ACS · (1 − EC) qt aij xj 6 ˜bi , i = 1, 2, ..., m (2) t ∼ j=1 X + GT − GS · (1 − EC) qt (5) and Crisp (Non-fuzzy) constraints. t The left side of ith fuzzy constraint in (2),  X  n CET · (1 − EC) · qt ˜ij xj is aggregated as a fuzzy set - utilizing P a Env = AC ·  t  (6) j=1 Zadeh's extension principle. Assuming a cred- − (CEcb − CEtp) ibility level ε, 0 < ε < 1+C B chosen by the such that, decision maker, as a risk is taken and all the X CEncbi,k · Dcbi,k membership degrees smaller than ε levels are CEcb = 2 X P Qi,k,t · ignored (Rommelfanger et al. [53]). All fuzzy i,k,t i,k LCcbi,k data b˜i ≡ S(b ˜ a , bb ) comprise of fuzzy variables i i   with the following logistic membership functions X X (Elamvazuthi et al. [22]), + P Qi,k,t · CEicbi,k Dcbi,k  (7) i,k,t i,k 1 if bi 6 bai       B a b X X CEntbi,k · Dbpi,k if bi 6 bi 6 bi CEtp = 2 SQi,k,t · µb˜i = b −ba α bi ia LCtpi,k b −b i,k,t i,k 1 + Ce   i i     if bi > bbi  0 X X (3) + P Qi,k,t · CEicbi,k Dcbi,k  . (8) i,k,t i,k where α = d/j . The fuzzy constraints are as follows: The fuzzy coecients B = 1, C = 0.1 and the X X α ∈ (0, 1). The following points are considered qt ∈ [Qmin , Qmax ] → ˜ min , Q q˜t ∈ [Q ˜ max ] when we replace a crisp system by a fuzzy system t t (Atanu et al. [73]): (9) 36 c 2020 Journal of Advanced Engineering and Computation (JAEC)
VOLUME: 4 | ISSUE: 1 | 2020 | March The crisp constraints for the biofuel supply chain Tab. 1: CS Settings model are given below: Parameters Values qt 6 qmax (10) Total Number of Eggs (N) 20 Number of nests, nests 4 Lévy ight step-size, λ 1.5 IQi,t > SIlbi (11) Relaxation factor, β 0.8 Maximum 300 X IQi,t 6 IQmax (12) number iteration, Tmax 3. Cuckoo search i X HVmin 6 HVi · BRi,t 6 HVmax (13) i CS is a population-based stochastic search and optimization algorithm (Mareli and Twala [41]; Joshi et al. [33]). It was initially inspired by X SQi,k,t > SQmin i,k (14) i brood parasitism which was often found among certain species of cuckoo birds. This parasitism X occurs when the cuckoo birds lay their eggs P Qi,k,t 6 P Qmax (15) in the nests of other bird species (non-cuckoo i birds). The heavy-tailed random walk proba- bility distribution, Lévy ights was used as a stochastic generator for the CS technique. The X P Qi,k,t 6 AQmax,i,t (16) i iterative expression at iteration, t for the candi- date solution i for the CS technique is: 1 − M Corii,t yit+1 = yit + β · Levy (λ) (20) W Ri,k,t 6 (17) 1 − M Cmax .i,t such that the Lévy distribution is given as fol- X lows: SQi,k,t · P Pi,k,t i,t Levy (λ) = t−λ (21) ≥ [E1 + E2 + E3 ] · (1 + ERk ) (18) wherett is the random variable, β > 0 is the where relaxation factor (which is modied based on the problem at hand) and λ ∈ (1, 3] is the Lévy E1 := F Cbk ight step-size. With t ≥ 1, λ is related to the AP fractal dimension and the Lévy distribution   i,k,t 1 becomes a specic sort of Pareto distribution. X E2 := SQi,k,t ·  T Ccbi,k · Dcbik  i,t + W Ri,k,t The CS algorithm is based on a few fundamental LCcbi,k X T Ctpi,k Dbpk philosophies. For instance each cuckoo bird lays E3 := SQi,k,t · a single egg at one time and randomly places LCtpik i,t the egg in a selected nest. The second being: via tness screening, the best egg (candidate so- such that, lution) is carried forward into the next iteration. i ∈ [1, 2], k ∈ [1, 10], t ∈ [1, 12] (19) The worst solutions are discarded from further iterations. The nests represent the objective The decision parameters are: space (or the optimization problem landscape). qt , IQi,t , SQi,k,t , P Qi,k,t and BRi,t . De- The parameter setting for the CS technique tails on the parameter settings of the biofuel used in this work is shown in Tab. 1 while supply chain model used in this work could be its respective algorithm is given in Algorithm 1: obtained in Tan et al. [61]. c 2020 Journal of Advanced Engineering and Computation (JAEC) 37
VOLUME: 4 | ISSUE: 1 | 2020 | March Algorithm 1: Cuckoo Search (CS) such that hsi = hλn+1 − λn i, where λn is the nth eigenvalue sequentially such that λ1 < ... < Step 1: Initialize algorithmic parameters; yi , β, λn < λn+1 . The rst type of random matri- ces are those that are modeled based on com- λ, N Step 2: Dene parameters in the constraints plex quantum systems (which have chaotic clas- sical counterparts). RMT consists of four ma- and decision variable Step 3: Via Lévy ights randomly lay a cuckoo jor ensembles to determine the spacing distribu- tions of the eigenvalues: the Gaussian Orthogo- egg in a nest Step 4: Dene tness function based on solu- nal Ensemble (GOE), Gaussian Unitary Ensem- ble (GUE) and Gaussian Symplectic Ensemble tion selection criteria Step 5: Screen eggs and evaluate candidate so- (GSE). In this work, the GUE distribution is considered: lution IF: tness criteria is satised 32 4 P (s) = 2 s2 exp(− s2 ) (24) Select candidate solution (egg) to be π π considered in the next iteration, n + 1 These ensembles describe the probability density ELSE: tness criteria is not satised functions governing the random matrix entries. Discard candidate solution (egg) from The constants, Ai and Bi are selected such that further iterations the following averaging properties are respected: Step 6: Rank the best solutions obtained dur- ing tness screening Z∞ Z∞ Step 7: If the tness criterion is satised and dsP (s) = 1 and dsP (s)s = 1 (25) t = Tmax halt and print solutions,else proceed to 0 0 Step 3. Metaheuristics are equipped with stochastic generators called the random generator - which 4. Random matrix theory randomly initializes the search operation of & stochastic generators the metaheuristic. This is done by positioning the starting point of the search operation prior to exploring the objective space. In the Random Matrix Theory (RMT) is a robust works of Ganesan et al. [25], Ganesan et al. mathematical framework which is very eec- [26] and Ganesan et al.[27], it was seen that tive for describing behavior of complex systems. variations in the type of stochastic generators RMT is known to exhibit universality a prop- have an inuence on the optimization results. erty of global symmetries shared by many sys- Therefore in this work the RMT is employed tems within a certain symmetry class. Details on as the stochastic generator to solve the fuzzy the application of RMT on a non-fuzzy (crisp) biofuel supply chain problem. Essentially RMT biofuel supply chain model could be seen in deals with systems with a complex network of Ganesan et al. [28]. In RMT there exists two many interlinked and interacting components probability distributions describing: the random - which are often encountered in real-world matrix entries and the eigenvalue spread. The settings. The proposed algorithmic framework nearest neighbor spacing probability distribu- for developing a random matrix generator is as tion of eigenvalues is given by Wigner's Surmise: follows: P (s) = Ai si exp(−Bi s2 ) (22) Algorithm 2: Random Matrix Generator where s is the eigenvalue spacing, Ai and Bi are constant parameters. The normalized spacing, s Step 1: Generate random eigenvalue spacings, and the mean spacing hsi is as follows: s from a GUE Step 2: Determine the average eigenvalue spac- λn+1 − λn ing, ∆λ s= (23) hsi Step 3: Set initial eigenvalue, λ0 38 c 2020 Journal of Advanced Engineering and Computation (JAEC)
VOLUME: 4 | ISSUE: 1 | 2020 | March Step 4: Set initial n × n matrix, Hij Step 5: Determine consequent eigenvalues λi+1 = λi + ∆λ Step 6: Determine n × 1 eigenvector, Ei : X Ei = Hij + λi j Step 7: Generate random variables from a Gaussian probability distribution function en- dowed eigenvector as the variance, σ 2 = Ei : ! 2 1 (x − µ) Pi (x) = p exp − 2πEi 2 2Ei 2 5. Results and discussion The following frameworks have been introduced in the past for tackling MO optimization prob- lems: Strength Pareto Evolutionary Algorithm (SPEA-2) (Zhao et al.[69]), Weighted sum ap- proach (Naidu et al. [46]), Normal-Boundary Intersection (NBI) (Ahmadi et al.[2]; Ganesan et al.[23]) and Non-Dominated Sorting Genetic Al- gorithm (NSGA-II) (Mousavi et al.[45]). Scalar- ization and NBI approaches involve the aggre- gation of multiple target objectives. Eectively transforming the MO problem into a single ob- jective one reduces its complexity to a high de- Fig. 1: Pareto frontier from the fuzzy RMT-CS tech- gree - making it easier to solve. In this work, nique the objective functions of the fuzzy MO bio- fuel supply chain problem was combined into a single function using the weighted sum ap- solutions were then used to construct the Pareto proach (Kalita et al.[34]). This procedure eec- frontier. - at which the individual solutions were tively transforms the fuzzy MO problem into a classied as best, worst and median. The mea- single-objective optimization problem which can sured solutions were ranked using the hypervol- be solved for dierent weight values. The com- ume indicator (HVI) (Bringmann and Friedrich putational experiments employed in this work [13]). Applying the HVI, the level of solution was done using the C++ programming language dominance for the fuzzy MO optimization prob- on a computer using a 64-bit Win 10 platform lem could be measured. A Nadir point is usually with an Intel Core i5-7200U CPU (2.50 GHz). employed as a basis (or a baseline value) while measurement using the HVI. In this work, the Due to the stochastic nature of the algorithms nadir point for the HVI is computed as follows: employed in this work, the computational tech- nique was executed multiple times (3 executions) (z1 − 106 )(z2 − 104 )(z3 − 102 ) and the best solution was taken. 28 solutions HV I = (26) were obtained for a variation of weights. These 1016 c 2020 Journal of Advanced Engineering and Computation (JAEC) 39
VOLUME: 4 | ISSUE: 1 | 2020 | March Tab. 2: Ranked individual solutions for the CS technique Description Best Median Worst w1 0.4 0.2 0.1 weights w2 0.2 0.4 0.3 w3 0.4 0.4 0.6 PR 246831000 352010000 342717000 Objective functions SW 1005810 1004490 1004550 Env 242631 10598.1 124.342 Iterations t 100 78 67 Metric HVI 5937.18 366.46 0.83 where z1 , z2 and z3 are individual candidate so- The variation of the objective function, PR lutions. The ranked weighted individual solu- with respect to the parameters in the fuzzy tions obtained using the fuzzy CS approach with membership function is given in Fig. 2: random matrix generators (fuzzy RMT-CS) is given in Tab. 2. The entire Pareto frontier con- structed 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) is xed: d = 0.2 and j = 0.2. The HVI value for the entire Pareto frontier is 27,957.73 and the total computational time taken for construction was 33.128 seconds. To observe the variation of the objective function with respect to the membership functions, the weights are xed to (PR, SW, Env) = (0.4, 0.2, Fig. 2: Objective function, PR with respect to the pa- d j 0.4) which is the best individual solution ob- rameters function and in the logistic membership tained (see Tab. 2). In Fig. 1, some prevalent trends could be ob- In Fig. 2, the maximum value of PR is served in the distribution of the solution points 365,021,000 obtained at d = 0.4 and j = 4 in the objective space. One of these trends is which corresponds to µ = 0.1 (see equation) the high concentration of solution points in spe- The median value of PR is 327,626,500 and it cic regions of the objective space: P R ∈ (3.0 × falls between µ = 0.05 and µ = 0.1. The mini- 108 , 3.8 × 108 ), SW ∈ (1, 004, 400, 1, 004, 800) mum value of PR (259, 954, 000) is obtained at and Env ∈ (0, 8 × 104 ). This high concentra- µ = 0.0667. The objective function, SW plot- tion could be attributed to the technique itera- ted with respect to the parameters in the fuzzy tively reaching the most optimal (local or near membership function is given in Fig. 3: optimal) region of the objective space. Despite those high concentrations, some solution points The maximum and minimum value of the could be observed to exist beyond those optimal SW objective function shown in Fig. 3 were regions. This shows that the proposed technique 1,005,660 and 1,004,490 with the membership generates a sparse distribution of solutions and value of µ = 0.0667 and µ = 0.1. The median hence has good exploration capabilities - where value obtained was 1,004,650 which was in be- the fuzzy RMT-CS approach explores regions in tween µ = 0.033 and µ = 0.05. The objective the objective space in search of other local op- function, Env with relative to the membership tima. function is presented in Fig. 4: 40 c 2020 Journal of Advanced Engineering and Computation (JAEC)
VOLUME: 4 | ISSUE: 1 | 2020 | March Nevertheless the proposed technique does have some disadvantages. The rst is the al- gorithmic complexity - where the addition of the RMT segment into the CS technique sig- nicantly increases the complexity of the al- gorithm. This in eect may considerably im- pact the computational time of the optimiza- tion 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 Fig. 3: Objective function, SW with respect to the pa- performance of the proposed technique. Finally rameters d and j in the logistic membership this work considers the uncertainty in the an- function nual electricity generation output of the biomass power plant [kWh/year] to be of type-1 fuzzy uncertainty. It is high possible that uctua- tions in the monthly/weekly electricity gener- ation output of the biomass power plant may more precisely capture the mentioned uncertain- ties - making the model more realistic. 6. Conclusion and recommendations Fig. 4: Objective function, Env In this work, the proposed MO biofuel sup- with respect to the pa- rameters d and j ply chain problem was reformulated by taking in the logistic membership function into account uctuations in the annual electric- ity generation output of the biomass power plant [kWh/year]. This eectively converts the prob- Figure 4 shows that the objective function, lem into a fuzzy MO problem; which is non- Env has a maximum of 266,045 at µ = 0.05 and linear, nonconvex and multivariate. To deal minimum of 5,326.36 at µ = 0.1. The median of with this MO problem the CS technique was 22,092.65 was obtained between the membership retrotted with the RMT approach to boost its values of µ = 0.075 and µ = 0.3. performance when faced with high levels of com- plexity. The proposed approach was eectively The fuzzy RMT-CS approach produced fea- applied and Pareto ecient solutions were at- sible solutions; where the constraints in the tained. The dominance of these solutions were fuzzy biofuel supply chain model was not bro- gauged using the HVI. ken. The computations performed during the numerical experiments were stable and the al- Further computational tests could be done by gorithm achieved convergence every time dur- using the GSE and GOE ensembles for the RMT ing execution. In terms of robustness, the fuzzy segment in the proposed approach. In addition, RMT-CS method performed stable computa- RMT - based generators could also be employed tions and converged towards a feasible solution to complement other metaheuristic techniques during each variation of the fuzzy membership such as PSO (Mousavi et al. [45]), dierential function. The random matrix segment success- evolution (Ganesan et al. [24]) as well as other fully complemented the CS technique to navi- computational approaches (Ganesan et al.[25]). gate through the objective space of the complex The accuracy of the fuzzy formulation proposed and multivariate biofuel supply chain problem. in this work could be further improved by ac- c 2020 Journal of Advanced Engineering and Computation (JAEC) 41
VOLUME: 4 | ISSUE: 1 | 2020 | March counting for monthly/weekly uctuations; by re- and sustainable supply chain management. formulating the MO biofuel supply chain prob- Journal of cleaner production, 52, 329-341. lem by utilizing a type-2 fuzzy framework. This work can also be extended by exploring other [4] Akemann, G. (2017). Random matrix approaches for handling uncertainty such as: ro- theory and quantum chromodynamics. bust optimization, (Bairamzadeh et al. [7]; Kara Stochastic Processes and Random Matrices: et al.[31]), stochastic optimal control (Vinod et Lecture Notes of the Les Houches Summer al. [63]) and chance constraint optimization School, 104, 228. (Cheng et al. [17]). This extension could include [5] Sengupta, A., Vasant, P., & Andeeski, C. emerging areas of applications such as complex J. (2008). Fuzzy optimization with robust networks in alternative energy systems (Syahpu- logistic membership function: A case study tra et al. [60]; Lot et al. [39]), social networks, in for home textile industry. IFAC Proceed- gene networks (Youseph et al. [67]) and phar- ings Volumes, 41(2), 5262-5266. maceutical supply chain networks (Zahiri et al. [68]). [6] Babazadeh, R. (2019). Application of Fuzzy Optimization to Bioenergy-Supply-Chain Acknowledgments Planning under Epistemic Uncertainty: A New Approach. Industrial & Engineering Chemistry Research, 58(16), 6519-6536. The authors would like to sincerely thank Pro- [7] Bairamzadeh, S., Saidi-Mehrabad, M., & fessor Dr. Ivan Zelinka (Technical University Pishvaee, M. S. (2018). Modelling dierent of Ostrava, Czech Republic) for his great help types of uncertainty in biofuel supply net- and support in this research work. This research work design and planning: A robust opti- project also supported by Modeling Evolution- mization approach. Renewable energy, 116, ary Algorithms Simulation and Articial Intelli- 500-517. gence (MERLIN), Faculty of Electrical & Elec- tronics Engineering, Ton Duc Thang University, [8] Bairamzadeh, S., Pishvaee, M. S., & Saidi- Ho Chi Minh City, 758307, Vietnam and Univer- Mehrabad, M. (2016). Multiobjective ro- sity Technology PETRONAS, 32610, Seri Iskan- bust possibilistic programming approach dar, Malaysia. to sustainable bioethanol supply chain de- sign under multiple uncertainties. Indus- trial & Engineering Chemistry Research, References 55(1), 237-256. [9] Balaman, . Y., Matopoulos, A., Wright, [1] Abdelsalam, H. M., & Elassal, M. M. D. G., & Scott, J. (2018). Integrated opti- (2014). Joint economic lot sizing prob- mization of sustainable supply chains and lem for a threeLayer supply chain with transportation networks for multi technol- stochastic demand. International Journal of ogy bio-based production: A decision sup- Production Economics, 155, 272-283. port system based on fuzzy ε-constraint method. Journal of cleaner production, 172, [2] Ahmadi, A., Kaymanesh, A., Siano, P., 2594-2617. Janghorbani, M., Nezhad, A. E., & Sarno, D. (2015). Evaluating the eectiveness [10] Balaman, . Y., & Selim, H. (2016). Sus- of normal boundary intersection method tainable design of renewable energy supply for short-term environmental/economic hy- chains integrated with district heating sys- drothermal self-scheduling. Electric Power tems: A fuzzy optimization approach. Jour- Systems Research, 123, 192-204. nal of cleaner production, 133, 863-885. [3] Ahi, P., & Searcy, C. (2013). A comparative [11] Beenakker, C. W. J. (2015). Random- literature analysis of denitions for green matrix theory of Majorana fermions and 42 c 2020 Journal of Advanced Engineering and Computation (JAEC)
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VOLUME: 4 | ISSUE: 1 | 2020 | March HVmax maximum heat value of mixed Dcbik average transport distance when fuel [kJ/kg] broker k collecting biomass fuel i [km] HVmin minimum heat value of mixed fuel [kJ/kg] Dtpk transport distance between broker k and biomass power plant [km] IQi,0 inventory quantity of biomass fuel i at the beginning E eciency of biomass power plant of month 1 [tonnes] [decimal fraction] CEcb carbon dioxide emissions during EC electricity consumption rate of collecting biomass fuel [kg] biomass power plant [decimal fraction] CEtp carbon dioxide emissions during transporting biomass fuel to extraY1 rst extra cost of excessive biomass power plant [kg] biomass power plant fuel inventory [yuan/month] EIC extra inventory cost of biomass power plant [yuan] extraY2 second extra cost of excessive biomass power plant fuel IQi,t inventory quantity of biomass inventory [yuan/month] fuel i at the end of month t [tonnes] ERk expected return of broker k [decimal fraction mass/year] PPi purchase price of biomass fuel i from brokers [yuan/t] FCbk xed cost of broker k [yuan/year] PQik,t purchase quantity of biomass fuel FCp xed cost of biomass power plant i by broker k in month t [t] [yuan/year] qt electricity generation of biomass GC unit generation cost of biomass power plant in month t power plant [yuan/kWh] [kWh/month] GS government subsidies to biomass Rt conversion rate from biomass fuel power generation to electricity in month [yuan/kWh] t [kg/kWh] GT government tax revenues from IQmax maximum inventory quantity of biomass power plant [yuan/year] biomass power plant [t] HVi heat value of biomass fuel IL rate of inventory loss i [kJ/kg] [decimal fraction/month] HVe heat value of electricity LCcbik load capacity of conveyance [kJ/kWh] when broker k collects biomass fuel i [t] c 2020 Journal of Advanced Engineering and Computation (JAEC) 47
VOLUME: 4 | ISSUE: 1 | 2020 | March LCtpik load capacity of conveyance ˜ min Q fuzzy minimum annual when broker k transports electricity generation of biomass fuel i to biomass biomass power plant power plant [t] [kWh/year] MCmaxi maximum moisture content of RIub1 rst upper bound of biomass fuel i required by reasonable fuel inventory [t] biomass power plant [decimal fraction mass] RIub2 second upper bound of reasonable fuel inventory [t] MCoriit original moisture content of biomass fuel i in month t SIlbi lower bound of safety [decimal fraction mass] inventory for biomass fuel i [t] MCaftik moisture content of biomass fuel i after processing by SC unit storage cost of broker k biomass power plant [decimal fraction mass] [yuan/month] P on-grid price of biomass SQminik minimum supply quantity of power plant [yuan/kWh] biomass fuel i from broker PQmax,k maximum purchasing quantity T Ccbik average unit transportation of biomass fuel by broker cost of broker k when k [t/month] collecting biomass fuel i [yuan/km] qmax maximum monthly electricity generation quantity of T Ctpik average unit transportation biomass power plant cost of broker k when [kWh/month] transporting biomass fuel i to biomass power plant Qmax maximum annual electricity [yuan/km] generation quantity of biomass power plant W Rik,t ratio of the weight of [kWh/year] biomass fuel i after processing to the weight Qmin minimum annual electricity before processing by broker generation of biomass power k in month t plant [kWh/year] [decimal fraction mass] q˜max fuzzy maximum monthly APik,t average price of broker electricity generation k buying biomass fuel i quantity of biomass power in month t [yuan/t] plant [kWh/month] ˜ max Q fuzzy maximum annual electricity generation quantity of biomass power plant [kWh/year] 48 c 2020 Journal of Advanced Engineering and Computation (JAEC)
VOLUME: 4 | ISSUE: 1 | 2020 | March BCi,t biomass fuel i consumption in month t [t] Random Matrix Theory and CS BRi,t blending ratio of biomass fuel i in mixed fuel P1 (s) Probabilistic Spacing in month t Distribution for Gaussian [decimal fraction mass] Orthogonal Ensemble (GOE) CER carbon dioxide emissions P2 (s) Probabilistic Spacing reduction [kg] Distribution for Gaussian Unitary Ensemble (GUE) CETeq carbon dioxide emissions P3 (s) Probabilistic Spacing of thermal power plant Distribution for Gaussian for power generation equal Symplectic Ensemble to biomass power plant [kg] (GSE) Po (s) Probabilistic Spacing CEB carbon dioxide emissions Distribution for of biomass power plant [kg] Poisson Distribution s Eigenvalue Spacing SQik,t supply quantity of biomass distribution fuel i by broker k in λ Eigenvalue month t [t] ∆λ Eigenvalue Interval Ei Eigenvector VCp total variable cost of Hij Initial Matrix biomass power plant σ2 Statistical variance [yuan/year] µ Statistical Mean wi Weights for the weighted Y 1t binary variable to sum method determine whether the Tmax Maximum limit of inventory is over RIub1 function evaluations at the end of month t m Maximum number of objective functions Y2t binary variable to determine HV I Hypervolume Indicator whether the inventory is over β Relaxation factor RIub2 at the end iter Number of algorithm of month t iterations t Random Variable B, C fuzzy coecients y ti Candidate solution Levy(λ) Lévy Distribution α fuzzy membership function N Total Number of Eggs d, j fuzzy membership values About Authors ε credibility level Timothy GANESAN is currently a Senior aij , ˜bi fuzzy constraints Analyst at the Royal Bank of Canada special- izing in computational intelligence and data PR Prots (yuan) analytics. He has experience working as a Prin- cipal Researcher for the Fuels and Combustion SW Social Welfare (yuan) Section in the research and development arm of the Malaysian power producer - Tenaga Env Environmental Benet (kg/power station) c 2020 Journal of Advanced Engineering and Computation (JAEC) 49
VOLUME: 4 | ISSUE: 1 | 2020 | March Nasional Berhad (TNB). In addition to having Lab in University Sains Malaysia. From March degrees in Chemical Engineering and Compu- 2008 to March 2010, he worked as a research tational Fluid Dynamics, he holds a Ph.D. in associate at the same University. Currently, Process Optimization. His research interests he is working with Machine Learning, Big include engineering/industrial optimization, Data, Data Analytics, Deep Learning, specially multi-objective/multi-level programming, evo- targeting on Convolutional Neural Networks lutionary algorithms, machine learning, chaos (CNN) and Bi-directional Recurrent Neural optimization, and swarm-based optimization. Networks (RNN) for image tagging with em- bedded natural language processing, End to end Pandian VASANT is a senior lecturer steering learning systems and GAN. His work at Department of Fundamental and Applied involves experimental research with software Sciences, Faculty of Science and Information prototypes and mathematical modelling and Technology, Universiti Teknologi PETRONAS design He is an editorial board member for in Malaysia. He holds PhD (UNEM, Costa the Journal of Energy Optimization and Engi- Rica) in Computational Intelligence, MSc neering (IJEOE), and invited guest editor for (UMS, Malaysia, Engineering Mathematics) Journal of Visual Languages Communication and BSc (2nd Class Upper-Hons in Mathe- (JVLC-Elsevier). He has published more than matics, UM, Malaysia) in Mathematics. His 30 papers in leading international conference research interests include Soft Computing, proceedings and peer reviewed journals. Hybrid Optimization, Holistic Optimization, Innovative Computing and Applications. He Igor LITVINCHEV received his M.Sc. has co-authored research papers and articles degree in Applied Mathematics from Moscow in national journals, international journals, Institute of Physics and Technology (Fizteh), conference proceedings, conference paper pre- Russia; Ph.D. in Systems Theory and Oper- sentation, and special issues lead guest editor, ations Research and Dr.Sci. (Habilitation) in lead guest editor for book chapters' project, Systems Modeling and Optimization from Com- conference abstracts, edited books, keynote puting Center, Russian Academy of Sciences, lecture and book chapters (257 publications Moscow. He held long term visiting positions indexed in Web of Science). In the year 2009, at universities in Brazil, Mexico, and Norway, Dr. Pandian Vasant was awarded top re- as well as positions at various universities and viewer for the journal Applied Soft Computing research centers in Russia. He is an author of (Elsevier), awarded outstanding reviewer in 4 books, editor of 5 more books and published the year 2015 for ASOC (Elsevier) journal numerous research papers focused on large-scale and Top reviewer for Sentinels of Science: systems modeling, optimization, and control. Computer Science (Oct. 2015 - Sept. 2016). His research was supported by more than 35 He has 26 years of working experience at the grants from NATO Scientic Aairs Division and various universities from 1989-2017. Currently European Community; ISF (USA) and RFBR he is Editor-in-Chief of IJEOE and member (Russia); CNPq and FAPESP (Brasil); BRFBR of American Mathematical Society, ACM, (Belarus); CONACYT, PROMEP and PAICYT SIAM, MERLIN, and NAVY Research Groups. (Mexico). He is a member of Russian Academy https://publons.com/researcher/499841/dr- of Natural Sciences and Mexican Academy of pandian-vasant-phd/ Sciences. Joshua THOMAS is a senior lecturer at KDU Penang University College, Malaysia since 2008. He obtained his PhD (Intelligent Systems Techniques) in 2015 from University Sains Malaysia, Penang, and Master's degree in 1999 from Madurai Kamaraj University, India. From July to September 2005, he worked as a research assistant at the Articial Intelligence 50 "This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited (CC BY 4.0)."